Chem. Senses 25: 293-311,
2000
© Oxford University Press 2000
Perireceptor and Receptor Events in Olfaction. Comparison of Concentration and Flux Detectors: a Modeling Study
ivan1
2
Unité de Biométrie, INRA, 78026 Versailles Cedex, France,
1 Institute of Entomology, Academy of Sciences of Czech Republic, Brani
ovská 31, 370 05 
eské Bud
jovice and
2 Institute of Physiology, Academy of Sciences of Czech Republic, Víde
ská 1083, 142 20 Prague 4, Czech Republic
Correspondence to be sent to: Jean-Pierre Rospars, Unité de Biométrie, INRA, 78026 Versailles Cedex, France. e-mail: rospars{at}versailles.inra.fr
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Abstract |
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Transduction in chemosensory cells begins with the association of ligand molecules to receptor proteins borne by the cell membrane. The receptor–ligand complexes formed act as signaling compounds that trigger a G-protein cascade. This receptor–ligand interaction, described here by a single-step or double-step reaction, depends on factors controlling the access of the ligand molecules to the cell membrane. Two basic mechanisms can be distinguished: concentration detectors (CD), in which the ligand can freely diffuse to the membrane, and flux detectors (FD), in which it accumulates irreversibly in a distinct perireceptor space where it is chemically deactivated. These two systems, plus their generalization, are investigated and compared. The transient and steady-state numbers of complexes are studied as a function of the external ligand concentration. The biological significance of the results is shown in a well-studied example of FD, the insect sex-pheromone olfactory receptor neuron. How the number of complexes can code for the intensity of stimulation is analyzed using the size, dynamic range and sensitivity of the steady-state responses, and the time needed to reach a predefined level of the transient responses. It is shown that the FD design affords a large increase in sensitivity (a shift of the threshold response towards low concentration) with respect to the CD design, which is paid for by a lesser ability to follow fast changes in stimulus intensity.
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Introduction |
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TopAbstractIntroductionThe models and their...Numerical resultsDiscussionAppendix: differential equationsReferences |
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Olfactory receptor cells are able to transduce chemical stimuli present in their environment into an electrical response [reviewed in (Ronnett and Snyder, 1992
; Shepherd, 1992; Lancet and Ben-Arie, 1993; Ache, 1994; Buck, 1996; Schild and Restrepo, 1998); see also (Lauffenburger and Linderman, 1993; Hildebrand and Shepherd, 1997)]. A crucial step in this transduction process is the interaction of the ligand molecules with receptor proteins borne by the cell membrane. Following the ligand–receptor binding and the subsequent receptor activation, a GTP-binding G-protein is activated which in turn acts on enzymes that generate second messengers (e.g. cyclic AMP). These second messengers open a large number of ion channels which change the membrane potential. This response can essentially be regarded as an amplified version of the weak signal resulting from the initial binding.
In our previous work (Lánsk and Rospars, 1993, 1995; Rospars et al., 1996a,b) we studied simple models of these transduction events. Concerning ligand–receptor interactions, we considered two basic types of interaction: in the first, the transduction cascade was triggered by mere binding of the ligand to the receptor to form a complex (single-step interaction); in the second, additionally, an activation of the receptor–ligand complex was required (two-step interaction). The simpler of these two descriptions dates back to Lasareff, who proposed a model in which the effect was proportional to the number of bound receptors at equilibrium state (Lasareff, 1922). Lasareff’s model led to several generalizations. Beidler considered a population of different receptors yielding a linear combination of the original characteristics, and also a mixture of different odor ligands (Beidler, 1962). Later, Tateda (Tateda, 1967; Ennis, 1991) suggested that several ligand molecules had to bind to the receptor to activate it. These models were investigated and implemented in a range of different applications (Getz and Akers, 1995; Malaka et al., 1995; Getz, 1999). All of them can be classified as concentration detectors (CDs), using the terminology introduced by Kaissling (Kaissling, 1998a), because the sensory membrane is assumed to be directly exposed to the external stimulus.
However, in several systems of interest, the cell membrane is only part of a larger system, so that the ligand–receptor interaction cannot be considered in isolation (Kaissling, 1974). Ligand molecules, being originally in the external environment, must be transported to the vicinity of the membrane, into the so-called perireceptor space. In insects, for example, olfactory receptor neurons are located within hairs. Odorant molecules are caught on the hairs, diffuse at their surface, enter the hair lumen through small holes in the cuticle and are transported by olfactory binding proteins (Pelosi, 1996) across the lumen to the dendritic membrane of receptor neurons. Because odorant molecules cannot leave the system (Kanaujia and Kaissling, 1985) they must be degraded to prevent their accumulation in the perireceptor space. This involves various enzymatic mechanisms acting on different time scales [slow (Kasang et al., 1988); fast (Ziegelberger, 1995)]. Recently Kaissling, studying olfactory mechanisms in insect sensilla, proposed a model taking into account the perireceptor space and the degradation reactions (Kaissling, 1998a,b). These flux detectors (FDs), which offer an elegant solution to the shortcomings of the previous approach (Lánsk and Rospars, 1998), might apply to olfactory receptors in both invertebrates and vertebrates.
In the present work we undertake a systematic comparison of the properties of these various perireceptor and receptor models, with the aim of better understanding their signal processing features and, therefore, their biological meaning. We not only consider the CDs and FDs in their pure forms, but also introduce a more general model, denoted GD, that admits the concentration and flux detectors as special cases. For all three detector models, the single- and double-step types of ligand–receptor interactions are studied. Consequently the results obtained can be used for modeling a wide range of chemosensory systems. The relative merits of these models depend on the specific system under examination. For example, CD models are more appropriate for describing systems in which the ligand can relatively freely access and leave the perireceptor space, as seems to be the case in insect carbon dioxide receptors (Stange, 1996), taste receptors, hormone receptor systems and unicellular organisms. FD models can be considered for systems in which the diffusion of the ligand to the perireceptor space is irreversible, as in insect olfactory sensilla. For systems in which the diffusion to and from the perireceptor space is asymmetrical, the more flexible GD model appears as a better choice than the extreme CDs and FDs. Other features differentiating the models from a signal-processing point of view are described and might suggest selecting one of them for further consideration. In all cases single-step models are simpler, which is a good reason to study them because they might give a sufficiently accurate description of some systems. In the present paper, the description is limited to the insect sex-pheromone receptor neuron (we show that CD cannot account for its known properties whereas FD, and its generalized version GFD, can). This is why the analysis of the GD model is not treated in its full generality. However, the approach followed is intended to be applicable to any chemosensory system.
