Chem. Senses 28: 509-522,
2003
© Oxford University Press 2003
Extracellular Transduction Events Under Pulsed Stimulation in Moth Olfactory Sensilla
3,5
ivan4,51 Unité de Phytopharmacie et Médiateurs chimiques, INRA, 78026 Versailles Cedex 2 Unité BIA, INRA, 78352 Jouy-en-Josas Cedex, France 3 Institute of Physiology, Academy of Sciences, Videnska 1083, 14220 Prague 4 4 Institute of Entomology, Academy of Sciences, USB, Branisovská 31, 37005 Ceské Budejovice, Czech Republic 5 Faculty of Biological Sciences, USB, Branisovská 31, 37005 Ceské Budejovice, Czech Republic
Correspondence to be sent to: J.-P. Rospars, Unité de Phytopharmacie et Médiateurs chimiques, INRA, 78026 Versailles Cedex, France. e-mail: rospars{at}versailles.inra.fr
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Abstract |
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 Top Abstract IntroductionModelResultsDiscussionAppendix A: differential...Appendix B: receptor...References |
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In natural conditions, pheromones released continuously by female moths are broken in discontinuous clumps and filaments. These discontinuities are perceived by flying male moths as periodic variations in the concentration of the stimulus, which have been shown to be essential for location of females. We study analytically and numerically the evolution in time of the activated pheromone-receptor (signaling) complex in response to periodic pulses of pheromone. The 13-reaction model considered takes into account the transport of pheromone molecules by pheromone binding proteins (PBP), their enzymatic deactivation in the perireceptor space and their interaction with receptors at the dendritic membrane of neurons in Antheraea polyphemus sensitive to the main pheromone component. The time-averaged and periodic properties of the temporal evolution of the signaling complex are presented, in both transient and steady states. The same time-averaged response is shown to result from many different pulse trains and to depend hyperbolically on the time-averaged pheromone concentration in air. The dependency of the amplitude of the oscillations of the signaling complex on pulse characteristics, especially frequency, suggests that the model can account for the ability of the studied type of neuron to resolve repetitive pulses up to 2 Hz, as experimentally observed. Modifications of the model for resolving pulses up to 10 Hz, as found in other neuron types sensitive to the minor pheromone components, are discussed.
Key words: intensity coding, neuron modelling, olfaction, pheromone, receptor, temporal coding
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Introduction |
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TopAbstractIntroductionModelResultsDiscussionAppendix A: differential...Appendix B: receptor...References |
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The present paper is based on two lines of thought. The first one comes from the realization that temporal discrimination is an essential feature of odorant perception in insects. It has been shown in natural conditions that air turbulence physically breaks the initially continuous pheromone plume into spatially and temporally discontinuous patches (Murlis et al., 1992
). For an insect flying in the plume the discontinuities
appear as a temporally structured signal. Males of Bombyx mori orient
upwind only when the signal is pulsed (Kramer,
1986,
1992). Behavioral (Kennedy
et al., 1980,
1981;
Willis and Baker, 1984;
Vickers and Baker, 1992) and
neurophysiological (Christensen and
Hildebrand, 1988; Rumbo and
Kaissling, 1989; Marion-Poll
and Tobin, 1992;
Kodadová, 1996)
experiments indicate that the optimum frequency of the periodic stimulation is
in the range 1–10 Hz.
The second line of thought comes from the recent progress in the
understanding of perireceptor and receptor events [reviewed in
(Stengl et al.,
1999)]. On this basis, Kaissling
(Kaissling, 2001) proposed an
integrated model of the network of reactions taking place up to receptor
activation, in the case of the receptor neuron of the moth Antheraea
polyphemus sensitive to the main component of the sexual pheromone.
Significant improvements have been brought with respect to previous models
(Kaissling,
1998a,b;
Rospars et al., 2000;
Lánsk et al.,
2001; Kivan et
al., 2002). These improvements are qualitative, with the
introduction of pheromone binding protein (PBP) and of a more realistic
deactivation of the pheromone molecules which can be removed by two enzymatic
reactions, the first one (enzyme E) degrading the free ligand, the other one
(hypothetical enzyme N) deactivating the PBP–ligand complex. They are
also quantitative with the estimation of all reaction rate constants involved
in the system.
Putting the two lines of thought together led us to determine the
properties of the activated receptor complex resulting from this realistic
network of reactions under periodic pulse stimulation. We investigate how the
concentration of the complex evolves in time during the transient and steady
states. The static and oscillating components in both states are distinguished
and their temporal and concentration characteristics are determined. We are
especially interested in the amplitude of the oscillations of the receptor
complex in the cell type sensitive to the main component of the pheromone
which follows pulses only up to 
2 Hz
(Rumbo and Kaissling, 1989;
Kodadová, 1996). We
extend this investigation to two other cell types sensitive to the minor
components (Meng et al.,
1989) which are able to discriminate pulses up to 10 Hz. Clearly,
intuitive understanding of the behavior of such systems is difficult and only
quantitative simulations can lead to definite conclusions.
