Primary events in odour detection

Bulletin o[ Mathematical Biology, Vol. 44, No. 3, pp. 411-423, 1982.

Printed in Great Britain.

PRIMARY •

EVENTS

IN ODOUR

0092-82401821030411-13503.00/0 Pergamon Press Ltd. © 1982Society for Mathematical Biology

DETECTION

W I M VAN DRONGELEN

Department of Animal Physiology, Agricultural University, Haarweg 10, Wageningen, The Netherlands • YVES PA6NOTTE I.U.T.I.-Informatique, Universit6 Claude Bernard, 43 Bd. ll novembre 1918, 69621 Villeurbanne, France •

MAT H . HENDRIKS

Department of Mathematics, Agricultural University, De Dreijen 8, Wageningen, The Netherlands

An analysis of the interaction between stimulus molecules and the olfactory receptor cell membrane is presented. The model is based upon a sequence of events, i.e. stimulus delivery at the olfactory epithelium, absorption of molecules in the mucus layer, diffusion of the molecules towards the receptor cells and molecule-receptor cell membrane interaction. The mathematical analysis considers the situation during electrophysiological experiments, where an odour puff is delivered at an exposed olfactory mucosa. Such a situation resembles sniffing of odour samples. The analysis is discussed in relation to experimental evidence.

1. Introduction. The arrival of odorants over the olfactory epithelium is the first step in the series of events that may finally lead to detection and recognition of odour. In terrestrial vertebrates the odorants are in a gaseous phase and must be absorbed in the mucosa, then they must tranverse the mucus layer before reaching the olfactory receptor cells (Ottoson, 1956). 411

412

WIM VAN DRONGELEN et al.

Prediction of the time course of the events that lead to the response of olfactory neurones might aid in our understanding of the processes underlying primary mechanisms in olfaction. Several general theories describing early events in olfactory perception have been presented (Wright et al., 1956; Amoore, 1962; Davies, 1965). Quantitative models for stimulus molecule-chemoreceptor cell membrane (receptor site) interaction have been described by Beidler (1971), Kaissling (1971), Poynder (1973) and De Simone and Price (1976). Bostock (1973) has presented a time-dependent model for the diffusion process. In the present study a model describing the time course of the absorption and diffusion processes combined with receptor site occupation is developed. 2. The Model. An odour puff delivered at the mucosa results in a modification of neural activity, i.e. excitation and/or inhibition of receptor cell's spike discharge frequency (Gesteland et al., 1965; Holley et al., 1974; van Drongelen, 1978). The olfactory cell response is thought to be a result of five sequential processes: 1. convection and diffusion of molecules from an odour source to the olfactory mucosa; 2. absorption of molecules in the mucus that surrounds the olfactory receptors; 3. diffusion of molecules from the mucus surface to the receptor cell; 4. detection of the stimulus molecules by a physico-chemical modification of substances in the receptor membrane, i.e. receptor site modification; 5. ion permeability changes in the olfactory cell membrane caused by the physico-chemical changes mentioned above. This process finally leads to modification of the membrane potential. In the case of excitation the membrane becomes depolarised, which causes spiking. The processes described are represented in Figure 1. In this figure an experimental set-up is shown. A stimulator (ST) delivers an odour puff with a certain, time-dependent concentration of odorant in gaseous phase c(0 ~, t). Absorption in, and diffusion through, the mucus results in a time- and distance-dependent concentration of substance [c(x, t)] in the mucosa. The detection of the stimulus at the cell membrane is modelled as a transition of receptor sites from state A to B without incorporation of stimulus molecules (Figure 1). 3. Odour Concentration Above the Mucus. In electrophysiological experiments an air stream led over an odour source delivers a number of

PRIMARY EVENTS IN ODOUR DETECTION

413

W • " "" ,.,,-

c(oat)

, '