The response of the model systems to stimuli of various strengths and time courses are analyzed. First, we analyze how the steady-state levels of the signaling (bound or activated) complex depend on a stimulation of constant strength. In addition to this steady-state response, we investigate the kinetics of the response when the systems are exposed to a stimulus varying in time. We determine and compare the times the systems need to approach equilibria (or to reach any predefined levels) using their response to a stepwise increase of the external stimulus concentration. The square wave stimulation enables us to describe their post-stimulus time evolution and the times they need to return to their resting state. Finally, we also consider periodic stimuli, which are better approximations to natural odorant stimuli than constant ones because, both in air and water, turbulence breaks odor plumes into discontinuous patches (Kramer, 1986; Murlis, 1997). The periodic stimuli allow us to compare the diverse ability of these systems to follow rapidly fluctuating concentrations of ligands. For analyzing the static and dynamic aspects of these responses, we use biologically meaningful characteristics. The static characteristics based on the steady-state conditions include the stimulus concentration at threshold, which measures the sensitivity of the system, and the range of concentrations discriminated. The time to reach a predefined level (e.g. half-maximum steady-state response) or latency is used to characterize the dynamic properties of the systems.
This comparison of various types of detectors is presented in two parts. In the first part (The models and their analytical description) equations describing their behavior are reviewed, and the problems that can and cannot (or not easily) be described analytically are indicated. In the second part (Numerical results), using the analytical results when available or numerical simulations otherwise, the properties of the detectors, including their sensitivity, dynamic range and latency, are compared. The comparisons illustrate the advantages and drawbacks of the CD and FD systems using the moth sex-pheromone olfactory sensillum as a case example, based on the classical investigations by Kaissling and co-workers and on recent numerical estimates of the main parameter values of this system (Kaissling, 1998b).
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The models and their analytical description |
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Stimulations, detectors and receptor–ligand interactions
All symbols used are defined in Table 1. Table 2 gives a synopsis of the six models investigated.
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Step, square and periodic stimulations
The ligand molecules L are diluted at time t in the carrier medium (water or air) at a concentration Lex(t). This external concentration can be constant or varying in time in a way that is controllable or at least measurable by an experimenter. Four types of stimulation are considered in this article:
Step stimulation. At time t = 0 the concentration of the ligand changes abruptly from zero to a constant level Lex and remains at this level (Figure 1a). This stimulation allows one to investigate the steady-state behavior of the system.
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Square pulse stimulation. The same change in concentration at time t = 0, from zero to level Lex, is followed at t = t0 by an abrupt reset to zero (Figure 1b). It permits one to study the effect of removing the stimulus and the effect of the stimulus duration t0. The total amount of ligand delivered in this type of stimulation is proportional to Lext0. The effects of square pulses, either strong and short or weak and long, delivering the same amounts of ligand can be compared.
Periodic stimulations. Two kinds of periodic stimulations were used. The first one (Figure 1c) consists of a series of square pulses with fixed length t0 and fixed height Lex, which were presented at different frequencies 
(number of pulses per time unit). It has the advantage of being directly comparable to real experimental situations. Unfortunately, in this kind of stimulation changing the frequency also changes the amount of ligand delivered per unit of time. For this reason, we also considered a stimulation described by a sinus wave initiated at time 0 (Figure 2d)
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 | (1) |
where the parameters are the angular frequency (in rad/s, i.e. the period is T = 2
/, expressed in s), the amplitude 
1 and the level of the constant component 0 (both in µmol/l). The amplitude 1 has to be smaller than or equal to 0, so that Lex(t) 
0. The constant component may be seen as the amount of ligand delivered per unit of time, which is independent of the frequency of stimulation. In the periodic stimulation the main variable of interest is its frequency, which may correspond to some external conditions, e.g. breathing in vertebrate animals or segmentation of the air plume in insects. Stimulation (1) for small t is interesting for studying the transient effects at onset; for large t the solutions converge to a periodic solution (quasi-steady state).
All four stimulation types permit the study of the response of the system at the onset of stimulation, thus solving the problem of initial detection. However, if an organism can detect only concentrations varying in time and loses its ability to perceive a constant stimulus, which is, for example, the case of male moths in a pheromone plume (Kennedy et al., 1980; Willis and Baker, 1984), a periodic stimulation is more appropriate and becomes one of the most important parameters.
Concentration, flux and generalized detectors
Ligand molecules L present in the external environment can reach the perireceptor space in the vicinity of the cell sensory membrane. Their concentration there is denoted by L(t). In accordance with Kaissling (Kaissling, 1998a,b), two kinds of transfer between the external space and the perireceptor space are investigated: fast (immediate) and reversible—concentration detection—in contrast with slow and irreversible—or flux detection (Figure 2).
Concentration detector. This is the classical system in which the odorant concentration L(t) in the perireceptor space is always equal to the concentration Lex in the external space, L(t) = Lex(t). The molecules can move freely and instantly between the two spaces, which are therefore indistinguishable. No parameters other than those used for characterizing the receptor–ligand interaction are needed for describing this fast and reversible system (Table 2, CD1 and CD2).
Flux detector. On the contrary, in this system the external and perireceptor spaces are physically distinct and the molecules can move inward only from the first into the second. It follows that the ligand molecules must be degraded in the perireceptor space to prevent their indefinite accumulation there. The influx in the perireceptor space is
 | (2) |
and the outflow (degradation flux) is koC(t) or koC*(t), where C(t) and C*(t) are the concentrations of the signaling receptor–ligand complex in one- and two-step interactions respectively (see below Single- and double-step ligand interaction with receptors). So, two parameters are needed, the rate constant ki, characterizing the velocity of the translocation, and the rate constant ko, characterizing the velocity of the transformation of the ligand molecules L into a degraded form 
that cannot interact with the receptors (Table 2, FD1 and FD2; note that the arrow 
denotes a translocation, in contrast to 
, which denotes a chemical reaction).
Generalized detector. However, the two detectors above can be considered as ideal (extreme) cases, whereas many real systems are likely intermediate. Moreover, the FD models can lead to unrealistic consequences if the stimulation is high or the degradation is not sufficient (see below), because the influx to the perireceptor space is independent of the ligand concentration there. This unrealistic behavior can be prevented if the influx 
(t) slows down when the concentration in the perireceptor space grows. This is the case if it obeys the following kinetics
 | (3) |
because the added term k–iL(t), where k–i is a positive constant, increases with L(t), which counterbalances the influx term kiLex(t). Equation (3) does not prevent a backward flow of ligand molecules from the perireceptor space if the external concentration decreases and kiLex(t) becomes less than k–iL(t). So, the influx of this generalized detector is reversible, as in the pure concentration detector.
A special case of GD, which we call generalized flux detector (GFD), prevents ligand outflow. It obeys the following kinetics
 | (4) |
Choosing the maximum of zero and kiLex(t) – k–iL(t) prevents a backward flow in the condition stated above. So, the influx of the GFD may be seen as irreversible, like the pure flux detector. This GFD model remains realistic even at high concentrations and, moreover, it behaves at stimulus onset like either the CD or the FD, depending on the values of ki and k–i (see Generalized flux detector GFD). Only GFD, not GD, is studied in the present paper.
