Validity Examination of the Dissipative Quantum Model of Olfaction
aa r X i v : . [ phy s i c s . c h e m - ph ] J a n Validity Examination of the Dissipative Quantum Model of Olfaction
Arash Tirandaz, Farhad Taher Ghahramani, and Vahid Salari
2, 3, ∗ School of Biological Sciences, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran Department of Physics, Isfahan University of Technology, Isfahan P.O. Box 84156-83111, Iran
The validity of the dissipative quantum model of olfaction has not been examined yet and thereforethe model suffers from the lack of experimental support. Here, we generalize the model and proposea numerical analysis of the dissipative odorant-mediated inelastic electron tunneling mechanism ofolfaction, to be used as a potential examination in experiments. Our analysis gives several predictionson the model such as efficiency of elastic and inelastic tunneling of electrons through odorants,sensitivity thresholds in terms of temperature and pressure, isotopic effect on sensitivity, and thechiral recognition for discrimination between the similar and different scents. Our predictions shouldyield new knowledge to design new experimental protocols for testing the validity of the model.
INTRODUCTION
Olfaction seems to be an immediate and intimate sense but surprisingly the mechanism is still not well understood.This is an important problem in its own right for both fundamental science and industry [1–5]. The olfactory system inhuman beings is triggered by binding the small, neutral, and volatile molecules known as odorants to specific sites onolfactory receptors (ORs) in the nasal cavity (see Figure 1). Despite considerable knowledge of structure of ORs, thedetailed molecular mechanisms for discrimination between different odorants are not yet fully understood [6]. In 1963,Amoore conjectured that such molecular mechanism is primarily related to the shape of the odorant and accordinglyit is initiated by a mutual structural fit between the odorant and ORs (i.e. lock and key model) [7]. In fact, the ideawas motivated from the molecular mechanism of the enzyme behaviour. The model can be modified by introducing adistortion of the whole system to induce a more appropriate mutual fit (i.e. hand and glove model). A more refineddemonstration of the idea requires that ORs respond to only one structural feature, such as a functional group, insteadthe main body of the odorant (i.e. odotope model) [8]. There is plenty of evidence for cases where the structure doesseem important to an odorant’s detection (e.g. see [9, 10]). Despite the predictive power of these structure-basedmodels, there are some evidence against them: odorants that smell similarly whilst being structurally different (e.g.benzaldehyde versus hydrogen cyanide), and odorants that smell differently whilst being structurally the same (e.g.ferrocene versus nickelocene) [11–13]. All such shape-based models are primarily based on mechanical mechanisms.The quantum model of olfaction, which was firstly proposed by Dyson [14] and refined by Wright [15], is based onthe idea that the signature of scent is due to the odorant’s unique vibrational spectrum not its structure. An uniquescent is attributed to its unique spectrum in the same way a colour is associated to its unique frequency of light.
FIG. 1: A scheme for the sense of smell in which odorants are absorbed by odorant receptors (ORs) in the olfactory receptorcells in the nasal cavity. In the quantum model, each odorant can be simulated as an asymmetric double-well potential forodorant recognition. The signal transduction relies on the success of an electron tunneling from a donor site of an OR to anacceptor site of the same or another OR, facilitated by a vibrational transition in the odorant according to the energy differencebetween the donor and acceptor sites.
