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FINAL REPORT CHEM F266 Page of1 22
CHEM F266
STUDY ORIENTED PROJECT
IISEMESTER 2015-16
PROJECT TITLE : QUANTUM TUNNELING TO EXPLAIN
OLFACTION IN ANIMALS
SUPERVISING FACULTY: Dr. Shamik Chakraborty
SUBMITTED BY : Siddharth Shankar Tripathi
2014B2A3365P

ACKNOWLEDGEMENTS
This being my ﬁrst major project I faced many challenges throughout the
project , ranging from deciding its scope , to accessing the material over the
web and otherwise but I would thus like to highly thank my mentor Dr.
Shamik Chakraborty who adequately helped me throughout the project and
was ﬂexible enough to let me do anything wayward which was not in any
close connection with the project . He guided me very subtly and encour-
aged me towards the right direction for which I am grateful.
Also I would like to take the opportunity to thank the
head of the department , Dr. Anil Kumar for giving his due permission to let
me do this project .

ABSTRACT
The project involves the study of the process of olfaction in mammals. As was
suggested by Luca Turin in 1996 , the smelling systems of our body utilise the
process of tunneling in order to differentiate between different kinds of smell, but
in principle this theory has seen a lot of abnormalities which cannot be explained .
Thus at the moment , smell is a physical process used by us all, but fully under-
stood by none. The kind of electron tunneling has been speculated to be IETS
( inelastic electron tunnelling ).
In the project I have tried to understand the process of tunneling in
detail and using it as the base have carried its use to try to understand the
process of tunneling occurring in humans. After reaching to a comfortable enough
position I have then tried to investigate into the physiological details of the
process.

TUNNELING :
It is very rare or almost impossible for a classical particle to
pass through a wall or any physical barrier of such sort. This is fairly obvious as the par-
ticle size is a very big constraint that makes such a happening difﬁcult task. But such a
process is performed by quantum particles very easily by a process known as quantum
tunnelling.
HOW DOES THIS WORK ? 
Imagine that an electron, for example, is a marble sitting in one of two depressions sep-
arated by a small hill, which represent the effects of a sculpted electric ﬁeld. To cross the
hill from one depression to the other, the marble needs to roll with enough energy. If it
has too little energy, then classical physics predicts it can never reach the top of the hill
and cross over it. 
Tiny particles such as electrons, however, can still make it across even if
they don't have enough energy to climb the hill. Quantum physics describes such parti-
cles as extended waves of probability—and it turns out that there is a probability that
one of them will "tunnel" through the hill and suddenly materialise in the other depres-
sion, even though the electron can't occupy the high ground between the two low spots.
Figure 1. Depicting tunneling of wave through a barrier
Quantum tunnelling (or tunneling) refers to the quantum mechanical phe-
nomenon where a particle tunnels through a barrier which is classically forbidden. The
phenomenon of tunneling is an important consequence of quantum mechanics. Consid-

er a particle with energy E in the inner region of a one-dimensional potential well V(x). (A
potential well is a potential that has a lower value in a certain region of space than in the
neighbouring regions.) In classical mechanics, if E < V (the maximum height of the po-
tential barrier), the particle remains in the well forever; if E > V , the particle escapes. In
quantum mechanics, the situation is not so simple. The particle can escape even if its
energy E is below the height of the barrier V, although the probability of escape is small
unless E is close to V. In that case, the particle may tunnel through the potential barrier
and emerge with the same energy E.
THE BARRIERS IN THE PATH OF A WAVE ( ELECTRON MOVING AS A WAVE )
1) THE POTENTIAL STEP: 
For a time independent potential, the wave function can be factorised as Ψ(x, t) = e−iEt/
ψ(x), where ψ(x) can be obtained from stationary form of the Schrodinger equation,
(time independent) .
Since here E and V(x) are assumed to be ﬁnite so must be the double de-
rivative of x . This condition forces the implication that : both ψ(x) and ∂xψ(x) must be
continuous functions of x, even if V has a discontinuity.
Figure 2 . This depicts the barrier extending from x=0 to x= infinity.