Sl-imulus

dehvery

,',

I D,f-~us,on' ' "-. c(x.l)

c l o 1-) ks~+ A ~ B

k6

Figure 1. A model for olfactory stimulation. A stimulus device (ST) delivers a number of molecules c(0a, t) above the mucosa. The molecules become absorbed in the mucus [c(0", t)] and diffuse in the x direction, with 0 ~
molecules over an e x p o s e d olfactory mucosa. A f t e r stimulus application, a second airstream withdraws the odour vapours. In order to avoid artifacts due to mechanical stimulation of the o l f a c t o r y receptors, both air streams are given a low flow rate of a few cm per sec. B e c a u s e of the experimental a r r a n g e m e n t , the stimulus c o n c e n t r a t i o n above the m u c u s displays a gradual time course as can be m e a s u r e d b y a flame ionisation detector (Bostock and P o y n d e r , 1972). T h r o u g h o u t the analysis the odour concentration is considered as a function of time (t) and depth in the mucus layer (x), as shall be indicated b y c(x, t). The distance x is defined at zero at the m u c u s - a i r b o u n d a r y layer (Figure 1). At x = 0 the indices a and m indicate w h e t h e r the zero depth refers to the air or mucus. We consider a rise of odour c o n c e n t r a t i o n in a time interval b e t w e e n to and tl and a fall of this c o n c e n t r a t i o n b e t w e e n t~ and t2. The formulas used to describe the rise and fall of the c o n c e n t r a t i o n in air are:

Cr(Oa, t) = C~t(1 -- e -k'')

(0 ~< t ~< t,)

ct(O ", t) = c ~ ( 1 - e -k''') e -k~('-'')

(& <~ t <- tz).

(la) (lb)

414

WIM VAN DRONGELEN et al.

The factor c~ represents the maximal concentration that can be reached in air. Time constants kl and k2 govern the rate of stimulus delivery and stimulus withdrawal processes. The indices r and f indicate the rise and fall of the concentration respectively.

4. Odour Absorption.

When the odorant is delivered, a fraction of the molecules becomes absorbed in the mucus. Absorption is considered as an exchange between air and mucus. This takes place at x = 0, absorption with a time constant k3 and desorption with a time constant k4 (Figure 1). If c(Oa, t) is expressed in a partial vapour pressure and c(Om, t) in molar, (k4[k3) is approximately equal to the coefficient of Henry (Holley and MacLeod, 1977). Local odorant concentrations in the air and mucus are represented by c(0a, t) and c(Om, t). The differential equation for an absorption process, similar to Langmuir adsorption, taking into account the molecules that diffuse into the mucosa, is: d¢(0 m, t) : kac(O a, t) - k4c(Om, t) - Ra.

dt

In the following we neglect the rate of removal by diffusion (Ra). We obtain the following boundary value problems (2), (3a) and (2), (3b): dc(0", t) = _ k4c(O m, t) + k:(O a, t) dt

(2)

Cr(O", 0) = 0

(3a)

c/(0 ", tl) = cr(0 ", tl).

(3b)

The solutions can be found by the method of variation of parameters. The solution satisfying (2) and (3a) is:

c~(Om, t) = ~ c~G(O, l, t, k., k4, D).

(4a)

The solution satisfying (2) and (3b) is:

c/(O", t) = ~ c~[ G(O, l, t . k~, k4, D) e -k"-'° k4(1 - e-k:')

-~Z~22

]

g(O,l,t-t~,k2, k,,O) ,

(4b)

PRIMARY EVENTS IN ODOUR DETECTION

415

where

G(x, 1, t, kl, k4, D) = 1

k4 e -k~' cos k4-

kl

(1 - x)x/(klD -~)

cos l~/(klD -1)

÷ k~ e -ht cos (l - X)~v/(k4D -') k 4 - kl cos Ix/(k4D-')

(5a)

and

g(x, 1, t, ki, kn, D) = e -k~' cos (l - x)~v/(k4D -1) cos IN/(k4D -1) _ e_k,t cos (l - x)N/(klD -~) cos lx/(kiD -~)

(5b)

5. Odour Diffusion.

Before being able to contact a receptor neurone's membrane, the absorbed odour molecules have to traverse a layer of mucus. This process may be described as diffusion in the x direction with 0 ~< x ~< l (see Figure 1) by the diffusion equation:

Oc(x, t) D ~2C(X' t) cgt = ~x 2

(6)

in which D denotes the diffusion constant. (i) The rise of stimulus concentration. In the domain defined by 0 ~< x ~< l and 0 ~< t ~< tl we have the following initial and boundary conditions: cr(x, 0) = 0

c'(O r', t) = ~4 c~G(O, l, t, k~, k4, D) act I = 0. aX x=t

(6a) (6b)

