Simon Hess

Christophe Pouzat
email: christophe.pouzat@math.unistra.fr

Peter Kloppenburg

A Simple Method for Getting Standard Error on the Ratiometric Calcium Estimator

Keywords: Calcium imaging, Fluorescence measurements, error propagation, Python, reproducible research Calcium measurements, Fura-2, propagation of uncertainty, propagation of errors, Monte-Carlo method, reproducible research

1. a precise estimation of the standard errors of the raw data is obtained first, 2. the standard error of the ratiometric calcium estimator (a function of the raw data values) is derived using both the propagation of uncertainty and a Monte-Carlo method.

Once meaningful standard errors for calcium estimates are available, standard errors on fitted model parameters follow directly from the use of nonlinear least-squares optimization algorithms.

Method name Standard error for the ratiometric calcium estimator

Abstract

The ratiometric fluorescent calcium indicator Fura-2 plays a fundamental role in the investigation of cellular calcium dynamics. Despite of its widespread use in the last 30 years, only one publication [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] proposed a way of obtaining confidence intervals on fitted calcium dynamic model parameters from single 'calcium transients'. Shortcomings of this approach are its requirement for a '3 wavelengths' protocol (excitation at 340 and 380 nm as usual plus at 360 nm, the isosbectic point) as well as the need for an autofluorence / background fluorescence model at each wavelength. We propose here a simpler method that eliminates both shortcommings:

Method overview 2.1. Rational

Since its introduction by [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF], the ratiometric indicator Fura-2 has led to a revolution in our understanding of the role of calcium ions (Ca 2+ ) in neuronal and cellular function. This indicator provides a straightforward estimation of the free Ca 2+ concentration ([Ca 2+ ]) in neurons and cells with a fine spatial and time resolution. The experimentalist must determin a 'region of interest' (ROI) within which the [Ca 2+ ] can be assumed uniform and is scientifically relevant. Fluorescence must be measured following excitation at two different wavelengths: typically around 340 and 380 nm; and, since cells exhibit autofluorescence or 'background fluorescence' at those wavelengths, the measured fluorescence intensity is made of two sources: the Fura-2 linked fluorescence and the autofluorescence. The measured intensity within the ROI is therefore usually corrected by subtrating from it an estimation of the autofluorescence intensity obtained from simultaneous measurements from a 'background measurement region' (BMR); that is, a nearby region where there is no Fura-2. At a given time the experimentalist will therefore collect a fluorescence intensity measurement from the ROI at 340 and 380 nm; we are going to write adu 340 and adu 380 these measurements, where 'adu' stands for 'analog to digital unit' and correspond to the raw output of the fluorescence measurement device, most frequently a charge-coupled device (CCD); if the experimentalist is careful not to saturate the sensor, the adu count is proportional to the number of photo-electrons present in the pixel, or in the group of pixels when on-chip binning is used, at the end of the exposure period. The experimentalist will also collect intensity measurements from the BMR, measurements that we are going to write adu 340,B and adu 380,B . If P CCD pixels make the ROI and P B pixels make the BMR and if the illumination time at 340 nm is T 340 , while the illumination time at 380 nm is T 380 (both times are measured in s), the experimentalist starts by estimating the fluorescence intensity per pixel per time unit following an excitation by a light flash of wavelengths λ (λ = 340 or 380 nm) as:

f λ = 1 T λ adu λ P - adu λ,B P B , with λ = 340 or 380 nm , (1)
where an assumption of autofluorescence uniformity is implicitly made. The following ratio is then computed:

r = f 340 f 380 . (2)
This is an important and attractive feature of the method as well as the origin of its name. Since only ratios are subsequently used, geometric factors like the volume of the Fura loaded region under the ROI do not need to be estimated.

The estimated [Ca 2+ ] that we will write Ca for short (the ' ' sign is used for marking estimated values) is then obtained, following [2, Eq. 5, p. 3447], with:

Ca = K ef f r -R min R max -r , (3)
where K ef f (measured in µM), R min and R max are calibrated parameters (the last two parameters are ratios and are dimensionless). R min is the ratio (Eq. 2) observed in the absence of calcium, while R max is the ratio observed with a saturating concentration. K ef f is the calcium concentration at which the ratio is half way between R min and R max . If a set of experiments is performed on a given cell type with the same batch of Fura, as in the companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF], the calibration errors on these three parameters will be the same for each experiment. If different cell types are considered and/or different Fura batches are used, the calibration errors should be taken into account before making comparison of estimated calcium dynamics parameters (see [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] for discussion).

If we now want to rigorously fit [Ca 2+ ] dynamics models to sequences of Ca, we need to get standard errors, σ Ca , on our estimates. This is where the ratiometric method gets 'more involved', at least if we want standard errors from a single transient as opposed to a mean of many transients. We typically work (e.g. [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF][START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF]) in a setting, using the so called 'added buffer approach', where we cannot get more than a single transient in given conditions since Fura is constantly diffusing into the recorded cell modifying thereby the time constant of calcium transients. It is worth pointing out that there is a more general interest in obtaining standard errors from a single transient: getting these fluorescence measurements requires shinning UV light on the neurons we are recording from and generates photodamage. Despite the ubiquity of ratiometric measurements in neuroscience and cell physiology, we are aware of a single paper-by some of us [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF]-where the 'standard error question' was directly addressed. The method proposed in [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] requires a 3 wavelengths protocol: measurements at 340, 380 and 360 (the isosbestic wavelength) nm; it drops, so to speak, the above advantage of working with a ratiometric estimator since it fits directly the adu 340 and adu 380 data (at the cost of estimating some geometry related parameters) and it requires a model of the autofluorescence dynamics if the latter is not stationnary. It therefore requires a slightly more complicated '3 wavelengths' recording protocol as well as a more involved fitting procedure. The dataset of the companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF] exhibits a clear but reversible autofluorescence rundown that cannot be ignored since autofluorescence accounts for half of the signal in the ROI. Rather that constructing / tailoring the accurate enough autofluorence models required by the 'direct approach' of [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] we looked for an alternative method providing standard errors for the ratiometric estimator.

Ratiometric estimator variance

Fluorescence intensity

As detailed in [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF][START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF], the fluorescence intensities giving rise to the adu 340 , adu 340,B , adu 380 and adu 380,B signals can be written as:

I 340 = [F ur a] total φ K F ur a + [Ca 2+ ] R min K ef f + R max [Ca 2+ ] + F 340B T 340 P , (4)
I 340B = F 340B T 340 P B , (5)
I 380 = [F ur a] total φ K F ur a + [Ca 2+ ] K ef f + [Ca 2+ ] + F 380B T 380 P , (6)
I 380B = F 380B T 380 P B , (7)
where F λB is the autofluorescence intensity per pixel per time unit at wavelength λ, K F ur a is the Fura dissociation constant (a calibrated parameter measured in µM), [F ur a] total , is the total (bound plus free) concentration of Fura in the cell (measured in µM) and φ is an experiment specific parameter (measured in 1/µM/s) lumping together the quantum efficiency, the neurite volume, etc (see [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] for details). As detailed and discussed in [START_REF] Van Vliet | Digital Image Analysis of Microbes; Imaging, Morphometry, Fluorometry and Motility Techniques and Applications, Modern Microbiological Methods[END_REF][START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF], the signal adu λ recorded with a CCD chip whose gain is G and whose read-out variance is σ 2 r ead-out can be modeled as the realization of a Gaussian random variable ADU λ with parameters:

Recorded signals adu

µ ADU λ = G I λ , (8)
σ 2 ADU λ = G µ ADU λ + G 2 P σ 2 r ead-out , (9)
with the obvious adaptation when dealing with the BMR signal: I λ is replaced by I λB and P is replaced by P B . Parameters G and σ 2 r ead-out are CCD chip parameters provided by the manufacturer. Calibration procedures are discussed in [START_REF] Van Vliet | Digital Image Analysis of Microbes; Imaging, Morphometry, Fluorometry and Motility Techniques and Applications, Modern Microbiological Methods[END_REF][START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] and a comprehensive exemple with data and codes can be found in [START_REF] Pouzat | christophe-pouzat/ENP2017: Course material for C Pouzat's ENP 2017 and 2018 lectures[END_REF]. Our experience is that the values provided by manufacturers are good starting points; the user calibrated read-out noise is sometime slightly larger than the one specified by the manufacturer.

3.3.

