Qiwei Shi

Liyao Jiao

Dominique Loisnard

Chengyi Dan

Zhe Chen

Haowei Wang

Stéphane Roux

Improved EBSD indexation accuracy by considering energy distribution of diffraction patterns

Keywords: EBSD calibration, High-angular-resolution EBSD, Integrated digital image correlation, electron energy distribution, error analysis

Registering experimental and simulated electron diffraction patterns is increasingly used for advanced electron backscatter diffraction indexation (EBSD) analysis, yet the accuracy of registration is limited by several effects not accounted for in pattern simulation, such as the Kikuchi band asymmetry, gray level reversal, optical distortion and non-uniform electron energy. Though some of these phenomena can be simulated by Monte Carlo method and dynamical simulation, the computation is highly demanding and their effects on EBSD calibration are seldom analyzed. Here a simple weighting of energy for simulated diffraction pattern is proposed based on several master patterns calculated before-hand, to effectively account for the electron energy distribution.

Integrated digital image correlation is alternatively applied to quantify the electron energy field and calibrate geometry parameters. A metric of the accuracy of indexed crystal orientation is proposed based on a large-area high-definition experimental EBSD acquisition on a (100)-face single crystal Si wafer. The consideration of inhomogeneous energy distribution reduces the crystal orientation discrepancy by 0.128°for the Si dataset.

Introduction

Electron BackScatter Diffraction (EBSD) technique has evolved as a technique of choice to obtain crystallographic orientation fields in scanning electron microscopes. This technique aims principally at obtaining orientation maps from the analysis of the Kikuchi diffraction pattern (thereafter denoted as EBSP for Electron BackScatter Pattern) by Hough-transformation [START_REF] Wright | Automatic analysis of electron backscatter diffraction patterns[END_REF] or the newly emerged convolutional neural network [START_REF] Jha | Extracting grain orientations from EBSD patterns of polycrystalline materials using convolutional neural networks[END_REF]. Standard EBSD usually provides results with an angular resolution of about 0.5°. Such an uncertainty is sufficient to evaluate local orientation maps but cannot reveal elastic strains or small angle grain boundaries, which are associated with low deformations of diffraction images.

High-(angular) Resolution EBSD (HR-EBSD) -exploiting high-definition EBSPs registered onto a reference one through Digital Image Correlation (DIC) -has been proposed to obtain relative crystallographic orientations with a much higher accuracy [START_REF] Wilkinson | High-resolution elastic strain measurement from electron backscatter diffraction patterns: New levels of sensitivity[END_REF][START_REF] Wilkinson | Strains, planes, and EBSD in materials science[END_REF]. It has been shown that HR-EBSD enables the elastic strains to be measured with acceptable uncertainties and excellent spatial resolution [START_REF] Plancher | Full-field measurements of elastic and total strains for the determination of the local behaviour in polycrystals[END_REF]. Both local DIC algorithm based on cross-correlation [START_REF] Wilkinson | High-resolution elastic strain measurement from electron backscatter diffraction patterns: New levels of sensitivity[END_REF] and (integrated) global DIC framework [START_REF] Vermeij | A consistent full-field integrated DIC framework for HR-EBSD[END_REF][START_REF] Ruggles | New levels of high angular resolution EBSD performance via inverse compositional Gauss-Newton based digital image correlation[END_REF][START_REF] Shi | Estimation of elastic strain by integrated image correlation on electron diffraction patterns[END_REF][START_REF] Ernould | Global DIC approach guided by a cross-correlation based initial guess for HR-EBSD and on-axis HR-TKD[END_REF] have been adopted in HR-EBSD. The latter method proves fast and precise for both simulated and experimental EBSPs. Global DIC can also measure grain-level absolute stress without simulated reference EBSP [START_REF] Vermeij | Demonstrating the potential of accurate absolute cross-grain stress and orientation correlation using electron backscatter diffraction[END_REF]. Yet, for both algorithms, the final results of HR-EBSD are limited by the accuracy of the projection parameters [START_REF] Maurice | Comments on the paper "Bragg's law diffraction simulations for electron backscatter diffraction analysis[END_REF][START_REF] Shi | Estimation of elastic strain by integrated image correlation on electron diffraction patterns[END_REF].