Motivated by the phenomena of inelastic electron tunneling (ET) in metals [16, 17], Turin proposed that themechanism of olfactory detection is an odorant-mediated biological inelastic ET [18]. The signal transduction relieson the success of an ET from a donor (D) site of an OR to an acceptor (A) site of the same or another OR, facilitatedby a vibrational transition in the odorant corresponding to the energy difference between D and A sites. Recently,Brookes and co-workers formulated the idea in a semi-classical model to show that such a mechanism fits the observedfeatures of smell [19]. They found that the rate of odorant-mediated inelastic ET and the elastic ET can be drasticallydifferent and surprisingly the inelastic ET is the dominant process for certain parameters of the model. The evidencesupporting the vibration-based mechanism was obtained from sophisticated quantum chemistry calculations [20, 21].Ch¸eci´nska and co-workers examined the dissipative role of the environment dynamically in vibration-based model andshowed that the strong coupling to the environment can enhance the frequency resolution of the olfactory system [22].The main evidence against this theory is given to be the differentiable smells of chiral odorants: they have identicalspectra in an achiral solution, but they have different smells [12]. Recently, we addressed this problem using themaster equation approach and showed that the chiral recognition in olfaction might rely on the detection of the chiralinteractions between the chiral odorant and ORs [23]. A major prediction of the vibration-based theory is the isotopeeffect: i.e. isotopes should smell differently. Recent behavioural experiments have revealed that fruit flies [24, 25],honeybees [26] and humans [27, 28] can distinguish isotopes. Yet, experimental evidence against isotopic discriminationkeeps the debate open [29, 30].In this paper, we examine the physical plausibility of the odorant-mediated inelastic ET model of olfaction. Wefocus on a typical vibrational degree of freedom of odorant molecule, known as contorsional vibration, in which anatom or a group of atoms oscillates between the two wells of the potential energy surface [31]. Such a vibration istypically modeled by the motion of a particle in a double-well potential. The biological environment is convenientlyrepresented as a collection of harmonic oscillators. We examine the dynamics of the odorant by using the time-dependent perturbation theory and thereby obtain the corresponding elastic and inelastic ET rates for all possibletransitions of the odorant. To test the physical limitations of the model, we analyze the rates in different limits ofmolecular and environmental variables. For simplicity we set ~ = 1 throughout the paper. MODEL
We focus on three parts of the system as the main components of the olfaction model: (1) the odorant , (2) the electron which tunnels through the odorant, and (3) the surrounding environment . In the original vibrational modelof olfaction, the relevant vibrational mode is represented by a simple harmonic oscillator [19]. Here, we consider amore realistic vibrational mode of non-planer odorant, known as contorsional mode, in which an atom or a group ofatoms oscillates between the left and right wells of a double-well potential. Unlike the harmonic mode the contorsionalmode can be used to charachterize the olfactory chiral reconition [23]. Thus, we model the odorant as an asymmetricdouble-well potential (see Figure 1). The minima of the potential correspond to the left- and right-handed states, | L i and | R i , of the odorant (see Figure 2). The handed states can be inter-converted by the quantum tunneling throughthe barrier V . In the limit V ≫ ω ≫ k B T ( ω is the vibration frequency at the bottom of each well), the statespace of the odorant is effectively confined in a two-dimensional Hilbert space spanned by two handed states. Such anapproximation works properly for a large class of odorants even in the high-temperature limit [31, 32]. The odorant’sHamiltonian can then be expanded by the handed states as ˆ H od = − ω z ˆ σ z + ω x ˆ σ x where ˆ σ i is the i -component of Paulioperator and ω x and ω z are the tunneling and asymmetry frequencies, respectively. The tunneling frequency ω x canbe calculated from the WKB method as ω x = Aq √ M ω exp( − BV /ω ) [33], where M is the molecular mass and q is the distance between two minima of the potential. The value of the parameters A and B depends on the explicitmathematical form of the potential, but it can usually be approximated by 1 [33, 34]. The asymmetry is due to thefundamental parity-violating interactions [35, 36] and the chiral interactions (i.e. interactions that are transformedas pseudoscalars [37]) between the odorant and environmental molecules. The former is typically small but the lattercan be significant especially between a chiral odorant and ORs. The eigenstates of the odorant’s Hamiltonian can bewritten as the superposition of handed states as | E i = sin θ | L i + cos θ | R i and | E i = cos θ | L i − sin θ | R i where wedefined θ = (1 /
2) arctan( ω x /ω z ).The electron tunnels through the odorant from a donor state | D i with energy ε D to an acceptor state ε A withenergy ε A . We then describe the electron with Hamiltonian ˆ H e = ε A | A ih A | + ε D | D ih D | . The biological environmentis typically modeled as a collection of harmonic oscillators with Hamiltonian ˆ H env = P i ω i ˆ b † i ˆ b i where b † i and b i arethe creation and annihilation operators for modes of frequency ω i in the environment.The interaction Hamiltonian has three contributions: between donor and acceptor of the receptor with tunnelingstrength ∆, between the donor (acceptor) and the odorant with coupling frequency γ D ( γ A ), and between the donor(acceptor) sites and i -the harmonic oscillator of the environment with coupling frequency γ iD ( γ iA ). Thus, theinteraction Hamiltonian of the whole system is given by [19]ˆ H int = ∆( | A ih A | + | D ih D | ) + ( γ D | D ih D | + γ A | A ih A | ) ⊗ ˆ σ x + X i ( γ i,D | D ih D | + γ i,A | A ih A | ) ⊗ (ˆ b † i + ˆ b i ) (1)The total Hamiltonian characterizes the time evolution of whole system by which the ET rates are calculated. Thedetails of the dynamics are explained in the Methods section. RESULTS
Tunneling Rates
When the electron tunnels through the odorant, three type of transitions might take place in theodorant (FIG.2): | L i → | E i , | R i → | E i and | E i −→ | E i . FIG. 2: Possible transitions of the odorant described as a particle in an asymmetric double-well potential.