Assuming that a beam of particles withe kinetic energy E move from right to left are be-
ing incident at x=0 position of the barrier. Now if the beam has amplitude of one unit and
the reflected and transmitted wave are marked by r and t we have the wave functions
given by :
here the definitions of k should be taken note of . for x<0 k= (2mE/h2)1/2 and for x>0
k=(2mE/h2)1/2 .Now to make this system continuous at x=0 we obtain the relation 1+r=t
and ik<(1 − r) = ik>t leading to reflected and transmitted amplitudes ,

The reflectivity, R, and transmittivity, T, are defined by the ratios,R = reflected flux/inci-
dent flux , T = transmitted flux/incident flux . 
Keeping the values for all kind of fluxes obtained , we solve for the solution and get


Figure 3. A graph depicting Reflectivity vs Transmittivity as the ratio of the energy of the wave
and the potential barrier values vary .
2) THE POTENTIAL BARRIER :
This is the most basic geometry of a scattering experiment to be discussed. In this phe-
nomenon a beam of particles is “deﬂected” by a local potential. Here the barrier is locat-
ed form x=0 to x=a , the wave function is of the form eik1x where k1= (2mE/h2)1/2 and
the wave functions only differ in their complex amplitudes after encountering the barrier
the transmitted wave function only goes in a change of amplitude and a phase shift. now
the relative phase changes can be parameterised as :

Applying the continuity conditions on the wave function, ψ, and its derivate , ∂xψ, at
the barrier interfaces at x = 0 and x = a, and then solving for four unknowns r , T , A and
B we get to the result :


which results into a transmittivity of
and the reflectivity , R=1-T . So, for barrier heights in the range E V0 0, the transmit-
tivity T shows an oscillatory behaviour with k2 reaching unity when k2a = nπ with n inte-
ger. When the energy of the incident particles falls below the energy of the barrier, 0 E
V0, a classical beam would be completely reflected. How- ever, in the quantum sys-
tem, particles are able to tunnel through the barrier region and escape leading to a non-
zero transmission coefficient. In this regime, k2 = iκ2 becomes pure imaginary leading to
an evanescent decay of the wave function under the barrier and a suppression, but not
extinction, of transmission probability.
Figure 4. Transmission probability of a finite potential barrier


Figure 5. Real part of the wave function for E/V0 = 0.6 (top), E/V0 = 1.6 (middle),
and E/V0 = 1 + π2/2 (bottom), where mV0a2/2 = 1. In the first case, the system shows tunneling
behaviour, while in the third case, k2a = π and the system shows resonant transmission. [Image
courtesy : advanced quantum chemistry ,chapter 3 of book published in
THE DIFFERENT THEORIES OF OLFACTION :
1) The lock and key model :
This was proposed by Axel and Buck . They proposed that the receptors belonged to the
class G of proteins (GPCRs) a type that will allow only speciﬁc odourants to bind to it on
the basis of their shape and size. Odourants and receptors can be tough to resemble as
a lock and key model . The theory was proposed with the view that like almost all other
systems of the body the process of olfaction wouldn't be much different and would cor-
respond to the same. The form of the pocket depends greatly on the sequence of amino
acids forming the protein and hence the corresponding three dimensional structure.
Thus, the sequence of amino acids that make the protein is crucial. A single change in
the order can change the shape of the pocket leading to changes in the chemicals that
ﬁt into the pocket. A sequence of chemical events is initiated within a cell that involve
molecules called second messengers by the conformational changes in receptor pro-

teins caused by their binding with a fitting chemical (ligand). The degree to which a large
number of ion channels are opened is affected by a single odour molecule through the
second-messenger signals produced when they bind to a receptor protein. Large
enough potentials are produced by these actions.
CHALLENGES TO THE MODEL : 
A lot of structured molecules have been discovered where differently structured
odourants create same olfaction senses while similar structured odourants have created
different olfaction senses.
Figure 6. Depicts the anomaly between shape similarity and smell [Image courtesy Google]
For example : 
Hydrogen Sulphide and Decaborane are different in structure but smell very similar , in
the same manner ferrocene and nickelocene are similar in structure but smell complete-
ly differently .
THE ALTERNATE SWIPE CARD MODEL :
This is the alternate model as developed by Turin to explain the the anomalies in the
lock and key model. 
The model speculates that shape is important for the odourant molecule to dock but
other specifications are also of utmost importance for olfaction. For e.g. a lot of credit
cards might fit into an ATM machine but only with an appropriate pin number can they be
used to withdraw cash from the machine.
The additional essential information that we use in our model is molecular vibrational
frequency of the odourant molecule. 
The Turin model (our main discussion topic) proposes that electron transfer through the