Full pattern matching (FPM) EBSD calibration methods have known a rapid progress in the recent years. The basic idea is to tune the projection parameters (in particular the three coordinates of the PC and the Euler angles of crystal orientation) in order to maximize the similarity between experimental and simulated EBSPs. Dynamical EBSP simulation proves to have a higher similarity with experimental pattern than kinematic simulations [START_REF] Jackson | Performance of dynamically simulated reference patterns for cross-correlation electron backscatter diffraction[END_REF]. To achieve the optimization of the projection parameters, many algorithms have been suggested, such as a Nelder-Mead algorithm [START_REF] Vespucci | Diffractive triangulation of radiative point sources[END_REF][START_REF] Nolze | Improving the precision of orientation measurements from technical materials via EBSD pattern matching[END_REF][START_REF] Pang | Global optimization for accurate determination of EBSD pattern centers[END_REF], cross-correlation with log-polar transform [START_REF] Foden | Indexing electron backscatter diffraction patterns with a refined template matching approach[END_REF], evolution algorithm [START_REF] Tanaka | Pattern matching analysis of electron backscatter diffraction patterns for pattern centre, crystal orientation and absolute elastic strain determination -accuracy and precision assessment[END_REF], SNOBFIT algorithm [START_REF] Pang | Global optimization for accurate determination of EBSD pattern centers[END_REF], integrated DIC algorithm [START_REF] Shi | Calibration of crystal orientation and pattern center of EBSD using integrated digital image correlation[END_REF] or non-disclosed ones used in commercial software [START_REF] Friedrich | Application of the pattern matching approach for EBSD calibration and orientation mapping, utilising dynamical EBSP simulations[END_REF]. IDIC is able to calibrate overlapped EBSPs at grain boundaries, to obtain multiple sets of Euler angles, one set of PC and the contribution ratio for each crystal orientation [START_REF] Shi | Indexation of electron diffraction patterns at grain boundaries[END_REF]. The PC calibration uncertainty has been shown to reach values as low as 10 -5 times the pattern width for simulated EBSPs. The reported Euler angles uncertainty is also very low: 0.03°for a 640×480 pixel EBSP used for the analysis of a tungsten carbide sample [START_REF] Winkelmann | Improving EBSD precision by orientation refinement with full pattern matching[END_REF]. The same crystal orientation uncertainty level is obtained at grain boundaries for both grains [START_REF] Shi | Indexation of electron diffraction patterns at grain boundaries[END_REF].

However, different groups studying FPM EBSD calibration on polycrystal samples have very recently reported grainwise discontinuity in PC fields [START_REF] Pang | Global optimization for accurate determination of EBSD pattern centers[END_REF][START_REF] Shi | Calibration of crystal orientation and pattern center of EBSD using integrated digital image correlation[END_REF]. The shift of PC components across grain boundaries could reach 1-2 pixels, that cannot be due to the sample topography nor to the elastic strain. This intriguing phenomenon highlights the imperfection of registering experimental EBSP with a simple projection of simulated master pattern. Researchers have pointed out several secondary effects in experimental EBSPs that need to be considered. For example, the brightness of Kikuchi band is closely linked to the sample tilt angle, and when the latter is smaller than 60°, it could even lead to reversed gray level distributions [START_REF] Winkelmann | Analysis of kikuchi band contrast reversal in electron backscatter diffraction patterns of silicon[END_REF]. Recent studies have also reported the inhomogeneous distribution of backscattered electron (BSE) energy [START_REF] Ram | Energy dependence of the spatial distribution of inelastically scattered electrons in backscatter electron diffraction[END_REF][START_REF] Winkelmann | Constraints on the effective electron energy spectrum in backscatter Kikuchi diffraction[END_REF], explicitly, that of electrons diffracted to the top of the detector is several keV lower than that of the bottom. Horizontal Kikuchi bands have brighter upper edges and darker lower ones, a phenomenon named excess-deficiency (abbreviated as ED hereinafter). This effect is due to the anisotropic diffracted electron distribution, a phenomenon well described in Ref. [START_REF] Winkelmann | Dynamical effects of anisotropic inelastic scattering in electron backscatter diffraction[END_REF]. If these secondary effects are not accounted for, systematic errors in PC and crystal orientation will be inevitable for EBSD calibration methods, regardless of the choice of optimization algorithm. Besides, this systematic error depends on crystal orientation, as ED effect depends on the tilt angle of each Kikuchi band.

The aforementioned phenomena, i.e., gray level reversal, non-uniform energy distribution and ED effect, can be included in dynamical simulations with specific simplification assumptions, and results in a better similitude with experimental EBSPs [START_REF] Winkelmann | Dynamical effects of anisotropic inelastic scattering in electron backscatter diffraction[END_REF][START_REF] Winkelmann | Analysis of kikuchi band contrast reversal in electron backscatter diffraction patterns of silicon[END_REF]. The effect of a better similitude on EBSD indexation has recently been addressed, and a difference in PC calibration about 1% of EBSP width could exist [START_REF] Winkelmann | Kikuchi pattern simulations of backscattered and transmitted electrons[END_REF]. However, only a few individual EBSPs have been analyzed, and the accuracy and precision of the indexed orientation are not addressed. The simulated EBSP of different crystal orientation or different acquisition condition requires lengthy and dedicated calculation, and cannot be efficiently projected from a pre-calculated master pattern. Besides, the complex theory of dynamical simulation and sizable coding works have limited its development to only a few groups, namely mostly those led by Winkelmann [START_REF] Winkelmann | Many-beam dynamical simulation of electron backscatter diffraction patterns[END_REF][START_REF] Winkelmann | Dynamical effects of anisotropic inelastic scattering in electron backscatter diffraction[END_REF][START_REF] Winkelmann | Analysis of kikuchi band contrast reversal in electron backscatter diffraction patterns of silicon[END_REF] and De Graef [START_REF] Wang | Modeling dynamical electron scattering with bethe potentials and the scattering matrix[END_REF][START_REF] Zhu | EBSD pattern simulations for an interaction volume containing lattice defects[END_REF].

The present work proposes to apply a weighted average to the simulated EBSPs in order to account for the effect of non-uniform electron energy, then rely on the IDIC framework to calibrate EBSD, including calibration of electron energy distribution, refinement of both Euler angles and PC position. Section 2 distinguishes the precision and accuracy of crystal orientation measurement, and stresses the importance of accuracy assessment. Section 3 introduces an IDIC algorithm for EBSD calibration that aims at retrieving electron energy distributions, crystal orientation and projection parameters. Section 4 analyzes high-resolution EBSPs of a single crystal Si wafer, and demonstrates the accuracy improvement of the algorithm. The application of energy weighting improves the similarity between experimental and simulated EBSPs and the accuracy of indexed Euler angles.