Variance estimates for adu 340 , adu 340,B , adu 380 and adu 380,B So, to have the variance of ADU λ we need to know I λ and for that we need to know [Ca 2+ ] (Eq. 4 and6) precisely what we want to estimate. But the expected value of ADU λ is G I λ (Eq. 8), we can therefore use as a first approximation the observed value adu λ of ADU λ as a guess for G I λ , so in Eq. 9 we plug-in adu λ for G I λ , leading to:

σ2 ADU λ = G adu λ + G 2 P σ 2 r ead-out ≈ σ 2 ADU λ . (10)
In other words, we will use the observed adu λ as if it were the actual fluorescence intensity times the CCD chip gain, ADU λ = G I λ , in order to estimate the variance. In doing so we will sometime slightly underestimate the actual variance (when the observed adu λ turns out to be smaller than ADU λ ) and sometime slightly overestimate it (when the observed adu λ turns out to be larger than ADU λ ). Since we are going to combine many such approximations, we expect-and we will substantiate this claim in Sec. 4-that overall the under-estimations will be compensated for by the over-estimations.

Variance estimate for Ca

Now that we have a σ2 ADU λ we can work with-that is, an estimate from the data alone-, we want to get σ2 r (Eq. 2) and σ2

Ca

. We can either use the propagation of uncertainty (also referred to as error propagation, compounding of errors or delta method) [START_REF] Rice | Mathematical Statistics and Data Analysis[END_REF][START_REF] Wilson | An Introduction to Scientific Research[END_REF] together with Eq. 2 and 3, or a 'quick' Monte Carlo approach. We drop any explicit time index in the sequel in order to keep the equations more readable, but it should be clear that such variance estimates have to be obtained for each sampled point.

Propagation of uncertainty

This method requires, in the general case, an assumption of 'small enough' standard error since it is based on a first order Taylor expansion (see Sec. A for details). It leads first to the following expression for the variance, σ2 f λ , of f λ in Eq. 1:

σ2 f λ ≈ 1 T 2 λ σ2 ADU λ P 2 + σ2 ADU λ B P 2 B . (11)
The variance σ2 r of r in Eq. 2 is then:

σ2 r ≈ 1 f 2 380 (σ 2 f 340 + r 2 σ2 f 380 ) (12)
and the variance σ2

Ca

of Ca in Eq. 3 is:

σ2 Ca ≈ K ef f R max -r 2 (1 + Ca) 2 σ2 r . (13)
A remark on σ2

Ca behavior

The last three equations 11, 12 and 13 can be used together with Eq. 8 and 9 to understand why σ2

Ca will increase with the calcium concentration and therefore why a weighted nonlinear least-square procedure is required [START_REF] Bates | Nonlinear Regression Analysis and Its Applications[END_REF][START_REF] Seber | Nonlinear Regression[END_REF][START_REF] Nielsen | Introduction to Optimization and Data Fitting[END_REF] in order to get proper confidence intervals on calcium dynamics model parameters. Eq. 9 tells us that the variance of the raw signals is an increasing linear function of their means. When the calcium concentration increases, the recorded signal at 340 nm increases while the one at 380 nm decreases (Fig. 1). So according to Eq. 11, σ2

f 340 increases while σ2 f 380 decreases in proportion to [Ca 2+ ]. From Eq. 12 we see that σ2 r also increases since σ2 f 340 does increase and r 2 σ2 f 380 is roughly proportional to f 2 340 /f 380 and increases. Then from Eq. 13 we see that r is getting closer to R max , therefore the denominator is decreasing, while we just argued that σ2 r increases.

The two together imply that σ2

Ca is an increasing function of [Ca 2+ ]. This can be seen on the bottom panel of Fig. 2 where the error bars on the left side (corresponding to larger [Ca 2+ ]) are about twice as large as the ones on the right side (corresponding to smaller [Ca 2+ ]).

Monte-Carlo method

Here we draw, k quadruple of vectors

adu [j] 340 , adu [j] 340B , adu [j] 380 , adu [j]
380B , j = 1, . . . , k , from four independent Gaussian distributions of the general form:

adu [j] λ = adu λ + z [j] λ σADU λ , (14)
where adu λ is the observed value and z

[j]
λ is drawn from a standard normal distribution. We then plug-in these quadruples into Eq. 1 leading to k couples:

f [j] 340 = 1 T 340 adu [j] 340 P - adu [j] 340B P B , f [j] 380 = 1 T 380 adu [j] 380 P - adu [j] 380B P B , j = 1, . . . , k .
These k couples are 'plugged-in Eq. 2' leading to k r [j] :

r [j] = f [j] 340 f [j] 380 j = 1, . . . , k ,
before plugging in the latter into Eq. 3 to get k Ca

[j]

:

Ca [j] = K ef f r [j] -R min R max -r [j] j = 1, . . . , k .
The empirical variance of these simulated observations will be our σ2

Ca : σ2 Ca = 1 k -1 k j=1 ( Ca [j] -Ca • ) 2 ,
where

Ca • = 1 k k j=1 Ca [j] . ( 15
)
Since the Monte-Carlo method requires milder assumptions (the variances do not have to be small) and is easy to adapt, we tend to favor it; to be on the safe side, users can use both methods and, if they disagree, plot a histogram of the Ca to make sure that the discrepancy source is the non-normality of the latter.

Comment

The present approach based on a σ2 Ca estimation is slightly less rigorous than the 'direct approach' of [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] but it is far more flexible since it does not require an independent estimation / measurement of [F ur a] total . In line with the discussion following Eq. 3, in the companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF] we chose to consider the calibrated parameters K ef f , R min and R max as fixed.

Empirical validation 4.1. Rational

Equations 4, 5, 6, 7, together with Eq. 8 and 9 can be viewed as a data generation model. This means that if we choose model parameters values as well as an arbitrary [Ca 2+ ] time course, we can simulate measurements (adu) at both wavelengths in the ROI as well as in the BMR. We can then use these simulated adu exactly as we used the actual data, namely get r (t i ) (Eq. 2) and Ca(t i ) (Eq. 3) as well as the (squared) standard errors σ2 Ca (t i ) (Sec.

3.4). Now if the σ2

Ca (t i ) are good approximations for the actual but unknown σ 2 Ca (t i ), the distribution of the normalized residuals:

Ca(t i ) -Ca(t i ) σ Ca (t i ) ,
should be very close to a standard normal distribution. This is precisely what we are going to check.

Simulated data

We are going to use the first transient of dataset DA 121219 E1 of the companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF]. The 'static' parameters-that is the parameters not link to the calcium dynamics-used for the simulation are the actual experimental parameters rounded to the third decimal (Table 1 The simulated calcium dynamics is a monoexponential decay mimicking the tail of the transient:

Ca(t) = Ca 0 + 0 if t < t 0 δ exp(-(t -t 0 )/τ ) if t ≥ t 0
and the parameter values (Table 2) are just a rounded version of the fitting procedure output (see companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF]). The simulated data obtained in that way are shown on Fig. 1 (blue traces) together with the actual data (red curves) they are supposed to mimic. At a qualitative level at least, our data generation model is able to produce realistic looking simulations.

Parameter

Software and simulation details

The methodological details of the measurements to which the analysis presented in the present manuscript was applied are described in the companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF].

The simulations, computations and figures of the present manuscript were done with Python 3 (https://www.python.org/), numpy (https://numpy.org/), scipy and matplotlib (https://matplotlib.org/). The Python codes and the data required to reproduce the simulations and figures presented in this manuscript can be downloaded from GitLab (https://gitlab.com/c˙pouzat/getting-se-on-ratiometric-ca-estimator).

The use of scipy was kept to a bar minimum to maximize code lifeduration (scipy tends to evolve too fast with minimal concern for backward compatibility). The random number generators used were therefore the ones of numpy: the uniform random number generator derives from the Permuted Congruential Generator (64-bit, PCG64) (https: //www.pcg-random.org/) [START_REF] O'neill | PCG: A Family of Simple Fast Space-Efficient Statistically Good Algorithms for Random Number Generation[END_REF] while the normal random number generator is an adaptation of the Ziggurat method [START_REF] Marsaglia | The Ziggurat Method for Generating Random Variables[END_REF] of Julia (https://docs.julialang.org/en/v1/); unfortunately one has to check the source code of both numpy and Julia to find that out.

Are the standard errors of ratiometric estimator accurate?

Since the two σ2 Ca estimation methods, propagation of uncertainty and Monte-Carlo, agree at each time point within 2%, we illustrate in this section the results obtained with the Monte-Carlo method.

We take next the simulated data (blue curves on Fig. 1) together with the simulated background signals (not shown) as if they were actual data and we compute the ratiometric estimator and its standard error as described in Sec. 3.4, using k = 10 4 replicates. Figure 2 shows the standardized residuals as well as the simulated data together with the true [Ca 2+ ], we know it since we used it to simulate the data!

