Chem. Senses 1997 Lamine 67 75

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Application of Statistical Thermodynamics to theOlfaction MechanismA. Ben Lamine and Y BouazraDepartement de Physique, Faculte des Sciences de Monastir, 5000 Monastir, Tunisia

Correspondence to be sent to: A. Ben Lamine, Departement de Physique, Faculty des Sciences de Monastir,5000 Monastir, Tunisia

AbstractThe application of the grand canonical ensemble in statistical thermodynamics to the stimulus adsorption on theolfactory receptor sites, assuming some simplifying hypotheses, leads us to an expression of the olfac tory response 9J,which is a function of various physico-chemical parameters involved in the olfaction mechanism, e.g. the stimulusconcentration, the saturated vapor pressure, the power law exponent and the partition coefficient. This expression of9? is in agreem ent w ith the olfactory response of the H ill mo del, but is more explicit. Stevens' law and the olfac torythreshold expression are easily deduced fr om 9?. The expression of the threshold we established from 91 enabled us toexplain some empirical relations in the literature between the parameters quoted above. The use of the grandcanonical ensemble with the chemical potential notion gave us an interpretation of Stevens' law and a betterunderstanding of the role of some parameters involved in the olfaction mechanism, such as saturated vapor pressureand power law exponent. Chem. Senses 22: 67-75, 1997.

IntroductionA number of investigations in olfaction consist in lookingfor significant correlations between an olfactory parameter,e.g. the odor intensity, the odor quality or the olfactorythreshold, and one or many physico-chemical parameters,e.g. the partition coefficient in octanol-water (Greenberg,1979; Wolkowski et al., 1977), the molecular volume(Wright, 1982), the saturated vapor pressure (Patte et al,1989) and the molecular section (Davies, 1971). Thesecorrelations are often empirical (Patte et al, 1989; Laffortand Patte, 1987).

A global formulation of the dynamic olfactory responseover a wide range of stimulus concentrations was proposedsuccessively by Beidler (1954), Tateda (1967), Laffort (1977)and Patte et al.,{\989 ). Molecu lar kinetics was introduced in

a semi-empirical model by Wright (1978), who tried todeduce directly the power law of the olfactory response.The olfactory threshold has often been independentlyformulated from the olfactory dyn amic response problem . Amathem atical model was proposed by Davies (1971), thoug hthis was based on the puncturing theory. Empirical modelswere also proposed by Randebrock (1971) and by Laffortand co-workers (Laffort et al., 1974; Patte and Laffort, 1989;Laffort, 1994). Some statements and question s have emergedfrom these models and investigations, such as: (i) the lowerthe power law exponent, the better the o lfactory power (po{)(Laffort, 1974, 1994); (ii) the models of Beidler, Tateda andLaffort have not yet been validated (Laffort, 1994); (iii) thesaturated vapor pressure is very important to the olfactory

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6 8 I A. Ben Lamine and Y. Booazra

dose-response curves in insects (Patte et al, 1989; Laffort,1994); (iv) what is the physical origin of the power law andthe meaning of its exponent n (Laffort, 1977)?

The purpose of the present work is to establish anolfactory response expression that is based on a statisticalthermodynamics model, using a minimum number ofphysico-chemical and physiological parameters. Thisexpression will allow us to deduce some known results andto answer some of the above questions.

Theoretical modelDifferent stages of the stimulusThe different stages which the odorous stimulus passesthrough can be reduced to the two following equilibriums:

Sg 1 and equili-brium (3) can be written as:

K (30We can see that ri is the number of receptor sites which areoccupied by one stimulus molecule. We will consider thenthat ri is the anchorage number of one molecule of theodorous stimulus. This is in agreement with expression (4)when, written in the following form:

K"N g = N,M1 - ^

where Afo is the num ber of olfactory receptor sites per unit

it becomes identical to the known expression (Howard andMe Connell, 1967): kC = [9/(1 - 8)]v. This latter equationwas experimentally established in the case of polymeradsorption and we identify ri with v, which is known to bethe average number of segments per polymer chain that areanchored to the surface.