Precision and accuracy of EBSD indexation

When a quantity is measured from experiment, it is natural to inquire about its precision and accuracy. Ram et al.

has provided a detailed comparison of these two notions [START_REF] Ram | Error analysis of the crystal orientations and disorientations obtained by the classical electron backscatter diffraction technique[END_REF]. The crystal orientation precision for Hough transform and dictionary approach has been assessed separately [START_REF] Ram | Error analysis of the crystal orientations and disorientations obtained by the classical electron backscatter diffraction technique[END_REF][START_REF] Ram | Error analysis of the crystal orientations obtained by the dictionary approach to EBSD indexing[END_REF].

Here, as defined by Ram et al., the accuracy describes the distance between the measured value and the true value (the ground truth), and the precision is the standard deviation of several repeated measurements of the same physical object. The true crystal orientation is generally unknown, and as EBSD is the most commonly adopted technique to obtain crystal orientation, its measurement is often used as a reliable ground truth to characterize the accuracy of other measurement techniques [START_REF] Gaskey | Optical characterization of grain orientation in crystalline materials[END_REF]. One way to circumvent the absence of ground truth in accuracy assessment is to exploit the known sample features, such as twins in metallic samples. The measured misorientation across twin boundaries can be compared to the theoretical value, thus provide a metric of the accuracy. However, an error up to several degrees on each side of the twin boundary may hardly affect the apparent twin misorientation.

Thus a favorable measurement of the latter is not a guaranty for an absolute precision of the indexation.

Figure 1a shows the inverse pole figure (IPF) map of a polycrystal Ni sample, which is provided by Wright et al. [START_REF] Wright | Introduction and comparison of new EBSD post-processing methodologies[END_REF] and shared with the community by Jackson et al., including the indexation results by Hough transformation and dictionary indexation [START_REF] Jackson | Dictionary indexing of electron back-scatter diffraction patterns: a hands-on tutorial[END_REF]. Many twin boundaries exist in the map, and a vertical profile across two boundaries is selected for misorientation analysis. Figure 1b plots the between-neighbor misorientation along the profile with three EBSD indexation algorithms: Hough indexation, dictionary indexation, IDIC EBSD indexation with floating or filtered PC values. As the Ni sample is not strained, between-neighbor misorientation could be attributed to indexation uncertainty. The former three indexation algorithms have comparable uncertainty levels around 0.40°, while the peak values reach 1.5°. The IDIC EBSD with filtered PC coordinates lead to the lowest uncertainty around 0.15°. Figure 1c-1d plot the misorientation to the central EBSP along the profile, with zooming on 60°, the theoretical misorientation across the FCC twin boundaries, and 0°respectively. Figure 1c shows that all 4 misorientation profiles fluctuate closely with the theoretical value, including those EBSPs that have betweenneighbor as high as 1.5°. This phenomenon demonstrates that the relative misorientation across twin boundaries is not a sufficient proof of sub-degree EBSD indexation accuracy. Besides, the low uncertainty of IDIC EBSD indexation especially with filtered PC values is confirmed in Figures 1b-1d. Another accuracy evaluation method is to use a standard sample with known crystal orientation to calibrate the EBSD system. It is vital to avoid or correct the experimental setup errors in this procedure, to name a few, the sample tilt angle, the misalignment of the tilt axis, the intentional or unintentional tilt of EBSD detector around different axes. These errors pose a severe threat on EBSD characterization, especially on the 'global optimization' approach of all EBSPs of an EBSD acquisition, which takes the EBSD geometry as a key input [START_REF] Pang | Global optimization for accurate determination of EBSD pattern centers[END_REF][START_REF] Zhu | Orientation, pattern center refinement and deformation state extraction through global optimization algorithms[END_REF]. There could exist unknown errors in all these aspects, and one remedy is to use the calibrated PC coordinates to correct all these potential sources of errors. To calibrate the sample surface plane more accurately, a large EBSD scan with high-definition EBSPs is preferred, the roughness of sample surface should be as low as possible, and ideally the sample should be unstrained, as the strain state influences the calibrated PC coordinates [START_REF] Maurice | Comments on the paper "Bragg's law diffraction simulations for electron backscatter diffraction analysis[END_REF][START_REF] Alkorta | Limits of simulation based high resolution EBSD[END_REF][START_REF] Shi | Calibration of crystal orientation and pattern center of EBSD using integrated digital image correlation[END_REF]. These requirements converge to a large-area EBSD scan with full definition on an unstrained single crystal sample, for example the single crystal Si wafer. The manufacturing of large high-purity Si wafer and its cut along a specific lattice plane is a mature technical question, thus it provides a cheap (the Si wafer piece used in this work costs 10 US dollars) but adequate standard sample to evaluate the accuracy of EBSD systems. Note that the uncertainty of Si wafer plane index is below 0.5°, as stated by most manufacturers. The IDIC EBSD algorithm is based on the assumption that only 6 projection parameters matter. Namely, the three coordinates of the projection center x * = (x * , y * , z * ) and the three Euler angles (φ 1 , ϕ, φ 2 ) of the crystal orientations which are collectively denoted as P . In the present study, it is argued that several additional parameters are to be included in the calibration to account for the BSE energy distribution.