The upper part of Fig. 2 is only a qualitative way of checking that the normalized residuals follow a standard normal distribution. A quantitative assessment is provided by the Shapiro-Wilk W statistic, that is here: 0.987; giving a p-value of 0.128. There is therefore no ground for rejecting the null hypothesis that the normalized residuals are IID draws from a standard normal distribution.

As an additional, visual but less powerful test, we plot the empirical cumulative distribution function (ECDF) of the normalized residuals together with the theoretical (normal) one and with Kolmogorov's confidence bands (Fig. 3). If the empirical ECDF arises from a normally distributed sample with mean 0 and SD 1, it should be completely contained in the 95% confidence band 95% of the time and in the 99% band, 99% of the time (these are confidence bands not collections of pointwise confidence intervals).

We conclude from these visual representations and formal tests that our normalized residuals follow the expected standard normal distribution, implying that our proposed method for getting the standard errors of the ratiometric estimator is fundamentally correct.

Discussion

We have presented a new and simple method for getting standard errors on calcium concentration estimates from ratiometric measurements. This method does not require any more data than what experimentalists using ratiometric dyes like Fura-2 are usually collecting: measurements at 340 and 380 nm both within a region of interest and within a background measurement region. Once the errors bars have been obtained, arbitrary models can be fitted to the calcium transients-by weighted nonlinear least-squares [START_REF] Nielsen | Introduction to Optimization and Data Fitting[END_REF]-and meaningful confidence intervals for the parameters of these models will follow as illustrated in the companion paper [START_REF] Hess | Analysis of neuronal Ca2+ handling properties by combining perforated patch clamp recordings and the added buffer approach[END_REF]. The present contribution is therefore best viewed a major simplification of the 'direct approach' of [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF]. In contrast to the latter, the new method does not require a '3 wavelengths protocol', it does not require either a precise fit of the autofluorescence dynamics at the three wavelengths and is therefore much easier to implement. We provide moreover two independent implementations, one in C and one in Python, they are open source and freely available. The rather verbose Python implementation of the heart of the method (Sec. 3.4) requires 25 lines of code and nothing beyond basic numpy functions. We are therefore confident that this method could help experimental physiologists getting much more quantitative results at a very modest extra cost.

A. Propagation of uncertainty

We outline in this section how to reach Eq. 11, 12 and 13 of Sec. 3.4.1. We first need to remember that is X and Y are two independent random variables with mean EX = µ X and EY = µ Y and variance VX = σ 2 X and VY = σ 2 Y , then if Z = a + bX + cY (a, b, c ∈ R) we have:

EZ = a + b µ X + c µ Y , VZ = b 2 σ 2 X + c 2 σ 2 Y .
Eq. 11 is a direct consequence of the last equality. If X is (approximately) normally distributed with (X ∼ N (µ X , σ 2 X )) as well as Y , we can write:

X ≈ µ X + Z 1 σ X and Y ≈ µ Y +Z 2 σ Y ,
where Z 1 and Z 2 are independent and follow a standard normal distribution, N (0, 1). If now Z = f (X, Y ) and the partial derivatives of f at (µ X , µ Y ) exist then:

Z = f (µ X + Z 1 σ X , µ Y + Z 2 σ Y ) ≈ f (µ X , µ Y ) + Z 1 σ X ∂f (µ X , µ Y ) ∂X + Z 2 σ Y ∂f (µ X , µ Y ) ∂Y .
This is just a first order Taylor expansion and that is where the 'small enough standard error' assumption is necessary. Z is then (approximately) a linear combination of two independent standard normal random variables and we immediately get:

EZ = f (µ X , µ Y ) , VZ = ∂f (µ X , µ Y ) ∂X 2 σ 2 X + ∂f (µ X , µ Y ) ∂Y 2 σ 2 Y .
Eq. 12 follows directly by computing the necessary partial derivatives, while Eq. 13 requires the computation of a single derivative.

B. Auto-fluorescence dynamics B.1. General features

The evolution of the aduλB is shown on Fig. 4. We see that the autofluorescence runs down when high frequency flashes are applied during the 3 transients, with a partial recovery between transients.

B.2. Within transient dynamics

The 'direct method' of [START_REF] Joucla | Quantitative Estimation of Calcium Dynamics From Ratiometric Measurements: A Direct, Nonratioing Method[END_REF] requires the knowledge of the autofluorescence value at each time point during a transient at both 340, 360 and 380 nm, since Eq. 4 and 6 are fitted directly to the recorded adu 340 and adu 380 and they depend on the total Fura concentration at transient time that is estimated from the difference of the 360 nm measurements in the ROI and the BMR. We therefore take a closer look a the autfluorescence dyanmics during the first transient (Fig. 5).

At that stage we can fit a straight line plus a cosine function whose period is the duration of a transient. That's a good way to capture the main structure in the transient, but is still does not account for the full signal variability (Fig. 6). As can be seen from the normalized residuals-the residuals divided by the standard deviation-that should be very nearly independent random draws from a standard normal distribution if the model is correct, there are finer structures left (like the double valley on the 380 nm residuals) meaning that those fits won't pass formal goodness of fit tests. Indeed if we apply Pearson's χ 2 tests to these stabilized residuals we get:

• at 340 nm a residual sum of squares (RSS) 326, leading to a P(χ 2 197 > 326) = 0.0,

• at 360 nm a RSS of 288, leading to a P(χ 2 197 > 288) = 2.6e-05,

• at 380 nm a RSS of 275, leading to a P(χ 2 197 > 275) = 0.000203.

We are then left with three possibilities:

1. try to refine the 'straight line plus cosine function' empirical model in order to get acceptable fits, 2. try to get a better understanding of the autofluorescence dynamics, 3. find another way to get error bars on our estimates.

Since we wanted to propose an 'as general and easy as possible' method we chose the third approach in the present manuscript.

  Figures

Figure 1

 1 Figure 1

Figure 1 Observed

 1 Figure 1

Figure 2

 2 Figure 2

Figure 2 Top:

 2 Figure 2 Top: Simulated ratiometric estimator -'actual' [Ca 2+ ] divided by ratiometric estimator standard error (if everything goes well we should see draws from a standard normal distribution); bottom: Simulated ratiometric estimator (with error bars given by the standard error) in black and 'actual' [Ca 2+ ] in red.

Figure 3

 3 Figure 3

Figure 3 Empirical

 3 Figure 3

Figure 4 Figure 4

 44 Figure 4 (Appendix)

Figure 5 (

 5 Figure 5 (Appendix)

Figure 5 Normalized

 5 Figure 5

Figure 6

 6 Figure 6 (Appendix)

Table 1 :

 1 ). 'Static' parameters used for the simulation.

	Parameter	Value
	R min	0.147
	R max	1.599
	K ef f	1.093 (µM)
	K F ur a	0.225 (µM)
	[F ur a] total φ 1.89e+05 (s -1 )
	T 340	0.01 (s)
	T 380	0.003 (s)
	P	3
	P B	448
	G	0.146
	σ 2 r ead-out F 340B	268.96 189512 (s -1 )
	F 380B	711589 (s -1 )

Table 2 :

 2 Calcium dynamics parameters used for the simulation. Time 0 is when seal is obtained.

		Value
	t 0	2283.415 (s)
	Ca 0	0.059 (µM)
	δ	0.114 (µM)
	τ	2.339 (s)


Guénaëlle Diler

; Cécile

Rannou Ab

Claire Guyon Ab

Carole Prost Ab

Alain Le-Bail
email: alain.lebail@oniris-nantes.fr

Use of amyloglucosidase in a soft wheat dough: impact of process and formulation on glucose production

Keywords: enzyme, model dough, baking, temperature, dough hydration, sweetness

The use of amyloglucosidase (AMG) in baked goods was studied to develop no added sugar products, by investigating the impact of baking process stages and ingredients on AMG glucose production. Their application in a simplified model dough was studied first. Glucose content was monitored during the various stages of the process (mixing, resting at 30, 50 and 60°C, and baking), considering a dough containing different amounts of water (35%, 50% flour basis) and oil (0%, 5.2%, 10.4%, 33.4% flour basis). The results indicated that AMG can produce significant amounts of glucose under the conditions encountered during the main stages of a standard baking product making process. Glucose production by AMG was favoured by higher dough hydration, low oil content and the integration of a high-temperature rest phase in the process.