(6c)

The solution of the problem defined by (6) up to (6c) can be found by using Laplace transform (see Appendix): Cr(X, t) = ~ c~[G(x, l, t, k,, k4, D) - S(x, l, t, k,, k4, D)], ~4

(7)

416

WIM VAN DRONGELENet al.

where G is defined by (5a) and S by

S(x, l, t, kl, k4, D) -

2klk414 ---~-y- ~

rl=O

( - 1) 2 cos (n + ½)Tr(! - x)1-1, e -("+½'2~2r2°' lx2 (n +½){(n + ~)~'ZTr2-k,12D-~}{(n + ~) ¢r2 _ k412D-1}.

(7a)

(ii) The fall of stimulus concentration. In the domain defined by 0 ~< x ~< ! and tl ~< t ~< t2 the initial and boundary conditions are (8a)

ci (x, tl) = c" (x, tl) o [ G(0, 1, tl, kl, k4, D) e -g4(t-tl) cI(0 ", t) = ~k3 c m

k4(1 - e-k'") ] k 4 - k: g(0, l, t - t~, k:, k4, D ) j tgCI [ = 0. aX I~=t

(8b)

(8C)

The solution of the problem defined by (6) and (8a) up to (8c), again using Laplace transform (see Appendix), is:

c'(x,t)=~c~[G(x,l,t,

kl, k4, D ) - S ( x , l , t ,

kl, k4, D)

+ g(x, 1, t - t~, O, k4, D)

k4 k 4e- -kl`' k~ g(x, l, t - t~, k~, k4, D)

k4(1 - e -klt~) . -k-~-- k2 g~x, l, t - tl, k2, k4, D) - s(x, l, t - tl, O, k4, D) + e-kltl s(x~ l, t - t~, k~, k4, D) + (1 - e-~'")s(x, l, t - t~, k2, k4, D)],

(9)

where G, g and S are defined by (5a), (5b) and (7a) and _ 2~k_k41z ~ ( - 1)"(n +½)cos (n +2)~r(I- x)l -~" e -~"+½~2"~'-~D' s(x, l, t, k~, k4, D) D ,=o {(n + 2)2~r2 - k,12D-1}{(n + ½)ZTr2- k,l:D -~} "

(9a) 6. Modification of Receptor Sites. In this study the detection of stimulus molecules is modelled by a hypothetical reaction represented in Figure 1;

PRIMARY EVENTS IN ODOUR DETECTION

417

the concentration of stimulus molecules at a certain depth catalyses the transition of receptor sites from A to B. The rate constant k5 characterizes the transition A ~ B, and the rate constant k6 determines the reverse process B ~ A. The question is how the modification of receptor sites has its impact on permeability changes of the cell membrane. Starting from a chemical model for receptor site occupation, several parameters can determine the neural response (Heck and Erickson, 1973): 1. the brute or net rate of the receptor site transition A ~ B; 2. the brute or net o c c u p a n c y of receptor sites; 3. rate as well as o c c u p a n c y of receptor sites. In case of net o c c u p a n c y of receptor sites, the following equation is obtained: dB dt = k : ( a , t ) A - k6B,

(10)

where A = T - B with T the total concentration of adequate sites. Equation (9) may be written as: dB dt ~-f ( t ) B + g(t) = O,

(11)

with f ( t ) = ksc(a, t) + k6 and g(t) = - ksTc(a, t). Recognising (11) as a linear differential equation, we may write: B = - e -mr)

(fo'

with F ( t ) = fo' f ( t ) dt and M = constant.

)

g(t) e ~") dt - M ,

418

WIM VAN DRONGELEN

et al.