Single- and double-step ligand interaction with receptors
A patch of sensory membrane uniformly covered with identical receptors R is considered (total concentration N). Ligand molecules L can bind to R and create various complexes. The number of different forms creating the bound class depends on the complexity of the model (Lauffenburger and Linderman, 1993). Here only two forms are considered. In the single-step interaction (Table 2, CD1, FD1 and GFD1), the neuron response is triggered by the binding of the ligand to the receptor, forming a ligand–receptor complex denoted by C. In the double-step interaction (Table 2, CD2, FD2 and GFD2), binding of the ligand is not sufficient to trigger the response; the bound complex must go through an additional step, which can correspond to an allosteric or covalent modification, to produce an activated complex C*. These interactions are characterized by two parameters (rate constants k1 and k–1; see Table 1) in the single-step case, where k1 characterizes the velocity of the association between the receptor and ligand and k–1 the velocity of the breakdown of the receptor–ligand complex C, and four parameters (rate constants k1, k–1, k2 and k–2) in the double-step case, where k2 characterizes the activation of C in C* and k–2 the deactivation of C* in C. The concentrations of C in single-step models and C* in double-step ones are the main variables studied in this paper. Both are referred to as the signaling complexes. The time 
t to reach any predefined level S after a stimulation onset, e.g. the concentration S of signaling complexes needed to trigger an action potential, is also studied.
When obtainable, the concise analytical forms describing the evolution in time and steady state of the signaling complex C(t) or C*(t) are given for the CDs (see Concentration detector CD), FDs (see Flux detector FD) and GFDs (see General flux detector GFD). In all subsections, the special case when Lex is small is studied separately because it is biologically relevant and leads to simpler results. In this case, to avoid any confusion with the exact solutions C(t), the approximated solutions are denoted M(t). Only solutions available in compact form are given here. The details of the mathematical treatment will be published elsewhere (P. Lánsk et al., in preparation). For the static CD2, see (Kaissling, 1987; Rospars et al., 1996a,b); for the non-periodic FD1 see (Kaissling, 1998a; Lánsk and Rospars, 1998). These solutions are illustrated in Figures 3– 6. The parameter values used to draw the curves and the comparisons of the curves are presented in Numerical results.
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0, complete model) (b) of receptor–ligand interaction. (a) Horizontal line, concentration detector CD; inclined line, flux detector fd (lowercase indicates incomplete model); exponential curve, generalized flux detector gfd. In fd, L(t) increases indefinitely, in contrast with CD and gfd (asymptote is at L = 102 µmol/l). In model fd, according to L(t) = kiLext, it takes 1 s to reach the level 10 µmol/l, which is reached instantaneously in model CD. This is the main reason for the slower response of FD with respect to CD. In (b), the curves for fd and gfd with k1 = 0 are compared with the complete models (k1
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Concentration detector CD
In CDs the ligand concentrations in the perireceptor space L(t) and in the external space Lex are the same (Figure 3, CD). A natural consequence of this assumption is that L(t) is not influenced by the interaction of L with R, i.e. neither binding to R nor release from R changes the concentration of L.
Model of binding (Table 2, CD1)
The transduction cascade is triggered by mere binding of L to R to form C (complex RL). The time rate of change of C is described by equation (A1) in the Appendix.
Step stimulation. The number C(t) of bound receptor proteins (Figure 4, CD1) rises exponentially from C(0) = 0 at time 0 according to
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 | (5) |
and reaches a steady-state level C
 | (6) |
The curve C as a function of Lex is an hyperbola. In the special case when Lex is small, which implies C << N, the approximated steady-state level M of the signaling complex C depends linearly on Lex (Figure 4, dashed line),
 | (7) |
Square stimulation. At the onset of stimulation at time 0, C(t) rises exponentially according to equation (5). After the stimulation that ends at time t0, the number of signaling complexes decays according to
 | (8) |
where C(t) is given by equation (5) (Figure 5, CD1). Equations (5) and (8) show that at time t0 all molecules of ligand are instantly removed (Lex = 0) from the vicinity of the membrane. Even the molecules released from the reaction are removed and cannot re-enter it. The time constant 1/(k–1 + k1Lex) of the rising phase is always larger than that 1/k–1 of the decaying phase. So, if k–1 << k1Lex, then the decay is much longer than the rising (Figure 5, CD1) and, after removing the stimulus, it can take a long time to achieve C(t) = 0. This feature may be significant if the stimulus is repeated (see below).
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If S bound receptors are needed to fire an action potential, then the ligand has to be present at concentration Lex for at least
t time units. For S << C, with C given by equation (6), the latency is
 | (9) |
The reaction time is inversely proportional to concentration Lex (Figure 6, CD1). In the low concentration case, the latency
 | (10) |
where M is given by equation (7), is shorter than that given by equation (9).
Periodic stimulation. In the low concentration case the solution after a sufficiently long time is
 | (11) |
The asymptotic period is 2/ and the asymptotic amplitude gets to zero proportionally to 1/, i.e. when the stimulus frequency increases, the response tends to be constant. This means that the system cannot follow very fast time variations of the stimulus, which is a rather intuitive result. For large values of the asymptotic levels of constant (equation 7) and periodic (equation 11) stimulations coincide, since in this case M is the level around which the concentration oscillates very fast with a very small amplitude. In other words, for , the constant component of stimulation, 0, causes the same behavior as Lex = 0 in the step stimulation.
Model of binding and activation (Table 2, CD2)
The receptors may appear in three states: unoccupied R; occupied and not activated C; and occupied and activated C*. The time rate of change of C and C* are described by equations (A3) and (A4) in the Appendix.
Step stimulation. This model has often been used for describing odorant–receptor interaction (Kaissling, 1969, 1971; Getz and Akers, 1995; Malaka et al., 1995; Rospars et al., 1996a,b). The kinetics C*(t) of the activated complex for a step stimulation is given by the sum of two exponentials
 | (12) |
where 
1 and 2 can be calculated from the parameters and C* is the steady-state level for the signaling complex
 | (13) |
So, the hyperbolic dependency of the number of activated receptors on odorant concentration Lex is obtained again as in equation (6) (Figure 4, CD2). However, by manipulating the parameters in equation (13), a larger variability of the curves can be achieved (Kaissling, 1987; Rospars et al., 1996b). In the low concentration approximation, the steady-state density of activated receptors is again linear (Figure 4, dashed line) as in equation (7)
 | (14) |
Square stimulation. An equation similar to equation (8) can be obtained for the time course of the number of activated receptors after odorant removal in a square wave stimulation (equation 2) (Figure 5, CD2). For the reaction time t, after which the threshold S is reached (Figure 6, CD2), the exact analytical solution is not available, but the approximate solution, comparable to equation (9), is
 | (15) |
Contrary to CD1, in which the time to reach threshold S tends to zero with increasing Lex, for CD2 there is a minimum time, which can be calculated exactly,
 | (16) |
Periodic stimulation. When the system is stimulated by a periodic signal (equation 1), an analytical solution can be obtained for M(t) and M(t), but it is notationally complicated and will not be given here.