The ET rates are obtained from the corresponding probabilities (see Methods) by using Γ i → j = dP r i → j /dt . If we set t = 0, in the high-temperature limit where the environment is presumably in thermal equilibrium, the inelastic ETrates are given byΓ D,L → A,E = ∆ r πk B T J λ cos( θ + υ ) h cos θ cos υ exp n − ( ǫ − J λ + ( η D − η A )) k B T J λ o − sin θ sin υ exp n − ( ǫ − J λ − ( η A + η D ))4 k B T J λ oi (2)Γ D,R → A,E = ∆ r πk B T J λ sin( θ + υ ) h sin θ cos υ exp n − ( ǫ − J λ − ( η D − η A )) k B T J λ o + cos θ sin υ exp n − ( ǫ − J λ + ( η A + η D )) k B T J λ oi (3)Γ D,E → A,E = ∆ r πk B T J λ ( sin ( υ ) h cos θ exp n − ( ǫ − J λ − ( η A + η D )) k B T J λ o + sin θ exp n − ( ǫ − J λ + ( η A + η D )) k B T J λ oi + 14 sin θ sin( υ ) h exp n − ( ǫ − J λ − ( η D − η A )) k B T J λ o − exp n − ( ǫ − J λ + ( η D − η A )) k B T J λ oi) (4)where υ = [tan − ( ω x + γ A ω z ) + tan − ( ω x − γ D ω z )]. The elastic ET coincides with the situation where the electron tunnelsfrom the donor site to the acceptor site, without any transition in the odorant. This situation can be considered to beequivalent to the ET in the absence of the odorant [22]. The elastic ET rate in the high-temperature limit is given byΓ D → A = ∆ r πk B T J λ exp n − ( ǫ − J λ ) k B T J λ o (5) Physical Parameters
To examine the obtained ET rates quantitatively we first analyze the parameters of the model.Since we aim to examine the model in experiment, the ET rates should be analyzed in terms of controllable parameters(aka variables). These variables include odorant’s parameters (tunneling frequency ω x and asymmetry frequency ω z ),and thermodynamical parameters of the environment (temperature and pressure). The parameters with interactioncharacter naturally depend on the odorant’s parameters. The energy conservation requires that the energy gap betweenthe donor and acceptor sites ε be close to the mean value of energy gap between odorant’s states. Thus we assumethat ε ≃ p ω x + ω z . The coupling between donor and acceptor sites of OR(s) is weak in comparison with the naturalfrequency of the odorant, so we estimate ∆ ≃ . p ω x + ω z Hz [23]. The coupling frequency between the DA pairand odorant, calculated from the Huang-Rhys factor [19], is approximated as γ D = − γ A ≈ . p ω x + ω z [23]. Sincethe biological environment is microscopically uncontrollable, the parameters of the corresponding spectral density isconsidered as mere parameters. The most common biological environment is water. The parameters of an aqueousenvironment can be estimated as J ≈ λ ≈ Hz [38, 39]. Odorant Analysis
We examine the ET rates for different odorants in terms of their molecular parameters, ω x and ω z . The magnitude of tunneling frequency ω x , ranging from the inverse of the lifetime of the universe to millions ofhertz, can be extracted from the spectroscopic data [40]. The asymmetry frequency of the odorant, ω z , representsan overall measure of all chiral interactions involved. For our system, these interactions are primarily due to theintermolecular interactions between the odorant and ORs. The magnitude of intermolecular interactions can inprinciple be determined by using quantum chemistry calculations. The dependence of the ET rates to ω x and ω z fordifferent transitions of the odorant is plotted in Figure 3. This clearly shows that the vibrational model based onodorant-mediated inelastic ET is improbable for a wide range of odorants. (a) (b) (c) FIG. 3: The inelastic-to-elastic ratio versus tunneling frequency ω x and asymmetry frequency ω x at biological temperature T = 310 K for transitions (a) L → E , (b) R → E , and (c) E → E . Since ORs are chiral structures, we assume that the contribution of chiral interactions is significant. At a fixed highmagnitude of asymmetry parameter ω z , the inelastic-to-elastic ratio against the tunneling frequency ω x are plottedin Figure 4 for different transitions of the odorant. Two different behaviors can be identified here; in the asymmetry-dominant limit, ω x < ω z , although different transitions exhibit different dependencies on ω x , the inelastic ET is notdominant. In the tunneling-dominant limit, ω x ≥ ω z , however, for all transitions the inelastic ET is dominant. In otherwords, for each transition there is a threshold of tunneling frequency ω x in the bottom limit in which the olfactorysystem cannot recognize the odorant. This fact can be used to examine the model in experiment. Temperature Analysis