7. Discussion. In the model presented the number of receptor molecules affected by a particular odour has been presented as a function of time. Because of the deductive character of the model it may be modified to represent the number of receptor sites affected as a function of stimulus concentration or as a function of previous stimulations. In this way theoretical expressions are obtained for a stimulus-response relationship or adaptation effects respectively. The amount of receptor molecules occupied can be compared with the time course of a single cell's spike activity. Alternatively, the electro-olfactogram (E.O.G.), which is considered to represent a summated receptor potential (Ottoson, 1956; Getchell and Getchell, 1977), can be related to the interaction process between odorant molecules and cell membranes. It should be noted that at the stage of a receptor potential chemo-electrical transduction has occurred. Theories concerning this transduction mechanism in chemosensory neurones were proposed by Davies (1965), De Simone and Price (1976) and van Drongelen et al. (1978). In our analysis the rate of removal by diffusion at x = 0 is neglected. Alternatively, one can postulate a surface layer with quite different properties from the rest of the liquid. In the olfactory mucosa such a surface layer is described by GetcheU and Getchell (1977). Here Ca, Cm and C refer to air, surface layer and internal values--h and h m given by the steady state relations C= ACa and Cm = hmCa. Then if 8 =layer thickness and A = area = 1, so that 8 = volume considered, one can state: (SCm) =

Po(AmCa-Cm)-PI[Cm-~-~C(x=O)]

together with

0C 02C c~t = D--~r and =T

--X- C ( x = o)

([Pohm/8] being k3 and [PoAm/8]+ P1 playing the role of /<4). This alternative complicates the obtaining of exact solutions but can be used if numerical solutions are sought. If it were possible to measure C(x, t) for small x, then the role of a surface layer could be established. The time course of the E.O.G. under the influence of prolonged

PRIMARY EVENTS IN ODOUR DETECTION

419

stimulation demonstrates an initial transient voltage peak followed by a lower tonic potential level (Ottoson, 1956). This might be due either to summation effects of individual receptor potentials or to a similar shape of these individual potentials and the E.O.G. Because of the time course of the receptor neurones of different modalities (Ottoson, 1971), we accept that the latter hypothesis may be realistic. Unless coefficients are added in the receptor occupation model (in general p A ~ q B ) the transiently and tonically composed time course does not follow from our model. However, when these coefficients p and q are added in an • appropriate manner the descending slope of the theoretical E.O.G. is also characterised by transient and tonic components. Such a characteristic was never observed in any recorded E.O.G. Presumably, the peculiar time course of the E.O.G. under the influence of sustained stimulation is mainly affected by local concentrations of some ion species during receptor potential generation. Takagi et al. (1968) demonstrated such effects of Na ÷ and K ÷ concentrations on the shape of the E.O.G. Recent studies on the structure of olfactory receptor neurone membranes demonstrated the presence of particles in these membranes and a correlation between the particle density and E.O.G. amplitude was found (Menco, 1977; Masson et al., 1977). It might be these particles that function as receptor molecules indicated by A and B in formula (10).

APPENDIX Let C(x, p) be the Laplace transform of c(x, t), i.e. C(x, p) = ~lc(x, t)](p). The problem defined by (6) up to (6c) becomes after transformation: 02C r pC r - 0 = D ax 2 Cr(Om'p)~-~4CM[1

,9C" [ OX

(6') ( k , - k , )k4 (p+kO

"4

( k 4 - k l )kl( p + k4)]J

=0.

(6'b)

(6'c)

x=l

The general solution of (6') is: Cr(x, p) = A e xw'D-I) + B e -xw'°-l).

The condition (6'c) leads to A = B e-2~v°'D-~). The condition (6'b) yields: (k4- kl)(p + kO

kl ] 1 (k4 - kO(p + k4)] e-2'v~-~) + 1"

420

WIM VAN DRONGELEN et al.

Finally, the solution is: k3 ~ [ 1 C'(x, p) = ~ c u [~

k4 kl ] cosh (l - x)~v/(pD -I) (k4- k~)(p + kl) 4 (k4 k0(p + k4) cosh l ~ / ( p D -~) -

(7')

-

To find the inverse Laplace transform we use: ~_~{

1 p

coshbx/p~

coshbx/(ic)_,,

ic cosh a x / p J = cosh a x / ( i c ) e

-2~- ~ (-1)n(n +½) cos (n + ½)~rba -l . e -~+~)2~2~-2' ,=o (n +½)2,r + ia2c ' where i = X/(- 1) and a, b and c are constants (see Roberts and Kaufman (1966), Table of Laplace Transforms). Substituting a = lx/(D-1), b = (l - x ) x / ( D - ' )

and respectively c = O, c = ikl, c = ik4

and using cosh iz = cos z yields: ..~-~ { l cosh (I - x ) x / ( P D -~) cosh I ~ / ( p D - ' ) = 1 -