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Model |
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Reaction network
Schematically the model involves two types of reactions: the activating
reactions and the deactivating ones (Table
1). The activating reactions form the main sequence. This sequence
starts with the pheromone molecules (ligand) in the air. The molecules that
are adsorbed on the cuticle can diffuse at the surface of the olfactory hairs
(Kaissling, 1987), cross the
cuticle through small pores (Steinbrecht,
1997) and finally reach the sensillum lymph. In
Table 1 this is modelled as
translocation from air (Lair) to perireceptor space (L). Pheromone
molecules then react with the reduced form Bred of the pheromone
binding protein PBP (Pelosi and Maida,
1995; Ziegelberger, 1995) that protects and transports them to the
neuron membrane where the PBP–ligand complex reacts with the receptor
protein in a two-step reaction (binding and activation). The deactivating
reactions are catalyzed reactions in which the active form L of the ligand is
deactivated into inactive forms
, LBox and
Box
, where
Box stands for the oxidized form of the PBP molecule
(Kaissling, 1998b).
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The set of differential equations describing this system is given in Appendix A. The variables of interest are L (free ligand in sensillum lymph), LBred (ligand bound to PBP, denoted P), LBredR (LBred bound to receptor, denoted O) and principally LBredR*, the activated form of the receptor protein. We denote C this activated (signaling) complex and C(t) its concentration at time t. The perireceptor and receptor network so defined is only part of a larger system which is preceded by cuticular adsorption and diffusion, and followed by biochemical transduction (amplifying stage) and electric phenomena (conduction of the signal along the dendrite and soma of the receptor neuron). The latter steps are not taken into account here.
The model given in Table 1 was studied in its complete form. However, simulations showed that, at least with used parameter values, some reactions have only a very small influence on concentration C(t). These are called `secondary reactions' in Table 1.
Parameter values
Kaissling was able to determine the values of all the parameters in the
model (Table 2) for the cell
type sensitive to the main pheromone component
(E,Z-6,11-hexadecadienyl acetate) of the saturniid moth Antheraea
polyphemus (Kaissling,
2001). These values come from three sources: (i) biochemical
experiments, as reported in the literature or done by the author and his
co-workers; (ii) electrophysiological experiments, based mostly on
measurements of the receptor potential and the elementary receptor potentials
(transient potential changes which are thought to result from single receptor
protein activations); and (iii) calculations based on the steady-state
solution of the model and several approximations. Obtaining such values is a
difficult task because it is based on a deep familiarity with the system. For
this reason and for the sake of comparison, we have taken the parameter values
as originally published. However, to account for the temporal characteristics
of cell types responding to the minor pheromone components
(E,Z-6,11-hexadecadienal and E,Z-4,9-tetradecadienal), we
studied also a modified set of parameter values.
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Constant stimulation
Although the subject of this paper is pulsed stimulation, the response to a
periodic stimulus cannot be understood without reference to the simpler
constant (or step) stimulation. When the system is stimulated at constant
ligand concencentration from time zero, the concentration
Cc(t) of the signaling complex initially
increases, then approaches a constant level. As shown in Appendix B, this
equilibrium response, denoted
c, was determined without
approximations as a result of a long but straightforward calculation and found
to be a hyperbolic function (equation B3 or B6) of the constant concentration
of ligand Lc in air surrounding the antenna. This
steady-state concentration of the signaling complex
c is not a quantity of
much behavioral interest, but is an important reference value from a
theoretical point of view. These symbols and all those introduced below are
defined in Table 3.
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Pulsed stimulation
The concentration of ligand in the air surrounding the antenna is described
by periodically repeated square pulses, in the form
 | (1) |
 | (2) |
Thus, the stimulation frequency (2) can be changed either by modifying
tL or tH. The temporal average of the
concentration of ligand in air is
 | (3) |
When the interpulse tL decreases and tends to zero,
increases and tends to the limiting value LH,
which corresponds to a permanent stimulation.
Concentration of ligand in air and flux from air to perireceptor space
Pheromone molecules in the air can cross the hair wall through pores and
reach the perireceptor space in the vicinity of the cell membrane. The rate at
which they enter the sensillum lymph can be considered, at any time
t, as proportional to the pheromone concentration in air
Lair(t) expressed in nM:
 | (4) |
(t) expressed in µM/s denotes the inward flux of
molecules at time t and ki expressed in
s–1 is the rate constant characterizing the translocation
(Rospars et al.,
2000). It follows from (4) that the intensity of stimulation can
be equivalently expressed as a concentration in air Lair
(molarity) or as a flux (molarity per time unit). For example, a
constant stimulation Lc generates a flux
c = kiLc and
during a pulse of height LH the flux is H
= kiLH. Experiments in
Antheraea with 3H-labeled pheromone lead to
ki = 2.9 x 104
s–1 (K.-E. Kaissling, personal communication). Note that this
value is smaller than the value ki =
106 s–1 used in our previous work
(Rospars et al.,
2000). We will use Kaissling's estimate in the present work. With
this rate a concentration of 1 nM corresponds to a flux of 29 µM/s. In the
following, concentration Lair is always expressed in nM
and flux in µM/s. |
Results |
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TopAbstractIntroductionModelResultsDiscussionAppendix A: differential...Appendix B: receptor...References |
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All numerical results are based on the constants given in Table 2, except those in the section on `Modified parameters for a higher temporal resolution'.