The temperature dependency of different transitions of the odorant are essentially similar.At a fixed high magnitude of asymmetry parameter ω z , for the transitions L → E , R → E , and E → E theinelastic-to-elastic ratio versus the tunneling frequency ω x are plotted in Figure 5 for different temperatures of theenvironment. For each odorant (with a fixed tunneling frequency ω x ) there is a threshold for temperature in thebottom limit in which the olfactory system cannot recognize the scent. This fact can also be used to examine themodel in experiment. → D ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o (a) → D ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o (b) → D ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o (c) FIG. 5: The inelastic-to-elastic ratio versus the tunneling frequency ω x at temperatures T = 300K, 310K, and 320K fortransitions (a) L → E , (b) R → E , and (c) E → E . All figures are plotted at ω z = 10 Hz. L → E R → E E → E A → D ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o FIG. 4: The inelastic-to-elastic ratio versus the tunneling frequency ω x at biological temperature T = 310 K for transitions L → E , R → E and E → E . All figures are plotted at ω z = 10 Hz . Pressure Analysis
The concentration of the odorant in the condensed environment is proportional to its pressure.The tunneling frequency of odorants is related to the pressure of the environment. To illustrate the pressure dependencyof the odorant’s dynamics, we focus on the ammonia molecule
N H as odorant. In the low-pressure limit, the tunnelingfrequency of ammonia, known as inversion frequency, is estimated as ω x ≃ . × Hz [41], and at P ≃ atm , ω x shifts to zero. This phenomenon is theoretically demonstrated in the context of the mean-field theory [42]. Thepressure dependency of the inversion frequency is given by ω ′ x = ω x p − P/P cr where the critical pressure P cr isapproximately 1 . atm at room temperature. If P → P cr , then ω ′ →
0, and accordingly the handed states becomeeigenstates of the molecular Hamiltonian. In the high-pressure limit, the relevant transition in the odorant would be | R i → | L i (see Figure 2). The inelastic ET rate according to this transition is obtained asΓ D,R → A,L = ∆ r πk B T J λ sin υ exp n − ( ǫ − J λ + ( η + η )) k B T J λ o (6)The inelastic-to elastic ratio versus the tunneling frequency ω x in the high-pressure limit is plotted in Figure 6 for ω z = 1 THz, 5 THz, and 10 THz respectively. L → RA → D (cid:3) (cid:4)(cid:5) (cid:7)(cid:8)(cid:9) ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o (a) L → RA → D (cid:10) (cid:11)(cid:12) (cid:13)(cid:14) (cid:15)(cid:16)(cid:17) ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o (b) L → RA → D (cid:18) (cid:19)(cid:20) (cid:21)(cid:22) (cid:23)(cid:24)(cid:25) ( THz ) I n e l as t i c - t o - E l as t i c Tunn e li ng R a t i o (c) FIG. 6: The inelastic-to-elastic ratio versus the tunneling frequency ω x in the high-pressure limit for transition DR → AL for(a) ω z = 1 THz, (b) ω z = 5 THz, and (c) ω z = 10 THz. In the high-pressure limit, the inelastic ET is always ineffective (see Figure 6). This provides another empirical testfor probing the validity of the tunneling model of olfaction. Specially, we predict that at P ≥ P cr the olfactory systemis unable to recognize the smell if the tunneling mechanism is at work. Isotopic Effect