2Ir ~ (-l)n(n + 1) cos (n + 1)zr(l - x)1-1" e (n+~)2"~2'-2°'

(i)

1 cosh (l - x)X/(pD-~)~ = e_k,, cos (l - x ) x / ( k ~ D -~) ~-1 p + kt" - ~ s h ~ J cos l~/(k~D -~) -2rr ~. (-1)"(n +½) cos (n +½)~r(l - x)1-1, e -c~÷~2'-2°~ (n +_~2~.2 12D-lkl n =0

Za_l {

-

2J

(ii)

--

1 cosh ( l - x)~v/(pD -1) e -k4t cos (! - x)~v/(k4D -1) P-~4" cosh I x / ( p D -1) = cos l % / ( k 4 D -1)

2~- ~.. (- 1)~(n + ½)cos (n + ½)~r(l - x)l -~. e -(n*~)2~2t-2°' ~o

(n + ½)2zr2 - 12D-lk4

(iii)

Using (6') and (i)-(iii), one obtains expression (7) for c'(x, t) in terms of the defined functions S and G. In the problem defined by (6) and (8a) up to (8c) we first substitute t = t, + r. Then cl(x, t) = ct(x, tl + z) = c{(x, ¢) 3t

= ac . l = 8¢ 0¢

PRIMARY EVENTS IN ODOUR DETECTION and

421

O:c~ O2c{ aU--~

The function c{ satisfies:

oc{ = n '~2c~ OT

(6-1)

"" OX2

(8a-l)

c((x, O) = cr(x, t,)

c{(om'~')=~4CM[(1

k4e-k'q-k'e-k'") e-k'k4-~

k'(1-e-k'q)" - k 4 " ~ 4 - - ~te- ~ 2 --e-k"]2)]

OC{

=0.

(8b-l)

(8c-1)

OX x=t

Application of the Laplace transform to this problem yields: 02C{0x 2 DPC{ = --~1 c r (x, tO

ltu , p ) = ~ c ~

1

k4 kl

/p+k3 oc; [ OX

k4-k2

(6'-1)

p+k4---

= o. x=l

The general solution of (6'-1) is:

C{(x, p) = ~(x) e xw"°-') + O(x) e -xw'°-'), where ~ and ~ satisfy the set of equations: tip' e xx/O'o-l) + ~b' e -x~/(po-l) = 0

~/(pD-1) • ~b' e xx/~°-l) - V'(pD-1)~b ' e -x~/°'D-~) = - D-Ic~(x, tO. Solving ~b and ~b we find 1 /" --x~/(pD-l) rz 2X/(p-~ J e c tx, tO dx

~b(X) = and

~(x) = 2 ~

f eXWpO-1)cr(x' h)dx.

After much calculation we find, using the boundary conditions (8'b-l) and (8'c-1), c o s (l-x)x/(klD -1) 1 k , - k~ " cos l~/(k~D-1) p + k~ + kl e -k4'l "cos (l - x)x/(k4D-1). 1 k4 - kl cos lx/(k4D -~) p + k4 k3

,~

[1

C{(x, p) = ~ cM ~

k4 e -klq

(8'b-l) (8'c-1)

422

WIM VAN DRONGELEN et al. 2k,k4l 4 ~ (-1) n cos (n + ½)Ir(l - x)1-1, e -(n+~)2~EI-2D'l 1 ~rD n~o (n + ½){(n+ ½)2~r2- kll2D-1}{(n + ½)21r2- k412D-'}" p + (n + ½)21r2/-2D

+

1 P+L

1 -P

k4 e-kit1 P L-k~ +L

P

Sk~

k4 - k2

,

p+

c--~hT ~

J

Taking the inverse Laplace transform and using r = t - tl we obtain the expression for c1(x, t) given in (9). T h e a u t h o r s t h a n k P r o f e s s o r s K . B. D ~ v i n g , A. H o l l e y , J. A. D e S i m o n e a n d L. M. S c h o o n h o v e n f o r v a l u a b l e s u g g e s t i o n s a n d M r s . C. A. R. B r e m e r Witteveen for typing the manuscript. This study was supported by the Netherlands Organization for the Advancement of Pure Research