Flux detector FD
The transfer Lex
L is unidirectional from outside to inside and compensated by a degradation C 
R + . Consequently, the concentration L(t) of the ligand in the perireceptor space is not the same as the concentration Lex(t) in the external space, but L(t) depends on Lex(t). The differential equations describing this model are given in Appendix A (equations A7–A16).
Model of binding (Table 2, FD1)
Step stimulation. The kinetics C(t) of the activated complex for a step stimulation cannot be given analytically because the corresponding differential equations (A7, A8) are nonlinear. Using these equations with Lex(t) = Lex, the steady-state solutions for C and L are
 | (17) |
and
 | (18) |
under the condition that
 | (19) |
which means that the influx to the perireceptor space given by equation (2) must be smaller than the maximum possible flow out of it. According to equation (17), the FD1 model does not result in the hyperbolic-like dependency of the receptor response on the external concentration found with the complete CD models, but in a linear dependency similar to that found with the approximate CD models (Figure 4, FD1; compare this curve with the dotted lines for CDs). Of course, for the limiting influx given by equation (19), Lex koN/ki, the steady-state concentration of ligand in the perireceptor space L tends to infinity while C tends to N. So, the linearity of FD1 is valid only for a limited range of Lex, and this was also the case for the steady-state approximate CDs (see equation 7 or 14). However, the limits of validity are not the same for CDs and FD1, being based on different reasonings; the validity of approximation (7) is restricted to the values of M far below N where few receptors are occupied, Lex << k–1/k1, whereas that of equation (17) is restricted by the fact that C cannot exceed N, Lex < k0N/ki. Note that the ratio of inward to outward rate constants appears in both relations.
In the low concentration approximation, the asymptotic levels are
 | (20) |
and M = kiLex/ko as in equation (17). It follows from this result that the simplification is less restrictive for the flux detector than for the concentration detector. In a way similar to that of the previous section, we obtain for the simplified model a monotonic growth to an asymptotic level described by two exponentials with negative exponents,
 | (21) |
where K, 1 and 1 can be calculated from the parameters.
Square stimulation. The results for the complete model are available only via numerical techniques. On the other hand, for the simplified model they directly follow from equation (21). Due to the similarity between equations (21) and (12), the approximation (15) for the time to response holds here also. The reaction time t, i.e. the time to reach threshold S, does not tend to zero with increasing strength of stimulation just because the stimulation is limited by equation (19).
Periodic stimulation. For the complete model the asymptotic periodicity and dependency of the amplitude on 1 and is available only by using the numerical methods. For the simplified low-concentration model the solution is available but notationally complicated.
Model of binding and activation (Table 2, FD2)
Step stimulation. For the step stimulation Lex(t) = Lex, the steady-state levels of the stimulus and the signaling complex can be easily obtained from equations (A11)–(A13) in the Appendix:
 | (22) |
 | (23) |
The formula for L shows that the condition of saturation of the system is
 | (24) |
It is more restrictive than the condition following from equation (18). On the other hand, if condition (24) is met, the concentration C* depends on Lex exactly like C in equation (17) (Figure 4, FD2); in both cases the concentration of the signaling complex depends only on the parameters related to the influx and degradation. From equation (24), the maximum of C* obeys the condition
 | (25) |
For the low-concentration approximation the steady-state levels are unchanged for M = C and M* = C*, as given by equation (23) and
 | (26) |
The time courses of LM, M and M* can be calculated but the solutions are notationally complicated.
Square and periodic stimulations. No analytical results are available.
Generalized flux detector GFD
In this model the flux (t) depends not only on Lex(t) but also on the concentration inside the perireceptor space L(t), according to relation (4). The differential equations describing this model are (A17)–(A20) in the Appendix. Interestingly, this model behaves like either CD or FD for special values of its parameters, and can also present intermediate features. First, if k–i 0, (t) reduces to kiLex(t), which defines the pure flux detector FD. In this case, the flux is independent of concentration L(t) in the perireceptor space, which can rise indefinitely according to equation (18). Second, if ki is large, ki = k–i (and ko = 0), and the concentration in the perireceptor space becomes immediately the same as outside when Lex is increased, so that GFD behaves in this case like the pure concentration detector CD (although this is not true when Lex is decreased because of the max condition; see below). Third, for other values of the parameters ki and k–i, the influx depends on L(t). If only the translocation is taken into account and no ligand–receptor interaction takes place, the GFD system is described simply by dL(t)/dt = (t), with (t) given by equation (4). Then, for a constant stimulation, the influx ceases when L(t) increases asymptotically to
 | (27) |
(Figure 3, gfd). This example illustrates how the unrealistic behavior of the pure flux model when Lex approaches koN/ki (see equation 24) is removed.
Model of binding (Table 2, GFD1)
Step stimulation. The steady-state values L and C are known explicitly; these expressions are relatively long and will not be given here. The dependency of C on Lex is, for increasing k–i, hyperbolic as the dependencies obtained for CDs (Figure 4, GFD1). Their low-concentration approximations LM and M are simpler expressions
 | (28) |
and
 | (29) |
As noticed, for step stimulation, model CD1 can be considered as a special case of the GFD model (equations 20 and 28) assuming ko 0 and ki . It can be verified that under these two conditions L tends to Lex and C is the same as equation (6). Such a comparison cannot be done for the pure flux detection models. For the same intensity of stimulation Lex, the steady-state level of the signaling complex is always lower for GFD (equations 28 and 29) than for FD (equations 17 and 20). Therefore, the model of proportional flux decreases the steady-state concentrations of the reactants obtained for the step stimulation.
Square and periodic stimulations. No analytical results are available.
Model of binding and activation (Table 2, GFD2)
For the step stimulation the equilibria can be calculated but are notationally complex. For the square and periodic stimulations no analytical results are available.
