Long-range donor-acceptor electron transport mediated by alpha-helices
Larissa S. Brizhik, Jingxi Luo, Bernard M.A.G. Piette, Wojtek J. Zakrzewski
LLong-range donor-acceptor electron transportmediated by α -helices L.S. Brizhik ∗ ,Bogolyubov Institute for Theoretical Physicsof the National Academy of Sciences of Ukraine,03143 Kyiv, UkraineJ. Luo † School of Mathematics, University of Birmingham,Birmingham B15 2TT, UKB.M.A.G. Piette ‡ , and W.J. Zakrzewski § ,Department of Mathematical Sciences, University of Durham,Durham DH1 3LE, UK Abstract
We study the long-range electron and energy transfer mediated bya polaron on an α -helix polypeptide chain coupled to donor and accep-tor molecules at opposite ends of the chain. We show that for specificparameters of the system, an electron initially located on the donorcan tunnel onto the α -helix, forming a polaron, which then travelsto the other extremity of the polypeptide chain where it is capturedby the acceptor. We consider three families of couplings between thedonor, acceptor and the chain, and show that one of them can leadto a 90% efficiency of the electron transport from donor to accep-tor. We also show that this process remains stable at physiologicaltemperatures in the presence of thermal fluctuations in the system. PACS numbers: 05.45.Yv, 05.60.-k, 63.20.kd, 71.38.-k ∗ e-mail address: [email protected] † e-mail address: [email protected] ‡ e-mail address: [email protected] § e-mail address: [email protected] a r X i v : . [ c ond - m a t . o t h e r] J a n ey words: long-range donor-acceptor electron and energy transfer, largepolaron, alpha-helix, donor-bridge-acceptor complex, soliton, self-trapping The mechanisms behind the highly efficient long-range electron transfer (ET)in redox reactions accompanying photosynthesis and cellular respiration havebeen intensively discussed over several decades [1, 2]. This transfer takesplace at macroscopic distances along the so-called electron transport chain inKrebs cycles in membranes of chloroplasts, mitochondria or cells, and occursat physiological temperatures. Conventional mechanisms, such as tunnelling,Forster and Dexter mechanism etc. [3, 4, 5], cannot provide such long-rangeET even at zero temperature, let alone 300 K. Nevertheless, it should benoted that the very structure of the ET chain can facilitate these processes.An ET chain consists of a spatially separated sequence of biological molecularcomplexes (peptides, enzymes, etc.), along which the sequential transport ofelectrons takes place via redox processes, so that every site in this chain playsthe role of an acceptor for the left neighbour and donor for the right one alongthe chain (see, e.g., [6]). The electron transport chain in mitochondria canbe schematically represented as the following sequence:NADH + H + → Complex I → Q → Complex II → Complex III →→ cyt c → Complex IV → O . Here NADH + H + is nicotinamide adenine dinucleotide, which serves asthe substrate; Complex I is NADH coenzyme Q reductase; Q is ubiquinonecoenzyme; Complex II is succinate dehydrogenase; Complex III is cytochromebc ; cyt c is cytochrome c ; Complex IV is cytochrome c oxidase; O ismolecular oxygen. Another example of the electron transport chain can befound in [7].In each elementary process, at the onset, there is a release of four electronsat the substrate, which then are carried along the chain with the reductionof molecular oxygen and hydrogen ions to a water molecule at the final stageof the process. This transport of electrons is so exceptionally efficient thatonly a tiny percentage of electrons leak out to reduce oxygen. The complexesin the ET chain can be conventionally divided into two groups: heavy andlight ones. In particular, in ET chains, such elements as ubiquinone or cy-tochrome cyt- c , have relatively small molecular weight which leads to theirhigh mobility. They can move outside the mitochondrial membrane, carrying2lectrons from a heavy donor to a heavy acceptor via a linear, e.g. the Forstermechanism [3, 5]. Some other complexes in the electron transport chain, suchas NADH-ubiquinone oxireductase, flavoproteids, cytochrome c -oxidase, cyt aa and cytochrome cyt bc are proteins with large molecular weight of upto several hundreds of kiloDaltons. Conventional linear mechanisms cannotprovide coherent transport of electrons across these heavy enzymes, eitheras a whole or internally between co-factors separated by macroscopic dis-tances, for instance porphyrins, metal clusters etc., that are separated bymacroscopic distance. Nevertheless, their regular crystal-like structure canfacilitate ET, as is discussed below. For instance, inside some large enzymeslike NADH ubiquinone oxidoreductase there can be several long pathwaysfor electron transport [2], where one can identify the alpha-helical part of theenzyme between the donor and the acceptor.A significant part of heavy macromolecules is in the alpha-helical confor-mation, whose regular structure results in the formation of electron bandsin their energy spectrum. The alpha-helical structure is stabilized by rela-tively weak hydrogen bonds resulting in strong electron-lattice interactions,and thus, in the polaron effect. An α -helical segment of a protein containsthree almost-parallel polypeptide strands bound by hydrogen bonds along thestrands, with weak interactions between these strands. An isolated strandis described by the Fr¨ohlich Hamiltonian, and this description leads to asystem of coupled nonlinear equations for the electron wavefunction and lat-tice variables, and admits soliton solutions. The possibility of self-trappingof electrons in an isolated one-dimensional molecular chain, like a polypep-tide strand, has been first shown in [8] (see also [9, 10]) and later it wasalso demonstrated in helical systems [11, 12, 13]. The soliton solutions ofthese models are particular cases of a large polaron. Such a polaron canbe described as a crossover between an almost-free electron and small po-laron states depending on the strength of the exchange interaction energy,electron-lattice coupling constant, the number of phonon modes, their typeand the corresponding Debye energies [14]. The soliton properties dependon the parameters of the system. Moreover, the helical structure of proteinswas shown to lead