(Z.W.O.). LITERATURE Amoore, J. E. 1962. "The Stereochemical Theory of Olfaction". Proc. Scient. Sect., Toilet Goods Ass. 37 (suppl.), 1-23. Beidler, L. M. 1971. "Taste Receptor Stimulation with Salts and Acids". In Handbook of Sensory Physiology Chemical Senses 2, Ed. L. M. Beidler, pp. 200-220. Berlin: Springer. Bostock, H. 1973. "Diffusion and the Frog E. O. G.". In Transduction Mechanisms in Chemoreception, Ed. T. M. Poynder, pp. 27-38. London: Information Retrieval Limited. and T. M. Poynder. 1972. "Apparatus for Delivering and Monitoring a Sequence of Odour Stimuli". J. Physiol. Lond. 224, 14-15. Davies, J. T. 1965. "A Theory of the Quality of Odours". J. theor. Biol. 8, 1-8. De Simone, J. A. and S. Price. 1976. "A Model for the Stimulation of Taste Receptor Cells by Salt". Biophys. J. 16, 869. Drongelen, W. van. 1978. "Unitary Recordings of Near Threshold Responses of Receptor Cells in the Olfactory Mucosa of the Frog". J. Physiol., Lond. 277,423--435. --, A. Holley and K. B. D¢ving, 1978. "Convergence in the Olfactory System: Quantitative Aspects of Odour Sensitivity". J. theor. Biol. 71, 39-48. Gesteland, R. C., J. Y. Lettvin and W. H. Pitts, 1965. "Chemical Transmission in the Nose of the Frog". J. Physiol. Lond. 182, 525-559. Getchell, T. V. and M. L. Getchell, 1977. "Early Events in Vertebrate Olfaction". Chemical Senses Flavor 2, 313-326. Heck, G. L. and R. P. Erickson, 1973. "A Rate Theory of Gustatory Stimulation". Behav. Biol. 8,687-712. Holley, A., A. Duchamp, M. F. Revial, A. Juge and P. MacLeod, 1974. "Qualitative and Quantitative Discrimination in the Frog Olfactory Receptors: Analysis from Electrophysiological Data". Ann. N. Y. Acad. Sci. 237, 102-114. - and P. MacLeod, 1977. "Transduction at Codage des Informations Olfactives chez les Vert6br6s". J. Physiol. Paris 73, 725-828. Kaissling, K. E. 1971. "Insect Olfaction". In Handbook of Sensory Physiology Chemical Senses 1, Ed. L. M. Beidler, pp. 351-431. Berlin: Springer. Masson, C., S. Kouprach, I. Giachetti and P. MacLeod, 1977. "Relation between Intramembranous Particle Density of Frog Olfactory Cilia and E. O. G. Response". In Olfaction and Taste 6, Eds J. Le Magnen and P. MacLeod, p. 195. London: Information Retrieval Limited. Menco, B. P. M. 1977. "A Qualitative and Quantitative Investigation of Olfactory and Nasal

PRIMARY EVENTS IN ODOUR DETECTION

423

Respiratory Mucosal Surfaces of Cow and Sheep Based on Various Ultrastructural and Biochemical Methods". Commun. Agric. Univ., Wageningen 77, 13-170. Ottoson, D. 1956, "Analysis of the Electrical Activity of the Olfactory Epithelium". Acta Physiol. Scand. 35, suppl. 122, 1-83. --. 1971. "The Electro-Olfactogram". In Handbook of Sensory Physiology Chemical Senses 1, Ed. L. M. Beidler, pp. 95-131. Berlin: Springer. Poynder, T. M. 1973. "Self-shunting and the Frog E. O. G." In Transduction Mechanisms in Chemoreception, Ed. T. M. Poynder, pp. 241-250. London: Information Retrieval Limited. Roberts, G. E. and H. Kaufman, 1966. Table of Laplace Transforms. Philadelphia: W. B. Saunders Co. Takagi, S. F., G. A. Wyse, H. Kitamura and K. Ito, 1968. "The Roles of Sodium and Potassium Ions in the Generation of the Electro-olfactogram". J. gen Physiol. 51,552-578. Wright, R. H. C. Reid and H. G. V. Evans, 1956. "Odour and Molecular Vibration. A New Theory of Olfactory Stimulation". Chem. Ind. 37, 973-977. RECEIVED 9-11-78 REVISED 2-19-81