An application in a sandwich bread with no added sugar was tested. AMG allowed the production of glucose under real baking conditions. The glucose content of 5.1% obtained with the use of 1100 AGU/kg flour was close to the average sugar content of commercial sandwich breads on the French market. The addition of AMG in no added sugar breads enhanced their overall quality compared to no added sugar bread. AMG favours sweetness perception but also softness and flavour improving the quality of breads to reach consumers' expectations.

INTRODUCTION

The overconsumption of sugars has evident health implications, by increasing caloric intake and largely contributes to the epidemic of obesity, considered by the World Health Organization as the epidemic of the 21 st century [START_REF] Merino | Intestinal Fructose and Glucose Metabolism in Health and Disease[END_REF][START_REF] Ter Horst | The FGF21 response to fructose predicts metabolic health and persists after bariatric surgery in obese humans[END_REF]. But not all sugars are equal. In the past, fructose was largely used as it is sweeter, more soluble, and less gluconeogenic than glucose and sucrose. Today, Humans commonly consume fructose together with glucose in the form of sucrose or high fructose corn syrup. However, concerns about the consumption of fructose have been raised and fructose is now considered to be a particularly harmful sugar [START_REF] Ter Horst | The FGF21 response to fructose predicts metabolic health and persists after bariatric surgery in obese humans[END_REF] Indeed, although fructose and glucose share the same molecular formula (C6H12O6) and caloric value (4 kcal/g), their intestinal absorption and metabolism are quite different [START_REF] Merino | Intestinal Fructose and Glucose Metabolism in Health and Disease[END_REF]. Excessive fructose consumption (and thus sucrose) promotes visceral adiposity, de novo lipogenesis, dyslipidemia, and hepatic insulin resistance, and more so than glucose [START_REF] Coello | Effectiveness of a low-fructose and/or low-sucrose diet in decreasing insulin resistance (DISFRUTE study): study protocol for a randomized controlled trial[END_REF][START_REF] Sahin | Sugar reduction in bakery products: Current strategies and sourdough technology as a potential novel approach[END_REF][START_REF] Ter Horst | The FGF21 response to fructose predicts metabolic health and persists after bariatric surgery in obese humans[END_REF]. In addition, fructose is less satiating than glucose, increasing food intake [START_REF] Coello | Effectiveness of a low-fructose and/or low-sucrose diet in decreasing insulin resistance (DISFRUTE study): study protocol for a randomized controlled trial[END_REF][START_REF] Merino | Intestinal Fructose and Glucose Metabolism in Health and Disease[END_REF]. A reduction in the consumption of fructose is therefore to be encouraged..

To tackle this epidemic, many products bearing the claim "with no added sugars" are being developed (European Parliament and Council of the European Union, 2006). However, cereal products are generally rich in added sugars (cake, brioche, biscuit, etc.), and the removal of added sugar (often sucrose) leads to a loss in sweetness that can compromise the consumer acceptance. The high carbohydrate content of these nosugar-added products is related to their high starch content, a glucose polymer that does not impart sweetness to the product.

A bulking agent is often used as a technical substitute for sugar to compensate the loss in texture, volume and sometimes sweetness. Polyols are often used as bulking agents since they can impart a sweet flavour with a sweetness between 30% and 90% relative to sucrose [START_REF] Sahin | Sugar reduction in bakery products: Current strategies and sourdough technology as a potential novel approach[END_REF]. However, they are sometimes associated with aftertaste and have laxative effects [START_REF] Ghosh | A review on polyols: New frontiers for health-based bakery products[END_REF][START_REF] Sahin | Sugar reduction in bakery products: Current strategies and sourdough technology as a potential novel approach[END_REF]. Indeed, foods containing more than 10% added polyols must be labelled with the mention 'excessive consumption may produce laxative effects'. Another drawback of polyols is their inability to participate in Maillard reactions [START_REF] Ronda | Effects of polyols and nondigestible oligosaccharides on the quality of sugar-free sponge cakes[END_REF] which prevents them from participating in the expected flavour and color. Long chain polysaccharides, such as polydextrose, maltodextrin and hydrocolloids, as well as dietary fibres, are also considered as bulking agents. However, they do not contribute to sweetness and often have a negative effect on sensorial characteristics [START_REF] Sahin | Sugar reduction in bakery products: Current strategies and sourdough technology as a potential novel approach[END_REF]. These bulking agents are often combined with high-intensity sweeteners (HIS) in order to increase sweetness and improve consumer acceptance of sugar reduced products [START_REF] Martínez-Cervera | Rheological, textural and sensorial properties of low-sucrose muffins reformulated with sucralose/polydextrose[END_REF][START_REF] Psimouli | The effect of alternative sweeteners on batter rheology and cake properties[END_REF][START_REF] Struck | Fiber enriched reduced sugar muffins made from isoviscous batters[END_REF][START_REF] Zoulias | Effect of sugar replacement by polyols and acesulfame-K on properties of low-fat cookies[END_REF]. However, the use of HIS in fine bakery products has been banned in Europe since 2018 (European Commission, 2018). Amyloglucosidase (AMG), also named glucoamylase, is an enzyme involved in the breakdown of starch chains. It belongs to the class of exo-acting β-amylases that act on the non-reducing ends of starch polymer chains and produce β-D-glucose. In contrast to other amylases, AMG can catalyse the hydrolysis of both (1→4) and (1→6) α-d-glucosidic bonds in starch, although the α-(1→4) linkage is hydrolysed approximately 600 times faster than the α-(1→6) linkage [START_REF] Robyt | Enzymes and Their Action on Starch[END_REF]. Thus, AMG can completely convert starch to Dglucose. Its main application is the industrial production of glucose syrups (from starch) since glucose is the principal carbon source in many biotechnological processes and of great importance for fermentative chemicals and fuel production such as succinic acid, bio-plastic and ethanol [START_REF] Han | Fermentative hydrogen production using wheat flour hydrolysate by mixed culture[END_REF][START_REF] Lam | Production of Fungal Glucoamylase for Glucose Production from Food Waste[END_REF][START_REF] Melikoglu | Stepwise optimisation of enzyme production in solid state fermentation of waste bread pieces[END_REF].

Endogenous AMG is rare in native flour [START_REF] Struyf | The effect of amylolytic activity and substrate availability on sugar release in non-yeasted dough[END_REF]; however, commercial AMG addition to soft wheat dough products is well documented for several reasons. In bread and Danish pastries, AMG is used to produce glucose with the objective of enhancing CO2 production from yeast, and consequently improve the volume of products [START_REF] Pomeranz | Use of Amyloglucosidase in breadmaking[END_REF][START_REF] Struyf | Investigating the impact of a-amylase, a-glucosidase and glucoamylase action on yeast-mediated bread dough fermentation and bread sugar levels[END_REF][START_REF] Valjakka | Studies on a raw-starch digesting enzyme. II. Replacement of sucrose in white pan bread[END_REF]. According to [START_REF] Hebeda | Baked Goods Freshness: Technology, Evaluation and Inhibition of Staling[END_REF], it is possible to use AMG in baked products to reduce the amount of added sugar and still have the desired colour and sweetness. In combination with other enzymes such as amylases, AMG can reduce the staling of breads [START_REF] Else | Use of an anti-staling enzyme mixture in the preparation of baked bread[END_REF] and improve the preservation and colouring of frozen Danish pastries after baking [START_REF] Larsen | Method of producing a frozen yeast dough product. Patent US005589207A[END_REF].

The activity of AMG has mainly been studied for industrial applications of glucose production in optimal conditions such as an acidic pH between 4.5 and 5, a temperature range of 50 -60°C [START_REF] Gonzalez Tello | Kinetics of the hydrolysis of soluble starch by glucoamylase[END_REF][START_REF] Kusunoki | A kinetic expression for hydrolysis of soluble starch by glucoamylase[END_REF][START_REF] Norouzian | Fungal glucoamylases[END_REF][START_REF] Oliveira | The intensification of amyloglucosidase-based saccharification by ultrasound[END_REF], and a pure substrate such as soluble starch (Pervez et al, 2019a) or maltodextrin solutions [START_REF] Aslan | Covalent immobilization of Aspergillus niger amyloglucosidase (ANAG) with ethylenediamine-functionalized and glutaraldehyde-activated active carbon (EFGAAC) obtained from sesame seed shell[END_REF].