to the existence of several types of soliton solutions of themodel with different properties and symmetries [13]. In such soliton stateselectrons can propagate along macromolecules almost without any loss ofenergy.The results mentioned above have been obtained for isolated strands orhelices, while in reality, the electron transport occurs in the system Donor-Bridge-Acceptor, as is the case for the ET chain in the Krebs cycles. Thesimple case when the bridge is modelled as a polypeptide strand had beenstudied in [15]. It was shown there that the long-range ET can be provided3y the soliton mechanism within a wide range of parameter values of donor,acceptor and polypeptide strands.In the present paper we study the possibility of a coherent long-rangeelectron transport in the system Donor- α -helix-acceptor. As one can expect,the formation of the soliton on the α -helix depends on the helix-donnor andhelix acceptor coupling, as well as on the parameters of the system understudy (see e.g. , [15, 16, 17]), and we can find conditions which lead to theformation of a soliton on the helix.There are two other aspects of the model developed in the present paper.The first one is related to the fact that the functioning of the ET chain istightly connected with the production of adenosine triphosphate (ATP): inmost organisms the majority of ATP is generated in ET chains (see, e.g. ,[18]). The energy of the hydrolysis of ATP into ADP is the basic unit ofenergy used in biological systems, in particular, in muscles to produce me-chanical work, to establish electrochemical gradients across membranes, inbiosynthetic processes, and in many other physiological and biochemical pro-cesses necessary to maintain life. The amount of energy released by ATP hy-drolysis is approximately 0.43 eV, which is only 20 times the thermal energyat physiological temperatures and is not enough for an electronic excitation.It is sufficient to excite some vibrations, such as an AMID I vibration, an ex-citation which requires an energy of 0.21 eV. AMID I is mainly (up to 80%)the stretching vibration of double C=O bond of the peptide group whichhas a relatively large dipole moment 0.3 Db oriented along the alpha-helixaxis. This excitation is registered in optical spectra of polypeptide molecules,its wavelength being 1650 cm − , and, according to [19], the ATP hydrolysisenergy is transferred along protein macromolecules in the form of AMID Ivibration. For more details see[20, 21, 9, 10, 22] or more recently [23, 24].As has been shown by Davydov, the Amide-I vibration can be self-trappedin macromolecule into a soliton state and carried along it to the place whereit is utilized for biochemical or mechanical needs [9, 10]. This process, fromthe mathematical point of view, is described formally by the same system ofequations as the ET. Therefore, the results obtained here are equally validfor such energy transfer processes.The second aspect of the model is related to the potential importance ofour results for micro- and nano-electronics where conjugated donor-acceptorcopolymer semiconductors with intra-molecular charge transfer on large dis-tances are widely used. A large number of such systems have been recentlysynthesized. They include donor-acceptor pairs mediated by salt bridges [25],thienopyrazine-based copolymers [26] and some others [27, 28, 29]. Donor-bridge-acceptor systems with efficient ET play an important role in elec-tronic applications [30, 31, 32, 33]: they can be used in photovoltaic cells434, 35, 36, 29], light-emitting diodes [37, 38, 39, 40] and field-effect transis-tors [41, 42, 43, 44], in particular, thin-film organic field effect transistors [45].Proteins and synthetic macromolecules have a great technological potential;one example is the improvement of efficiency and UV-photostability of planarperovskite solar cells using amino-functionalized conjugated polymers as ETmaterials [27, 29].Recent novel applications in bioelectronics such as organic photovoltaics,fuel cell technology and other, are based on metalorganic frameworks orstructures, that are complexes of electroconducting compounds/substratesand polypeptides (see, e.g., [46, 47, 48] and references therein). It has beenshown that both the peptide composition and structure can affect the effi-ciency of electron transport across peptides [47]. Moreover, long-range con-ductivity and enhanced solid-state electron transport in proteins and peptidebioelectronic materials has been proven experimentally [49, 50]. The effec-tiveness of electron transport processes in living systems is already used innovel electronic devices, e.g., in Shewanella Oneidensis MR-1 Cells, basedon multiheme cytochrome mediated redox conduction [51], or in synthesizedsupramolecular charge transfer nanostructures based on peptides [52], syn-thetic biological protein nanowires with high conductivity [53], self-assembledpeptide nanotubes used as electronic aterials [54] and many others. We quote[49]: The ability of such natural and synthetic protein and peptide materialsto conduct electricity over micrometer to centimeter length scales, however,is not readily understood from a conventional view of their amino acid build-ing blocks. Distinct in structure and properties from solid-state inorganicand synthetic organic metals and semiconductors, supramolecular conduc-tive proteins and peptides require careful theoretical treatment. This is oneof the factors which have motivated our interest in the problems discussedin the present paper and we hope that our study will shed some light on thisproblem.In the first section of the paper we derive a model of the α -helix coupledto a donor molecule and an acceptor molecule. This model is a combinationof the models derived in [13] and [15]. We then perform a parameter scalingto make all the parameters dimensionless and derive the equations in suchunits. After selecting the parameters that best describe the α -helical protein,we compute the profile of a static self-trapped electron state (soliton-like or,in other words, large polaron state, which for simplicity we call from now on a‘polaron’) by solving the model equations numerically. We then study variousconfigurations where the electron density has been set to 1 on the donor and0 elsewhere and let the system