Chem. Senses 27: 511-520,
2002
© Oxford University Press 2002
Application of the U and 
' Models in Binary Sweet Taste Mixtures
CNRS, Centre Européen des Sciences du Goût, Dijon, France 1 Chemistry Department, National University of Ireland, Galway, Ireland
Correspondence to be sent to: Paul Laffort, CNRS, Centre Européen des Sciences du Goût, Dijon, France. e-mail: laffort{at}cesg.cnrs.fr
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Abstract |
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 Top Abstract IntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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The U and
' models of sensory interactions, successfully
applied in olfaction for several years, are tested here using data from
published studies on sweetness. The models are subsequently tested on new data
obtained in studies of binary mixtures of four sodium sulfamates. The U model
allows for the estimation of a global interaction, whereas the '
model allows for the distinction between that which is due to an intrinsic
interaction in the mixture itself and that which may be due to the power
function exponents in the mixture. The models give satisfactory predictions
for observed phenomena of sweet taste suppression, synergism or pure
additivity. Additionally, they appear to be more suitable than other models
recently applied in taste, particularly the equiratio model. Application of
the models to the sulfamate mixtures, reveals additivity for sodium
cyclohexylsulfamate (cyclamate)/potassium cyclohexylsulfamate and sodium
cyclohexylsulfamate/sodium exo-2-norbornylsulfamate, respectively; whereas for
sodium cyclohexylsulfamate/sodium 3-bromophenylsulfamate, the models revealed
a slight hypo addition which is simply due to the dissimilarity values of the
power function exponents of the components. |
Introduction |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Based on analysis of several sets of data, the U model (Patte and Laffort, 1979
)
appears to be a satisfactory predictive tool for olfactory binary mixtures.
While being very similar to the vector model of Berglund et al.
(Berglund et al.,
1973), the U model can be considered as an improvement on the
latter. In both models, each pair of odorants is characterized by a parameter
of interaction, called cos 
, which is almost constant irrespective of
the respective concentrations of the components of the mixture. Without
entering into the mathematical details, it appears that both the experimental
basis and theoretical considerations support the fact that the scattering of
cos values is less widespread using the U model rather than the vector
model (Laffort and Dravnieks,
1982).
The family of models (Laffort
et al., 1989) is based on the U model, but it allows one
to consider separately (from the observed synergies and suppressions) what is
due to an intrinsic interaction within a given pair of odorants and what is
due to the odorants themselves taken separately (their power function
exponents). Among all the tested members of the family, the model
called ' appears to be the most suitable for studies on
olfactory, gustatory and pharmacological mixtures
(Sérée and Laffort,
1997; Thomas-Danguin and
Laffort, 1998).
It has also been shown that other models applied in taste are particular
cases of the ' model (Laffort
et al., 1989;
Laffort, 1994), for example
the substitution model (Hyman and Franck,
1980) or the equiratio model
(Frijters et al.,
1984; De Graaf and Frijters,
1986). Their validity is therefore less general (see Appendix
A).
The models U and ' are applied here to published studies on
sweet mixtures, i.e. glucose/fructose, sucrose/aspartame and sucrose/sodium
cyclamate. The models are also tested on unpublished data on sulfamate
mixtures, including sodium cyclohexylsulfamate (cyclamate) mixtures, which are
also sweet.
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The U and ' models
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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The U model is very simply defined by the following:
 | (1) |
U is the
parameter of interaction (not related with a real angle , as in the
vector model, but named by analogy in this manner).
The ' model is given by:
 | (2) |
' is an index of intrinsic interaction. ' =
1 corresponds to an absence of intrinsic interaction; ' > 1
corresponds to an intrinsic synergy; ' < 1 corresponds to an
intrinsic suppression; and cos U is directly derived from
equation (1):
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UPL2 is the parameter of interaction due to the power
function exponents of components (UPL2 stands for the second version of the
model, in which the power law was applied to the U model). The definition of
cos UPL2 requires the prior definitions of P (as
proportion) and cos A and cos B:
 | (3) |
The mathematical proof of relation (3) is reported in Appendix B. When the
above equations are applied to a given set of data, predicted values of
perceived intensities are easy to obtain for chosen concentrations. However,
the reverse is not true: if one attempts to obtain concentrations
corresponding to given perceived intensities, the ' model lead to
analytically intractable equations. To solve this difficulty, Laffort and
coworkers use an iterative computer program called MIG (mixtures intensities
generation) [first application in 1989
(Laffort et al.,
1989)]. The program allows, for example, the construction of
iso-intensity curves. Several examples are provided below.