Few studies have been conducted on this enzymatic action in baking applications, particularly when a more complex substrate such as a real dough matrix composed of several ingredients is used [START_REF] Eynard | Modifications of starch during baking: studied through reactivity with amyloglucosidase[END_REF][START_REF] Struyf | Investigating the impact of a-amylase, a-glucosidase and glucoamylase action on yeast-mediated bread dough fermentation and bread sugar levels[END_REF][START_REF] Diler | Assessment of amyloglucosidase activity during production and storage of laminated pie dough. Impact on raw dough properties and sweetness after baking[END_REF]. As AMG use appears to be a promising strategy for the formulation of non-added sugar bakery products, without a laxative effect or the generation of offflavours, it could be interesting to study its addition to better understand its activity. This work explores the AMG's glucose production in real breadmaking conditions, in order to produce no added sugar foodstuffs with satisfying sweetness. First, the activity of AMG will be studied in simplified model dough, via the amount of glucose produced. The effect of the dough water content (2 levels), the fat content (4 levels), and the dough resting temperature (3 levels) will be characterized. In a second step, the addition of AMG will be studied in a no added sugar sandwich bread whose organoleptic properties will be studied through sensory tests.

MATERIALS AND METHODS

Raw material

The flour used was an organic wheat flour T65 (Moulins Bourgeois, Verdelot, France) which contained 66.5% starch (enzymatic method, NF EN ISO 15914), 14.8% water (ISO 712:2009), and 8% protein (Dumas, NF EN ISO 16634-1). Water absorption was 56.8% (Farinograph, Brabender, Germany, NF V 03 717-1), damaged starch content 22.4 UCD (SDmatic, Chopin Technologie, France, ISO 17715) and falling number 352 s. The fat used was rapeseed oil, purchased at the local market

The amyloglucosidase preparation used (AMG 1100BG, Novozymes A/S, Bagsvaerd, Denmark) presented an activity of 1100 amyloglucosidase units (AGU).g -1 . One unit is defined as the amount of enzyme required to release 1μmol of glucose per minute under the conditions of the assay (temperature 25.0°C; pH 4.3). The optimum activity for this enzyme was obtained at pH=4 and 70°C (according to supplier information).

Model dough preparation and sampling

Process study

For this study, the simplified model doughs contained only flour, 50% water on flour basis (fb) and 0.15% fb of AMG (1650 AGU/kg flour). Flour and AMG were mixed together for 30 seconds at 6 rpm (Kitchen Aid, Puteaux, France). Water was gradually added and mixed for an additional 1 minute 30 seconds. A stopwatch was started when the first drop of water was added (t=0min) and the time of AMG action recorded for each sample. At the end of mixing, the dough was divided into 20 g dough pieces distributed in small silicon pans and placed in a heat chamber at 30°C for 2 hours. Rested dough was then baked in a deck oven (Condo, MIWE, Sarreguemines, France) for 30 min at 180°C. To better qualify the baking, part of the dough was baked in the same conditions without a resting period. To better qualify resting, two additional temperatures were studied: 50°C and 60°C. The different steps of preparation and sampling of the model dough are shown in Figure 1.

Samples (of 20 g) were taken at different intervals of the process to monitor the glucose content: at the end of mixing, after dividing the dough, after 10, 20, 30, 60, 90 and 120 minutes of resting, after baking the unrested dough and the rested dough. Each sample was immediately frozen at -18°C to stop AMG activity.

The tests were conducted in triplicate. A control dough that did not contain AMG was tested in each condition to confirm the origin of the glucose produced for each recipe.

Dough formulation study

For this part, the water and oil contents of the model dough were adjusted at different levels to represent the formulations of a wide range of cereal products (from biscuits to soft bread). Two levels of added water were tested: 35% and 50% fb, corresponding to the quantity of water necessary to obtain a free water content comparable to that contained in biscuit dough, and bread dough (data not shown), respectively. Four oil contents were tested in dough containing 50%fb water: 0%, 5.2%, 10.4% and 33.4% fb. 1650 AGU/kg flour (0.15% fb of AMG) was added to each recipe. Doughs were prepared in the same conditions as those described in paragraph 2.2.1. Oil was added in one go, just before the gradual addition of water. Glucose content was monitored by sampling after mixing, after dividing the dough, and after 10, 20, 30, 60, 90 and 120 minutes of resting at 30, 50 and 60°C. Each sample was immediately frozen at -18°C to stop AMG activity.

The model doughs were coded Wxx/Fxx-xx°C to specify their water content (W), fat (oil) content (F), and resting temperature.

Sandwich bread preparation and sampling

Six sandwich bread recipes were used: a standard bread (reference) with sucrose, a reduced sugar bread with 30% sucrose reduction, a "no sugar added" bread without sucrose, and 3 breads with different amounts of AMG added to the "no sugar added" recipe (details of composition shown in Table 1). For all the recipes, ingredients were added in this order: water, then flours (wheat flour, soybean flour and fibres), AMG, sourdough (Böcker France), oil, sucrose and yeast (L'hirondelle fresh yeast, Lesaffre, France). Mixing was done in a spiral mixer SPI11 (VMI, Montaigu, France) for 3 min at 50rpm, 12 min at 100 rpm, and then 4 min at 200 rpm. The salt was added 3 min after mixing began. The first sample of dough of 30g was taken at the end of mixing and immediately frozen at -18°C. The rest of the dough was left to rest for 30 minutes.

Then the dough was divided into pieces of 285 g ± 1g, shaped and placed in a mould, in a fermentation chamber at 28°C and 85% relative humidity. The time between the end of mixing and the start of fermentation was set at 45 ± 2 min. Fermentation time was adjusted (between 90 and 120 min) to increase the volume of the dough by a factor of 2.5. (90 min for AMG breads, 105min for no added sugar bread, 115 min for sugar reduced bread and 120 min for reference bread). This allowed obtaining the same dough porosity before baking. A second dough sample of 30g was taken at the end of fermentation and frozen at -18°C. The breads were baked in a deck oven for 20 minutes at 210°C, then left to chill for 2 hours. The last sample was taken from a baked bread and frozen at -18°C.

Glucose quantification

About 2.5 g of frozen sample (weighed with a precision of 0.0001 g) were blended for 1 min at 11000 rpm with an Ultra Turrax (Ultra Turrax T25 basic, IKA Labortechnik, Imlab, Lille, France) in 45 mL of water.

Then, 2.5 mL of Carrez I reagent (solution of Potassium Hexacyanoferrate(II) Trihydrate in deionized water (150 g/l)) and 2.5 mL of Carrez II reagent (Solution of Zinc Sulfate Heptahydrate in deionized water (300 g/l) were added to the mixed sample solution. This solution was homogenized and filtered through a folded filter (Whatman no. 41 ashless) and then through a 0.45 µm pore size cellulose acetate membrane filter, before injection.

The HPLC system (1260 Agilent Technologies, Santa Clara, United-states) was equipped with an amino HPLC column Supelcosil LC-NH2-NP (Supelco-USA) maintained at 35°C in the oven. A refractive index detector was used for sugar detection. The sugar separation was conducted in isocratic mode, with a mobile phase composed of 80% acetonitrile and 20% ultrapure water, at a flow rate of 1.5 mL/min. The injected volume of extracted sample solution was 10 µl. The glucose was quantified by the injection of glucose solutions at different concentrations in the same conditions. The limit of quantification was 0.5% of glucose on flour basis (0.3% on sample basis). The results of the glucose produced were expressed in g for 100g of flour (noted % fb for "flour basis") and the results of the sample glucose content in g for 100g of sample.

The analyses of the detailed sugar composition of sandwich breads were subcontracted to Eurofins Analytics (Nantes, France).

Sensory analysis 2.5.1. Ethics

The sensory tests were conducted in accordance with the Declaration of Helsinki. All applicable institutional and governmental regulations concerning the ethical use of human volunteers were complied with during this research study. The assessors gave their written consent after reading detailed information about the study.

The sensory tests performed in this study were approved by the ethics evaluation committee of INSERM (IRB00003888, IORG0003254, FWA00005831).

Panel

Panellists were recruited among the staff and students of Oniris. The panel was composed of 78 persons aged between 23 and 60 years. The panellists were regular consumers of soft breads.

Sample preparation and evaluation conditions

The sensory analysis took place in the sensory analysis room of Oniris (ISO NF EN 8589). Five breads were presented to the panellists: a standard bread (reference) with sucrose, a sugar reduced bread, a "no sugar added" bread, and two "no sugar added" breads supplemented with AMG (825 AGU/kg flour and 1100 AGU/kg flour, respectively). The sensory analysis took place 24h after bread production. Bread loaves were sliced mechanically (18mm thickness). The 5 bread samples were coded with three-digit numbers. The presentation of the samples was randomized.

Hedonic test

A half slice of each sample was served to the panellists who had to rank the 5 bread samples according to their overall preference from the least liked (=1) to the most liked (=5).