The isotopic dependency of ET rates can be followed from the mathematical form of criticalpressure as P cr = p ω x mk B T /σ ( P ) [43] where we defined m = µm t / ( µ + m t ) with µ as the reduced mass of theodorant, m t as a measure of average mass of the odorant and environment, and σ ( P ) is the decoherence cross-section.Using the hard sphere model for the odorant at room temperature, the decoherence cross-section can be approximatedas 150 a b ( a b is the Bohr radius). Isotopic substitution alters m and its effect is significant for heavier odorants.Replacing an isotope with another with larger mass causes increasing P cr . According to ω ′ x = ω x p − P/P cr if P cr is increased then ω ′ x approaches to ω x and we arrive the limit in which the inelastic ET is effective. As a result, thesubstitution of massive isotopes amplifies the sensitivity of possible quantum mechanism of odorant discrimination.For large odorants, the sensitivity is affected considerably. Exact values of P cr can be calculated for different odorantsusing computational methods. So, we expect that the ability to distinguish a specific odorant is decreased whenexposed to the pressures near P cr . Moreover, when we replace atoms of the odorant with their heavier isotopes, wepredict that the same behavior appears in pressures larger than the P cr of the original odorant. Chiral Recognition
The existence of enantiomers, which have different smells, are usually used as an argumentto reject the quantum model of olfaction [21]. Since such enantiomers have the same vibrational spectrum, it seemsthat the shape-based parameters should be included in the quantum model to distinguish them from each other. Themodel presented here can be used to generalize the vibrational model for chiral recognition. Chiral molecules can beeffectively modeled by a double-well potential [32]. The shape factor is the configuration of the chiral odorant. Ourmodel predicts that the ET rates of inelastic ET are different for the two enantiomers for transitions | L i → | E i and | R i → | E i (see equations (2) and (3)). Our results here are in agreement with the Born-Markov master equationapproach [23]. The values of elastic and inelastic ET rates are given in TABLE 1 for a series of odorant parameters.The enantiomers with similar smells lie in the limit ω x ≪ ω z (i.e. the first block in the TABLE 1). But the inelasticET is ineffective in this limit. The enantiomers have different smells in the limit ω x ≈ ω z (i.e. the second block in theTABLE 1), but inelastic ET is still ineffective. The inelastic ET is effective for all transitions in the limit ω x > ω z = λ (i.e. the third row of third block in the TABLE 1). In this limit, the ratio of the inelastic ET rate for the left-handedenantiomer to that of the right-handed one increases with the ratio of the tunneling frequency to the asymmetryfrequency. Typical times for electron transfer in proteins are of order 10 − − − s [44]. Although the differencebetween inelastic ET rates of transitions may be appeared insignificant, however, in comparison with similar processesin biology it can be possible for the system to discriminate between two enantiomers under quantum constraints. TABLE I: Elastic and inelastic ET rates for some parameters of the chiral odorants at biological temperature T = 310 K . ω x ω z Γ − D → A Γ − D,L → A,E Γ − D,R → A,E Hz Hz s s s10 . × . × − . × − . × − . × − . × − . × − . × . × . × . × − . × − . × − . × − . × − . × − . × − . × − . × − SUMMARY
Our analysis can be summarized as follows for the main ingredients of the olfactory system: • Odorant : In the original vibrational model of olfaction, the relevant vibrational mode is represented by a simpleharmonic oscillator [19]. Here, we focused on a more realistic vibrational mode of non-planer odorant, known as contorsional mode, in which an atom or a group of atoms oscillates between the left and right wells of a double-well potential. Unlike the harmonic mode the contorsional mode can be used to charachterize the olfactorychiral reconition [23]. The eigenstates of the double-well potential are essentially doplets. The first doplet isenergetically available for most molecules at room temperature [31, 32]. The two-dimensional Hamiltonian ofthe mode is expressed by the tunneling frequency ω x and asymmetry frequency ω z . Our expressions for the(in-)elastic ET rates are reduced to the corresponding expressions for a harmonic mode in the limit ω x → • Electron : The detailed biological origin of the electron which tunnels through the odorant is not known but itmay be due to redox agents in the cell fluid [1]. According to the original model [19], we considered donor (D)and acceptor (A) sites of traveling electron as single molecular orbitals with energies ε D and ε D , coupled to eachother by a weak hopping integral ∆. In order to satisfy energy conservation during the tunneling process, theelectron’s parameters should be consistent with the odorant’s paramaters. Thus, they cannot be considered asvariables. • Environment : We modeled the biological environment as a harmonic bath with an ohmic spectral densityaccording to the original model [19]. Such an environment is characterized by its microscopic parameters (e.g.coupling frequency J and cut-off frequency λ ) and macroscopic parameters (e.g. temperature and pressure).Unlike the microscopic parameters, the macroscopic parameters can be controlled in experiment and thus theyare considered as variables. CONCLUSION