EBSD calibration algorithm with a non-uniform BSE energy distribution

Monte-Carlo simulations

A way to study the energy distribution of BSEs is stochastic simulation such as Monte-Carlo methods. The software CASINO v2.5.1 [START_REF] Drouin | CASINO v2. 42-a fast and easy-to-use modeling tool for scanning electron microscopy and microanalysis users[END_REF] is used to simulate the interaction of 100,000 electrons of acceleration voltage 20 kV with a 70°tilted Si sample. The incident beam hits the sample at the calibrated PC coordinates, and the trajectories of BSEs inside the Si sample is shown in Figure 3a. The last segment of the trajectories are straightly prolonged until hitting the EBSD detector. The deviation angle α between the scattered BSE and the incident beam direction is also defined in Figure 3a. The relationships between scattering time, deviation angle and BSE energy are shown in Figure 3b-d. A linear relationship exists between BSE energy loss and its scattering time inside the sample, as shown in Figure 3b. This is as anticipated, as more scattering times imply more inelastic collision and thus more energy losses. Figure 3c plots the data points of BSE scattering time and deviation angle, together with the mean profile and the errorbar. For BSE deviation angles lower than 60°, the scattering time and BSE energy loss is also lower. Yet when the deviation angle reaches 60°, the accompanying mean energy does not change. The results of BSE energy distribution for EBSP are summarized in Figure 4. The locations of the hitting, plotted in Figure 4a, spread all over the detector with sufficient densities. The average energy distribution is shown in Figure 4b, and the energy at the bottom is 1 keV higher that at the top, a similar distribution as reported in

Accounting for energy distribution in EBSPs

Let G(u, e) be the dynamically simulated master pattern by EMsoft projected on the equatorial plane u, as illustrated in Figure 5 for two values of BSE energy, e = 16 keV and 26 keV. Note that this large interval is chosen to highlight the Kikuchi band width differences. Without considering the energy distribution, the EBSP pattern is obtained by projecting the master pattern, and is denoted g(x, P ), with

g(x, P ) = G(u(x, P ), e) (1)
The Monte-Carlo simulations prompt one to account for the fact that the energy e should no longer be considered as a constant, but rather as a field. Hence the previous equation, Eq. 1, is generalized to

g(x, P ) = G(u(x, P ), e(x)) (2)
The selected region for projection, u(x, P ), is plotted on both patterns in Figure 5. Note that u does not depend on the BSE energy distribution by nature. This property helps the rapid construction of simulated EBSP of any energy distribution from the pre-calculated master patterns for a predetermined set of energies e i , according to the following approximation

g(x, P ) = i η i (e(x))G(u(x, P ), e i ) (3)
where η i are weights designed to allow for an interpolation of the patterns for any value of energy e. More precisely, if e i ≤ e(x) < e i+1 , then

η i = (e i+1 -e(x)) (e i+1 -e i ) η i+1 = (e(x) -e i ) (e i+1 -e i ) (4)
In the present study, the chosen energies are 17, 18, 19, 20, 21 keV, and the corresponding master patterns are calculated and stored beforehand, so that the final pattern can be obtained from the above weighted sum. It is acknowledged that this BSE energy modeling is a simplification of the real case, as in experimental EBSPs each pixel does not correspond to a single energy level, but rather a spectrum of continuous BSE energy levels. Nonetheless, the attribution of different BSE energy levels to an EBSP is already a progress in EBSD analysis. Note that η i (x) depends on the EBSD experimental setup, such as sample tilt angle and acceleration voltage.

In the analysis of the experimental EBSP, it is desirable not to impose a predetermined e(x), but rather to evaluate it from the observed pattern. This task is basically out of reach without any further a priori information, because of the large number of unknowns, and the expected poor sensitivity. However, as shown in the energy distribution simulated by Monte-Carlo method, (Figure 4b) the spatial distribution of e(x) is very regular [START_REF] Ram | Energy dependence of the spatial distribution of inelastically scattered electrons in backscatter electron diffraction[END_REF],

thus can be fitted by a low-order (such as 2nd order) polynomial to reduce the number of degrees of freedom in calibration. In the following, the BSE energy field e(x, E) is expressed as

e(x, E) = E 1 + E 2 x + E 3 y + E 4 x 2 + E 5 xy + E 6 y 2 (5)
where E 1,2...6 are collectively denoted as a vector E, the BSE energy parameters. These parameters can be seen as supplementary to the classically considered ones P . The inverse problem of determining the optimal E will be seen to be accessible. To summarize the above formulation, the simulated EBSP is written

g(x, E, P ) = i η i (x, E)G(u(x, P ), e i ) (6)

Formulation

Then a registration procedure similar to the IDIC algorithm described in Ref. [START_REF] Shi | Calibration of crystal orientation and pattern center of EBSD using integrated digital image correlation[END_REF] is proposed on the combined images with limited changes. To distinguish from the normal IDIC EBSD, the method described in the current paper is named as IDIC-E EBSD (Integrated Digital Image Correlation with Energy weighting).