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Application to several published data sets on sweet taste mixtures |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Three data sets were selected from published studies on sweet taste mixtures in order to underline the various equisweet types of curves which can be observed.
Glucose/fructose
These experimental data (De Graaf and
Frijters, 1986) involve five levels of sweetness, respectively
equivalent to molar concentrations of glucose of 0.125, 0.250, 0.500, 1.000
and 2.000. Each of these levels of intensities includes five points: two pure
sugars and three mixtures. The exponent of glucose being fixed at 1, the
exponent for fructose was found equal to 0.82 and the ' value
equal to 1.08 ± 0.04 (mean ± SD), which corresponds to a slight
intrinsic synergy. Figure 1
reproduces the 25 experimental points, as well as the five superimposed
equisweet curves obtained by using the MIG computer program.
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It is clear that, in this particular case, the ' value is
close to 1 and the exponent values are also close to 1; the equisweet curves
are very close to the diagonal dotted line and therefore the equiratio model
is approximately applicable. That is true only in this particular case.
Sucrose/aspartame
The data for the mixture sucrose/aspartame
(Schifferstein, 1995) also
include 25 experimental points. The mixture of molar
concentrations—0.0016 of aspartame and 0.1376 of sucrose—is fixed
at a sweetness intensity of 10. The subjects' responses correspond to
multiples of this reference. The power function exponents found by the author
were, under these conditions, 0.915 for aspartame and 1.275 for sucrose. The
' value then obtained is 1.03 ± 0.34), i.e. not
significantly different from 1. The five equisweet curves obtained for
perceived intensities from 1 to 50 by using the MIG computer program are
superimposed. They are shown in Figure
2, as well as the experimental points of Schifferstein.
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The interesting result displayed by the theoretical curve according to the
' model is that an observed global synergy appears at the top of
the figure and a global partial suppression at the bottom, in spite of the
absence of an intrinsic interaction (' # 1). Both results are due
only to the difference in the power function exponents of the components. The
treatment cannot, however, be applied to the present experimental points, due
to the widespread scattering in relation to the curve, as is indicated by a
rather high value of the standard deviation for the ' index for
these data. Probably the only method by which the shape of this figure could
be verified would be using equisweet experimentation and not magnitude
estimation as used by Schifferstein, after fixing the power function exponent
of sucrose to 1.
Sucrose/Na cyclamate
A mixture sucrose/Na cyclamate was studied previously
(Nahon et al., 1998)
at only one level of sweetness: 10 w/v% sucrose solution, denoted as 10 % SEV
(sucrose equivalent value). Exponents of the power function cannot therefore
be obtained. The curve reported in Figure
3 is, however, consistent with equal exponent values for the two
components, taken as equal to 1, and to a ' value equal to 1.32,
i.e. strongly synergistic (indicating good agreement between the theoretical
curve and the experimental points).
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This third type of curve observed in taste is, of course, not at all in
agreement with Beidler's mixture equation and the equiratio model, since the
dotted straight line corresponding to that model does not fit the experimental
points, whereas the curve derived from the ' model fits them
well.
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New experimental data on sulfamate mixtures |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Studies were carried out on three sets of binary mixtures: sodium 3-bromophenylsulfamate/sodium cyclohexylsulfamate (cyclamate); potassium cyclohexylsulfamate/sodium cyclohexylsulfamate; and sodium exo-2-norbornylsulfamate/sodium cyclohexylsulfamate.
These mixtures were chosen because they gave a wide range of relative
sweetness (RS) values against 3% sucrose as standard and it was hoped the
study of their mixtures would yield interesting results (see
Drew et al., 1998).
The relative sweetness is defined, according to Paul
(1921), as the ratio of
concentrations, in mg/l, of sucrose solution and a compound equisweet with the
sucrose solution. The RS values for the selected sulfamates are: sodium
3-bromophenylsulfamate, 11.2; sodium cyclohexylsulfamate, 39.8; potassium
cyclohexylsulfamate, 40.2; and sodium exo-norbornylsulfamate, 70.5.
Methods
The previously published procedure
(Frank et al., 1989)
was followed using a panel of eight tasters. All stimuli were made by
combining solutes, not solutions. Stimuli for the binary mixtures and
component solutions were chosen so as to approximately match the taste
intensity of the sodium cyclohexylsulfamate at millimolar concentrations of
2.5, 4.5 and 7.5, respectively. This matching was accomplished during
preliminary testing with each component. In addition, distilled water was also
used as stimulus.