Response as a function of time
When periodic pulses of ligand molecules in air are applied to the system, the concentration L(t) of free ligand in the perireceptor space follows the stimulus without much distortion (Figure 1A). This is not the case of the concentrations of the three next species along the activation pathway, P(LBred), O(LBredR) and C(LBredR*), which all initially increase for several periods, then reach steady-state oscillations (Figure 1). Now, these `steady' states are actually periodic, P(t), O(t) and C(t) fluctuating around constant values with different amplitudes and the same period T. Period T is always equal to that, tL + tH, of the stimulation (Figure 2). This means that the intermediate species and the signaling complex merely follow, with a time lag and a more or less severe deformation, the time course of the stimulus. The responses oscillate between a lower bound (trough) and an upper bound (crest). During each pulse the response increases and tends to a horizontal asymptote that can be closely approached only if the pulse is of sufficiently long duration. Then, the response decays and tends to the zero level, which similarly can be reached only if the interpulse is sufficiently long. For all stimulations shown in Figure 2, the response of the complete system, with all 13 reactions, is practically identical to the response of the simplified 10-reaction system shown in Table 1 (main reactions).
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Because the concentrations fluctuate around average levels they can be analyzed as the sum of a monotonic component (average of response over one period), which is studied in the next two sections, and an oscillating component, examined in the two final sections. For the sake of simplicity only the concentration C(t) of the signaling complex LBredR* is studied in detail. It is called here the response of the system.
Monotonic component of the response during the steady state
Once the steady state is reached, the (constant) monotonic component can be
estimated by averaging C(t), over one period T. It
can be shown numerically that this average
is equal to the equilibrium response
c of the system to a
constant stimulation Lc delivering each period the same
amount of ligand as the periodic
pulses (Figure 3), i.e.
Lc = , where
is given by equation (3). This is a
noteworthy simplification for the analysis of the system because it implies
that, for studying the monotonic component at steady state, attention can be
restricted to constant stimulations without any loss of information. The
response c can be
determined exactly (see Appendix B); this is the only feature of the response
to a periodic pulsed stimulation that can be derived analytically. It follows
from equation (3) that the same mean steady-state level of the signaling
complex can be achieved in several ways with pulses of different heights,
durations or interpulse lengths.
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When increases,
also increases as shown in
Figure 4. In the standard plot
vs.
, identical to
c vs.
Lc (Figure
4A), is a branch of hyperbola that tends to an asymptotic maximum
Cmax given by equation (B4). The ligand concentration at
half-maximum response, C/Cmax=0.5, given by
equation (B5), is KD = L50 
1 nM
(corresponding to a flux 50 = 30.2 µM/s) for parameters of
Table 1. In the semilog plot
vs. log
, identical to
c vs. log
Lc (Figure
4B), is a logistic curve with an inflection point at
KD. The dynamic range of the curve between
L1 at 1% saturation
(/Cmax = 0.01)
and L99 at 99% saturation is 4 log units, which is a
general property of logistic curves
(Rospars et al.,
1996). For Ù
L10, i.e. 0.12 nM, the hyperbola can be very well
approximated by the straight line given by equation (B7). In this range, the
system is practically linear.
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Monotonic component of the response during the transient state
The transient response to a periodic stimulus, like the steady state, can
be described as the summation of a monotonic component (that yielded by the
corresponding step stimulation) and an oscillating one
(Figure 5). Thus the
characteristics of the monotonic component are those derived from a step
stimulation of intensity Lc =
. How the transient state changes
into the steady state for different
values of , or equivalently
c for different values of
Lc, is shown in Figure
5A. It is difficult to judge from this figure whether the time
needed to come close to the steady state is the same or not. This can be
better seen using the ratio
C(t)/, which allows
one to compare the kinetics of the signaling complex at different stimulation
strengths independently of the asymptotic concentrations
(Figure 5B). Then it appears
that the time to reach the steady state
of the signaling complex decreases
when increases. For all ligand
concentrations <
L1 the
C(t)/ curves are
very similar, which means that the duration of their transient states are very
close, 5.58 s to reach 99% of
. For KD the
transient time is noticeably shorter, 4.55 s to reach the same 99% level.
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Oscillating component of the response during the steady state
The oscillating component of the steady state can be shown in isolation by substracting the response Cc(t) to a step stimulation from the response C(t) to a pulsed stimulation. The oscillating component calculated in this way is shown in Figure 6. The main characteristic of interest, the amplitude of the oscillations, i.e. half of the distance between extrema of C(t), has to be calculated numerically.
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Contrary to the average magnitude
, the amplitude A of
the response during the steady state does not depend on the quantity
but on the individual characteristics LH,
tH and tL of the pulses. Before
studying these dependencies it is useful to introduce the notion of the
`natural pulse' of the system; it reverses the usual point of view of
observing the response yielded by a specified stimulation by modifying the
stimulation in order to obtain a specified response. The periodic stimulus is
`natural' for the system if the pulse duration is such that
C(t), can rise to 99% of its maximum asymptotic value
c, and the interpulse long enough for
C(t) to decay to 1% of
c.