The EBSP screen is chosen as the reference defining the (x, y) plane, or z = 0. The unit vector w linking the pattern center to each pixel x = (x, y, z = 0) of the screen can be expressed as

w = (x -x * ) |x -x * | (7)
In order to rotate this vector from the screen frame of reference w to that of the spherical master pattern, v, one should multiply it by a rotation operator

Q v = Qw (8)
whose expression, using the Euler angles, reads in Bunge notations:

Q =      cos φ 1 cos φ 2 -sin φ 1 sin φ 2 cos ϕ sin φ 1 cos φ 2 + cos φ 1 sin φ 2 cos ϕ sin φ 2 sin ϕ -cos φ 1 sin φ 2 -sin φ 1 cos φ 2 cos ϕ -sin φ 1 sin φ 2 + cos φ 1 cos φ 2 cos ϕ cos φ 2 sin ϕ sin φ 1 sin ϕ -cos φ 1 sin ϕ cos ϕ      (9)
The spherical projection of v onto the master pattern plane, u, is written

u = v -n (1 -v.n) + n (10)
where n = (0, 0, 1) ⊤ denotes the unit vector pointing at the north pole. Thus the projected position, u(x), on the master pattern can be expressed explicitly as a function of x and of the projection parameters P = (φ 1 , ϕ, φ 2 , x * , y * , z * ).

Optimization

In the IDIC-E EBSD algorithm, experimental EBSP is taken as f (x). The dynamical simulation of the EBSP master pattern for the i th energy level is G i (x, P ). The algorithm involves matching at best f (x) and g(x, E, P ), through the minimization of a quadratic norm of the residual r summed over the entire ROI. The cost function to minimize is written

Θ = ROI ω(x) 2 [f (x) -g(x, E, P )] 2 (11)
where ω(x) is the weight associated to each pixel x, g u (x, E, P ) is the current estimate of simulated EBSP during iterative algorithms. The weights ω are introduced here as they may be used to make this functional optimal with respect to the handling of noise. The minimization of the cost function leads to successive corrections of the transformation u(x) or η(x, E) estimation until convergence [START_REF] Hild | Digital image correlation[END_REF]. Starting from an approximate solution, the transformation u(x, P ) and η(x, E) are progressively corrected with linear combinations of sensitivity fields constituting the kinematic basis. The cost function ( 11) is alternatively minimized by optimizing P and E, each of which consists of iterative minimization with a Gauss-Newton algorithm.

A slight modification of any parameter δP j with given energy parameters E, induces a modification of the corrected deformed image as δg(x, E, P

) E = j i η i (x, E)∇ u G(u, e i ) • ∂u(x, P ) ∂P j δP j ≡ j Φ j (x, E, P )δP j (12)
where Φ j is a compact notation for the sensitivity with respect to parameter P j . The column vector {δP } gathering all corrections to P is obtained from the linear system

[M ] {δP } = {γ} (13)
where [M ] is the Hessian matrix of size 6 × 6 at iteration n -1 in P optimization.

M (n-1) jk = ROI ω(x) 2 [Φ j (x, E, P )] [Φ k (x, E, P )] (14)
and the second member {γ j } includes the residual field in P optimization.

γ (n) j = ROI ω(x) 2 f (x) -g (n) (x, E, P ) Φ j (x, E, P ) (15)
When ∥{δP }∥ < ϵ P , ϵ P being chosen equal to 10 -6 for all the calculations of this paper, the minimization stops and P is stored. Otherwise, P is updated

P (n) = P (n-1) + δP (n) ( 16
)
Once P is obtained, the simulated EBSP, with the prescribed BSE energy field, resembles the reference experimental EBSP at best.

A similar IDIC procedure could be designed to calibrate the BSE energy field. The sensitivity vector of g(x, E, P ) with any energy parameter E j with given projection parameters P , is denoted Ψ j (x) δg(x, E, P

) P = j i G(u, e i ) ∂η i (e) ∂e ∂e(x) ∂E j δE j ≡ j Ψ j (x, E, P )δE j (17)
The column vector {δE} gathering all corrections to E is obtained

[N ] {δE} = {κ} (18)
where [N ] is the Hessian matrix at iteration n -1

N (n-1) jk = ROI ω(x) 2 Ψ j (x, E, P )Ψ k (x, E, P ) (19)
and the second member {κ j } in E optimization,

κ (n) j = ROI ω(x) 2 f (x) -g (n) (x, E, P ) Ψ j (x, E, P ) (20)
The determination of δE (n) allows to update E as

E (n) = E (n-1) + δE (n) (21)
The same stopping condition for E optimization is adopted, though the convergence criteria ϵ E is set to 10 -4 .

The above detailed IDIC-E algorithm is summarized in the flowchart shown in Figure 6a. Generally, only a few outer iterations are necessary in IDIC-E EBSD before convergence as shown in Figure 6b, as the crystal orientations and projection center values given by single-energy IDIC EBSD are already close enough to the IDIC-E EBSD results. Besides, the cost function Θ drops both in E calibration and P calibration, stressing the necessity to include all these parameters into the EBSD analysis.

Start

Figure 7a shows the energy field calibrated by IDIC-E EBSD, and the corresponding energy weights are given in Figure 7(b-d). The above energy field e(x) as obtained from IDIC-E EBSD, can be compared to one simulated by CASINO (Figure 8. The latter energy field has been generated with the same code and parameters as the one shown in Figure 4b. Their only difference is the deliberately introduced alignment error that will be explained in Section 4.

The simulated energy field with tilt is comparable with the calibrated one, though a difference of about 1kV exist between them. This 1kV contrast is presumably due to the gray level profiles across Kikuchi bands, such as the excess-deficiency effect or the smooth gray level transition at band edges, not perfectly simulated in dynamical simulations. Nevertheless, this similarity validates the proposed IDIC-E EBSD methodology.