Panelists were asked to judge the sweetness of a number of stimuli on a 21 point category scale which was labeled as follows: 0, no taste; 5, weak; 10, medium; 15, strong; and 20, very strong. Prior to tasting, the panelists were presented with either three (binary mixtures) or four (self-mixtures) stimuli and were told that these stimuli encompassed the range of sweetness levels they would recounter. The test volume was set at 5 ml and tasters were required to sample the entire volume. Panelists were told to ignore all other sensations and just concentrate on the sweetness of the compounds.
Results
The experimental results and the calculations are summarized in Tables
1 and
2, respectively. A category
scale can be considered as a logarithmic scale of the perceived intensity
(Patte et al., 1975).
Therefore, the experimental responses for pure components and mixtures
(termed, respectively, Ra, Rb and
Rab) were transformed into R' values,
according to the equation:
 | (4) |
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The regression equations applied to log R' values versus log Ca values provide power function exponents (slopes) n with satisfactory R2 values, as can be seen in Table 1.
The main comment concerning the data in Table 2 is that, using the U model, the correlation coefficients between predicted and perceived sweetness of mixtures (Rab) are rather satisfactory; furthermore, the averages of the ratios of predicted/experimental sweetness are also very close to 1. This rather good prediction is shown in Figure 4.
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The second comment concerning Table
2 is that ' values are almost equal to 1 in each case
(absence of intrinsic interaction). A slight negative value of cos
U for the first mixture and an almost zero value for the two
other mixtures is also observed. This means that when data are considered in
terms of perceived intensities, a slight hypoaddition occurs in the first case
only—due to differences of power function exponents—and additivity
is observed for the two other mixtures.
A possible further use of the data in
Table 2 would be to draw
equisweet curves, similar to those of Figures
1,
2 and
3 for data already published.
However, it can be seen in Table
2, that the values of the standard deviations for '
are intermediate between those obtained by magnitude estimation and the
equisweet procedure respectively. Thus, in this case there was poor agreement
between the experimental points and the theoretical curves and therefore these
are not shown.
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General discussion |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Classically, in studies of taste and olfactory mixtures, there are normally two theoretical references of additivity which can be considered in the analysis of experimental data: (i) additivity of relative concentrations (Beidler, 1971
) in taste and
generalized with the equiratio model; and (ii) additivity of perceived
intensities (Berglund et al.,
1973) in olfaction.
In the first case, when additivity is not observed, the divergences will
change with the proportions of the two components; i.e. the mutual interaction
(synergism or suppression) cannot be characterized by a single index,
preventing any generalization from a limited number of experiments. This
difficulty is overcome by using the second method, which in turn allows us to
come back to the first one with the use of the ' index.
In several studies (Frank et
al., 1989) a third way has been tried: additivity of
responses using a category scale. This is not an appropriate method, since the
responses obtained can be considered to be proportional to logarithmic values
for the perceived intensities (Patte
et al., 1975). Here, therefore, a change of variable must
to be made before applying tests for additivity. This procedure is applied in
the present study to the new experimental data on sulfamate mixtures.
Regarding the present study, it can be concluded that U and '
models, previously applied in olfaction, can be satisfactorily applied in
sweet taste also. The U model, from a limited number of experiments for a
given binary sweet taste mixture, allows for the generalization of predicted
perceived intensity values, irrespective of the experimental method (magnitude
estimation, category scaling or equisweet procedure).
The principal usefulness of the ' model is that it can be used
to draw equisweet theoretical curves from a limited number of experiments. In
sweet taste, the curves produced for several levels of perceived intensity
often appear to be superimposed (this superimposition is not observed in
olfaction). Equisweet curves allow observation of interactions in terms of
concentrations, instead of perceived intensities. Equisweet curves adopt
shapes according to the phenomenon which is occurring: additivity, synergy, or
synergy and partial suppression at the same time, according to the relative
concentrations. Only the equisweet experimental procedure appears to be
precise enough to verify the theoretical curves obtained. Neither the
magnitude estimation nor category scaling procedures seem sufficiently
accurate, even if the latter is the slightly better of the two.
Many published experimental studies cannot be tested comparatively with the
U and ' models as presented here, usually due to an absence of
tables reproducing the experimental data in full. That is the case, for
example, with all the binary mixtures of nine sweet taste compounds
(Frank et al., 1989)
already quoted. Be that as it may, in Appendix C several examples are given in
which the ' model is applied to published data on taste
mixtures.