(v) Although many kinds of receptor sites exist (Laffort,1977, 1994; Wright, 1978), in a first approach we willconsider only one kind of receptor, as w as done in previous

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Application of Statistical Thermodynamics to the Olfaction Mechanism I 6 9

models (Davies, 1971; Laffort, 1977; W right, 1978). Thisdoes not imply that n is near 1, as was mentioned by Laffort(1977). On the contrary ri (ri = \lri) can be different from 1,since it is an anchorage number of the stimulus molecule onreceptor sites. For a given ads orption ri is normally a w holenumber (Randebrock, 1971). However, the stimulus mole-cule can be adso rbed with many different configurations,depending on the exposed face geometry, so ri is an averageand does n ot need to be a whole number.

(vi) An anchorage of the stimulus molecule on anadsorbing site is supposed to have only one adsorptionenergy level (-e). The free enthalpy of gaseous stimulusadsorption AF.a is then:

AEa = Norie where No is Avogadro's numberThis can be written as:

\ - l (5)

On the other hand, from the equilibrium (3) wemay express the conservation of stimulus quantity by:Nd + NJri = constant. At equilibrium the Helmoltzfunction F is such that:

dFdN,dF

We then obtain: u^ = ri\i = \Un, which is equivalent to thelaw of mass action.

In the case of non-dissociation of the stimulus moleculein equilibrium (2) we have u.g = \i^, where ng and u^ arerespectively the partial chemical potential in gaseous anddissolved states.

If we note Em the adsorption energy of a whole stimulusmolecule, then we have:

where AE? is the free enthalpy of vaporization of purestimulus, which approaches the free enthalpy of dissolutionin the case of an ideal solution (hypothesis iii), A is thefree enthalpy of stimulus adsorption from the dissolvedstate.

(vii) Finally, we assume the non-impoverishment of thestimulus into the mucus at every equilibrium state. Thissuggests we can use the grand canonical ensemble in thestudy of this adsorption mechanism.

Statistical treatm entCombining the Langmuir model with that of moleculardissociation (Nagai and H irashima 1985; Diu et al., 1989b),the grand canonical partition function of one receptor sitecan be expressed as:

- rie - e/n and so e =

where u is the chemical potential of one site.Although a stimulus molecule could have many anchor-

ages, a given receptor site cannot have more than oneanchorage. So Ni is either 0 or 1.

We obtain: z = 1 + e E +^, where p = MkT, and T is thehuman or animal body temperature. The grand partitionfunction of NM sites is then: Z^. = zN" and the averagenumber of occupied sites is given by (Diu et al., 1989b):

The number of occupied sites is:M

The p artial chemical potential (ig of the stimulus assumedto be an ideal gas is given by:

Ne r" = =

where Z g is the stimulus molecule partition function perunit volume.In general, the stimulus molecule is polyatomic and so Z gis given by: Z g = Z^.Z^. The first term Zgt,. corresponds tothe translation partition function:

Taking the fundamental state in the gas as the origin ofenergy levels, we get:


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7 0 I A. Ben Lam ine and Y. Booazra

mkT]: h2 )

3/ 2

where m is the stimulus molecule mass.From the Rankine formula for the saturated vapor

pressure (SVP) of an ideal gas, expressed in statisticalthermodynamics (Couture et al, 1989), we obtain:

= pT(2ronA:77/!2)3/2 kTe-AE"/RT L+ _ o

where R is the universal ideal gas.The second term Zg contains all the internal degrees of

freedom of the stimulus molecule. Because electronic androtational energy levels are not excited in the mucussolution, they do not contribute to Z^. As a first approach,we also assume the abscence of vibrational contribution.Thus we take Zp = 1 and get:

"M 'M

1 + x 3 /2 1 +p-SVP-e"

2nmkT/h2) ywhich may be written in the following form:

Na

where

-AE'/RT

is the gaseous stimulus concentration in air, for which halfof the receptor sites are occupied (Na = N^IT).

Finally we assume that the olfactory response 9t isproportional to the number of occupied sites Na and not tothe number of adsorbed molecules {NJny.

1+ ( ^ fThe proportionality coefficient a included in 91

incorporates transduction and other effects such as aneventual porosity of the adsorbing surface. This coefficientis denoted h by Wright (1978).