These pulse
characteristics define the `natural frequency' of the system because it is the
highest frequency which gives the (almost) maximum possible amplitude. For any
higher frequency, the response cannot go so close to the asymptotes and thus
the amplitude of the oscillations is only a fraction of the distance between
the asymptotes. The natural frequency depends on LH but in
the range of stimulus concentration
up to 0.1 nM in air (i.e. a flux of 3 µM/s), in which the system is
practically linear, it is constant, f = 0.082 Hz and the
corresponding `natural period' is T = 12 s
(Figure 7). The natural
frequency of the whole system is 10 times lower than that of the
receptor–ligand interaction reactions considered in isolation (0.88 Hz),
which indicates that the transport and degradation reactions are responsible
for this slowing down.
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LBredR only, one-step CD; dotted line), double-step
receptor–ligand interaction (R + LBred
In the simplest case, where both tH and
tL are longer than in the `natural pulse',
C(t) can first approach the asymptote
=
c corresponding to a
constant stimulation of intensity
(upper bound), then approach zero (lower bound), so that the amplitude is
(almost) fully expressed and A
/2. Then equation (B3) applies:
amplitude A grows hyperbolically with
and tends to
Cmax/2 when the concentration of activated receptors
approaches its maximum. However, for any frequency higher than the natural
frequency, the amplitude of the oscillations is limited to a fraction of this
range because the pulses stop before the upper bound is reached and start
again before decay to the lower bound.
The basic protocol for studying how the amplitude depends on the input variables consists in maintaining the third variable constant (e.g. tH) and plotting A vs. the first variable (e.g. LH) for a series of fixed values of the second variable (tL), in practice different frequencies. Using this approach we investigated how amplitude changes when pulse height LH is increased for extreme values of tH (1 and 50 ms; Figure 8) and frequency (1 and 10 Hz; Figure 9).
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Figure 8 shows that, for a
biologically meaningful range of values of tH and
tL, logA first increases linearly with
logLH, reaches a maximum, then decreases linearly
(Figure 8A). Equivalently, the
relative amplitude A/
remains constant then decreases linearly
(Figure 8B). The latter
representation confirms that, in the left part of the curve, A
depends linearly on LH and that the slope of this straight
line is not a constant but depends on f and tH.
The ligand concentration in air at which the maximum amplitude occurs depends
on tH, it is 3.4 nM (i.e. flux 100 µM/s) for
tH = 100 ms and 3.4 x 103 nM (i.e.
flux 105 µM/s) for tH = 1 ms.
Figure 8 shows that the
amplitude of oscillations depends first of all on pulse height, secondarily on
frequency and marginally on pulse duration; the effect of
LH is greater by several orders of magnitude than that of
tH. Practically, only LH is important
since the curves of A for extreme values of tH
and f are very similar (Figure
8A). On the contrary, for relative amplitude
A/ only frequency is
decisive, LH and tH being negligible
(Figure 8B). The final decline
of A for high LH results from the fact that the
`saturation' of the receptors prevents the upper bound from moving further up
while the lower bound of the oscillations still moves up with
LH. It follows from these simulations that the optimum
LH of the system, that which gives the greatest amplitude,
is close to L99 for tH À 10 ms
and moves towards KD for tH > 100
ms.
The amplitude decreases steeply with respect to stimulation frequency
(Figure 9B, thick line). This
behavior follows from the observation above that, when frequency increases,
the signaling complex has not enough time first to reach the upper bound then
to return to the lower bound. At the highest frequency resolved by the cell
type modeled (2 Hz) (Rumbo and Kaissling,
1989), with 20 ms pulses, the amplitude is 2.42 x
10–6 µM for LH = 3.4 x
10–3 nM, i.e. flux 0.1 µM/s. It means that the signaling
complex fluctuates by 3.8 molecules/cell around its average level which is 50
molecules/cell. These numbers can be considered as compatible with periodic
threshold crossing and firing of the cell. However, because the average level
of signaling complex grows linearly
with frequency (Figure 9A), the
relative amplitude declines still faster than amplitude with pulse frequency
(Figure 9B, dashed line), so
that the oscillations become progressively lost on top of a growing
steady-state number of activated molecules. For example, amplitude is reduced
four times (0.95 activated molecules/cell for an average of 12 activated
molecules/cell) with a 4-fold decrease of pulse height and about 4 times (0.86
molecules/cell for an average of 200) with a 4-fold increase of frequency. So,
the cell becomes unable to follow pulses either because the amplitude falls
below one activated receptor molecule per cell, at low ligand concentration,
or because the relative amplitude becomes too small, at high
concentration.
Oscillating component of the response during the transient state
The oscillating component of the transient state can be shown in isolation
by substracting the response Cc(t) to a step
stimulation from the transient response C(t) under pulsed
stimulation. As seen before, response Cc(t) grows
to constant c, whereas
C(t) finally oscillates around the same level C =
c. The oscillating
component calculated in this way is shown in
Figure 5. It can be seen that
the oscillating component is not identical during the transient and the steady
states. However both fluctuate with the same period and with a similar
amplitude. So, contrary to the impression given by
Figure 3, for example, the
oscillations of the ligand concentration are immediately encoded in the
oscillations of the signaling complex, although in a distorted way.