Improved indexation accuracy of a Si wafer

The EBSD dataset of a commercial Si wafer, first reported in Ref. [START_REF] Zhong | Residual-based pattern center calibration in high-resolution electron backscatter diffraction[END_REF], is used again to characterize the accuracy of IDIC and IDIC-E indexation. The sample does not show regular engravings on its surface. No strain is introduced in the sample, thus the local misorientation is expected to be negligible. The sample is observed in Tescan MIRA3 scanning electron microscope via the Bruker EBSD technique, with an acceleration voltage 20kV and beam current 10nA. The sample is placed in the chamber using a stub pre-tilted at 70°. In order to limit the charging effect, a large piece of copper conductive tape is 'sandwiched' between the wafer and the EBSD stage. No alignment of the wafer surface by rotation was achieved due to the lack of regular marking on the surface. The step size is deliberately set to a very large value, 50 µm, as the Si wafer lacks fine structures. A total of 1200 high-resolution (1140×1600 pixels) diffraction patterns are recorded using a Bruker e -Flash HD EBSD detector, covering an nominal area of 1.5×2 mm 2 .

An example of high quality experimental EBSP of Si wafer is shown in Figure 9a. The Si master patterns of 17, 18, 19, 20 and 21 kV simulated by EMsoft, selectively shown in Figure 5, are used for the IDIC EBSD indexation. A simulated EBSP is shown in Figure 9b, whose residual compared with the experimental EBSP is shown in Figure 9c.

The simulated EBSP successfully predicts the position and width of Kikuchi bands, but the intensity distribution is not fully captured, as repetitively stated before [START_REF] Britton | Factors affecting the accuracy of high resolution electron backscatter diffraction when using simulated patterns[END_REF][START_REF] Fullwood | Validation of kinematically simulated pattern HR-EBSD for measuring absolute strains and lattice tetragonality[END_REF]. Besides, the noise level in the experimental EBSP is not negligible, and the weighting field ω obtained by inverting the second-order polynomial fitting of the standard noise deviation is shown in Figure 9d. Here ω is higher in the center-left area, which coincides with the very bright Kikuchi bands shown in Figure 9a. Si or Ge single crystal has frequently been used for testing EBSD calibration methods [START_REF] Mingard | Towards high accuracy calibration of electron backscatter diffraction systems[END_REF][START_REF] Alkorta | Improved simulation based HR-EBSD procedure using image gradient based DIC techniques[END_REF][START_REF] Friedrich | Application of the pattern matching approach for EBSD calibration and orientation mapping, utilising dynamical EBSP simulations[END_REF][START_REF] Foden | Indexing electron backscatter diffraction patterns with a refined template matching approach[END_REF]. Alkorta et al. [START_REF] Alkorta | Improved simulation based HR-EBSD procedure using image gradient based DIC techniques[END_REF] have compared the normal of fitted plane for different positions of Si single crystal, yet their precision is of order 1°, and the calibrated Euler angles are not verified by the [001] sample surface. When one aims at assessing the accuracy of EBSD indexation, the 'ground truth' must be reliably known. However, the true orientation is mostly unknown. The crystal plane of Si single crystal wafer is known precisely thanks to the surface energy minimization, yet the uncertainty of experimental setup of EBSD, such as sample tilt angle and the positioning of EBSP detector add to the error of the true crystal orientation. In this case, one could rely on the fitting results of calibrated PC fields to get the ground truth. We hereby propose the following procedure to quantify the crystal orientation accuracy and PC coordinates accuracy based on the Si wafer of known cutting surface:

1. Calibrate all the crystal orientations (φ 1 , ϕ, φ 2 ) and projection centers of EBSPs acquired in the 2D EBSD test.

2. Apply a plane fitting based on the PC point cloud, to get the surface normal N s of the sample.

3. Based on the known Si wafer surface lattice index s = [0, 0, 1] ⊤ , apply the rotation matrix Q corresponding to (φ 1 , ϕ, φ 2 ) to obtain the rotated surface lattice index S r ,

S r = QS (22)
Note that the explicit expression of Q is provided in Equation 9. The misorientation between vectors S r and N s , arccos(S r • N s ), is a metric of the orientation discrepancy between the indexed orientation and the nominal orientation. This discrepancy comprises of 2 parts: the error of the crystal orientation indexation and the error of Si wafer cutting along the wanted plane. In fact, a full 2D orientation discrepancy map can be drawn this way.

It is important to note that Equation 22holds true because the crystal orientation is defined as the rotation from the EBSD detector to the crystal lattice in IDIC EBSD, rather than the rotation from the sample coordinates to the crystal lattice.

All the 1200 EBSPs are analyzed by our in-house IDIC EBSD and the current IDIC-E EBSD calibration methods, and the resulting pattern center coordinates are shown in Figures 11a-11c respectively. An interesting point is that while the x * field varies along the X direction, the gradient of y * and z * fields are not along Y . This effect is not due to the charging effect, which would also influence x * . In fact, this is due to the bad alignment of sample surface and the EBSD scanning direction. PC coordinate shifts between IDIC EBSD and IDIC-E EBSD are shown in Figures 11d-11f. PC values are shifted less than 1 pixel when the BSE energy distribution is considered in the calculation, and y * has a most significant upward shift, resulting from the fact that BSE energy inhomogeneity mainly exist in the y direction. Figure 11g shows the misorientation between the calibration results of IDIC and IDIC-E. This orientation discrepancy level, up to 0.05°, several times higher than the reported uncertainties of FPM indexation, shows that the systematic errors of calibration results are non-negligible. Pattern center data for this large-field EBSD acquisition provide valuable information about the sample surface geometry.