In this paper, we have generalized and proposed an analysis for examination of the dissipative quantum model ofolfaction in experiments. In fact, it has been suggested that inelastic electron tunneling through the odorant potentialis a dominant process in olfaction. Here, we have suggested a region of easy measurable parameters in the lab (e.g.temperature and pressure) in which we can discriminate between the elastic and inelastic tunneling through thepotential. We have shown that the dissipative odorant-mediated inelastic electron tunneling mechanism of olfactionis ineffective for a wide range of odorants, and the range of ineffectiveness depends on the type of transition in thecontortional mode. Moreover, our results indicate that there are thresholds in the bottom limit for both temperatureand pressure in which the olfactory system cannot recognize the scent. Additionally, the substitution of massiveisotopes amplifies the sensitivity of olfactory odorant discrimination. Perhaps the most relevant part of our analysisis related to the chiral recognition of the odorants in which enantiomers with similar smells lie in the asymmetry-dominant limit of dynamics and enantiomers with different smells lie in the tunneling-dominant limit of dynamics.We expect that our analysis can be used in experiments to examine the validity of the dissipative quantum modelof olfaction.
METHODS
Polaron Transformation
The unperturbed Hamiltonian ˆ H = ˆ H od + ˆ H e + ˆ H env is diagonalized by a polarontransformation as [23] ˆ H ′ = X R = A,D ( ε R + η R ˆ σ z ) | R ih R | + X i ω i ˆ b † i ˆ b i (7)where η A = − ω z (cid:8) tan − (cid:0) ω x + γ A ω z (cid:1)(cid:9) η D = − ω z (cid:8) tan − (cid:0) ω x − γ D ω z (cid:1)(cid:9) (8)Similarly, the interaction Hamiltonian is transformed toˆ H ′ int = ∆ | A ih D | exp n i (cid:2) tan − (cid:0) ω x + γ A ω z (cid:1) + tan − (cid:0) ω x − γ D ω z (cid:1)(cid:3) ˆ σ y o ) exp n X i (cid:0) γ i,D − γ i,A ω i (cid:1) (ˆ b † i − ˆ b i ) o + h.c (9) Dynamics
Time evolution operator in the interaction picture at weak-coupling limit can be written asˆ U I ( t ) = 1 − i Z t dt ˆ H int ( t ) − Z t dt ′ Z t ′ dt ˆ H int ( t ′ ) ˆ H int ( t ) (10)We assume that initially the electron is located at the donor site | D i , the odorant is found in the ground state ofenergy or left- or right-handed states (see FIG.1), all denoted by | i i , and the environment is described by the densitymatrix ρ env (0). The initial state of the whole system is then ρ (0) = | D, i ih D, i | ρ env (0). The density matrix of thewhole system at time t is given by ρ I ( t ) = ˆ U I ( t ) ρ (0) ˆ U † I ( t ). The probability of finding the electron at time t on theacceptor site and the odorant in the state | j i is obtained as P r
D,i → A,j = T r env h A, j | ˆ U I ( t ) ρ (0) ˆ U † I ( t ) | A, j i = ∆ Z t dt ′ Z t ′ dt e − iǫ ( t − t ′ ) h j | ˆΩ( t ) | i ih i | ˆΩ( t ′ ) | j i f ( ω, t , t ′ ) (11)where ˆΩ( t ) is a matrix with elementsΩ ( t ) = Ω ( − t ) = cos n (cid:2) tan − (cid:0) ω x + γ A ω z (cid:1) + tan − (cid:0) ω x − γ D ω z (cid:1)(cid:3)o e − it ( η − η ) Ω ( t ) = − Ω ( − t ) = sin n (cid:2) tan − (cid:0) ω x + γ A ω z (cid:1) + tan − (cid:0) ω x − γ D ω z (cid:1)(cid:3)o e − it ( η + η ) (12)and f ( ω, t , t ′ ) is the correlation function of the environment, defined by f ( ω, t , t ′ ) = T r env (cid:8) ˆΘ( t ) ρ env (0) ˆΘ † ( t ′ ) (cid:9) (13)where ˆΘ( t ) is a displacement operatorˆΘ( t ) = exp n X i γ i,D − γ i,A ω i (cid:16) e iω i t ˆ b † i − e − iω i t ˆ b i (cid:17)o (14)To calculate the environmental correlation function (13) we should specify the initial state of the environment. Re-garding the environment in the thermal equilibrium, the corresponding correlation function is obtained as f ( ω, t , t ′ ) = T r env (cid:8) ˆΘ( t ) (cid:16)