Modified parameters for a higher temporal resolution
In order to follow pulses up to 10 Hz, as the cell types responding to the minor pheromone components, the model parameters of Table 2 must be modified. It seems reasonable to modify the parameters so that the amplitude of 3.8 molecules of signaling complex per cell, found at 2 Hz with the previous parameters, is now reached at 10 Hz. This implies that the amplitude of the new system at 2 Hz should be considerably increased. To have larger amplitude, at least one of two conditions must be fulfilled: (i) the system must react faster, i.e. transient state must be shortened; or (ii) the steady-state level must be increased. In the activation pathway (first line of Table 1) these conditions are achieved by changing the forward kj and backward k–j rate constants (with j = 2, 3, 4) for which the ratio Kj = k–j/kj is large. If Kj is small, the change is not efficient. In the present case, with K2 = 0.0638, K3 = 37.46 and K4 = 5.667, the best results are obtained by increasing k3, which controls the binding of the loaded PBP to the receptor. In the deactivation pathway (second line of Table 1) the reaction that controls the binding of the loaded PBP to the enzyme N is the most important. Increase of amplitude is achieved by decreasing K5. Since in differential equations (Appendix A) k5 always appears in the product k5N, where N is the constant concentration of enzyme N, it is equivalent to increase k5 or N. However, any increase of k5N lengthens the transient state. This slowing down can be compensated by increasing k6 but then the amplitude is decreased.
Due to these complex effects, the simplest candidate for improving the
temporal resolution of the extracellular reaction network is
k3. A 6-fold increase of k3 with no
change of the other parameters results in an amplitude of 3.7 molecules at 10
Hz, with pulses of 3.4 x 10–3 nM (flux 0.1 µM/s) in
height (see Discussion) and 20 ms duration as used in experiments
(Rumbo and Kaissling, 1989;
Kodadová, 1996) (see
Table 4). This is approximately
the same amplitude as that obtained at 2 Hz with the original value of
k3 without much slowing down the system (the natural
period remains approximately the same, see
Figure 7). Of course, the
average level of the signaling
complex does not remain the same; it grows from 50 molecules/cell for
k3 = 0.209 (Figure
9A, thick line) to 1500 molecules/cell for
k3 = 1.254 s–1 µM–1
(Figure 9A, thin line).
Interestingly, Figure 9B
(dashed line) shows that the resulting fast decline of the relative amplitude
A/
is not influenced by k3. Therefore improving
relative amplitude can only be attained through the modification of constants
other than k3.
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Discussion |
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Basic properties
Studying how the pheromone sensory system responds to a periodic train of identical pulses offers an idealized equivalent of how a moth `sees' a pheromone plume when it flies through it. We investigated this problem at the level of the sensory membrane of receptor neurons. All events affecting pheromone molecules up to receptor activation, i.e. translocation from external air to sensillum lymph, transport via PBP, enzymatic deactivation and receptor interactions, were taken into account. However, to simplify the interpretation of the results the transduction of activated receptors into receptor potential was not considered.
We show, using computer simulation of a reaction network model proposed by
Kaissling (Kaissling, 2001),
that, after a transient state, the concentration of activated receptor
proteins (signaling complex) at the surface of receptor neurons oscillates in
time around a constant value, with a constant amplitude and with the same
period as the stimulus. This is an important feature of the model that the
response can be considered at each instant as the sum of two components, one
monotonic (average on time) and the other oscillating, whose properties are
very different.
The characteristics of the monotonic component, i.e. the time-averaged
concentration of signaling complex and the duration of the transient state,
are the simplest to describe because they depend only on the time-averaged
amount of ligand delivered to the
system, as given by equation (3). Therefore the time-averaged response to a
train of pulses is identical to the response to a constant (step) stimulation
at concentration Lc of pheromone molecules in the air
surrounding the antenna equal to . It
implies that many different pulse trains differing in their intensity
(LH) and temporal structure (tH and
tL) yield the same monotonic component of the response,
provided they have the same . For
this reason the quantity of equation
(3) appears as a major feature of the pulse trains.
On the contrary, the properties of the oscillating component depend on the detailed characteristics of the pulses, i.e. their height LH, duration tH and separation tL. Although the period of the response is the same as that of the stimulus, T = tH + tL, its amplitude A is a more complex quantity that depends simultaneously on all three pulse characteristics. However, analysis of the model (see Figure 8) showed that amplitude is influenced much more by period T than by tH and tL taken individualy, so that what is important is again the stimulation frequency f = 1/T. For example, an increase of tH compensated by a decrease in tL which leaves frequency unchanged, is much less influential than the same increase of tH with unchanged tL, which increases frequency. This conclusion is valid only for concentrations of the signaling complex far from zero and saturation (see natural period below).
Comparison with other models
Kaissling (Kaissling, 1998a)
distinguished two types of chemosensory sensilla, `concentration detectors'
(CDs) and `flux detectors' (FDs). CDs
(Kaissling, 1998a;
Lansky et al., 2001;
Kivan et al.,
2002) are characterized by unrestricted back and forth access of
external ligand molecules to the receptor layer, which is not the case of the
various FDs in which there is only a forward flux into the perireceptor space
compensated by a deactivation of the ligand. The CD model is a realistic
description of CO2-sensitive sensilla, for example, but not of
pheromone-sensitive sensilla, which are better described by a FD model. The
present 13-reaction system is an example of relatively complex FD. However, it
behaves qualitatively like much simpler FDs in which the perireceptor space
and reactions taking place in it are considered in a simplified form, and even
like CDs for certain properties. For this reason it is interesting to compare
the system studied here to these much simpler variants, which differ by the
receptor–ligand interactions (CDs and FDs) and the ligand deactivation
(FDs only).