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Numerical results |
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Parameter values
In the following the detectors are compared numerically and graphically, based on the solutions presented above or, when not available, on the numerical solutions of the differential equations given in the Appendix. To make the comparisons easier, the same set of parameter values as given in Table 3 were used throughout. Most of these values were taken from the Kaissling’s inspiring study where they are derived from an extensive set of experimental data (Kaissling, 1998b). The parameters were fit to the flux detection model (Kaissling, 1998a) (see above) under the assumptions that the ligand–receptor interaction is a single-step reaction and that the receptor itself catalyzes the oxidation of pheromone molecules into an inactivated form (Ziegelberger, 1995). The measurements, which represent more than two decades of effort [reviewed in (Kaissling, 1986, 1987, 1996)], were performed in the male moth Antheraea polyphemus on specialized olfactory receptor neurons that are sensitive to the major component (a 16-carbon acetate) of the sexual pheromone emitted by females. The principles on which the parameter estimation are based are summarized below (a detailed presentation is given in Kaissling, 1998b).
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The range of effective pheromone concentration Lex to which the response of the models can be compared is 10–12
Lex 10–4 µmol/l. At the behavioral (Kaissling and Priesner, 1970) and electrophysiological (Zack, 1979) thresholds (10–12 µmol/l) there are 300 molecules of pheromone per cm3 of air and the total uptake is only 15 molecules per antenna in the standard experimental conditions (airflow ~60 cm/s and stimulus duration ~1 s). At the upper end of the range (10–4 µmol/l), which corresponds to saturation of the receptor potential, there are 3 x 1010 molecules/cm3 and the uptake is 1.2 x 105 molecules per hair.
The fictive concentration N of the receptor proteins was expressed with respect to the volume of the hair lumen (~10–12 l) (Keil, 1984), under the assumption that they are as tightly packed in the membrane as the rhodopsin molecules in visual cells.
The estimate of the rate constant ko was based on the assumption that the pheromone degradation reaction saturates for a concentration Lex almost equal to that yielding the maximum electrophysiological response (receptor potential).
The rate constants k1 and k–1 were deduced from the kinetics of the receptor potential (Zack, 1979; Meng et al., 1989), under the assumption that it is governed primarily by the reversible pheromone–receptor binding reaction. The rate constant k–1 of the deactivation reaction was deduced from the decaying phase of the elementary receptor potential, obtained with weak stimulation, which is thought to be triggered by a single pheromone molecule acting on a receptor; k1 was obtained from the decaying phase of the receptor potential, knowing the other rate constants.
The rate constants for activation k2 and deactivation k–2 of the ligand–receptor complex were not used in Kaissling’s studies (Kaissling, 1998a,b) and, to our knowledge, are not known for the pheromone–receptor interaction. We chose a value (0.25) of the deactivation equilibrium constant k–2/k2 slightly below that (0.5) for which the concentration L0.5 at half-maximum steady-state response of the two-step concentration detector is equal to that of the single-step concentration detector, i.e. for which both systems have the same sensitivity. Then, the absolute values of k2 and k–2 were chosen to be in agreement with that of ko.
The input rate constant ki was estimated from measurements of the adsorption of radiolabeled pheromone molecules by the antenna (Kanaujia and Kaissling, 1985; Kasang et al., 1988; Kaissling, 1995), knowing the number and volume of the sensory hairs (Gnatzy et al., 1984; Keil, 1984). In (Kaissling, 1998b) the flux kiLex is used instead of Lex, so that the value of ki is not explicitly given. It can be derived from the flux (102 µmol/l/s) at saturation of the receptor potential, which leads to ki = 106 s–1.
The rate constant k–i limiting the influx in the GFD models was chosen so that the counteracting effect it introduces remains relatively small. Then, the GFD is closer to a FD than to a CD, its response being no different from that of the corresponding FD on most of the Lex scale. With the k–i value chosen (0.1 s–1), according to equation (27), L cannot be greater than 107Lex. In particular, at the Lex intensity for which L becomes infinite in the pure FD model, L reaches 103 µmol/l.
The numerical comparison of the models is divided into three parts with respect to the type of stimulation (step, square wave and periodic). The effect of changing the parameter values, especially the rate constants, is not considered. So, Lex is considered here as the only variable in the system, in accordance with our approach in terms of intensity discrimination. Because only the level of the signaling complex (bound or activated receptors) plays a role in further information transfer, the results for the other compounds are not presented. Comparisons of the simplified (approximation at low concentration) and complete models were performed only in the steady-state case.
Steady-state level of the response
The curves of the concentration of the signaling compound at steady state C and C* versus the external concentration Lex, or more appropriately logLex, depend dramatically on the model considered (Figure 4). From an information coding point of view each of these curves can be characterized by three independent quantities giving the size, dynamic range and sensitivity of the steady-state response (Rospars et al., 1996a,b). The size (vertical dimension) is measured by the maximum steady-state concentration C,max of the signaling complex for large Lex. The dynamic range (horizontal dimension) gives the range of Lex over which the fraction of receptors in the signaling state grows from a small value 
, e.g. 1% of C,max, which defines threshold Lex,0.01, to almost complete saturation 1 – , e.g. 99% of C,max, which defines saturation Lex,0.99. The sensitivity indicates the position of the curve on the horizontal axis Lex, using concentration threshold Lex,0.01 or the concentration at half-maximum response Lex,0.5. The values of these characteristics are given in Table 4.
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The CD models, whether with one- or two-step ligand–receptor interactions, are characterized by branches of hyperbola that are transformed to logistic curves in log scale. The curves for the GFD models are also sigmoid, whereas those for the one- and two-step FD models are not. As expected from the choice of k–i, FD and GFD curves are superimposed at low concentration and become increasingly different at higher concentration. In this zone, the influence of the influx saturation feature built into GFD becomes dominant and prevents the infinite accumulation of the ligand in the perireceptor space which characterizes FDs when approaching the limiting value Lex,max = koN/ki = 10–4 µmol/l (one step; see equation 24) and 0.57 x 10–4 µmol/l (double-step; see equation 23). Beyond this point the system behavior is no longer described by the model [and not merely saturated, as might be wrongly concluded from figure 7 in (Kaissling, 1998b
)], although such concentrations can be used experimentally, saturation of air with pheromone being reached only above 1 µmol/l at 20°C. The most striking feature of Figure 4 is the 6 log units increase in sensitivity afforded by the FD design over the simple CD one, which is seen as a shift of the FD and GFD curves to the left of the CD curves. The shift depends on the value of ki and consequently on the relative speed of the flying insect with respect to wind. If the relative speed is so low that ki = 1, FD becomes similar to the CD approximation for low concentrations. Then GFD is even closer to CD. So, under this low airspeed condition all models work similarly at steady state.
The maximum size of the response depends on the presence or absence of the second (activation) step. In all single-step models the fraction of receptors in the signaling state at large Lex is 100%. This stands in contrast with double-step models, whose asymptotic levels are lower (80% for CD, 57% for FD and GFD with the parameter values chosen), so that there is no stimulus intensity for which all receptors are activated. Choosing smaller values of k2 and k–2 while keeping their ratio constant would lead to a still smaller maximum (if k2 = 2 and k–2 = 0.5 s–1 it is 16% for FD and GFD). In CD models, the effect of the activation step is to reduce the maximum size of the response and to offer a gain in sensitivity, as the whole curve moves to the left with respect to the single-step reaction curve for k2 > k–2. In contrast, in FDs and GFDs the activation step, when present, entails only a loss in response size without a gain in sensitivity. This can be intuitively justified by the fact that the FD offers the maximum possible sensitivity, which prevents any shift of the GFD response curve to the left.