odourant preferentially occurs on when the molecule has the right vibrational frequency.
The model places the use of INELASTIC ELECTRON TUNNELING which we shall talk
about now in the further discussion :
IETS is a non optical form of vibrational spectroscopy relying on the interaction between
electrons tunneling across a narrow gap between metallic electrodes and molecule in
the gap. 
Inelastic tunneling describes tunneling between two states with different
energies . In order for energy to be conserved , something else has to pick up some en-
ergy from or deposit it with the electron. The typical energy source/sink is the crystal lat-
tice of the material , whose vibrations are called sound or , in quantum language ,
phonons.

a) shows energy band diagrams for tunneling while b) shows corresponding I vs V dia-
gram.
Figure 7. Depicts IETS model as used in inorganic systems
Turin's model almost like all normal tunneling models uses two
points of contact between the odourant and the receptor i.e. namely the donor (D) and
the acceptor energy level (A). 
The favourable condition for the inelastic tunneling to occur is that the electronic differ-
ence between D and A sites must match the vibrational energy taken up by odourant
(M).


Figure 8. This is the kind of tunneling desired by us under the circumstances as stated in
the model.
Figure 9. Another type of electron tunneling which can happen but not useful to the model that
we use.
THE DIFFERENT ROUTES TAKEN BY THE ELECTRON IN THE
COURSE OF TUNNELING

So far, we have established that as an electron transfers from D to A, it alters forces on
M, so causing a change in the odourant’s vibrational state. This behaviour has parallels
in a lot of other solid state systems. Until we have more information about the receptor
structure, doubts must remain as to precisely which groups D and A correspond. We
have assumed that D and A are relatively localised, and that the odourant molecule M in
the receptor is close to either D or A or it is perhaps equidistant and equally localised be-
tween them. The donor and acceptor species will have discrete energies, unlike the
electrodes in most inorganic inelastic tunnelling experiments.
This would be realistic if we believe the likelihood that these electron
source/sinks are amino acids and if we compare to distances between important
residues for rhodopsin. Site-directed mutagenesis studies have determined that for
odourant recognition in MOR-EG (mouse olfactory receptor) there are nine amino acids
involved directly at the binding site, with Ser113 (amino acid) being a crucial H-bond
donor for odourants with aliphatic alcohols. It is noteworthy that none of the nine is
strongly conserved and indeed some are at sites that are highly variable. Thus they can
only be associated with binding or modifying the donor and acceptor characteristics.
Figure 10. A configuration coordinate diagram to show the initial state (the left curve) and the
final state (the right curves) where there are two options : the inelastic (n=1) versus the elastic
(n=0) route. [Image courtesy : google images]
Two different kinds of routes have been proposed to be taken by an electron during the
tunneling process in the receptor protein.

Figure 11. A scheme for the proposal of electron transfer in the olfactory receptor
with intra-protein electron transfer. Only 5 transmembrane helices for the olfactory receptor are shown
(cylinders) here for clarity. (a) The odourant approaches the receptor, meanwhile an electron is present at
donor site D; (b) The odourant docks at the ligand binding domain, the overall conﬁguration of receptor
and odourant changes (c) The electron jumps from D to A, causing the odourant to vibrate (d) The
odourant is expelled from the ligand binding domain.
Figure 12. A scheme for the proposal of electron transfer in the olfactory receptor. Only
5 transmembrane helices (of the 7 in total) for the olfactory receptor are shown (cylinders) here for clarity.
(a) The odourant approaches the receptor, meanwhile an electron moves to position RD on a helix; (b)
The odourant docks at the ligand binding domain, the overall conﬁguration of receptor and odourant
changes, meanwhile the electron tunnels within the protein to D and it spends some time there; (c) The
electron jumps from D to A causing the odourant to vibrate; (d) The odourant is expelled from the ligand
binding domain and the electron tunnels within the protein to site RA. Signal transduction is initiated with
the G-protein release. [ Image courtesy : Sensors 2012, 12, 15709-15749 ] .