evolve. We do this for three different types ofcouplings between the donor and acceptor to the α -helix and we determinenumerically the donor and acceptor coupling parameters that lead to the best5ransfers of the electron. We end the paper by describing the solutions wehave found and draw some conclusions. α -Helix –Acceptor’ We consider a polypeptide chain in an α -helical configuration made out of N peptide groups (PGs), with a donor molecule attached to one end andan acceptor molecule attached to the other end. The peptide chain formsa helical structure in which each molecule is coupled by chemical bonds toits neighbours along the chain as well as to the PG 3 sites away from it byhydrogen bonds. With this 3-step coupling, the α -helix can also be seen as 3parallel chains [55] which we refer to as strands in what follows. This modelis depicted in Fig. 1.We label the PGs with the index n along the polypeptide chain, and use n = 0 for the donor and n = N + 1 for the acceptor. This means that PGswith an index difference which is a multiple of 3 belong to the same strandof the α -helix. Donor Acceptor1 2 3
Figure 1: The model of α -helix with a donor and an acceptor. The con-tinuous line represents the helix backbone formed by chemical bonds, thedash lines represent the hydrogen bonds that are links along the strands andthe dash-dot lines the links between the donor/acceptor and the differentstrands. The numbers 1, 2 and 3 label the 3 strands.The donor and the acceptor can, a-priori , be coupled respectively to thefirst 3 and the last 3 peptides, i.e. , with the nodes n = 1 , , N =6 − , N − , N . In our study, we will consider 3 different types of couplingsbut for now, we assume that all the coupling parameters are different.The Hamiltonian of the system is given by H = H e + H p + H int , (1)where H e , H p and H int are respectively the phonon, electron and interactionHamiltonians given by H e = E d | Ψ | + E a | Ψ N +1 | + E N (cid:88) n =1 | Ψ n | − J N − (cid:88) n =1 (cid:16) Ψ n Ψ ∗ n +3 + Ψ n +3 Ψ ∗ n (cid:17) + L N − (cid:88) n =1 (cid:16) Ψ n Ψ ∗ n +1 + Ψ n +1 Ψ ∗ n (cid:17) − (cid:88) (cid:96) =1 D d,(cid:96) (Ψ Ψ ∗ (cid:96) + Ψ (cid:96) Ψ ∗ ) − (cid:88) (cid:96) =1 D a,(cid:96) (Ψ N +1 Ψ ∗ N − (cid:96) + Ψ N − (cid:96) Ψ ∗ N +1 ) , (2) H p = 12 (cid:104) P d M d + P a M a (cid:105) + 12 (cid:88) (cid:96) =1 (cid:104) W d,(cid:96) ( U − U (cid:96) ) + W a,(cid:96) ( U N +1 − U N − (cid:96) ) (cid:105) + 12 N (cid:88) n =1 P n M + 12 N − (cid:88) n =1 W ( U n +3 − U n ) , (3) H int = | Ψ | (cid:88) (cid:96) =1 χ d,(cid:96) ( U (cid:96) − U ) + | Ψ N +1 | (cid:88) (cid:96) =1 χ a,(cid:96) ( U N +1 − U N − (cid:96) )+ (cid:88) (cid:96) =1 | Ψ (cid:96) | (cid:2) χ d,(cid:96) ( U (cid:96) − U ) + χ ( U (cid:96) +3 − U (cid:96) ) (cid:3) + (cid:88) (cid:96) =1 | Ψ N − (cid:96) | (cid:2) χ a,(cid:96) ( U N +1 − U N − (cid:96) ) + χ ( U N − (cid:96) − U N − (cid:96) ) (cid:3) + χ N − (cid:88) n =4 | Ψ n | ( U n +3 − U n − ) . (4)In these expressions, E describes the on-site electron energy, J the reso-nance integral along the strands, L the resonance integral along the helix, M the mass of the unit cell, χ the electron-lattice coupling and W the elasticityof the bond along the strands. The constants with subscript d and a refer toparameters of the donor and the acceptor respectively.The functions Ψ n describe the electron wave function (and so | Ψ n | de-scribe the electron probability of being at the site n ) and U n describe the7isplacement of molecule n along the strands. P n are the canonically con-jugated momenta of U n . Of course, the electron wave function satisfies thenormalization condition N +1 (cid:88) n =0 | Ψ n | = 1 , (5)where, following our convention, Ψ = Ψ d and Ψ N +1 = Ψ a .Our model is meant to describe the case in which the principal chain canbe sufficiently well approximated by one phonon band corresponding to anacoustical phonon mode which describes the longitudinal displacements ofthe unit cells from their positions of equilibrium along the helix’s strands.The electron-lattice interaction Hamiltonian induces a dependence of theelectron Hamiltonian on the lattice distortions. We also assume here thatthe dependence of the on-site electron energy on the lattice distortion is muchstronger than that of the inter-site electron interaction energy.The model we present here is a combination of the polaron model of the α -helix which was described in detail in [13] and of the donor-acceptor modeldescribed in [15]. The first model describes polarons on an α -helix, insteadof using the traditional single chain, proposed by Davydov [9, 10], whichcorresponds to what we call a strand in this paper. In fact, it was shownin [13] that the polaron is spread over the 3 strands hence the relevance ofusing a more realistic helical model. The second paper describes a model ofthe transfer of an electron from a donor molecule to an acceptor one via thecoherent propagation of a polaron along a simple chain (a single strand in thepresent model). The model we describe here is a combination of these twomodels in which the donor and the acceptor are coupled to a proper α -helixinstead of to a single strand. To facilitate the analysis of the model solutions, it is convenient to scalethe parameters so that they become dimensionless. Thus, following [13], weperform the following scalings: d = 10 − m , u n = U n d , τ = tν,E = E ¯ hν , E d = E d ¯ hν , E a = E a ¯ hν ,J = J ¯ hν , D a = D a ¯ hν , D d = D d ¯ hν ,W = Wν M , W d,(cid:96) = W d,(cid:96) ν M , W a,(cid:96) = W a,(cid:96) ν M ,χ = d χ ¯ h ν , χ d,(cid:96) = d χ d,(cid:96) ¯ h ν , χ a,(cid:96) = d χ a,(cid:96) ¯ h ν ,L = L ¯ hν , K d = MM d , K a = MM a . (6)8s a result, the Hamiltonian takes the form H p = M ν d H p , H e = ¯ h ν H e and H int = ¯ h ν H int where the dimensionless terms are H e = E d | Ψ | + E a | Ψ N +1 | + E N (cid:88) n =1 | Ψ n | − J N − (cid:88) n =1 (cid:16) Ψ n Ψ ∗ n +3 + Ψ n +3 Ψ ∗ n (cid:17) + L N − (cid:88) n =1 (cid:16) Ψ n Ψ ∗ n +1 + Ψ n +1 Ψ ∗ n (cid:17) − (cid:88) (cid:96) =1 D d,(cid:96) (Ψ Ψ ∗ (cid:96) + Ψ (cid:96) Ψ ∗ ) − (cid:88) (cid:96) =1 D a,(cid:96) (Ψ N +1 Ψ ∗ N − (cid:96) + Ψ N − (cid:96) Ψ ∗ N +1 ) , (7) H p = 12 (cid:104) K d (cid:18) du dt (cid:19) + 1 K a (cid:18) du N +1 dt (cid:19) (cid:105) ++ 12 (cid:88) (cid:96) =1 (cid:104) W d,(cid:96) ( u − u (cid:96) ) + W a,(cid:96) ( u N +1 − u N − (cid:96) ) (cid:105) + 12 N (cid:88) n =1 (cid:18) du n dt (cid:19) + 12 N − (cid:88) n =1 W ( u n +3 − u n ) , (8) H int = | Ψ | (cid:88) (cid:96) =1 χ d,(cid:96) ( U (cid:96) − U ) + | Ψ N +1 | (cid:88) (cid:96) =1 χ a,(cid:96) ( U N +1 − U N − (cid:96) )+ (cid:88) (cid:96) =1 | Ψ (cid:96) | [ χ d,(cid:96) ( U (cid:96) − U ) + χ ( U (cid:96) +3 − U (cid:96) )]+ (cid:88) (cid:96) =1 | Ψ N − (cid:96) | [ χ a,(cid:96) ( U N +1 − U N − (cid:96) ) + χ ( U N − (cid:96) − U N − (cid:96) )]+ χ N − (cid:88) n =4 | Ψ n | ( U n +3 − U n − ) . (9)We must thus have M ν d = ¯ h ν and so ν = ¯ h/ ( M d ). With M =1 . × − kg [15] and, as ¯ h = 1 . × − Js, we have ν = 5 . × s − .Before deriving the dimensionless equations it is also convenient to mul-tiply the wave function by a time-dependent phase and so we define ψ ( t ) = Ψ( t ) exp (cid:18) − it ¯ h ( E + 2 L − J ) (cid:19) . (10)Following [15] we also add to the acceptor equation a term of the form i (cid:80) (cid:96) =1 A a,(cid:96) | ψ N − (cid:96) | ψ N +1 , which describes the transfer of the electron from9he alpha-helix to the acceptor and has a clear physical meaning: the higherthe probability of the electron localization at the terminal end of the helix, thehigher the probability of its transfer to the acceptor. It is easy to check thatthis extra term does not violate conservation of the total electron probability.From the above Hamiltonian (1),(7)-(9) one can easily derive the followingequations for U n and Ψ n : i d Ψ dτ = ( E d − E − L + 2 J )Ψ − (cid:88) (cid:96) =1 D d,(cid:96) Ψ (cid:96) + Ψ (cid:88) (cid:96) =1 χ d,(cid:96) ( u (cid:96) − u ) ,i d Ψ (cid:96) dτ = (2 J − L )Ψ (cid:96) − J Ψ (cid:96) +3 + L (Ψ (cid:96) +1 + Ψ (cid:96) − (1 − δ (cid:96), )) − D d,(cid:96) Ψ + χ d,(cid:96) Ψ (cid:96) ( u (cid:96) − u ) + χ Ψ (cid:96) ( u (cid:96) +3 − u (cid:96) ) , l = 1 , , ,i d Ψ n dτ = (2 J − L )Ψ n − J (Ψ n +3 + Ψ n − ) + L (Ψ n +1 + Ψ n − ) + χ Ψ n ( u n +3 − u n − ) ,n = 4 . . . N − ,i d Ψ N − (cid:96) dτ = (2 J − L )Ψ N − (cid:96) − J Ψ N − (cid:96) + L (Ψ N − (cid:96) + Ψ N − (cid:96) (1 − δ (cid:96), )) − D a,(cid:96) Ψ N +1 + χ a,(cid:96) Ψ N − (cid:96) ( u N +1 − u N − (cid:96) ) + χ Ψ N − (cid:96) ( u N − (cid:96) − u N − (cid:96) ) − iA a,(cid:96) | Ψ N +1 | Ψ N − (cid:96) , l = 1 , , ,i d Ψ N +1 dτ = ( E a − E + 2 J − L )Ψ N +1 − (cid:88) (cid:96) =1 D a,(cid:96) Ψ N − (cid:96) + Ψ N +1 3 (cid:88) (cid:96) =1 χ a,(cid:96) ( u N +1 − u N − (cid:96) )+ i (cid:88) (cid:96) =1 A a,(cid:96) | Ψ N − (cid:96) | Ψ N +1 ,d u dτ = K d (cid:16) (cid:88) (cid:96) =1 W d,(cid:96) ( u (cid:96) − u ) + (cid:88) (cid:96) =1 χ d,(cid:96) ( | Ψ | + | Ψ (cid:96) | ) (cid:17) ,d u (cid:96) dτ = W ( u (cid:96) +3 − u (cid:96) ) + W d,(cid:96) ( u − u (cid:96) ) − χ d,(cid:96) ( | Ψ | + | Ψ (cid:96) | ) + χ ( | Ψ (cid:96) | + | Ψ (cid:96) +3 | ) ,(cid:96) = 1 , , ,d u n dτ = W ( u n +3 + u n − − u n ) + χ ( | Ψ n +3 | − | Ψ n − | ) , n = 4 . . . N − ,d u N − (cid:96) dτ = W ( u N − (cid:96) − u N − (cid:96) ) + W a,(cid:96) ( u N +1 − u N − (cid:96) )+ χ a,(cid:96) ( | Ψ N +1 | + | Ψ N − (cid:96) | ) − χ ( | Ψ N − (cid:96) | + | Ψ N − (cid:96) | ) , l = 1 , , ,d u N +1 dτ = K a (cid:16) (cid:88) (cid:96) =1 W a,(cid:96) ( u N − (cid:96) − u N +1 ) − (cid:88) (cid:96) =1 χ a,(cid:96) ( | Ψ N +1 | + | Ψ N − (cid:96) | ) (cid:17) , (11)where δ i,j is the Kronecker delta function. We now need to select the param-10ter values that best describe the α -helix. For the numerical modelling we need to use some numerical values of theparameters. We recall that, in particular, the parameter values for thepolypeptide macromolecules are: J Amide-I = 1 . × − Joules ≈ − eV; J e ≈ . − .
01 eV ≈ − − − Joules; χ = (35 −
62) pN; w = 39 − V ac = (3 . − . × m/s [10]. The molecular weights of large macro-molecules which participate in the electron transport chain in redox processesare: NADH-ubiquinone oxidoreductase - 980 kDa; cytochrome bc complex- 480 kDa; cytochrome c − aa oxidase - 420 kDa. The mass of Cyt-c is 12kDa, in which the hem-A group has a molecular weight 852 Da, and hem-Bgroup has 616 Da, which are 3-5 times larger than the molecular weight,100-200 Da, of amino-acids that form macromolecules. Studies of the mito-chondrial ET chain shows that the electrochemical potential for the transferof an electron is E e − c = +1 .
135 V [56, 57].For completeness of the study we also summarize the data on the param-eter values of other relevant compounds in accordance with the discussionin the introduction. The molecular weights of many conjugated polymersemiconductors vary in the interval (10 - 176) kDa, and the hole mobility is4 × − − . × − cm /(V s). The ionization potential and electron affinitypotential for some donor-acceptor copolymer semiconductor molecules are:(2.5-4.5) eV and (1.5-3.1) eV, respectively [58]. The electrochemical bandgap is E ( el ) g = E IP − E EA is 1.5 eV for BTTP, 1.84 eV for BTTP-P, and2.24 eV for BTTP-F, which are 0.4-0.6 eV larger from the optically deter-mined ones E ( opt ) g = 1 . − . E ex ≈ . − . . × − cm /(V s) in BTTP-T to 1 . × − cm /(Vs) in BTTP-F (see [26]). The reduction potentials of BTTP, BTTP-P, andBTTP-F are -1.4, -1.73, and -1.9 V (vs SCE), respectively. The oxidationpotentials of the copolymers are in the range 0.29-0.71 V (vs SCE). Theonset oxidation potential and onset reduction potential of the parent copoly-mer BTTP are 0.2 and -1.3 V, respectively, which give an estimate for theionization potential (IP, HOMO level) of 4.6 eV ( E IP = E onsetox + 4 .
4) and an11lectron affinity (EA, LUMO level) of 3.1 eV ( E EA = E onsetred + 4 . E IP value of BTTP is 0.3 eV lower than that of poly(3- hexylthiophene)(4.9 eV), whereas its E EA value (3.1 eV) is 0.6 eV higher than that reportedfor the poly(2,3-dioctylthieno[3,4-b]pyrazine) homo-polymer ( ≈ . E IP value of 4.64 eV and EEA value of 2.8 eV were found in the case ofBTTP-P [26].In what follows, we set the on-site electron energy level as the zero ofenergy, hence we take E = 0 . We are also using a set of model parametersclose to those encountered in polypeptide macromolecules or to the bridge-mediated donor-acceptor systems summarized above i.e. J = 8 . × − J , L = 1 . × − J , W = 10 .