The dynamic range of the CD curves, whether of one or two steps, is 3.99 log units for = 1% or 2.56 for = 5%, which is a general property of logistic curves. This means that the concentration of signaling receptor complexes increases almost 10 000-fold from threshold, at which 1% of receptors are in the signaling state, to saturation, at which only 1% of receptors remains non-signaling. For FD curves the range is almost 100 times smaller, being 2.00 and 1.28 log units for = 1 and 5%, respectively. Finally, the dynamic range for GFD curves is intermediate, of 2.99 and 1.71 log units respectively in the single-step case. Choosing larger values of k–i, that is, increasing the resistance to influx of ligand into the perireceptor space, would widen the dynamic range and be accompanied by a very slight decrease in sensitivity. However, the range of the GFD model can never become greater than that of the CD model.
It follows from their relatively narrow dynamic ranges and their widely distinct sensitivities that the studied examples of CD and FD–GFD work in non-overlapping domains of concentrations. So, the efficiency of one or the other design depends on the natural concentrations of the ligand to be measured.
The low-concentration approximations, for which the solutions are often analytical, are non-sigmoid curves that give reasonable approximations for responses up to 10 or 20% of the maximum response.
Time-course of the response to square wave stimulation
The time-course of the response during stimulus presentation and its decay after stimulus removal can be studied in the case of a square pulse stimulation (Figure 5). Although this is not visible in all cases in Figure 5, the duration of the rising phase is always different from that of the decaying one.
The time t needed to achieve any preassigned level S of the response as a function of Lex reflects the response latency in a biologically meaningful way since this time may represent a significant part of the total reaction time of a neuron to stimulation. The t versus Lex curves are L-shaped whatever the value of S (Figure 6). In single-step CD and GFD models, t goes to zero when Lex is sufficiently large. In double-step models, t tends to a minimum time, which is given for double-step CD by equation (16). For S = 1 µmol/l, which is likely to be a relatively large value (10% of N) with respect to the number of activated receptors at maximum response of the neuron, tmin is 5 ms for CD2 and GFD2, 160 ms for FD1, and 320 ms for FD2.
From an information coding point of view, it is useful to adopt the reverse formulation, i.e. to have fixed Lex and variable S. For each curve, we chose Lex giving a half-maximum response (50% of C,max or C*,max). Then, each curve can be characterized by the time needed to reach a small value , e.g. 1% of C,max and almost complete saturation 1 – , e.g. 99% of C,max. These reaction times t are given in Table 5. They lead to the following conclusions:
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- CDs and FDs substantially differ in their reaction times. Apart from the steady-state levels already presented above, this is the most obvious difference between them. For example, 95% of the steady-state response is reached in 0.15 s for CD1, 6 s for GFD1 and 8 s for FD1. So, the CD’s detector follows the square wave with minimum distortion, whereas the responses of FDs and GFDs rise and decay more slowly, introducing a noticeable distortion. This means that there is a trade-off between the large increase in sensitivity provided by the FD design and the significant decrease in its capacity to reflect reliably the rapid fluctuations of the signal.
- For low response levels ( < 50%) double-step models are slower than their single-step counterparts. The difference is striking for CDs, the reaction time being 1 ms for CD1 and 16 ms for CD2 at = 1%; it is lesser for FDs and GFDs, ~30 ms for one step and 60 ms for two steps. However, for relative levels above 50%, FD2 and GFD2 become slightly faster than their counterparts FD1 and GFD1.
- For very low response levels, which are significant for stimulus detection (see Discussion), the response times can become very short also for FDs and GFDs. For example, with Lex eliciting a half-maximum response and = 10–5 (~10–4 µmol/l for the signaling complex), the reaction times predicted by the models are in the 1–5 ms range. This is a very short time compared with transduction time, suggesting that the perireceptor and receptor events are not the limiting steps in the odorant detection process.
Time-course of the response to periodic stimulation
The temporal features of the response observable with square-wave stimulations are emphasized in the case of stimulations with periodic intensity. For all types of detectors the frequency and amplitude of the stimulus must be considered simultaneously.
All systems achieve, after a certain delay, a periodic response which follows the stimulus with the same period. The level around which the response is modulated is the steady-state level observed under constant stimulation. Since the CD models are fast responding, the periodic regime is achieved more quickly than in the FD and GFD models.
For sufficiently low frequency, the amplitude of the response approaches a maximum which is equal to the level that would be achieved by a constant stimulation Lex = o + 1. With increasing frequency, the response, although still periodic, does not reproduce the upper and lower parts of the signal, so that the amplitude of the oscillating part of the response decreases. Finally, the system averages almost completely the quickly alternating signal which evokes a practically constant response. Figures 7 and 8 show that the amplitude of the response decreases at a slower rate for CDs than for FDs. When the frequency increases from 1 to 10 Hz, the amplitude decreases by 20% from 0.64 to 0.5 µmol/l for single-step CD, less than four times from 0.76 to 0.24 for double-step CD, and more than five times from 0.22 to 0.03 for single- and double-step FDs and GFDs. Although the same qualitative behaviors with roughly three domains (periodic with large amplitude, periodic with small amplitude, non-periodic) can be observed for both detector types, the dependency of these domains on stimulus frequency are quite different. Similarly, as seen in Figure 7, the phase shift with respect to the stimulation is larger for slowly responding systems.
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Figure 9 illustrates the response of the model to the periodic stimulation with square pulses as used in actual experiments (see Discussion for details). It shows clearly the effect of periodicity, although in this kind of stimulation the amount of ligand delivered per unit of time is changed when the frequency changes. The ratio of the amplitude of the periodic part of the signal to its mean constant level falls from 69% at 1 Hz to 15% at 3 Hz and 2% at 10 Hz.
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Discussion |
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TopAbstractIntroductionThe models and their...Numerical resultsDiscussionAppendix: differential equationsReferences |
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Concentration and flux detectors
The distinction between concentration and flux chemosensory detectors has been recently proposed by Kaissling (Kaissling, 1998a). It puts the conditions in which chemosensory systems operate in a new light and is of major importance in understanding their design principles. The significance of these concepts is illustrated here by comparing three models, the concentration detector (CD), the pure flux detector (FD) and the generalized flux detector (GFD). Each of these models was studied at two levels of complexity, the subsequent transduction cascade being triggered either by mere binding of the ligand to the receptor, in a single step, or by binding plus activation, in two steps.