DEVELOPMENT OF THE HAMILTONIAN FOR THE MODEL :
Electron transfer between the donor and acceptor can be described through or modelled
by the hamiltonian :
this accounts for the essential physical interactions in the system . Here HR denotes the
olfactory receptor hamiltonian , HR-O describes the coupling interaction between the ol-
factory receptor and an odourant molecule, HR-env and HT accounts for electron tunneling
from donor to acceptor sites of the olfactory receptor.
Thus the hamiltonian describes each and every interaction of the components in olfac-
tion.
Simple Huang–Rhys Factor Model :
• QUESTION — WHAT IS THIS MODEL ?
• ANSWER — The Huang-rhys factor gives a measure of the coupling of the olfactants
to the rapid movement of the electronic charge from the donor D to the acceptor A.
The model may be applied to any one of the ways in which the electron charge is sup-
posed to tunnel through D to A whose sudden energy change (force change on the
odourant molecule because of the electron tunneling through it in principal) causes M to
change vibrational states.
Huang-rhys factor comes out to be S = E2q2/(2Mh︎ω3) .
The energy of Donor and Acceptor energy levels is then calculated and also needs to be
decided what effective mass M and charge q should be taken.
Now when the electron passed from donor to acceptor change in force =

The Huang–Rhys factors can be calculated using a full electronic struc-
ture code. This is normally done by combining total energies from four calculations;
namely, for the system relaxed for the charge at R
⃗
D , calculate the energies when the
electron is on D and when it is on A, and correspondingly, for the system relaxed for the
charge at R
⃗
A, calculate the energies when the electron is on D and when it is on A.
There is some redundancy in these calculations, but this compensates somewhat for
working close to the limits of accuracy of electronic structure codes.
MATRIX ELEMENTS IN CALCULATION OF THE FACTOR :
The important electronic states are those where the electron is on the donor, represent-
ed by the notation |D⟩, and when the electron is on the acceptor, this is represented by
the notation |A⟩. Here making the assumption that the electronic wave functions evolve
adiabatically, as electron motion is very rapid compared with the nuclear motion during
transitions. The Hamiltonian to describe these energetic states is:
where t is very small (we get T1=87 ns , the time taken in the movement of electron
when odourant is absent and ︎ T2=1:3 ns, the time taken by the electron when it tunnels
through the odourant ; which satisﬁes the condition ︎ T2T1 , and shows that the overall
time for odour recognition is not limited by the discrimination process) the donor and ac-
ceptor are weakly coupled. Introducing an odourant M into the equation then the elec-
tron can go from D to A via the molecule (or via a different route) with state |M⟩. The
Hamiltonian for this scenario is:

In order to generalise to an effective two state Hamiltonian, the following determinant
was produced :
Making the secular equations and solving of the coefficients we finally derive the time
period of tunneling in the process which come out to be after calculation as
Approximating ε = εD as , it is not known whether the energy eigenstate ε, but the as-
suming ε = εD or εA, since εD and εA differ very little (meV), as compared with the dif-
ference between εD and εM (10’s eV). Thus the initial electronic state will involve an
admixture of D and M due to the presence of the odourant, and the final electronic state
is similarly an admixture of A and M. This implies the presence of an odourant M is inte-
gral to an electron transfer process.
Figure 13. A table depicting time scales of different processes in different processes. [Table cour-
tesy Sensors 2012, 12, 15709-15749]