Consequences and interpretationsConsequencesOlfactory response formFirst we note that the expression of SR is in agreement withthe Hill model contained in the expression (4), which wasestablished by a classical treatment (Tateda, 1967; Laffort,1994).

In addition we obtain the explicit expression of theequilibrium constant K of the stimulus adsorption:

K =.+3/2

h2 ) = P-SVP-e"

This analytical expression of K cannot be derived fromprevious models (Beidler, 1954; Tateda, 1967; Laffort, 1978;Wright, 1978).

Different behaviours(i) Low concentrations: When A g ./Vgi/2, the olfactoryresponse 9? takes the Stevens' law form:

a/V,M( o r O ^ M ^ g V 2 .which is written in the literature (Laffort, 1994) as:

(ii) High concentrations: When Ng iVgi/2, the olfactoryresponse curve 9J takes a hyperbolic form (Patte et al, 1989;Laffort, 1994):

M 1 - . J : hyperbolic lawThe hyperbolic asymptote, representing the case when ah1

sites are occupied (the saturation response), is 9lmflx = OLNM.

General behaviourThe olfactory response curve 91 = 9J(iVg) depends on thethree parameters JVM, #gi/2 and n. The first parametercontrols the odour strength at saturation. The higherthe more intense the saturation odour. The secondparameter Ng\ri (or SVP) controls the odour strength overthe whole range of stimulus concentrations. For a givenconcentration Ng, the greater Ng]/2 (or SVP), the weaker the

yg

p

j

g
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Application of Statistical Thermodynamics to the Olfactioo Mechanism I 71

0.010.001

0.0001

gi/2 m< xeases

0.0001 0.001 0.01 0.1

1 :

0.1 :

0.01 :

0.001 :

.0001

//

n = 0.2

_ /' n =l

^ ^

i i nun i i nun

10 10 0 0.0001 0.001 0.01 0.1 10 100Figure 1 Olfactory response curve 9? versus stimulus concentration A/g inair for different values of the parameter Ngm (0 .1 , 0.2, 0.3) in log-logcoordinates.

olfactory response SR all things {n and N^) being equal (seeFigure 1).Finally n or ri controls the strength of the odour but not

in the same way as A^gl/2. The lower n, the stronger the odo urfor Ng < Ngm and the weaker the odour for Ng > Niin (seeFigure 2).

Olfactory thresholdIn this section we are going to derive the olfactory thresholdfrom the dynamic response 9?. Because the olfactory signalis controlled by an action potential, the threshold is set bythis action potential. We may consider that the olfactorysignal becomes detectable only when the occupied sitesnumber iVa exceeds a given number iVat. Therefore theconcentration of gaseous stimulus must exceed thethreshold N& related to A at by:

A/ ^ M1 + N,gl/2

Then we get:N,gl/2

f-'Since

1, we must have 1 (6)

Figure 2 Olfactory response curve 9? versus stimulus concentration Wg inair fordifferent values of the exponent n (0.2, 0.6, 1) in log-log coordinates.

We thus obtain:N,at

"MFinally,replacingN^n by its expression we get:

logA^=logSVP- RT (7)It is clear that expression (7) of the threshold we obtained

is more general than the following relation:

logC0=\ogSVP +1 ^ - 2 3 3 (8)This relation was empirically obtained from electro-

antennographical responses in the honey bee (Patte et ai,1989; Laffort, 1994), where Q corresponds to the threshold,noted Ng, in our work. In this relation (8), the authors didnot take into account the adsorption energy eam or A*which should improve the correlation between calculatedand experimental values of the threshold. Note that in ourderivation we used Naperian logarithms inplace of decimallogarithms. This does not alter the correlation.

Physical interpretationsThe adsorption treatment by statistical thermodynamicswill clarify the role of some parameters in the expression of9t, and will give us interpretations of some empiricalrelations which should help to improve our understandingof the olfactory process.
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7 2 I A. Ben Lamine and Y. Bouazra