The CD models are described by only three (single step: k1, k–1 and N; see Table 1) or five (double step: as for single step, but adding k2 and k–2) parameters in their versions studied here, which involve only ligand–receptor interactions and no ligand-degrading reaction. In such systems, the signaling receptor complex responds directly to the external ligand concentration. The models apply to chemosensory membranes that are almost directly exposed to their environment, as might be the case for taste receptors, insect carbon dioxide receptors (Stange, 1996), hormone receptor systems and unicellular organisms.
The flux detectors, in either their original (FD) or modified (GFD) form, involve two or three more parameters that describe, on one hand, the adsorption process (ki in FD, ki and k–i in GFD) responsible for accumulating ligand molecules in the perireceptor space and, on the other hand, the symmetric degradation reaction (ko) responsible for their deactivation. They apply to chemosensory systems that collect the ligand of interest in a periplasmic space using, for example, either binding proteins attached to the membrane, a mucus layer or a physical barrier. Therefore, any chemosensory cell might belong to this category, and most odorant receptor systems probably belong to it.
Some models can be transformed to another by selecting appropriate reaction rate constants. For k2 = 0, the models with a two-step reaction simplify into the models with a one-step reaction. For large ki, ki = k–i and ko = 0, GFDs, but not FDs, are transformed into CDs. However, this is true only at stimulation onset because ligand outflow is not permitted in GFDs, so that at offset their behavior is different from that of CDs. The condition ko = 0 is needed in the present form of the CD models; however, this is only a formal change because the degradation reaction governed by ko might have been included also in the CD models. Note that this degradation mechanism has been kept as simple as possible, the receptor acting as an enzyme. A separate enzyme may be also considered; its effect in FD is to change both steady-state curves shown in Figure 4 in sigmoid curves [see figure 7 in (Kaissling 1998a)] and the time-course of the signaling complex.
The equations can be linearized for Lex(t) small enough, i.e. for a number of signaling complex C or C* much smaller than the total number of receptors. This low-concentration situation is often met in reality (see the last section of this paper) and is probably the only physiologically meaningful one (Rospars et al., 2000). Otherwise full non-linear versions must be considered.
Comparison of concentration and flux detectors in the case of moth olfaction
The behavior of the various detector models has been illustrated in the special case of the reception of sexual pheromone by moths, as originally studied by Kaissling. The most basic conclusion resulting from this illustration is that, using the parameter values that describe the actual system (as given in Table 3), the high sensitivity of the system cannot be accounted for by a concentration detector, which falls short by six orders of magnitude. Only a flux detector can account for this high sensitivity. Now, the relative merits for describing the system of the different versions of flux detectors, the original FD or its generalized version GFD, whether one step or two, must be assessed on other grounds. The two-step versions, having two more parameters than single-step versions, are more flexible. The same is true for the GFD version, which has one more parameter than FD (k–i) and thus can fit experimental data more precisely and, moreover, can describe any intermediate detector between the two extremes that are CDs and FDs.
Now it can be instructive to adopt an engineering point of view on this system and to wonder why sex-pheromone reception is not based on a CD device. This question arises because it is a priori possible to design a CD which is sensitive enough to monitor the sex-pheromone concentration range actually observed. Consider, for example, the steady-state response of the single-step CD. The problem to solve consists in shifting the CD1 curve as shown in Figure 4 to the left by 6 log units so that it coincides approximately with the FD1–GFD1 curves. This can be done if the dissociation equilibrium constant k–1/k1, which gives the ligand concentration at half-maximum response, is multiplied by 10–6, i.e. from its actual value of 5 x 10–5 mol/l (see Table 3) to 5 x 10–11 mol/l. This corresponds to a shift from the low-affinity end to the high-affinity end of the various known receptors (Lauffenburger and Linderman, 1993). So, the idea of a sex-pheromone CD does not appear unrealistic as far as the receptor-ligand is concerned.
So, the reason why the actual sex-pheromone system is a flux detector must be found elsewhere. A possible interpretation is that the flux-detector design is the only one compatible with the lipophilic nature of the bombykol molecules, which sticks them to the cuticle, and the multiporous cuticular structure, which entails irreversible entrance into the perireceptor space. Both features preclude free return of the ligand to air as in the (ideal) case of the CD. Then the low affinity of the receptor to its ligand would be a secondary adaptation. It is such that it permits monitoring the actual concentration range of bombykol, given the unavoidable accumulation of these molecules in the vicinity of the receptors. So, contrary to what might be thought a priori, the affinity of the receptors to bombykol would not be a limiting factor for the global sensitivity of the system.
Properties of flux detectors in the case of moth olfaction
The parameter values used in Table 3, which describe the sex-pheromone receptor system, can be changed in relatively large ranges without altering the qualitative conclusions presented. For example, the input rate constant ki influences the magnitude of the gain in sensivity to concentration and the corresponding gain in temporal acuity, afforded by FDs (including GFDs) with respect to CDs. Only the magnitude, not the existence of these two effects of relative gain in sensitivity and time resolution, which are the major properties of the systems from the signal processing point of view we adopt here, can be influenced by manipulating the value of ki. These are structural features of the models that are insensitive to the values of the parameters. Unfortunately, the exact value of ki cannot be easily determined experimentally because the perireceptor space, which is the volume in which the pheromone molecules have easy access to the receptors, is not well defined. If the whole internal hair volume were considered as the actual perireceptor space, which corresponds to a perireceptor layer ~1 µm thick, a lower bound of ki = 1.5 x 10–4 s–1 would hold. Then, according to equation (17), i.e. assuming the one-step FD model gives a correct description, and using ko = 10 s–1, one finds that at the stimulus intensity giving the maximum value of the steady-state receptor potential (Lex = 10–4 µmol/l), only 1.5% of the receptors are bound to pheromone molecules. Another possible assumption (but unlikely to be correct) (Rospars et al., 2000) is that all receptors are occupied (C = N = 10 µmol/l) for Lex = 10–4 µmol/l. Then, one finds from equation (17) that ki = 106 s–1, corresponding to a perireceptor layer only 0.2 µm thick.
The long time constant of flux detectors suggests that the steady-state level is not of primary importance in intensity coding. One possibility is that these systems are not primarily working as sensors of stimulus intensity with graded response, but as sensors of stimulus presence with binary (yes or no) response. Indeed, if the spiking threshold is low enough, which is actually the case in the moth sexual pheromone receptor neuron, it is crossed with very small latency (see Table 5), so that spike firing can reliably follow the fast temporal variations of the signal even if the steady-state level of the signaling receptor complex (and consequently the receptor potential) cannot. For example, a moth flying through a pheromone plume subdivided into small clumps of molecules would never stay long enough in a clump to measure its concentration, which might even be meaningless in this case. However, without considering such an extreme case, another possibility of fast measurement of odor intensity is possible based on the slope of the dynamic response (signaling complex or receptor potential) with respect to time. This is a straightforward interpretation of Figure 6, which represents an expression of the slope S/t via time t needed to reach a constant predefined level S. The regular decrease of t as a function of Lex shows that it can provide a reliable measure of Lex. This hypothetic mechanism would call for a spike generating mechanism in which the first spike would signal the onset of the stimulation and the subsequent interspike intervals would allow the system to estimate Lex via t.