59 kg / s , χ = 1 . × − J / m(12)corresponding to the following adimensional values of the parameters in ourequations J = 0 . , L = 0 . , W = 1 . , χ = 0 . . (13)The order of magnitude of these parameters values is close to the param-eter values for the electron transport in polypeptides and for other systemsdescribed above. Our aim, for these systems, is to establish a proof of conceptof the soliton mediated long-range ET rather than a performing a detailedstudy of their actual fine properties.Before studying the transfer of an electron from the donor to the accep-tor we have computed the profile of the static polaron on the helix for theparameters given in (13). This profile is shown on Fig. 2. To obtain thisprofile, we have relaxed the equations (11), using donor-acceptor parametervalues so that they do not interact with the chain.One sees clearly from Fig. 2, where the index i runs along the polypeptidehelix and where each curve corresponds to a different strand, that the staticpolaron is a broad localised lump which winds around the polypeptide chainrather than a single soliton located on a single strand or three identicalsolitons located on each of the strands. Having so far defined a model with a general set of couplings between the α -helix and the donor and acceptor, we will now restrict ourselves to 3 familiesof couplings.In the first set, the donor and the acceptor are coupled to all 3 strandsof the helix using identical coupling parameters. So we have D d, = D d, = D d, (cid:54) = 0 , D a, = D a, = D a, (cid:54) = 0 ,
25 50 75 100 125 150 175 i | i | i | i | a bFigure 2: Polaron with E = E d = E a = 0, J = 0 . L = D d, = D a, =0 . D d, = D d, = D a, = D a, = 0, W = W d, = W d, = W d, = W a, = 1 . W a, = W a, = 0, χ d,(cid:96) = 0 . χ d,(cid:96) = χ a,(cid:96) = 0, A a,ell = 0, K d = K a = 1. a) The electron probability densities are plotted versus theindex on the polypeptide chain. The 3 strands profiles are shown as separatecurves. b) The electron density along the α -helix backbone. W d, = W d, = W d, (cid:54) = 0 , W a, = W a, = W a, (cid:54) = 0 . (14)We call such a configuration the ‘full homogeneous’ coupling.The second configuration describes the case in which the donor and theacceptor are coupled to only one strand, so that D d, (cid:54) = 0 , D d, = D d, = 0 , D a, (cid:54) = 0 , D a, = D a, = 0 ,W d, (cid:54) = 0 , W d, = W d, = 0 , W a, (cid:54) = 0 , W a, = W a, = 0 ,A a, (cid:54) = 0 , A a, = A a, = 0 . (15)We call this the ‘single strand’ coupling. Notice that the donor is coupled tothe first peptide of the helix, i.e. , to the first peptide group of the first strand,but the acceptor is coupled to the second but last peptide of the helix, i.e. ,to the last peptide group on the same strand.For the third configuration we consider the case when the donor and theacceptor are coupled only to the first and last peptides on the alpha-helix so D d, (cid:54) = 0 , D a, (cid:54) = 0 , , W d, (cid:54) = 0 , W a, (cid:54) = 0 , A a, (cid:54) = 0 , (16)while the other parameters are equal to zero. We call this case the ‘end toend’ coupling.To find the best parameter values for the transfer of the electron fromthe donor to the acceptor, we have integrated the system of equations (11)13umerically on a lattice of 180 PGs. As the initial condition we have setthe electron probability density to 1 on the donor and to 0 everywhere else.We then integrated the equations (11) numerically up to τ = 500. Thistime was so chosen because it is roughly 3 times longer than it takes forthe polaron to reach the end of the 180-peptides chain. The value of | Ψ N +1 | varies with time, but tends to increase modulo some oscillations. To evaluatemax | Ψ N +1 | we have tracked its value during the evolution and recorded thelargest value obtained before τ ≤ α -helix. We then scanned a very large range ofparameter values for the acceptor to determine the one for which the maxi-mum value of the electron probability density on the acceptor, max | Ψ N +1 | ,reaches the largest value.We will now describe the results we have obtained for each type of cou-pling. The best parameter values we have found to generate a transfer of electronfrom the donor to the acceptor are (assuming all the values of A a,(cid:96) , D d,(cid:96) , D a,(cid:96) W d,(cid:96) , W a,(cid:96) , χ d,(cid:96) and χ a,(cid:96) are the same for (cid:96) = 1 , , E d = 0 . , D d,(cid:96) = 0 . J, W d,(cid:96) = 0 . W, χ d,(cid:96) = 0 . χ. (17) A a,(cid:96) = 0 . , E a = 0 . , D a,(cid:96) = 0 . J, W a,(cid:96) = 0 . W, χ a,(cid:96) = 0 . χ, and we have found that max | Ψ | = 0 .
896 for τ ≤ k = 0, while the other two non-symmetric lower energy bands are degener-ate and have their minima at respectively k = ± L/ ( √ J + L )). As aresult, solitons of the first type are formed by the electron from the higher14
25 50 75 100 125 150 175 i || i || i || a b c i || i || i || d e fFigure 3: Profile of | Ψ | for the full homogeneous coupling during the transferfrom donor to acceptor. a) τ = 25, b) τ = 50, c) τ = 100, d) τ = 150, e) τ = 200, f) τ = 500.energy band and have an energy, which is split from the higher energy bandbottom. On the other hand, the solitons of the second type have energieswhich are split from the degenerate energy band bottoms and are lower thanthe energy of the first type of soliton. More importantly, there is an ’hybrid’soliton formed by the entanglement (hybridization) of electron probabilitiesin the two lowest bands due to the Jan-Teller effect and this soliton has thelongest life-time. For the alpha-helix parameter values, which we use in oursimulations, this hybrid soliton has an energy which is almost 50 times lowerthan the energy of the first type of soliton. We can expect, and indeed wewill see in what follows, that the full homogeneous coupling provide the bestconditions for launching the hybrid soliton in the helix as it has the low-est energy and hence leads to the highest probability for the electron to betransported to the opposite end of the helix.The complex soliton-like wave generated in the helix after the electronhas tunnelled into it from the donor molecule corresponds, in our numericalsimulations, to this hybrid soliton. This hybrid soliton is not localized on asingle strand, instead it is distributed between the strands and propagatesalong the helix with some intrinsic oscillations, rather than along a particu-lar strand, a fact which reflects its hybrid nature. The propagation of this15ocalized polaron is followed by what looks like incoherent ripples. Theseripples describe the radiated sound waves in the helix. This is because oursystem is not completely integrable and while most of the initial electronenergy is transferred to the soliton, some of it is converted into oscillatingsoliton ’tails’.We have then studied how max | Ψ N +1 | varies when the acceptor param-eters are varied around their optimal value. This is shown in Figs. 4-9.To perform these simulations, we have defined the following parameters D a S = D a,(cid:96) J , W a S = W a,(cid:96) W , X a S = χ a,(cid:96) χ , (18)which relate the different parameters of the donor and the acceptor to thecorresponding ones on the peptide chain. A a m a x || Ea=-3Ea=-2Ea=-1Ea=-0.5Ea=0Ea=0.194Ea=0.25Ea=0.5Ea=1Ea=2Ea=3