The sulfamate studies presented show that, in terms of perceived
intensities, the mixtures of potassium cyclohexyl-sulfamate/sodium
cyclohexylsulfamate and sodium exo-2-norbornylsulfamate/sodium
cyclohexylsulfamate are almost additive, whatever the levels of perceived
intensities and the relative concentrations. By contrast, the mixture of
sodium 3-bromophenylsulfamate/sodium cyclohexylsulfamate presents a slight
hypoaddition (slightly negative value of cosU).
Expressed in terms of concentration S, the experimental results on sulfamates give satisfactory equisweet curves for the potassium cyclohexyl/sodium cyclohexyl and sodium exo-2-norbornyl/sodium cyclohexyl mixtures (all the power function exponents being close to 1), but not for the sodium 3-bromophenyl/sodium cyclohexyl mixture.
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Appendix A |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Relations between the substitution, equiratio and
'
models
This clarification is drawn from previous work
(Laffort et al.,
1989; Laffort,
1994).
The substitution model
The mixture discrimination index (MDI), also called the substitution model,
is defined according to Hyman and Frank
(Hyman and Frank, 1980), as
follows:
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Applying the power function, R =
(C/C0)n, to the above definition, two
equations result: the first expression of R'
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These two equations are only equivalent when the exponents for the two
components are the same. That is a restrictive view of the reality (even in
taste, but perhaps less dramatically than in olfaction). No rule is suggested
when the exponents differ, but one could make an arithmetic mean of the two
values of R'. In the particular case where the two exponents
are the same, it can be easily demonstrated that MDI is equivalent to the
first expression in the family indices (called , not
') for equally strong components. By contrast, no equivalence is
found between MDI and the ' index currently used and judged more
suitable.
Therefore, from the initial definitions, we have:
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index, in the particular
case where nA = nB and where
RA = RB, i.e. they are equally strong
components in the mixture, we have:
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The equiratio model
This model (Frijters and Ooude Ophuis, 1983) was proposed for the study of
taste mixtures. Even if the name was new, the method had been proposed
previously (Sales, 1958) for
odorous mixtures and used in the gas industry over a period of many years
(Borelli and Angleraud, 1965;
Blanchard, 1976).
The principle of the equiratio model is quite simple: the
concentration—response straight line in log—log coordinates for a
series of dilutions of a mixture is expected to lie between the straight lines
corresponding to the components of the mixtures, based on the ratio of the
mixture's components. Angleraud (personal communication, 1968) and later
Blanchard (Blanchard, 1976)
demonstrated that this assumption was not experimentally verifiable. The
non-applicability of this model to olfaction has been confirmed
(Schiet and Frijters,
1988).
By contrast, the model has been successfully applied in studies of sweet
taste mixtures (Frijters et al.,
1984; De Graaf and Frijters,
1986; Frijters and De Graaf,
1987).
In the particular case where the values of the exponents are very similar
and near to 1, the equiratio model for iso-intensity mixtures (equisweet for
sweet taste) is simplified as follows:
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).
The graphic expression of the equation is shown by the dotted line in Figures
1,
2 and
3.
Similarly, in that particular case the ' model is also very
simplified:
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Therefore, cos UPL2 also equals zero and the
' index becomes:
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On the other hand, we have:
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When equation (9) of De Graaf and Frijters
(De Graaf and Frijters, 1986)
is verified, ' = 1 (numerator equal to zero in the above
equation) and it becomes possible for the equiratio model to be successfully
applied. In all other cases, the ' index value may provide an
estimation of the divergence between the equiratio model and the experimental
results (the case shown in Figure
3). Once again, the comparisons are only possible when
nA = nB = 1.
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Appendix B |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Mathematical proof of relation (3)
This demonstration is from Callegari
(Callegari, 1998).
The aim of the family models is to separate what is due to the
interaction itself and what is due to characteristics of the components. For
this purpose, the power function is considered to be effective in the range of
studied concentrations, i.e. a straight line in log—log coordinates.
[When that fact is not experimentally verified, it is just necessary to make a
change of variables until one obtains a straight line in log—log
coordinates. Once the calculations with the U and ' models are
done, a reverse change of variables allows one to obtain calculated data to be
compared with the original data. Such a procedure was followed in the present
work for the reported calculations on sulfamates.] The challenge is to
transform exponential expressions into multiplicative coefficients.