In the models studied the dynamic range 1–99% never exceeds 4 log units (CD1) and it can be as narrow as 2 (FD1) (see Table 4). This stands in contrast with the >8 log-unit range of the receptor potential (Zack, 1979) under the same conditions. Consequently, the difference must be accounted for by the transduction cascade and the conductance-to-voltage conversion. The latter conversion can account for part of the widening effect, provided there is a long enough single sensory dendrite and a small input resistance of the non-sensory part of the neuron, especially a large soma (Vermeulen and Rospars, 1998), a situation that is reminiscent of the sex-pheromone receptor neuron. In this case the dynamic range of the receptor potential is always >4 (Rospars et al., 1996a,b).
The lower region of the range is the most important in practice. The conventional threshold defined by = 1% corresponds to a concentration of 0.1 µmol/l of the signaling complex, i.e. 1.6 x 105 molecules per dendrite. This is a large number since a single complex, which corresponds to a concentration of 6.3 x 10–7 µmol/l (with the values selected in Table 3), is sufficient to elicit a spike (Kaissling and Priesner, 1970; Kaissling, 1971). What is the stimulus intensity which elicits this picomolar concentration of C or C*? Since all FD and GFD models have the same behavior at low Lex concentration, equation (17) can be used to calculate this extreme threshold. One finds Lex 
10–11 µmol/l, which is in good agreement with the observed electrophysiological threshold (10–12 µmol/l). This confirms that the models and parameter values offer a satisfactory description of the experimental data, as pointed out by Kaissling (Kaissling, 1998b). Two remarks must be added at this point. First, the definition of the dynamic range used here must not be interpreted as measuring the true dynamic range of the system studied. It is merely a convenient measure of the shape of the curve which removes the subjectivity of appreciating at which points the asymptotes are reached and permits one to compare responses of different systems. Second, the calculation presented above for S equal to the picomolar concentration of C or C* is only a first approximation because with such a small number of molecules in interaction the deterministic description on which the present paper is based is no longer valid and must be replaced by a stochastic description. We have started to develop such a description for the CD models (Lánsk and Rospars, 1993, 1995) and will extend these results in the future.
Odorant molecules released from a point source are carried away by the wind. The odorant plume formed is distorted by atmospheric turbulence and the concentration at a stationary point has been found to fluctuate at a frequency of some tens of cycles per second (Murlis et al., 1992). Flying insects use this fluctuation to orient towards an odor source (Kramer, 1986). Successful orientation and source location is elicited in moths at frequencies of 2–10 s–1 (Vickers and Baker, 1992). Olfactory receptor neurons in A. polyphemus were shown to follow trains of short pulses (20 ms) at relatively high concentration (2.5 x 10–9 µmol/l) up to 0.5 and 10 s–1 depending on the neuron type (Christensen and Hildebrand, 1988; Rumbo and Kaissling, 1989; Marion-Poll and Tobin, 1992). The model studied is in remarkable agreement with these data. Our numerical simulations (Figure 9) show that in the critical region (1–10 Hz) the amplitude of the periodic part of the signaling-complex response falls from 75 to 2% of the mean amplitude. For higher frequencies, which are accompanied by a larger amount of odorant delivered, the periodic part is practically smoothed out.
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Appendix: differential equations |
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TopAbstractIntroductionThe models and their...Numerical resultsDiscussionAppendix: differential equationsReferences |
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Using principles of mass action kinetics, the reactions for the six models studied, summarized in Table 1, can be readily translated into differential equations giving the time rate of change of the various chemical species of interest. In all models a relation of conservation for the total number N of receptors holds; in single-step models (CD1, FD1 and GFD1) the relation is
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and in double-step models (CD2, FD2, GFD2) it is
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with N constant at any time. Due to these relations, the number of independent differential equations is decreased by one with respect to the number of chemical species (or states) involved.
Model CD1 (single step)
Only one independent equation can be written. We select the equation for C as it is the quantity of interest,
 | (A1) |
If Lex(t) is small, i.e. k–1 >> k1Lex(t), equation (A1) can be simplified into the form
 | (A2) |
Model CD2 (double step)
Only two independent equations can be written,
 | (A3) |
and
 | (A4) |
If Lex(t) is small, i.e. k–1 and now additionally k–2 prevail over k1Lex(t), formally when k–1 >> k1Lex(t) and k–2 >> k1Lex(t), equation (A3) becomes
 | (A5) |
whereas equation (A4) remains unchanged (only with different notation),
 | (A6) |
The conditions leading to equation (A5) imply that the number of interacting receptors is far below their total number, C(t) + C*(t) << N. Again, the external stimulation appears in equation (A5) as an additive term only.
Model FD1 (single step)
The system can be described by two independent equations, e.g.
 | (A7) |
 | (A8) |
If kiLex(t) << ko, which implies C(t) << N, then equations (A7) and (A8) can be reduced to a simpler system
 | (A9) |
 | (A10) |
where the upper index in LM(t) makes the distinction from L(t) in the complete model. The system of equations (A9) and (A10) is formally the same as that of equations (A5) and (A6), but with different interpretation of the parameters.
Model FD2 (double step)
Three independent equations can be written
 | (A11) |
 | (A12) |
 | (A13) |
If the number of interacting receptors C(t) + C*(t) is sufficiently small compared with N, then equations (A11)–(A13) simplifies to
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| (A14) |
 | (A15) |
 | (A16) |
Model GFD1 (single step)
Only two independent equations can be considered, (A7) and
 | (A17) |
If C(t) << N, which is again equivalent to kiLex << ko, the simplified system is formed by equation (A9) and
 | (A18) |
Model GFD2 (double step)
This model is described by equations (A12), (A13) and
 | (A19) |
and for the simplified situation it is defined by equations (A15), (A16) and
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| (A20) |
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Acknowledgments |
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We thank Karl-Ernst Kaissling and Gert Stange for helpful discussions. This work was partly supported by joint cooperation project Barrande no. 972SL between France and the Czech Republic, by grant from Département Santé des Plantes et Environnement of INRA, by Grant Agency of the Czech Academy of Sciences (grant no. A7011712), by Grant Agency of the Czech Republic (grant no. 201/98/0227) and by M\vSMT (grant no. VS 96086).
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Accepted December 20, 1999
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