Figure 4: Full homogeneous coupling. The plot of max( | Ψ N +1 | ) for τ ≤ A a for different values of E a and the parameters values (17).From Figs. 4 and 5, we first note that the value of the acceptor electronenergy E a has to be relatively small for the electron to be transferred to theacceptor, and that the values of E a and A a must be finely tuned for a good’capture’ of the electron. The parameters D a and W a and χ a , on the otherhand, offer a much broader tolerance when E a and A a are correctly tuned(see Figs. 6 to 9). This result has a clear physical meaning, since at this laststage of the electron transport the dominant parameters are the strength ofthe exchange interaction of the acceptor with the helix and the value of theon-site energy level on the acceptor, while on the other hand, the electron-lattice coupling and the elasticity of the acceptor-helix bond are much lessimportant. This being said, the last stage of the transport process is onlypossible if a proper soliton has been launched on the helix, carrying mostof the initial energy and electron probability to the acceptor with minimalenergy dissipation into the lattice vibrations and heat generation.16 E a m a x || Aa=0Aa=0.25Aa=0.5Aa=0.62Aa=0.75Aa=1Aa=2Aa=4
Figure 5: Full homogeneous coupling. The plot of max( | Ψ N +1 | ) for τ ≤ E a for different values of A a, = A a, = A a, = A a and theparameters values (17). D a S m a x || Aa=0Aa=0.25Aa=0.5Aa=0.62Aa=0.75Aa=1Aa=2Aa=4
Figure 6: Full homogeneous coupling. The plot of max( | Ψ N +1 | ) for τ ≤ D a S = D a,(cid:96) /J for different values of A a, = A a, = A a, = A a and the parameters values (17).As we will see in the next subsections, the effectiveness of the solitonformation and its parameters are determined by (i) the helix parameters,mainly by the electron-lattice coupling and strand elasticity, and (ii) by thenumber of helix strands coupled to the donnor. In this section we couple the donor only to the first node of the chain: D d, = D d, = W d, = W d, = χ d, = χ d, = 0. We obtain the best transfer from thedonor to the chain for the following donor parameters: E d = 0 . , D d, = 0 . J, W d, = 0 . W, χ d, = 0 . χ. (19)17 .00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 D a S m a x || Ea=-3Ea=-2Ea=-1Ea=-0.5Ea=0Ea=0.194Ea=0.25Ea=0.5Ea=1Ea=2Ea=3
Figure 7: Full homogeneous coupling. The plot of max( | Ψ N +1 | ) for τ ≤ D a S = D a,(cid:96) /J for different values of E a and the parametersvalues (17). W a S m a x || Aa=0Aa=0.25Aa=0.5Aa=0.62Aa=0.75Aa=1Aa=2Aa=4
Figure 8: Full homogeneous coupling. The plot of max( | Ψ N +1 | ) for τ ≤ W a S = W a,(cid:96) /W for different values of A a, = A a, = A a, = A a and the parameters values (17).Such conditions for creation of the soliton in the helix are not the optimalones, since in the soliton formation there will be two contradicting tendencies:redistribution of the electron between the three peptide groups within thesame unit cell and its dispersion to the nearest unit cell. These processes willbe accompanied by energy dissipation much stronger than in the case of thefull homogeneous coupling, and thus will result in the generation of a muchweaker soliton i.e., in a less efficient transport of the electron along the helix.Similarly, the type of coupling between the helix and the acceptor playsan important role in the electron transport, as we will see from this and thenext sub-section. First, we consider the coupling of the acceptor to the samestrand as the one to which the donor is coupled (single-strand coupling),setting A a, = A a, = D a, = D a, = W a, = W a, = χ a, = χ a, = 0.18 .00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 X a S m a x || Aa=0Aa=0.25Aa=0.5Aa=0.62Aa=0.75Aa=1Aa=2Aa=4
Figure 9: Full homogeneous coupling. The plot of max( | Ψ | ) for τ ≤
500 asa function of χ a S = χ a,(cid:96) /χ for different values of A a, = A a, = A a, = A a and the parameters values (17).We found that the best parameters to obtain a transfer of the electron tothe acceptor are E d = 0 . , D d, = 0 . J, W d, = 0 . W, χ d, = 0 . χ, (20) A a, = 6 . , E a = 0 . , D a, = 0 . J, W a, = 0 . W, χ a, = χ. As one could expect, the maximum value of the electron probability onthe acceptor | Ψ N +1 | for τ ≤
500 in this case is much lower, than in the fullhomogeneous case, and is equal to 0 . | Ψ N +1 | aroundthese optimal values of the parameters. To couple the acceptor to the last peptide of the helix, we take A a, = A a, = D a, = D a, = W a, = W a, = χ a, = χ a, = 0. In this case we have obtainedthe best transfer using the following parameters: E d = 0 . , D d, = 0 . J, W d, = 0 . W, χ d, = 0 . χ, (21) A a, = 1 . , E a = 0 . , D a, = 0 . J, W a, = 0 . W, χ a, = 0 . χ, and, with this choice, we have found that max | Ψ N +1 | = 0 . | Ψ N +1 | varies when the acceptor param-eters are varied around their optimal value. This is shown in Figs. 10 to15. 19 A a , 3 m a x || Ea=-3Ea=-2Ea=-1Ea=-0.5Ea=0Ea=0.1Ea=0.27Ea=0.276Ea=0.28Ea=0.5Ea=1Ea=2Ea=3
Figure 10: End-to-end coupling. The plot of max | Ψ N +1 | for τ ≤
500 as afunction of A a, for different values of E a and the parameters values (22). E a m a x || Aa3=0Aa3=1Aa3=1.98Aa3=3Aa3=4Aa3=5
Figure 11: End-to-end coupling. The plot of max | Ψ N +1 | for τ ≤
500 as afunction of E a for different values of A a, and the parameters values (22).As with the full homogeneous coupling, we have found that the absorptionis mainly controlled by a fine tuning between A a, and E a but that there isa broader tolerance for the values of D a , W a and χ a .Having analysed the parameter stability of our model we now turn to thestudy of its thermal stability. 20 .0 0.2 0.4 0.6 0.8 1.0 D a S m a x || Aa3=0Aa3=1Aa3=1.98Aa3=3Aa3=4Aa3=5
Figure 12: End-to-end coupling. The plot of max | Ψ N +1 | for τ ≤
500 as afunction of D a S = D a,(cid:96) /J for different values of = A a, and the parametersvalues (22). D a S m a x || Ea=-2Ea=-1.5Ea=-1Ea=-0.5Ea=0Ea=0.1Ea=0.27Ea=0.276Ea=0.28Ea=0.5Ea=1Ea=1.5Ea=2
Figure 13: End-to-end coupling. The plot of max | Ψ N +1 | for τ ≤
500 asa function of D a S = D a,(cid:96) /J for different values of E a and the parametersvalues (22). So far, in the study of our model, we have not taken into account any thermalfluctuations. To include them we have modified the model by adding thefollowing Langevin terms to the equations for U n : L n = F n ( τ ) − Γ du n dτ , (22)where Γ is an absorption parameter and F n ( τ ) represents the thermal noisemodelled as a Gaussian white noise of zero mean value and variance given21 .000 0.002 0.004 0.006 0.008 0.010 W a S m a x || Aa3=0Aa3=1Aa3=1.98Aa3=3Aa3=4Aa3=5
Figure 14: End-to-end coupling. The plot of max | Ψ N +1 | for τ ≤
500 as afunction of W a S = W a,(cid:96) /W for different values of A a, and the parametersvalues (22). X a S m a x || Aa3=0Aa3=1Aa3=1.98Aa3=3Aa3=4Aa3=5
Figure 15: End-to-end coupling. The plot of max | Ψ N +1 | for τ ≤
500 asa function of χ a S = χ a,(cid:96) /χ for different values of A a, and the parametersvalues (22).by (cid:104) F n ( τ ) F m ( τ ) (cid:105) = 2Γ kT δ ( τ − τ ) δ n,m , (23)where, k is Boltzmann coefficient, and for the dimensional thermal energy kT , we have kT = kT / ¯ hν . To implement this numerically, F ( τ ) has to bekept constant during each time step dτ and so we have used δ ( τ − τ ) = 1 /dτ .For each temperature, we have performed 100 simulations and computedthe mean values of max | Ψ a | , for τ ≤ kT ≈ . eV which in our adimensionalunits corresponds to 7 .