Just About Right (JAR) scales

Another half slice of each sample was served to the panellists with different codes from those of the hedonic test and in a different randomized order. The panellists had to assess the sweetness intensity of all the samples on a 5-point scale (1=not sweet enough, 3=just-about-right sweetness, 5=too sweet). Other attributes were also evaluated such as colour, softness, pastiness, global aroma intensity, saltiness and sourness.

Statistical analysis

The effects of resting temperature, water content and oil content were analysed by ANOVA with Statgraphics Centurion, version XVII (Statistical Graphics Corp, Princeton, New Jersey, USA). The results of the hedonic ranking test were analysed with a Friedman test.

RESULTS AND DISCUSSION

AMG action in model dough

3.1.1. Glucose production from AMG during the main stages of a standard baking product making process

AMG action during mixing and dough shaping

During mixing, the ingredients were homogenized and the flour was hydrated to form the dough. As shown in Table 2, in the flour/water model dough W50/F0 without AMG, the glucose content after mixing was below the quantification limit. The glucose produced by the flour's endogenous enzymes during mixing was negligible, as shown by [START_REF] Struyf | The effect of amylolytic activity and substrate availability on sugar release in non-yeasted dough[END_REF]. This implies that the glucose production observed in the other samples was the result of AMG activity. The addition of 1650 AGU/kg flour to the W50/F0 model dough produced 1.45% ± 0.30% fb of glucose during the 2 min of dough mixing. After shaping/moulding (10 min later), the total glucose produced increased only slightly (1.53% ± 0.38% of glucose produced during mixing + shaping). Thus, during the dough preparation, most of the glucose was produced during the mixing process. This shows that at 20-21°C, during a short amount of time, AMG seemed able to produce a small amount of glucose (0.97% ± 0.16% dough basis). This rapid production of glucose corresponded to the rapid hydrolysis of directly available substrates such as damage starch or maltose. Indeed, in a non-yeasted dough (without AMG added) prepared with two different flours, [START_REF] Struyf | The effect of amylolytic activity and substrate availability on sugar release in non-yeasted dough[END_REF] showed a fast release of maltose during mixing, due to endogenous amylase activity, and a more gradual and rather small increase of glucose.

Conversely, the results of the W50/F0 model dough containing AMG showed a rapid increase of glucose and a decrease of maltose (data not shown) during mixing, suggesting a breakdown of maltose into glucose.

AMG action during resting

For many dough processes, a resting period at controlled temperature follows mixing and shaping stages to let the dough ferment. For this work, the standard resting period was fixed at 2 hours at 30°C. In these conditions, the flour's endogenous enzymes did not produce glucose in the dough without AMG. On the contrary, in the presence of AMG, 2.87 ± 0.11% fb of glucose were produced, leading to a glucose content of 2.87% ± 0.09% on dough basis at the end of resting (Table 2).

The evolution of the glucose content in the model doughs was continuously monitored during the resting period (Figure 2). A linear relationship (R²=0.99) was observed between the amount of glucose produced and the resting time at 30°C in W50/F0 containing AMG (Figure 2, W50/F0-30°C). [START_REF] Struyf | Investigating the impact of a-amylase, a-glucosidase and glucoamylase action on yeast-mediated bread dough fermentation and bread sugar levels[END_REF] obtained comparable curves and glucose levels (2.4 ± 0.2%) in similar wheat dough after 3 h of incubation at 30°C with 15U/g of AMG. Indeed, [START_REF] Warren | The interplay of α-amylase and amyloglucosidase activities on the digestion of starch in in vitro enzymic systems[END_REF] specified that enzymatic starch breakdown is a pseudofirst order kinetic process, producing a curve that was initially linear with a constant rate at early time points, as the substrate is not significantly depleted. In our results, no plateau appears; the substrate was not limiting.

In its native state, starch is not readily available for AMG [START_REF] Kimura | Reaction of enzymes with starch granules: kinetics and products of the reaction with glucoamylase[END_REF], 1996); only damaged starch is available for AMG [START_REF] Eynard | Modifications of starch during baking: studied through reactivity with amyloglucosidase[END_REF] but its amount is limited in the flour (between 14 and 25 UCD in baking applications [START_REF] Dubat | Importance de l'endommagement de l'amidon et évolution des méthodes de mesure[END_REF], 22.4 UCD in our case). However, starch is not the only substrate for AMG. [START_REF] Struyf | The effect of amylolytic activity and substrate availability on sugar release in non-yeasted dough[END_REF] showed that in the presence of AMG, the amount of maltose decreased in a resting dough while it increased without AMG because of the endogenous amylase activity of the flour. [START_REF] Warren | The interplay of α-amylase and amyloglucosidase activities on the digestion of starch in in vitro enzymic systems[END_REF] even declared that AMG action is much faster on the products of α-amylase than when it acts directly on starch. The linear and unrestricted activity of AMG during the process was probably due to synergy between the AMG (added) and amylase (endogenous from the flour) activities in the dough [START_REF] Else | Use of an anti-staling enzyme mixture in the preparation of baked bread[END_REF][START_REF] Labout | Conversion of Liquified Starch into Glucose Using a Novel Glucoamylase System[END_REF][START_REF] Larsen | Method of producing a frozen yeast dough product. Patent US005589207A[END_REF][START_REF] Valjakka | Studies on a raw-starch digesting enzyme. II. Replacement of sucrose in white pan bread[END_REF][START_REF] Warren | The interplay of α-amylase and amyloglucosidase activities on the digestion of starch in in vitro enzymic systems[END_REF].

The slope of the plot showing glucose production as a function of resting time (Figure 2, W50/F0 -30°C) indicates a production rate of around 25 mg of glucose per minute (0.15 mg/min/AGU).

The increase of resting temperature from 30°C to 60°C increased AMG action (P-value= 0.0000). The glucose productions were respectively 2.87%, 4.75% and 7.09% fb and the production rates 25 mg/min, 44 mg/min and 62 mg/min for W50/F0-30°C, W50/F0-50°C and W50/F0-60°C (Table 2 andFigure 2). Glucose production was still linear during resting at higher temperatures. Many glucoamylases function at thermophilic temperatures, usually 50 to 60°C [START_REF] Norouzian | Fungal glucoamylases[END_REF]. [START_REF] Aslan | Covalent immobilization of Aspergillus niger amyloglucosidase (ANAG) with ethylenediamine-functionalized and glutaraldehyde-activated active carbon (EFGAAC) obtained from sesame seed shell[END_REF] and Pervez et al. (2019b) described an increase in the relative activity of AMG from around 45% at 30°C, to 70-80% at 50°C and a maximum obtained between 55 and 65°C on maltodextrin and starch solutions. Indeed, the number of collisions between molecules increases when the temperature increases, leading to an increase of the reaction rate [START_REF] Palmer | Enzymes: Biochemistry, Biotechnology, Clinical Chemistry[END_REF][START_REF] Punekar | Enzymes: Catalysis, Kinetics and Mechanisms[END_REF]. This increase stops when the enzyme is denatured and catalytic activity is lost. Temperatures above 60°C were not tested because they caused a drop in AMG activity, and they would have caused the dough to start baking [START_REF] Rouillé | Dough/crumb transition during French bread baking[END_REF][START_REF] Vanin | Biaxial extensional viscosity in wheat flour dough during baking[END_REF].

The optimization of resting time and temperature is thus essential to favour glucose production within the limits of the available substrate. In the case of fermented wheat dough, resting temperatures above 50°C cannot be used since the yeasts are inactivated above this temperature [START_REF] Roussel | Les pains français : e volution, qualite , production[END_REF]. For these doughs, lower temperatures (ideally 25-30°C) should be applied but over longer periods. Glucose consumption by yeast during fermentation could slightly accelerate the rate of the reaction by limiting the substrate inhibition effect, as observed by [START_REF] Humbert | Co-encapsulation of amyloglucosidase with starch and Saccharomyces cerevisiae as basis for a long-lasting CO2 release[END_REF] on the action of amyloglucosidase coencapsulated with starch and Saccharomyces cerevisiae. For products containing chemical leavening agents, the temperature chosen must be lower than the reaction temperature.