As the single crystal Si wafer is unstrained, its surface is assumed strictly planar. Thus, a plane fit is applied to the PC data clouds, minimizing

T (n, d) = (x * • n -d) 2 (23)
The vector n from the planar fitting is the normal to the plane in the reference of EBSD detector, which is evaluated to (0.4025 0.1779 0.8979) ⊤ by IDIC-E EBSD. Note that this direction is 25.2°from the nominal vector (0, cos(70°), sin(70°)) ⊤ . This large misorientation, originated from the difficulty to align the 5 cm-wide wafer without regular marking on the surface correctly in EBSD position, does not compromise the EBSD analysis as long as the inclination error is corrected by the planar fitting. Besides, the mis-positioning of the Si wafer causes directly the tilt in BSE energy fields shown in Figure 8. As the 1200 EBSPs are generated with the same EBSD set-up, the BSE energy fields should be constant. The mean value and variability of the 1200 BSE energy fields calibrated by IDIC-E EBSD are shown in Figure 12. The tilt of energy field is preserved in the mean field shown in Figure 12a. The median of the standard deviation field is 0.05 keV, though the variability of calibrated BSE energy is higher at the corners (especially the right-bottom one). This is due to the polynomial fitting of energy field e(x) tends to have higher residuals at end data (as for other polynomial fittings). This consistency of BSE energy calibration demonstrates the robustness of the proposed method. To sum up, the IDIC EBSD calibration on 1140×1600-pixel large step-size EBSD acquisition on Si single crystal wafer has an uncertainty in PC calibration of 0.187 pixel, and median crystal orientation discrepancy of 0.222°.

The accuracy of IDIC-E indexation is deemed improved as the orientation discrepancy is effectively reduced, as compared to the result (0.350°) with a homogeneous energy distribution. Another proof is the continuous reduction of cost function Θ when the BSE energy distribution is considered.

Full pattern match with simulated diffraction pattern has drawn increasing attention in advanced EBSD analyses, mainly thanks to its high precision. Some aspects of experimental EBSPs are however difficult or costly to simulate, such as Kikuchi band brightness asymmetry, gray level reversal for small tilt angles and non uniform diffracted electron energy. This paper proposes to overcome the inhomogeneous energy obstacle by relying on a dedicated integrated digital image correlation algorithm that quantifies the electron energy distribution and projection parameters.

Precision, or uncertainty of EBSD crystal orientation indexation has been frequently addressed, yet its correctness, or accuracy, is seldom analyzed, as the ground truth of crystal orientation is generally unknown. In this paper the (100)-face single crystal Si wafer is used as a standard sample to calibrate the EBSD system. By a large-area (2×1.5 mm 2 ) high definition (1140×1600) EBSD acquisition and analyzing full field projection center values, the sample surface inclination relative to the EBSD detector is precisely calibrated, thus the error of sample tilt and EBSD detector tilt are corrected in the accuracy assessment. As a result, the contrast between EBSD-indexed orientation and manufacturer-indicated orientation is assessed precisely.

By considering the inhomogeneous electron energy distribution of EBSPs, the orientation contrast has improved by 0.128°as demonstrated in the Si wafer dataset. Failing to account for the energy distribution effect induce a deviation of the PC by near 1 pixels, especially for y * . The remaining crystal orientation contrast, estimated around 0.222°for the Si wafer, could be attributed to other effects, such as Si wafer cutting error, Kikuchi band brightness asymmetry and optical distortion of EBSD detector. The further processing of these effects are left for future studies.

Figure 1 :

 1 Figure 1: Analysis of misorientation across twin boundaries. (a) IPF map of a polycrystal Ni, where a black line marks the EBSPs across twin boundaries analyzed for misorientation; (b) Neighbor misorientation evaluated by different algorithms along the profile marked in (a); The misorientation between all EBSPs in the profile and an EBSP inside the red twin stripe is shown in (c) and (d), where the former focuses at 60°misorientation and the latter at 0°.

Integrated

  Digital Image Correlation (IDIC) has been introduced to measure directly and optimally the geometric transformation of interest (or parameters quantifying it) from image registrations. The projection geometry of EBSP is recalled in figure 2 together with the reference frame used in the paper.

Figure 2 :

 2 Figure 2: EBSD setup, with the projection center marked as a red star. The coordinate systems (x, y, z) associated to the EBSP detector and (X, Y, Z) associated to the sample are also defined here.

Figure 3 :

 3 Figure 3: BSE energy analysis by Monte-Carlo simulation. (a) Examples of trajectories of BSEs inside the tilted sample, whose surface is marked with a dashed line; (b) The data points collecting the BSE energy and the number of scattering events inside Si sample. (c) The number of scattering events as a function of BSE deviation angle. (d) The BSE energy against the BSE deviation angle. The mean profiles and errorbar are plotted in (c) and (d).

Figure 4 :

 4 Figure 4: Monte-Carlo simulation of the backscattered electrons. (a) The hitting point of BSEs on the EBSD detector; (b) The average electron energy field of the EBSP, e(x), expressed in keV.