11 + ˜ n ∞ X n =0 (cid:0) ˜ n n (cid:1) n | n ih n | (cid:17) ˆΘ † ( t ′ ) (cid:9) = e iImζ ( t ) ζ ( t ′ )
11 + ˜ n ∞ X n =0 ( ˜ n n ) n h n | e P i χ i ˆ b † i − χ ∗ i ˆ b i | n i (15)where ˜ n = 1 / ( e ω/k B T − | n i is the number state and χ i := ζ ( t ) + ζ ( t ′ ). To obtain a closed mathematical form forcorrelation function we use the following relation for spectral density J ( ω ) of the environmental particles J ( ω ) = X i ( γ i,D − γ i,A ) δ ( ω − ω i ) ≡ J ωe − ωλ (16)where J is a measure of the coupling between the system and environment, and λ is the cut-off frequency of theenvironmental particles. Inserting (16) in (15), correlation function is summed up as f ( ω, t , t ′ ) = exp n − Z ∞ J ( ω ) ω h − cos (cid:0) ω ( t − t ′ ) (cid:1) f ( ω ) − i sin (cid:0) ω ( t − t ′ ) (cid:1)io (17)where for the environment in the ground state f ( ω ) = 1 and for the thermal environment f ( ω ) = coth( ω/ k B T ). ACKNOWLEDGEMENT
F.T.G acknowledges the financial support of Iranian National Science Foundation (INSF) for this work. V.S. thanksM. Aslani for illustrating the olfactory system in figure 1.
AUTHOR CONTRIBUTIONS
All authors contributed to developing the proposal and writing the manuscript.
ADDITIONAL INFORMATION
The authors declare no competing financial interests. ∗ [email protected][1] D. J. Rowe, Chemistry and technology of flavors and fragrances , Blackwell, Oxford, 2005.[2] R. Axel, Scents and Sensibility: A Molecular Logic of Olfactory Perception (Nobel Lecture),
Angew. Chem. Int. Ed. ,6110, 2005.[3] L. B. Buck, Unraveling the Sense of Smell (Nobel Lecture), Angew. Chem. Int. Ed. , 6128, 2005.[4] S. H. Lee et al. Mimicking the human smell sensing mechanism with an artificial nose platform,
Biomaterials , 1722, 2012. [5] R. H. Farahi, A. Passian, L. Tetard and T. Thundat, Critical Issues in Sensor Science To Aid Food and Water Safety, ACSNano , 4548, 2012.[6] M. Zarzo, The sense of smell: molecular basis of odorant recognition, Biol. Rev. , 455, 2007.[7] J. Amoore, The stereochemical theory of olfaction, Nature , 912, 1963.[8] K. Mori and G. Shepard, Emerging principles of molecular signal processing by mitral/tufted cells in the olfactory bulb,
Semin. Cell. Biol. . 65, 1994.[9] F. Yoshii, S. Hirono and I. Moriguchi, Relations between the odor of (r) ethyl citronellyl oxalate and its stable conformations, Quant. Struc-Act Rel. Nature Neuroscience , , 1248, 2000.[11] L. Turin and F. Yoshii, Struture-odor relations: a modern perspective, in Handbook of Olfaction and Gustaion , R. Doty,Marcel Dekker, New York, 2003.[12] R. Bentley, The nose as a stereochemist: Enantiomers and odor,
Chem. Rev. , 4099, 2006.[13] J. C. Brookes, A. P. Horsfield, and A. M. Stoneham, Odour character differences for enantiomers correlate with molecularflexibility,
J. R. Soc. Interface , 75, 2009.[14] G. Dyson, The scientific basis of odour, Chem. Ind. , 647, 1938.[15] R. Wright, Odor and molecular vibrations: neural coding of olfactory information, J. Theor. Biol. , 473, 1977.[16] J. Lambe and R. C. Jaklevic, Molecular Vibration Spectra by Inelastic Electron Tunneling, Phys. Rev. , 821, 1968.[17] C. J. Adkins and W. A. Phillips, Frequency shifts in inelastic electron tunnelling spectroscopy of adsorbed species,
J. Phys.C , 1313, 1985.[18] L. Turin, A spectroscopic mechanism for primary olfactory reception, Chem. Senses , 773, 1996.[19] J. C. Brooks et al. Could humans recognize odor by phonon assisted tunneling?,
Phys. Rev. Lett. , 038101, 2007.[20] I. A. Solov’yov, P.-Y. Chang and K. Schulten, Vibrationally assisted electron transfer mechanism of olfaction: myth orreality?, Phys. Chem. Chem. Phys. , 13861, 2012.[21] E. R. Bittner et al. Quantum Origins of molecular recognition and olfaction in drosophila,
J. Chem. Phys. , 22A551,2012.[22] A. Ch¸eci´nska et al. , Dissipation enhanced vibrational sensing in an olfactory molecular switch,
J. Chem. Phys. , 025102,2015.[23] A. Tirandaz, F. Taher Ghahramani and A. Shafiee, Dissipative vibrational model for chiral recognition in olfaction,
Phys.Rev. E 92 , 032724, 2015.[24] M. I. Franco et al.