Physical origin of the power lawThe two equilibriums (1) and (2) of the stimulus dis-solution and adsorption in the olfactory mechanism arecontrolled by the partial chemical potential. Pressure andtemperature are constant. The partial chemical potential ofthe gaseous stimulus Ug is imposed by A^ according to:Hg = kTlog(N^Zg). In the beginning of the stimulus dis-solution into the mucus, the stimulus partial chemicalpotential u g is different from u^. The entry of gaseousstimulus molecules into the mucus raises u^ to the level ofUg in the absence of a stimula ting m olecular dissociation.Then the change of the stimulus molecules from thedissolved to ad sobed state raises \i to the level of nu^, whichdepends on the anchorage number.General behaviour. For an equilibrium concentration theoccupation number of olfactory sites is given by equation(5). This equation represents the population rate of thesingle energy level (-e), which is considered to be NM timesdegenerated. This equation (5) has the same expression asthe Ferm i-Dirac distribution, where u = nu g corresponds tothe 'Fermi level'. The olfactory response curve, which is theadso rption isotherm, behaves like the Fermi-Dirac distribu-tion. Usually the Fermi-Dirac distribution is plotted as afunction of energy level (here assumed to be fixed for agiven stimulus). In olfaction, this distribution (responsecurve) is often plotted in semi-log coordinates (sigmoidcurve) as a function of chemical potential, which is indeedproportional to log N g since u = n\ig = nk T log(./Vg/Zg).Local behaviour Stevens' law. For sufficiently low con-centrations (iVg Ng\fi) \ig or u is very negative: u


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Application of Statistical Thermodynamics to the Olfaction Mechanism I 7 3

we can see that the constant of equilibrium K dependsexplicitly on SVP, since the two equilibria (1) and (2)represent an adsorption in two stages of the gaseousstimulus and this adsorption is indeed a particularcondensation. From eq uation (7) we deduce the expressionof the exponent n:

(9)p o \ + log SVP - AEa IRT + log Pwhere po\ - -log A^We are now able to give an interpretation of thecorrelations established by Patte et al. (1989) and Laffortand Patte (1987). In order to predict exponent values fromexperimental olfactory thresholds, Laffort and Patte (1987)have used, both in psychophysics and electrophysiology, thefollowing empirical expressions to calculate n:

n, = -> =n, =

1Po l

Po \

Pol

1-F1- I C E

Pol

Po \

1+ l o g t f p + / /1+ log SVP or

where F = log ^V uc us + H = log Kp + H (Laffort et al.,1974), ICE = -log SVP, (Laffort and Patte, 1987) orICE = AE*, where AF? is the free enthalpy of vaporization(Laffort and Patte, 1976).

ElectrophysiologyBy recording honey bee electrophysiological responses,Laffort and co-workers have established that the correlationbetween calculated and experimental values of n is better for3 than n\ , which is better than ni - (See figure 2 in Laffortand Patte, 1987.) The Spearman p coefficient is 0.87, 0.33and 0.22 respectively for /13, n\ an d ri2-

This expression (9) can perfectly explain the establishedcorrelations in electrophysiology. From (9) we can see indeedthat /i3 approaches n better than nx or n2. For n2 thecorrelation is not as good because of Kp intervention whichis different from SVP in the case of a pola r so lution. We canconfirm the previous interpreta tions (Keil, 1982; Laffort andPatte, 1987; Patte et al, 1989): this behaviou r is explainableby considering the mucus to be an ideal solution, or moreprobably by a dry route of the stimulus, the gaseousstimulus is adsorbed by receptor sites directly from air. The

derivation of the expression of 9* in these two cases leads tothe same formula in the absence of interaction.PsychophysicsIn the absence of interaction (hypothesis iii), the parameterSVP can be replaced by the partition coefficient Kp (Henry'slaw). The expression (9) is then completely equivalent to thefollowing expression:

(10)p o X + log Kp - A a / RT + log PIn the case of stimulus-mucus interaction, expression (9)

is no longer valid. On the other hand, expression (10) canimplicitly incorporate the interaction effect through thepartition coefficient Kp, which is affected by this interac tion.Therefore, expression (10) can perfectly explain theestablished correlations in psychophysics.

In this case the auth ors have used m any different sets ofexperimental data, which are noted K, D, B, S (Laffort andPatte, 1987). The correlation coefficient p and its variationsare not the same for each set of data.