For each type of detector, one-step and two-step receptor–ligand
interactions were studied. In one-step interaction only the binding of
pheromone to receptor, with reaction rates k3 and
k–3, is considered (Kaissling, 1988a,b;
Lánsk et al.,
2001; Kivan et
al., 2002), whereas in two-step interaction, binding is
followed by activation with reaction rates k3,
k–3, k–4 and
k–4 (Rospars
et al., 2000;
Kaissling, 2001). As shown
below, both one-step and two-step systems are qualitatively, but not
quantitatively, equivalent.
Ligand deactivation in FDs was considered as taking place only after the
ligand is released from the receptor, the receptor acting as an enzyme, or
before the interaction with the receptors, via enzymes present in the
sensillum lymph. The first mechanism (receptor-enzyme) was studied by
Kaissling (Kaissling,
1998a,b)
and Rospars et al. (Rospars
et al., 2000) and the second one (separate enzymes) by
(Kaissling, 1998a,
2001) and in the present paper.
A variant of these models, which uses both receptor-enzyme and a flux-limiting
process equivalent to separate-enzyme deactivation, is the `generalized flux
detector' (GFD) (Rospars et al.,
2000).
Consider first the static properties of these systems, i.e. the dependence
of the steady-state concentration on
. As shown in Appendix B, contrary to
Kaissling's conclusion, in both the one-step FD with separate enzyme
(Kaissling, 1998a) and the
present 13-reaction network (Kaissling,
2001), this dependence is hyperbolic. So, these systems behave
qualitatively like CDs, in which the static curve
c vs.
Lc is also a hyperbola
(Kaissling, 1998a;
Rospars et al.,
2000). From a quantitative point of view, the closest of the
simple models to the present model is the two-step GFD. This is manifest for
the position of the static curve along the concentration axis as quantified by
the ligand concentration in air at half-maximum response, i.e. the apparent
equilibrium dissociation constant KD. For the one-step CD,
it is KD =
k–3/k3 = 37.8 µM, and for the
two-step CD, KD =
k–3k–4/k3(k–4
+ k4) =32.3 µM, from equation (13) in Rospars et
al.(Rospars et al.,
2000), whereas it is KD À 1 nM for both
GFDs and for the present system (equation B5 and
Figure 4, with
ki = 2.9 x 104
s–1). So, at steady state, the curve
c vs. logLc of
all flux detectors is shifted to the left by log
ki, i.e. 4.5 log units, with respect to that of
concentration detectors.
The dynamic responses C(t) are qualitatively equivalent
in the concentration and flux detectors as can be easily judged by comparing
Figure 9 in Rospars et
al. (Rospars et al.,
2000) for the two-step CD,
Figure 2 in Kivan
et al. (Kivan et
al., 2002) for the one-step CD and
Figure 2 in the present
article. But again the quantitative differences are conspicuous, because the
13-reaction system appears slower than those we studied previously. This is
best illustrated by the `natural' period of the system, i.e. the minimum
period tH + tL such that `on' and `off
' durations allow for maximum amplitude of the response (see
Figure 7). This period is
12 s, compared with 1.15 s for both the one-step and two-step CDs,
so that the respective `natural' frequencies are 0.08 and 0.9 Hz. Thus, using
the same parameter values, the present complete FD system is 10 times
slower than the corresponding CDs.
Transient phase
However, the slowness of the 13-reaction system to reach steady state and to return to resting level is not necessarily an important feature as far as pheromone detection is concerned.
First, the rise of the transient monotonic state (dashed line in Figure 3) can be detected without waiting for the steady state. This is probably important in natural conditions where the pulses are irregular in height, duration and separation.
Second, with a a pulse train, the individual pheromonal pulses are also detectable right from the beginning of the transient response, much before the steady-state oscillations are reached. In practice, as shown in Figure 6, the period T and amplitude A can be approximately determined from the first oscillation and, again, there is no need to wait for the steady state. Then the following analysis of amplitude during the steady state holds also true for the amplitude during the transient period.
Amplitude
The dynamic properties of a periodic stimulation are reflected in the amplitude of the signaling-complex oscillations. This is the number of molecules of LBredR* that are added at each pulse to the time-averaged number of molecules of activated receptor complex.
In the biologically relevant range of pheromone concentrations, amplitude
A (see Figure 8C) and
relative amplitude A/, where
is the time-averaged concentration
of signaling complex at steady state over one period, have similar properties
with respect to stimulation frequency: both decrease quasi-exponentially when
frequency is increased. However, they differ radically with respect to pulse
height LH because A increases with
LH (Figure
8A), whereas A/ is
independent of it (Figure 8B).
This implies that amplitude reflects both pulse height and pulse frequency,
whereas relative amplitude reflects only frequency.