12. We have thus varied kT between 0 and 10 tocapture the physiological conditions when J is smaller than 0 . eV . kT m a x || N=15N=18N=21N=24N=27N=30
Figure 16: Full homogeneous coupling. The plot of max | Ψ N +1 | for τ ≤ kT for different values of the chain length N . Γ = 0 . | Ψ N +1 | as a function of tem-perature for different chain lengths. We see that for short chains, the tem-perature has a minimal effect while for longer chains, its influence is morepronounced.It is worth to recall that although large proteins in electron transportchains can consist of up to a few thousand aminoacids, the α − helical partsof their globular structure consist of up to N = 50 −
70 peptide groups.In trans-membrane proteins the α -helices are even shorter, with N = 30 oreven smaller. Moreover, in the biggest enzymes of electron transport chain,like NADH ubiquinone oxidoreductase, which is the first and the biggestprotein complex of the respiratory chain, there is a whole pathway for theelectron transport prior to the ubiquinone reduction via several iron sulfurclusters, connected by relatively short α -helices (see [8]). So we conclude,that under physiological conditions, the transfer of the electron from a donorto an acceptor is thermally stable.Looking at the data in Fig. 17 we see that the probability of an electrontransfer from the donor to the acceptor is relatively constant when Γ < m a x || kT=0.5kT=1kT=2kT=5kT=7kT=10 10 m a x || kT=0.5kT=1kT=2kT=5kT=7kT=10 a b m a x || kT=0.5kT=1kT=2kT=5kT=7kT=10 10 m a x || kT=0.5kT=1kT=2kT=5kT=7kT=10 c dFigure 17: Full homogeneous coupling. The plot of max | Ψ N +1 | for τ ≤ kT . a) N = 15, b) N = 18, c) N = 21, d) N = 24.0 . In this paper we have presented a model describing the long-range transportof an electron from a donor molecule to an acceptor one via the nonlinearstate of a large polaron (soliton-like state) formed in a α -helical proteinin a ‘Donor – α -helix – Acceptor’ system. Conventionally, we model the α -helix as a polypeptide chain, twisted in a helix, in which each peptide24roup is coupled to its nearest neighbours by a chemical bond and to every3 rd neighbour by a hydrogen bond. The helix can thus be described as3 parallel strands coupled to each other. We have found that the staticpolaron on such a helix, for the parameters that describe AMID I vibrationin α -helical protein, is a relatively broad localized hump extended over thepolypeptide macromolecule in agreement with other studies (see, e.g., [11,13]). In our model we have only taken into account one phonon mode, whilein real proteins there are many other phonon modes, the interaction withwhich results in bigger value of the effective electron-lattice coupling, and,hence, in stronger soliton localization than obtained here.We have then studied the transfer of an electron from a donor moleculeto the acceptor by initially placing the electron on the donor. For the properparameters of the couplings, the electron was, within a very short time in-terval, transferred onto the polypeptide chain where it was self-trapped ina polaron state, and then moved towards the other extremity of the chainwhere it was absorbed by the acceptor.We have considered three types of couplings between the donor and thepolypeptide chain as well as between the acceptor and the polypeptide chain.In the first case, the donor and the acceptor where coupled, respectively, tothe first 3 and the last 3 nodes of the chain, using the identical parametersand we called such a configuration the ‘fully homogeneous’ one. In the secondconfiguration, the donor was coupled to the first node of the chain and theacceptor to the last node of the same strand or to the last node of the helix.We called such couplings ‘single-strand’ and ’end-to-end’ ones, respectively.The fully homogeneous coupling is the one that leads to the best donor-acceptor electron transport with an efficiency of 90% or more depending onthe length of the chain. The ‘end-to-end’ coupling did not work so well, butstill led to a transfer probability of up to 60% while the ‘single-strand’ onewas the worst leading only to a 20% probability transfer. These results canbe explained from the dependence of the efficiency of the soliton generationnot only on the actual parameter values of the system, but also on the type ofcouplings between the helix and the donor and the acceptor. If one uses theinverse scattering theory for integrable system, applied to the time evolutionof certain initial conditions for the nonlinear Schr¨odinger equation that ap-proximates Davydov solitons [16, 17], these different couplings translate intospecific initial conditions which leads to families of solitons with differentefficiencies.Our study has shown that an electron in the polaron (soliton-like) statecan easily propagate as a travelling wave along the α -helical chain. Thepolaron that is generated in the helix in the vicinity of the donor molecule hasa complex internal structure: it is not just a clean simple polaron localized on25 single strand; instead, it is distributed between the strands and propagatesalong the helix with some intrinsic oscillations, rather than along a particularstrand, which reflects its collective hybrid nature.Unfortunately, the exact value of the exchange interaction for extra elec-tron in proteins is not known, but can be roughly estimated at 0.05 - 0.1 eV,comparable with J , and the other parameters are the same as for AMID Ivibration, which were used in our model. Other related manufactured Donor- α -helix Acceptor systems, described in the introduction, have parametervalues close to the considered here, so we can conclude that our model forthe long-range electron transport describes these systems as well.Our results explain the experimental evidence that the donor and acceptorparameters, as well as the type of their coupling affect the electron transportin ’Donor – α -helix – Acceptor’ systems (see [61, 62]).We have also shown that when we add thermal fluctuations to the model,the long range electron transfer in the ‘Donor – α -helix – Acceptor’ system isstable at physiological temperatures. We have thermalized a mixed quantumclassical system in a way which makes the quantum part behave classicallyas well. According to [63] this results in a broadening of the soliton wavefunction in the helix and in a decrease of the binding energy of the soli-ton. This, in turn, results in a lower stability of the soliton with respectto any perturbation, including thermal fluctuations, compared to a properanalysis of the thermal stability. Moreover, as shown in [64], accounting fortemperature fluctuations in the equation for lattice displacements within aquantum-mechanical description results in an effective decrease of the reso-nant interaction energy by the exponential Debay-Waller factor. This thenleads to a decrease of the spatial dispersion of the electron and an increase ofthe electron-lattice coupling which itself results in an increase of the bindingenergy of the soliton and, as a result, a higher thermal stability compared toour model. A proper analysis of thermal stability of electron transport wouldrequire a more rigorous treatment and be the topic of a paper on its own. Inthis paper we have decided to restrict ourselves to the simplest analysis. One of us, LSB, acknowledges the partial support from the budget programKPKVK 6541230 and the scientific program 0117U00236 of the Departmentof Physics and Astronomy of the National Academy of Sciences of Ukraineand thanks the Department of Mathematical Sciences of the University ofDurham for the hospitality during her short-term visit. WJZ thanks theLeverhulme Trust for his grant EM-2016-007.26 eferences [1] J. Jortner, M. Bixon, eds
Electron Transfer From Isolated Molecules toBiomolecules . Adv. Chem. Phys., . John Wiley and Sons, Inc., NewYork, NY, (1999)[2] Voet D., Voet J.G .