DISCUSSION OF THE DONOR AND ACCEPTOR SPECIFICA-
TIONS :
For an inelastic tunnelling mechanism (IETs) to work,
the molecular units D and A have to satisfy certain well deﬁned conditions. Just what D
and A are is not at the moment clear. They have been speculated to be common units
among the likely receptor structures. They must be able to occur in two charge states,
which can be assumed as full and empty (so the transition results from D(full)A(empty)
to D(empty)A(full) { here full is the situation when electron is in the particular energy
state } ), though that is possible for many possible molecular units. Transition metals, of-
ten found in living systems, are among the species that can occur in several charge
states. The D and A units should have the capability to be able to revert back to their
original states many times, i.e., D and A should not be destroyed in the one cycle of ol-
faction process. It must be possible to feed an electron into D and remove an electron
from A (inter- chain model) or return the electron to D (intra-chain model). To detect
odourants within milliseconds, through tunneling via an odourant can be much faster, the
replenishment of D and A should be within ms but not longer.
Whilst that is not a strong constraint as regards timescale, it
does require other reactions outside the receptor to maintain electrochemical equilibria
that drive these motions. It is to be noted also , that D and A must be sharp energy lev-
els, which means only weak interactions to cause broadening. This is consistent with the
calculated results, where all relevant interactions appear weak. The most important re-
strictions on D and A (energy levels ) is the need for a small energy splitting εD − εA that
is almost equal to the small (but in principal approximate) vibrational quantum ︎ω0. Most
olfactants M and many possible molecular units of the receptor are closed shell systems,
and in which the gap between the highest occupied molecular orbital (HOMO) and low-
est unoccupied molecular orbital (LUMO) levels is two orders of magnitude too large.
Electron transfer from the HOMO of one unit to the LUMO of another is ruled out by their
large energy difference, perhaps even 10 eV.

A PHYSIOLOGICAL DISCUSSION :
Journey of the odourant to the receptor :
The odorant’s journey begins as sniffed through the nose and meets the olfactory ep-
ithelium at the top of the nasal cavity. At the epithelium it meets the cilia which extend
from the neuron’s main body into the mucus layer. At this interface between mucus and
inside the cell, the odorant meets the olfactory receptor. Whatever happens at this point
to produce a discriminant signal results in ionic cascades that depolarise the cell.

Figure 14. The figure depicts diagrammatically what happens when we inhale a possible odourant
molecule. [Image courtesy : Anal Bioanal Chem (2003) 377 : 427–433 ]
The olfactory receptors deﬁnition has been one of the most tricky part of the physiologi-
cal understanding of the process. It has been proved by Axel and Buck to be GPCR
class of proteins for which discovery they received the nobel prize in 2004. This GPCRs
model has been proposed by Breer . The N-terminal extend into the extracellular layer,
the C-terminaI extend into the olfactory sensory neuron (OSN). The seven trans mem-
brane helices are represented here as cylinders connected by ﬂexible loops. They are
surrounded by lipids, the hydrophobic tails forming a barrel around the receptor, within
the barrel the odourant binds, the polar heads point towards the ‘wet’ layers.

Figure 15. The model showing the representation of the GPCR as perceived by Breer. [image
courtesy : Anal Bioanal Chem (2003) 377 : 427–433 ]
These proteins work via release of a G-protein into the cytoplasm which causes sec-
ondary messengers initiates a signalling process.

CONCLUSION
The process of smell is still one of the most ambiguous process that has yet not been
completely described by scientists. As we see in the report , the Donor levels , Acceptor
levels , path of the electrons tunneling before and after the process , any extra energy
produced or used by the process is not entirely deﬁned. The lock and key theory fails
several tests and the swipe card model provides an alternate which solves many issues.
Despite being a viable alternate it leaves many questions unanswered
and many solutions undelivered on many fronts which have a lot of scope to be investi-
gated in. We also see that how , just by a change of coordination factor a lot of things
might change and how even for the new theory the basic requirements for the body to
recognise odour ( shape , size of pocket etc) remain the same. The biggest question
which still remains is that is really vibrational theory a solution out of the crisis ?

BIBLIOGRAPHY
1. Jennifer C. Brookes (2011) : Olfaction: the physics of how smell works?, Contempo-
rary Physics, 52:5, 385-402
2. Review The Swipe Card Model of Odourant Recognition , Jennifer C. Brookes , An-
drew P. Horsﬁeld and A. Marshall Stoneham Sensors 2012, 12, 15709-15749;
3. Could Humans Recognize Odor by Phonon Assisted Tunneling?( Jennifer C. Brookes,
Filio Hartoutsiou, A. P. Horsfield, and A. M. Stonehamx , PRL 98, 038101 (2007))
4. Manuel Zarzo ,The sense of smell: molecular basis of odorant recognition , Biol. Rev.
(2007), 82, pp. 455–479.
5. Phys. Chem. Chem. Phys., 2012, 14, 13861–13871

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