However, an average trend can be drawn : from four sets ofdata, three of them, B, K, S, show a better correlation ofexperimental n with n2 than nx. This is easily explainablewith the analytical expression (10) of exponent n, which isbetter approached by the n2 than the nx expression. Thedecreasing of the correlation in the case of n 3 is due to thefact that Henry's law is no longer valid where there is astrong interaction. The parameter ICE (-log SVP or 6J?)does not incorporate the interaction effect. To improve thiscorrelation, ICE should be replaced by the free enthalpy ofdissolution in a polar solution.

Adsorption energyThe term A or AEa can affect the olfactory threshold: thehigher the adsorption energy, the better the ad sorption, thelower the threshold. To our best knowledge, there have notyet been any investigations about this subject in olfaction.

ConclusionThe statistical thermodynamic model, which we haveapplied to stimulus adsorption in the olfactory mechanism,turns out to be quite powerful. The established expressionof the olfactory response by this model clearly brings therelation between the dynamic response and the threshold.


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7 4 I A. Ben Lamin e and Y. Bouazra

The use of stimulus chemical potential in this modelresulted in a physical interpretation of Stevens' law and of'line convergence' behaviour. Furthermore, using thismodel, we derived the empirical relation of the threshold,which was established by Laffort and co-workers, and alsointerpreted the correlations between empirical expressionsof the law exponent and the experimental values, whichwere also obtained by Laffort and co-workers. Thus thismodel confirms the role of the adsorption phenomenon inthe olfaction mechanism.

However, we notice that this model can be furtherimproved by changing the starting hypotheses in order totake into account some experimental effects such as thedependence of the saturation olfactory response on eachstimulus, or the fact that the olfactory responsecorresponding to the inflexion point in the semi-logcoord inates is higher than the half-saturation response.

Appendix: glossary of symbolsK$ equilibrium constant of stimulus dissolution

from air into mucus.Ka equilibrium constant of stimulus adsorptionfrom mucus to receptor sites.K equilibrium constant of stimulus adsorption

from air to receptor sites.Kp stimulus partition coefficient, i.e. equilibrium

cons tant of stimulus vaporization from m ucus toair.ri stimulus molecule anchorage number on

receptor sites.

NaNatN\NAE aAE a

AEV2

ZgcZZ g

u g ,

stimulus molecule number per unit volume ingaseous state.gaseous stimulus concentration of half satura-tion olfactory response.threshold stimulus concentration at gaseousstate.receptor site number per unit surface,adsorbed stimulus molecule number per unitsurface.adsorbed stimulus molecule number per unitsurface at olfactory threshold,stimulus molecule number which are anchoredon a receptor site.total number of stimulus molecules,free enthalpy of stimulus adsorption fromgaseous state.free enthalpy of stimulus adsorption fromdissolved state.free enthalpy of vaporization of pure stimulus,grand canonical partition function of the energylevel (-) of one site.grand canonical partition function of NM sites,canonical partition function of N stimulusmolecules at gaseous state,canonical partition function of N g stimulusmolecules.translation canonical partition function,canonical partition function of internal degreesof N% stimulus molecules.

j , |i chemical pota ntial of gaseous, dissolved andadsorbed stimulus.

ACKNOWLEDGEMENTSTh e au th or s are inde bted to Mr A . Jo obeu r for fruitful discussions and for his help in revising the English version and toC N U D ST for provinding us with useful references.

REFERENCESBeidler, L.M. (1954) A theory of taste stimulation. J. Gen. Physiol.,38 , 133-139

Couture, L, Chahine, Ch. and Zitoun, R. (1989) Thermodynamique,Paris, Bordas.

Davies, J.T. (1971). Olfactory theories. In Handbook of SensoryPhysiology. Springer Verlag, Berlin, Vol. IV, Part 1 , Chap. 1 3.

Diu, B., Gu thm ann , C, Lederer, D. and Roulet, B. (1989a) PhysiqueStatistique. Hermann, Paris, p. 41 0.Diu B., Guthmann C, Lederer D. and Roulet B., (1989b) PhysiqueStatistique, Hermann, Paris, pp. 67 6-6 87 .Greenberg, M.J. (1979) Dependence of odor intensity on thehydrophobic properties of molecules. A quantitative structureodor intensity relationship. J. Agric. Food Chem., 27, 347-352.

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Application of Statistical Thennodywunics to the Olfaction Mecfaanisni I 7 5

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