Biochemistry (3rd ed.). John Wiley and Sons. (2004).ISBN 978-0-471-58651-7.[3] Frster, Theodor (1948). Annalen der Physik (in German). 437 (12):5575. Bibcode:1948AnP...437...55F. doi:10.1002/andp.19484370105[4] M. Kasha. In: M. Kasha ed.,
Physical and Chemical Mechanisms inMolecular Radiation Biology . Springer, p 231-255. (1992).[5] Jones, Garth A; Bradshaw, David S (2019). Frontiers in Physics. 7:100.doi:10.3389/fphy.2019.00100[6] Murray, Robert K.; Daryl K. Granner; Peter A. Mayes; Victor W. Rod-well (2003). Harper’s Illustrated Biochemistry. New York, NY: LangeMedical Books/ MgGraw Hill. p. 96. ISBN 0-07-121766-5.[7] T. Althoff, D. J. Mills, J.-L. Popot, and W. Khlbrandt. The EMBOJournal, (2011) Solitons in Molecular Systems (Dordrecht, Reidel,(1985)).[10] A.C.Scott.
Phys. Rep. ,
1, (1992).[11] A. S. Davydov, A. A. Eremko, and A. I. Sergienko, Ukr. J. Phys. 23,983, (1978)[12] V. K. Fedyanin and L. V. Yakushevich, Int. J. Quantum Chem. 21, 1019(1982).[13] L.S. Brizhik, A.A. Eremko, B. Piette W.J. Zakrzewski. Phys. Rev. E 70,031914, p.1-16, (2004).[14] L.S. Brizhik, A.A. Eremko, Z. Phys. B, 104, 771-775, (1997).[15] L. Brizhik, B. Piette, W. Zakrzewski. Phys.Rev. E 90 052915, (2014)DOI: http://dx.doi.org/10.1103/PhysRevE.90.0529152716] L.S. Brizhik, A.S. Davydov, Phys. Stat. Sol. (b), Bioenergetics 3 . Academic Press.ISBN 978-0-12-518121-1.[19] C.W.F. McClare. Ann. N.Y. Acad. Sci. (30), pp80518052, (1995). DOI: 10.1021/ja00135a038[26] Y. Zhu , R. D. Champion , S.A. Jenekhe , Macromolecules, (25),87128719, (2006); DOI: 10.1021/ma061861g[27] D. Li, C. Sun, H. Li, H. Shi, et al Chem. Sci., , 4587-4594, (2017),DOI:10.1039/C7SC00077D[28] Q. Van Nguyen, P. Martin, D. Frath, et al. J. Am. Chem. Soc,1403210131-10134, Publication Date: July 30, (2018)[29] L. Tian, Z. Hu, X. Liu,et al, ACS Appl. Mater. Interfaces. (5) 5289-5297, (2019). doi: 10.1021/acsami.8b19036.[30] H. Li, F.S. Kim, G. Ren, and S.A. Jenekhe, J. Am. Chem. Soci. 135(40), 14920-14923, (2013) 2831] H. A. M. van Mullekom, J. A. J. M. Vekemans, E. E. Havinga, and E.W. Meijer, Mater. Sci. Eng. 32, 1, ( 2001).[32] Y. Zhu, R. D. Champion, and S. A. Jenekhe, Macromolecules, 39, 8712,(2006).[33] G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, Science270, 1789, (1995).[34] L.M. Campos, A. Tontcheva, S. G¨unes, G. Sonmez, Neugebauer, N. S.Sariciftci, and F. Wudl, Chem. Mater. 17, 4031, (2005).[35] M. Svensson, F. Zhang, S. C. Veenstra, W. J. H. Verhees, C. Hummelen,J. M. Kroon, O. Inganas, and M. R. Andersson, Adv. Mater. 15, 988,(2003).[36] S. Admassie, O. Inganas, W. Mammo, E. Perzon, and M. R. Andersson,Synth. Met. 156, 614 (2006).[37] A. P. Kulkarni, Y. Zhu, and S. A. Jenekhe, Macromolecules 38, 1553,(2005).[38] C. Ego, D. Marsitzky, S. Becker, J. Zhang, A. C. Grimsdale, Mullen, J.D. MacKenzie, C. Silva, and R. H. Friend, J. Am. Chem. Soc. 125, 437,(2003)[39] Thompson, B. C., Madrigal L. G., Pinto M. R., Kang T.-S., SchanzeK. S., Reynolds J. R., J. Polym. Sci., Part A: Polym. Chem. 43, 1417,(2005).[40] Wu W.-C., Liu, C.-L., Chen, W.-C., Polymer, 47, 527, (2006).[41] Babel, J. D. Wind, and S. A. Jenekhe, Adv. Funct. Mater. 14, 891,(2004).[42] T. Yamamoto, T. Yasuda, Y. Sakai, and S. Aramaki, Macromol. RapidCommun. 26, 1214, (2005).[43] T. Yasuda, Y. Sakai, S. Aramaki, and T. Yamamoto, Chem. Mater. 17,6060, (2005)[44] Sh. Chen, K. C. Lee, Z.-G. Zhang, et al., Macromolecules
30 9617-9626, (2015).[48] P. G. M. Mileo, K. Adil, L. Davis, et al., J. Am. Chem. Soc. (33), 4481-4485,(2016). doi.org/10.1002/smll.201601112[54] B. Akdim, R. Pachter, R.R. Naik. Self-Assembled Peptide Nanotubes asElectronic Materials: An Evaluation from FirstPrinciples Calculations.Appl. Phys. Lett. (18), 183707, (2015).[55] A.S. Davydov and A.D. Suprun. Ukr. J. Phys.
44, (1974).[56] G.S. Engel, T.R. Calhoun, E.L. Read, T.K. Ahn, T. Mancal, Y.C.Cheng, R.E. Blankenship and G.R. Fleming. Nature α - helical membrane protein, pharaonis halorhodopsin,depends on thickness of gold films utilized for surface-enhanced infraredabsorption spectroscopy. Chem. Phys.56