We start with the definition of the power function for a given compound A,
at two concentrations A and A'. We have:
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In other respects, for an addition of an odorant to itself, we have from
the definition of P:
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Appendix C |
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TopAbstractIntroductionThe U and {Gamma}'...Application to several published...New experimental data on...General discussionAppendix AAppendix BAppendix CReferences |
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Applications of the
' model to additional published data
on binary sweet taste mixtures
The aim of global models such as the U and the vector models is to
characterize binary olfactory or taste mixtures with a single index, in order
to extend (predict) the perceived intensity of mixtures at any concentrations
from a limited number of experimental data. To be valid, the models must
present indices of interaction as constant as possible (low value of standard
deviation). In previous work we have demonstrated that the scattering of cos
is smaller by using the U model in place of the vector model
(Laffort and Dravnieks, 1982)
and also smaller than that obtained when using an alternative model called the
V model (Patte and Laffort,
1979). We have pursued the same goal with the so-called
family models. Here, the respective contributions of the intrinsic interaction
of the mixture on the one hand and of the power function exponents of the
components on the other are separated. As shown previously (Seree and Laffort,
1996; Thomas-Danguin and Laffort,
1998), the so-called '-vector model is not at all
suitable. Note that equation (2) for the '-vector model is
similar to that for the ' model, with cos VECT
instead of cos U and cos VCPL instead of
cos UPL2. The chain of equations (3) for cos
VCPL is similar to that for cos UPL2, with
the definition of cos A and cos B being:
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' model is suitable and also it is slightly
better than the original model.
Once the more appropriate model is developed, its suitability can be tested
and assessed on the basis of the standard deviation of its index of
interaction, in this case the ' index.
The sweet mixture aspartame—acesulfame-K has been studied
(Schifferstein, 1996) by
applying magnitude estimation. The results show exponent values equal to 0.994
for aspartame and 1.095 for acesulfame-K, with a strong synergistic
' value: 1.72 (' > 1, corresponding to a synergy
and ' < 1 to a suppression). Unfortunately, the value for the
standard deviation of ' is high (0.78). This standard deviation
value, greater than that found for the sucrose/aspartame mixture reported in
Figure 2
(Schifferstein, 1995) does not
allow reasonable predictions. Furthermore, it seems to confirm that magnitude
estimation may not be a suitable tool for the study of sweet taste mixtures.
This is surprising since, for example, for overall intensity estimations of
sucrose/citric acid mixtures, also using magnitude estimation, a relatively
constant suppression ' index value has been observed by several
authors, with low standard deviations and not too great a difference from that
obtained with other techniques (difference estimation procedure or 150 mm line
scale): by magnitude estimation, ' = 0.82 ± 0.11
(McBride, 1989) and 0.73
± 0.05 (Schifferstein and Kleykers,
1996); by a difference estimation procedure, ' = 0.81
± 0.03 (Schifferstein and Frijters,
1990); and by a 150 mm line scale, ' = 0.83 ±
0.11 (Schifferstein,
1997).
The sweet taste mixtures fructose/saccharin and xylitol/saccharin have been
studied (Hyvönen et al.,
1978) by successively applied magnitude estimation, paired
comparison and the triangle test. According to the authors, the power function
exponents are supposed to be equal to 1 for these three compounds. Under these
conditions the ' indices obtained are 1.53 ± 0.08 for
fructose/saccharin and 1.53 ± 0.14 for xylitol/saccharin.
The scattering for ' is comparable to that obtained with the
data reported in Figure 1 and
Table 2. The equisweet diagram
would be analogous to that shown in Figure
3 and in that case it is clear that the equiratio model would not
be suitable either. It should be noted that the authors studied these mixtures
at three different temperatures (5, 23 and 50°C) and that they found that
temperature had a slight influence on the observed synergy. By separating the
three temperatures, slightly different ' values are observed,
with reduced values of the standard deviations (the mean ' value
is almost equal to that at 23°C).
The sweet taste mixture saccharin/dulcin has been studied
(Täufel and Klemm, 1925)
by using an equisweet procedure with sucrose. According to the experimental
data of the authors, the power function exponents are 0.578 for saccharin and
0.464 for dulcin. On the basis of these exponents, the saccharin/dulcin
' value is ' = 1.16 ± 0.04.
These results for the saccharin/dulcin mixture also generates an equisweet diagram comparable to that of Figure 3, but with experimental points only in the zone of relative concentrations: 10% of dulcin and 50-70% of saccharin.
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Acknowledgments |
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This work was supported by a grant from Galway County Council and the TOSTQ European Commission Concerted Action Programme (FAIR-CT98-4040). The authors sincerely thank Dominique Valentin (University of Dijon) for her help with the data processing. They are also indebted to Pascal Callegari and Thierry Thomas-Danguin for their contribution to Appendices B and C, respectively. They also wish to acknowledge the valuable suggestions of two referees.
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