AMG action during baking

For the baking stage, two cases were considered: a direct baking process (without preliminary resting) and post resting baking. The results obtained with the W50/F0 model dough (Table 2) indicate a low increase in glucose content during baking. In the case of direct baking, there was only 0.86% ± 0.31% (fb) more glucose at the end of baking than at the beginning. Concerning the baking of the rested dough, there was even less glucose (-0.15 ± 0.18% fb) after baking than before. This is explained by the fact that during baking two phenomena occur: glucose production by the enzyme on the one hand and glucose consumption through browning reactions on the other hand. Indeed, at around 60°C, starch gelatinization starts and therefore becomes more accessible to enzyme action [START_REF] Eliasson | Effect of Water Content on the Gelatinization of Wheat Starch[END_REF][START_REF] Gough | Effect on the Gelatinization Temperature of Wheat Starch Granules of Prolonged Treatment with Water at 50°C[END_REF][START_REF] Robyt | Enzymes and Their Action on Starch[END_REF]. Robyt (2009) indicated differences in the hydrolysis of gelatinized starch and starch granules by amylases of the order of 100-to 1000-times greater. Moreover, the optimum condition for AMG activity is around 60-65°C, consequently the action of AMG is boosted at the beginning of baking, leading to an increase in glucose content. Above 70°C, AMG is probably inactivated and glucose production decreases gradually until the enzyme is denatured and catalytic activity is lost [START_REF] Palmer | Enzymes: Biochemistry, Biotechnology, Clinical Chemistry[END_REF][START_REF] Punekar | Enzymes: Catalysis, Kinetics and Mechanisms[END_REF]. When the temperature exceeds 105-120°C (in the outer region), the Maillard reaction and caramelisation take place simultaneously and consume reducing sugars, such as glucose [START_REF] Purlis | Browning development in bakery products -A review[END_REF]. In our case, it seemed that the amounts of glucose produced and consumed during baking were close, and the resulting glucose produced was close to 0. In the case of fermented dough, a third phenomenon occurs: the consumption of glucose by yeast. [START_REF] Hebeda | Baked Goods Freshness: Technology, Evaluation and Inhibition of Staling[END_REF] specified that AMG will remain active after the yeasts have been killed (at 50°C) because of the relatively high thermostability of glucoamylase [START_REF] Lam | Production of Fungal Glucoamylase for Glucose Production from Food Waste[END_REF] found a half-life (t1/2) of AMG extract of 315 min at 55 °C and 32.5 and 31.9 min at 60°C and 65 °C respectively).

The resulting glucose content of products after baking was logically higher for the rested samples. Samples with and without resting contained respectively 3.55% and 1.85% of glucose (sample basis).

It can also be mentioned that reproducibility was better when AMG had more time to work. The variation coefficient of the final glucose content evolved from around 13% without a resting period, to 3% with 2h resting at 30°C. A process with resting time, if the resting time is properly controlled, will give more repeatable glucose contents in the final products.

Impact of dough formulation on AMG glucose production

Impact of water content

Two water contents were tested: 50% and 35% fb. No production of glucose (or little, below the quantification limit) by the enzymes initially present into the flour occurred with these two water contents.

Considering the whole process, glucose production was lower (P=0.0001) in the less hydrated dough (Table 2, Figure 2). The glucose content of the final products was around 2.16% in W35/F0-30°C vs 3.55% in W50/F0-30°C (Table 2). Indeed, AMG have an inverting reaction mechanism, consisting of proton transfer from a general acid catalyst to the glycosidic oxygen, followed by the nucleophylic attack of a deprotonated water molecule assisted by a general base catalyst [START_REF] Larsen | Method of producing a frozen yeast dough product. Patent US005589207A[END_REF][START_REF] Norouzian | Fungal glucoamylases[END_REF].

As mentioned by [START_REF] Rezaei | Effects of water on enzyme performance with an emphasis on the reactions in supercritical fluids[END_REF], when water is used as a reactant in hydrolytic reactions, an increase in its concentration level has a positive effect on the reaction velocity within a certain limit imposed by the enzyme functionality, even if, as specified by De [START_REF] De Pilli | Effect of processing variables and enzymatic activity on wheat flour dough extruded under different operating conditions[END_REF], increasing moisture feed makes the enzyme more sensitive to denaturation.

More specifically, for a short process without resting period, the hydration of the dough will not significantly affect glucose production by AMG (P=0.2985). Glucose production during mixing and shaping was equivalent in W50/F0-30°C and W35/F0-30°C (respectively 1.53 ± 0.38% and 1.22 ± 0.21% fb). The hydration effect occurred during the resting period. During this period, whatever the resting temperature, glucose production fell when dough hydration was reduced. Furthermore, hydration effects occurred faster when the resting temperature increased. The glucose production curves for 50% and 35% water (fb) diverged after 1h of resting at 30°C, 30 min at 50°C and after 10 min at 60°C. Water, like temperature, promotes molecular mobility. The presence of higher amounts of free water promotes collisions between molecules and thus increases the number of effective collisions. Moreover, [START_REF] Sanromán | The influence of substrate structure on the kinetics of the hydrolysis of starch by glucoamylase[END_REF] considered that the limitation of mass transfer (dough consistency) can also be responsible for the decrease in the rate of hydrolysis.

Impact of oil

Oil, commonly found in soft wheat flour dough products may also influence AMG activity.

Globally, considering mixing and resting periods, the addition of oil has very little impact (P=0.3057) on the glucose produced by AMG (Table 2). At the end of 2 hours resting at 30°C, the glucose contents of the doughs were 2.86%, 2.75% and 2.49% in doughs W50/F5.2-30°C, W50/F10.4-30°C and W50/F33.4-30°C, respectively, versus 2.87% for dough W50/F0-30°C. A trend towards reduced glucose production with increased oil content in the dough appeared to emerge but could not be considered as significant according to the ANOVA on glucose production.

The addition of oil to the dough modified the appearance of the curve of glucose production over time (Figure 3), particularly during the resting period. The addition of oil at 5.2% and 10.4% (fb) seemed to generate a latent phase in glucose production by AMG. During the first 15 minutes of resting, glucose production was comparable to that without oil but thereafter the glucose content remained constant up to 30 min for the dough at 10.4% oil and to 60 min for the dough at 5.2% oil. Beyond these points glucose production restarted, reaching after 2 hours approximately the same amount of glucose produced (2.90% and 2.54% fb for doughs W50/F5.2-30°C and W50/F10.4-30°C, respectively) than in the dough without oil (2.87% fb for W50/F0-30°C). Concerning dough W50/F33.4-30°C, the rate of glucose production decreased continuously during the resting time, leading to the production of glucose during the resting period of 2.3% fb. However, the amount of glucose remained higher than that obtained in a less hydrated dough (W35/F0-30°C) without oil, suggesting that the addition of oil has less impact on AMG activity than the reduction of water content in the dough. Other authors found similar results showing decreased hydrolysis rates for corn starch [START_REF] Ai | Effects of lipids on enzymatic hydrolysis and physical properties of starch[END_REF] and foxtail millet starch [START_REF] Jin | Interactions between protein, lipid and starch in foxtail millet flour affect the in vitro digestion of starch[END_REF] after the addition of free fatty acids or oil. These authors attributed this reduction of the in vitro digestibility of starch to an amylose-helical complex formation between amylose and the lipids that is more resistant to amylase hydrolysis.

Possible applications in bakery products: the case of sandwich bread

The previous results obtained using a model dough indicated that glucose production by AMG is favoured by higher hydration, low oil content and the presence in the process of a resting period before baking. An application in the case of sandwich bread therefore appeared to provide a set of conditions favourable to the application of AMG.

The addition of AMG was tested in a sandwich bread with no added sugar. The glucose content was measured at different process stages (mixing, resting and baking), in bread containing different quantities of AMG: 1650 AGU/kg on the flour basis, which is similar to the test presented previously in the model dough of section 3.1, and two lower quantities (1100 and 550 AGU/kg flour). The glucose content in the product increased during the process (Figure 4). This confirmed that AMG is active under actual conditions of sandwich bread production, as found in a previous study of [START_REF] Struyf | Investigating the impact of a-amylase, a-glucosidase and glucoamylase action on yeast-mediated bread dough fermentation and bread sugar levels[END_REF].

A comparison of the results obtained for sandwich bread (case of 1650 AGU/kg flour), with those obtained in the model dough (W50/F0-30°C) showed similar glucose productions during resting (2.1% in sandwich bread vs. 2.9% in model dough), but much higher production during baking (2.4% in sandwich bread vs.

-0.15% in model dough). This increased AMG action could be linked to the lower pH of bread dough (sourdough bread, pH of sourdough 3.7) which promoted AMG activity [START_REF] Lam | Production of Fungal Glucoamylase for Glucose Production from Food Waste[END_REF].

Consequently, the resulting glucose contents obtained in bread (6.8% sample basis) were much higher than those obtained in the model dough (3.55% sample basis).

On the other hand, the amount of glucose in sandwich bread increased linearly with the amount of AMG (Figure 4). This suggests that AMG is the limiting factor and that the glucose content of the bread can be regulated by the amount of AMG added in the recipe. The 3 quantities of AMG tested, i.e. 550, 1100, and 1650 AGU/kg flour, yielded breads containing 3.3%, 5.1% and 6.8% of glucose, respectively (sample basis).