Molecular vibration-sensing component in Drosophila melanogaster olfaction,
Proc. Natl. Acad. Sci.USA , 3797, 2011.[25] E. R. Bittner, A. Madalan, A. Czader and G. Roman, Quantum origins of molecular recognition and olfaction in drosophila,
J. Chem. Phys. , 22A551, 2012.[26] W. Gronenberg et al.
Honeybees (Apis mellifera) learn to discriminate the smell of organic compounds from their respectivedeuterated isotopomers,
Proc. Biol. Sci. , 20133089, 2014.[27] L. J. W. Haffenden, V. A. Yaylayan and J. Fortin, Investigation of vibrational theory of olfaction with variously labelledbenzaldehydes,
Food Chem. , 67, 2001.[28] S. Gane et al. Molecular Vibration-Sensing Component in Human Olfaction,
PLoS One , e55780, 2013.[29] A. Keller and L. B.Vosshall, A psychophysical test of the vibration theory of olfaction, Nat. Neurosci. , 337, 2004.[30] E. Block et al. Implausibility of the vibrational theory of olfaction,
Proc. Natl. Acad. Sci. , E2766, 2015.[31] C. H. Townes and A. L. Schawlow,
Microwave Spectroscopy , McGraw-Hill, New York, 1955.[32] G. Herzberg,
Molecular Spectra and Molecular Structure: Electronic Spectra and Electronic Structure of PolyatomicMolecules , Krieger, Malabar, 1991.[33] A. J. Leggett et al.
Dynamics of Dissipative two state Systems,
Rev. Mod. Phys. , 1, 1987.[34] U. Weiss, Quantum Dissipative Systems , World Scientific, Singapore, 2008.[35] M. Quack, How important is parity violation for molecular and biomolecular chirality?,
Angew. Chem. Intl. Ed. , 4618,2002.[36] T. D. Lee and C. N. Yang, Question of Parity Conservation in Weak Interactions, Phys. Rev. , 254, 1956.[37] L. D. Barron, Chirality at the sub-molecular level: True and false chirality, in
Chirality in Natural and Applied Science ,edited by W. J. Lough and I. W. Wainer, Blackwell Publishing, Oxford, 2002.[38] J. Gilmore and R. H. McKenzie, Spin boson models for quantum decoherence of electronic excitations of biomolecules andquantum dots in a solvent, J. Phys.: Condens. Matter , 1735 (2005).[39] A. Tirandaz, F. Taher Ghahramani and A. Shafiee, Emergence of molecular chirality due to chiral interactions in a biologicalenvironment, J. Bio. Phys. , 369, 2014.[40] M. Quack, J. Stohner and M. Willeke, High-resolution spectroscopic studies and theory of parity violation in chiralmolecules, Annu. Rev. Phys. Chem. , 741, 2008.[41] B. Bleaney and J. H. Loubster, Collision broadening of the ammonia inversion spectrum at high pressures, Nature
Phys. Rev.Lett , 123001, 2002.[43] R. C. Weast et al. CRC handbook of chemistry and physics , Boca Raton, FL: CRC press, 1988.[44] K. Brettel and M. Byrdin, Reaction mechanisms of DNA photolyase, Curr. Opin. Struct. Biol.20