The neuron type studied, which is sensitive to the main pheromone
component, can follow periodic 20 ms pulses up to 2 Hz
(Rumbo and Kaissling, 1989;
Kodadova, 1996). The
stimulating apparatus used in these experiments, loaded with 5 x
10–3 µg of pheromone, was different from the standard one
which produced the 2 s step stimulations used for determining the parameters
of Table 2. Although the pulse
height LH and corresponding flux delivered by the pulse
stimulator were not determined, they are known to be about two orders of
magnitude higher for the same load than those of the standard apparatus
(Kaissling, personal communication). With this 100-fold factor, the 5 x
10–3 µg load would yield a concentration in air
LH = 3.4 x 10–3 nM and a
corresponding flux of 0.1 µM/s. This means, if the present model is
essentially correct, that this cell type can detect oscillations of the
concentration of signaling complex LBredR* with
amplitude as low as 2.4 x 10–6 µM for a steady-state
level of 3.2 x 10–5
(Table 4;
Figure 2D). These molarities
correspond, for a perireceptor-space volume of 2.6 x
10–12 l (Keil,
1984), to 3.8 receptor molecules oscillating on top of an average
level of 50 molecules (Table 4, Figure 9B). These results
suggest that the present model, whose parameters were determined from
responses to step stimulations of 2 s duration, can also account for periodic
pulse stimulations. It constitutes an encouraging independent test of the
essential validity of these parameters.
Extensions of the model
The other cell types, sensitive to the minor pheromone components, can
resolve pulses up to 10 Hz (Rumbo and
Kaissling, 1989). Assuming the same network of extracellular
reactions is present in these cells as in the cell type sensitive to the main
component, the difference in temporal resolution of pulses can be ascribed to
changes in rate constants or initial concentrations of reactants whose effect
would be to increase the amplitude of the periodic oscillations of the
activated receptor (signaling complex), expressed in both absolute, i.e.
number of molecules taking part to the oscillations, and relative terms, i.e.
with respect to the total time-averaged number of activated receptors.
Investigation of the sensitivity of the absolute amplitude to parameter values showed that a key factor is the balance between the two pathways that the ligand–PBP complex (LBred) can follow, i.e. either the activation or the deactivation pathway (see Table 1). Although this balance can be modified in several ways, the rate constant k3, which controls the binding of LBred to the receptor, was found to be a major factor. A 6-fold increase of k3 from 0.209 to 1.254 s–1 µM–1 (Table 4), with no change of the other parameters, increases 6-fold the amplitude of the signaling complex at 2 Hz and yields the same amplitude at 10 Hz as the original system at 2 Hz without slowing down of the system (the duration of transient state and the natural period remain the same).
However, modifying k3 has no influence on relative amplitude, so that the amplitude remains the same small fraction (0.25%) of the average number of activated receptors whatever k3. It seems unlikely that such a small relative variation can be detected by the cell. The determination of parameters improving both amplitude and relative amplitude which would permit the system to resolve 10 Hz pulses remains an open problem.
The observed decrease of the amplitude of the concentration of the
signaling complex when the stimulation frequency increases can be discussed in
a wider context. Indeed, Samoilov et al.
(Samoilov et al.,
2002) proved that this is a general property of any linear
chemical network driven by a single external oscillatory input signal. All
such networks act as low-pass frequency filters. Moreover, their theoretical
results show that if a reaction network is selectively sensitive to some
stimulation frequencies (band-pass filtering), then either it has another
source of oscillations or it is nonlinear. Since our system is linear only at
low ligand concentration, this rises the question of knowing whether the
present model could behave as a band-pass filter at high concentration, i.e. a
model in which the signaling complex reaches its highest amplitude at some
preselected stimulation frequency (e.g. 2 Hz). An alternative approach, used
by Lánsk et al.
(Lánsk et al.,
2001) for a simplified version of the present network, is to
consider not only the amplitude of the signaling complex but the product of
the amplitude and another variable (speed of change of the amplitude). This
product presents a maximum and thus an optimum frequency. This effect, which
evokes a band-pass filter, is certainly present also in the network studied
here.
Experimental vs. natural conditions
The results reported here are useful for interpreting laboratory
experiments in which identical pulses are periodically applied. For example,
Figures 2 in Rumbo and
Kaissling (Rumbo and Kaissling,
1989) and Kodadová
(Kodadová, 1996),
showing receptor potential and spike recording of the aldehyde cell under such
conditions, are qualitatively comparable to our
Figure 3, assuming the receptor
potential reflects the level of signaling complex and a spike is fired when
this level crosses a certain threshold. However, caution must be exerted in
applying them to natural conditions in which the pulses are known to be quite
irregular (Murlis et al.,
1992). We intend to develop the present approach to study the
response of the perireceptor and receptor system to pulses with stochastic
characteristics.
|
Appendix A: differential equations |
|---|
|
TopAbstractIntroductionModelResultsDiscussionAppendix A: differential...Appendix B: receptor...References |
|---|
To avoid notational complexity, all multiple-letter symbols for the various chemical species are replaced by single-letter symbols, i.e. P = [LBred], O = [LBredR], C = [LBredR*], ß = [LBox],
= [LE],
= [LBredN], 
= [LBredE], 
=
[LBoxE]. The other symbols (L, Bred,
Box, R, N, E) are kept unchanged. The 10
time-variable concentrations are L(t), (t),
P(t), R(t), O(t),
C(t), (t), ß(t),
(t), (t). All other concentrations
R0, Bred, Box,
E, N are constant. With this notation and omitting variable
t the system of differential equations describing the model given in
Table 1 is
 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
 | (A8) |
 | (A9) |
 | (A10) |
All time-variable concentrations are equal to zero at time zero, except R(t) for which R(0) = R0.