In comparison, a commercial sandwich bread found on the French market contains around 6% total sugars (mono and disaccharides). The quantity of AMG required is consistent with the 80 to 23000 AGU/kg of flour required for the anti-staling application recommended in the patent registered by [START_REF] Else | Use of an anti-staling enzyme mixture in the preparation of baked bread[END_REF].

In order to assess the impact of AMG on the organoleptic qualities of bread, sensory tests on bread characteristics and overall acceptance were conducted (Figure 5 (a) to (h)). Sandwich bread containing 825 and 1100 AGU/kg flour were compared to reference bread, no added sugar and sugar reduced breads, whose sugar profiles are shown in Table 3.

The most preferred products were the reference bread and the sugar reduced bread followed by the 825 AGU/kg flour, the 1100 AGU/kg flour breads and, finally, the least preferred bread was the no added sugar bread (Figure 5a). The sums of ranks ranged from 186 to 258 out of 390, which shows that the panellists liked the breads moderately.

More specifically, sweetness was highly impacted by the changes of formulation (Figure 5b). The no added sugar bread was clearly not sweet enough whereas the reference bread was slightly too sweet. The sugar reduced bread and the two AMG breads were sweet enough for approximately 2/3 of the panellists and not sweet enough for 1/3 of them. The right sweetness balance seemed to be with sugar content between these recipes and the full sugar bread.

The formulation also influences the colour of breads (Figure 5c). The sugar-reduced bread had the most acceptable colour. The reference bread was slightly not enough coloured whereas the AMG breads and the no added sugar bread were too much coloured. Generally, the Maillard reaction gives an appealing colour to the food products [START_REF] Rannou | Mitigation strategies of acrylamide, furans, heterocyclic amines and browning during the Maillard reaction in foods[END_REF] but in our case, the no added sugar breads had a too intense colour, which was not appreciated by the panellists.

Two parameters were evaluated for texture: softness and pastiness (Figures 5d and5e). The three no added sugar breads were less soft than the reference bread and the sugar-reduced bread. Sugar seems to play an important role for the softness of breads. However, the addition of AMG slightly improved the acceptance of softness of the no added sugar breads by the panellists compared to the no added sugar bread. Concerning pastiness, all the breads were relatively well accepted except the no added sugar, which was perceived as too pasty.

The breads were generally described as not enough aromatic (Figure 5f). The no added sugar breads had the highest proportion of 'too much aromatic' followed by the AMG breads. This result could be surprising but we can hypothesised that the removal of sugar favour the perception of taste and aroma from sourdough as sweetness generally counterbalance acidity. Sourdough is known to bring acidity to the products as well as volatile compounds contributing to the aroma of bread [START_REF] Pétel | Sourdough volatile compounds and their contribution to bread: A review[END_REF]. The evaluation of acidity (Figure 5h)

show that the no added sugar bread was effectively perceived as 'too much acid' for 28% of the panellists, which is the highest score. The acidity of the other breads was mainly 'just about right'. Thus, an increase of the amount of AMG, and the related sugar content, improves the perception of acidity compared to the no added sugar. Finally, saltiness was well accepted for the reference bread, the sugar reduced bread and the no added sugar + 1100AGU/kg flour. The no added sugar bread was the least accepted (Figure 5g).

As expected, the no added sugar bread, which had a very low sugar content (1.8g/100g) unacceptable colour and texture and the worst flavour, was that least preferred. The addition of AMG improved the preference, the sweetness, the flavour and the texture of a no added sugar bread. However, despite the total sugar content being close to those of breads formulated with sucrose, products containing AMG were still less preferred than products with sucrose. Sweetness perception is thus complex and not only linked to the total amount of sugar. We can hypothesise that a different sugar profile (table 3) or that a modification in the texture, colour or flavour of the product may lead to a different sweetness perception. In the same way, preference is the result of the evaluation of the overall qualities of bread. Sweetness is a driver of preference but not the only one.

CONCLUSION

This work showed that under conditions of hydration and oil content ranging from those of a biscuit to those of a sandwich bread, AMG was able to produce a significant quantity of glucose. The production of glucose is favoured by a high water content, a low oil content, and a process integrating a rest phase at a high temperature. The effect of hydration was greater than the effect of oil on the activity of AMG. The resting phase was the key to controlling the glucose production of AMG during the process. Adjusting the resting time and temperature conditions allowed optimising the desired glucose level. This work showed that the addition of AMG (1100AGU/kg of flour) in a sandwich bread with no added sugar made it possible to obtain a sugar content and a sweet taste comparable to a standard sandwich bread. (0, 5.2, 10.4 and 33.4 % flour basis, recipes F0, F5.2, F10.4 and F33.4, respectively), and for different resting 572 temperatures (30°C, 50°C and 60°C). ND means that the glucose content is below the detection limit of the analysis method.

Figure 2 :

 2 Figure 2: Evolution of the amount of glucose produced by 1650 AGU/kg flour of AMG, in dough containing

Figure 3 :

 3 Figure 3: Evolution of the amount of glucose produced by 1650 AGU/kg flour in dough containing flour and

Figure 4 :

 4 Figure 4: Impact of the quantity of AMG added to the recipe on the amount of glucose quantified in bread

Figure 5 :

 5 Figure 5: (a) Results of the hedonic ranking test (α=5%). A different letter reflects significant differences

Table 1 :

 1 Details of soft bread recipes used to test the AMG application for glucose production and for sensory comparison.

	Ingredients	No added sugar	Reference	Sugar reduced	No added sugar
	(% flour basis)	+ AMG bread	bread	bread	bread
	Flour	100	100	100	100
	Sourdough	10.0	10.0	10.0	10.0
	Fat	5.5	5.5	5.5	5.5
	Water	52.6	45	48	52.6
	Sugar	0	6.1	4.2	0
	Salt	1.5	1.5	1.5	1.5
	Fresh yeast	2.1	2.1	2.1	2.1
	Soybean flour	5.2	0	1	5.2
	Fibres	1.05	0.05	1.05	1.05
	AMG	0.05 to 0.15	0	0	0

573 glucose produced during the different stages (g for 100g flour) glucose content at the end of the different stages (g for 100g sample)

	Model dough recipe	AMG content (AGU/kg flour)	Resting temperature (°C)	during mixing	during dough preparation (mixing and shaping)	during resting	during baking without resting	during baking after resting	in dough before resting	in dough after 2h resting	in baked product without resting	in baked product after resting
	W50/F0	0		ND	ND	ND	ND	ND	ND	ND	ND	ND
	W50/F0	1650		1.45 ± 0.30	1.53 ± 0.38	2.87 ± 0.11 0.86 ± 0.31	-0.15 ± 0.18	0.97 ± 0.16 2.87 ± 0.09	1.85 ± 0.25	3.55 ± 0.11
	W50/F0	0		ND	ND	ND	-	-	ND	ND	-	-
	W50/F0	1650		2.08	2.07	4.75	-	-	1.38	4.54	-	-
	W50/F0	0		ND	ND	ND	-	-	ND	ND	-	-
	W50/F0	1650		1.20	1.37	7.09	-	-	0.91	5.63	-	-
	W50/F5.2	1650		1.34	1.39	2.90	-	-	0.93	2.86	-	-
	W50/F10.4	1650		1.19	1.59	2.54	-	-	1.06	2.75	-	-
	W50/F33.4	1650		1.11	1.43	2.30	-	-	0.96	2.49	-	-
	W35/F0	0		ND	ND	ND	ND	ND	ND	ND	ND	ND
	W35/F0	1650		1.17 ± 0.24	1.22 ± 0.21	1.80 ± 0.23	0.57	-0.44	0.85 ± 0.09 2.12 ± 0.16	1.59	2.16
	W35/F0	0		ND	ND	ND	-	-	ND	ND	-	-
	W35/F0	1650		1.59	1.71	4.11	-	-	1.14	3.88	-	-
	W35/F0	0		ND	ND	ND	-	-	ND	ND	-	-
	W35/F0	1650		1.91	1.50	5.35	-	-	1.00	4.57	-	-

Figure 1: Processing and sampling flowchart used during this study.

ACKNOWLEDGMENTS

This research was co-funded by BPI France and ONIRIS (Higher education and research institute affiliated with the French Ministry of Agriculture). The companies "ETS MOULINS" and "ABCD Nutrition" are thanked for their contribution in setting up the project and coordinating the project between the different partners. The authors would like to thank all the panellists for their participation in the sensory tests.