Species E, N and B are external species, i.e. with constant concentrations
despite entering in different reactions, in contrast to R, whose free amount
is decreased by bound and activated states. The reaction network is incomplete
for
, which accumulates, and for B since Bred is not regenerated
from the end product MBox which consequently also accumulates.
|
Appendix B: receptor concentrations at equilibrium |
|---|
|
TopAbstractIntroductionModelResultsDiscussionAppendix A: differential...Appendix B: receptor...References |
|---|
Differential equations (A1)–(A9) at equilibrium, i.e. for dL/dt = 0, d
/dt = 0 etc., result in a system
of nine algebraic equations with nine unknowns which can be solved exactly.
The calculations are long but straightforward. We give here the solutions
(B1)–(B3) obtained for the pheromone receptor.
At any time the receptor proteins are in three states, as given by equation
(A10): free R, occupied LBredR (denoted O) and activated
LBredR* (denoted C). For a step stimulation
Lc, the conservation equation at equilibrium can be
written R0 =
c +
c +
c. For a periodic pulsed
stimulation, the same equation holds at steady state, R0 =
+ +
, where
, and
are time-averaged concentrations
over one period. It can be shown numerically (see Results) that R =
c, O =
c and C =
c, for
= Lc, where
L is the time-averaged concentration of ligand in the air surrounding
the antenna, see equation (3). Therefore the following results established for
a constant stimulation are also true for the time-averaged response to the
equivalent periodic pulsed stimulation. The concentrations of the three
receptor states are given by
 | (B1) |
 | (B2) |
 | (B3) |
 |
 |
and ß simplify
as follows:
 |
 |
The concentration of the signaling complex at saturation and the ligand
concentration at half-saturation are given by
 | (B4) |
 | (B5) |
 | (B6) |
Equations (B3) and (B6) are hyperbolas with respect to
Lc (or equivalently c). Consequently the
curve c vs.
logLc (or log c) is a logistic whose
dynamic range is 
Lc –2 log 
decades (Rospars et al.,
1996), where is a fraction of Cmax. For
example, the horizontal distance between 1% (close to threshold) and 99% of
Cmax is always Lc = 4 log units
(see Figure 4), so that the
ratio of concentrations L99/L1, (respect.
fluxes 99/1) at 99% saturation and at 1%
threshold is always 104, whatever the values of
Cmax and KD.
With the parameter values given in Table
2, /ß = 3468.4 µM/s2,
Cmax = 0.24 µM and KD = 1.042 nM
(i.e. flux 50 = 30.21 µM/s). For a hair volume of 2.6 pl,
the total number of molecules per neuron is R0 = 2.56
x 106 and Cmax 3.75 x
105. The level
c = 1 receptor molecule
activated (threshold) is reached for a pheromone concentration in air
Lc = 2.8 x 10–6 nM corresponding to
flux c = 8 x 10–5 µM/s, i.e. 125
molecules/s.
For Lc << KD, hyperbola (B6) can be
approximated by the linear expression
 | (B7) |
.
The slope of line (B7) is
 | (B8) |
With the parameter values of Table
2, Cmax/KD = 0.23 with
KD and Lc in nM, and
Cmax and Cc in µM, and for the flux
equation the slope is 50/Cmax = 125.9
with all terms in µM. With these parameters the hyperbola is rather flat
(see Figure 4A) so that, in
practice, the line Cc = c/126 (i.e.
0.23Lc) is an excellent approximation in the range 0 
c 1 µM/s. The line Cc =
c/150 (i.e. 0.19Lc), as used by Kaissling
(Kaissling, 2001), is a better
approximation for c > 1 µM/s.
Note. The equation relating c to
c was also derived as
equation (4) in Kaissling (Kaissling,
2001) for the present 13-reaction system, and as equation (18) in
Kaissling (Kaissling, 1998a)
for a simplified three-reaction network with one-step activation and a
separate enzyme. In both cases the system of differential equations describing
the networks are correct but the equations giving
c at equilibrium are not.
The exact solutions for c
are hyperbolic functions of the form (B6), i.e. the –1 term appearing in
the denominator of both equations (18) and (4) must be removed. Because of
this term, it was wrongly concluded that the
c vs. c
curves were not hyperbolic and that there was a limiting flux
sat. Other consequences will be examined elsewhere (K.-E.
Kaissling and J.-P. Rospars, in preparation).
|
Acknowledgments |
|---|
The authors thank K.-E. Kaissling for unpublished data, helpful discussions and constructive criticisms on earlier versions of this manuscript. This work was partly supported by joint cooperation project Barrande No. 972SL between France and the Czech Republic, by NATO linkage grant LST CLG 976786, by Grant Agency of the Czech Republic (309/02/0168), by Grant Agency of ASCR (Z5007907) and by MSMT (12300004).
|
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Accepted May 31, 2003
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