Figure 5 :

 5 Figure 5: Examples of G(u, e) for electron energy of e = 16 keV (a) and e = 26 keV (b) respectively. The selected region for projection, a function of projection parameters P , is plotted in both figures. The Kikuchi bands are narrower for higher electron energies.

Figure 6 :

 6 Figure 6: (a) Flowchart of the IDIC-E algorithm. (b) The evolution of the cost function Θ during the IDIC-E EBSD calibration of an EBSP of Si wafer.

Figure 7 :

 7 Figure 7: Examples of η fields, the weighting ratio defined at each EBSP pixel, for EBSD analysis with considerations of BSE energy. (a) Mean energy of scattered electrons in the detector, e(x); (b-d) η weight fields for master patterns of 18, 19 and 20 keV respectively in IDIC-E EBSD calibrated BSE energy model.

Figure 8 :

 8 Figure 8: An example of Monte-Carlo simulation of the energy field obtained after tilting the Si wafer sample to mimic a possible imperfect set-up.

Figure 9 :

 9 Figure 9: (a) Experimental EBSP of the single crystal Si wafer. (b) The numerical EBSP projected from the master pattern. (c) Residual between (a) and (b). (d) Weights of EBSP ω used in IDIC EBSD indexation.

Figure 10 :

 10 Figure 10: Comparison of experimental EBSP and a series of simulated EBSPs of different energy levels. (a) Zoomed experimental EBSP of the single crystal Si wafer. The numerical EBSP projected from the 18 kV master pattern (b), 19 kV pattern (c), 20 kV pattern (e) and 21 kV pattern (f). (d) corresponds to the simulated EBSP with IDIC-E calibrated energy distribution fields. Readers are invited to compare the black trough stretching from the lower-left to the upper-right of all figures.

Figure 11 :

 11 Figure 11: Results of IDIC-E EBSD calibration on Si wafer. Retrieved pattern center coordinates x * (a), y * (b) and z * (c) by IDIC-E EBSD. The definition of these coordinates is given in Figure 2. (d-f) Pattern center coordinate shifts (in pixels) between IDIC EBSD and IDIC-E EBSD. (g) The misorientation (in °) map between IDIC and IDIC-E indexation.

Figure 12 :

 12 Figure 12: The mean field (a) and standard deviation field (b), both expressed in keV, of BSE energy calibrated by IDIC-E EBSD.

Figure 13

 13 Figure 13 plots the different BSE energy models used in this study and their corresponding crystal orientation indexation error. Three BSE energy models are adopted, namely a uniform 19 keV BSE energy field shown in Figure 13a, a linear stratified energy distribution between 18 and 20 keV shown in Figure 13b and the calibrated BSE energy distribution by IDIC-E EBSD shown in Figure 13c. Note that most existing EBSD indexation software packages assume a uniform BSE energy distribution [1, 18]. The estimated indexed crystal orientation error fields are shown in Figures 13d-f, corresponding to the three BSE energy fields respectively. The histogram profiles of the three orientation contrast fields are plotted in Figure 13g. The median crystal orientation discrepancy is 0.350°for the uniform 19 keV energy model, 0.250°for the linear 18-20 keV energy model and 0.222°for the IDIC-E calibrated energy model. The remaining crystal orientation contrast could be attributed to the imperfect Si wafer (non-zero misorientation between the sample surface and [001] direction), and other systematic errors causing inaccuracy in EBSD calibration, such as the optical distortion in EBSD detector or excess-deficiency effect. The overall reduction of crystal orientation error demonstrates the improvement of crystal orientation indexation when BSE energy distribution is considered.

  uniform 19 keV BSE energy Calibration with linear 18-20 keV BSE energy Calibration with IDIC-E EBSD (g)

Figure 13 :

 13 Figure 13: Crystal orientation discrepancy of Si wafer based on different BSE energy distributions. (a-c) BSE (in keV) distribution models used in IDIC EBSD calibration. (a) a uniform 19 keV BSE energy field. (b) Linear and stratified energy distribution between 18 and 20 keV. (c) The calibrated BSE energy field by IDIC-E EBSD. (d-f) show the orientation discrepancy (in °) for BSE energy fields (a-c) respectively. (g) plots the histogram profiles of (d-f).

Acknowledgements

This work is financially supported by the National Natural Science Foundation of China [No. 51901132]. We are also grateful to the financing of TESCAN CHINA, Ltd.

Data Availability

The 40×30 EBSPs of resolution 1140×1600 pixels are available at Zenodo (DOI: 10.5281/zenodo.5803072), together with the indexed crystal orientations and calibrated PC coordinates with a single BSE energy and the calibrated BSE energy fields. We invite interested colleagues to use this dataset to further improve the accuracy of EBSD indexation.


P Soudier

; Faure

; Kushwaha

; Faulon

Cell-Free Biosensors and AI integration

Cell-free biosensors hold a great potential as alternatives for traditional analytical chemistry methods providing low-cost low resource measurement of specific chemicals. However, their large scale use is limited by the complexity of their development.

In this chapter we present a standard methodology based on Computer aided design (CAD) tools that enables fast development of new cell-free biosensors based on target molecule information transduction and reporting through metabolic and genetic layers, respectively.

Such systems can then be repurposed to represent complex computational problems, allowing defined multiplex sensing of various inputs and integration of artificial intelligence in synthetic biological systems.

Introduction :

Metabolite biosensors emerged as a major application of synthetic biology. Repurposing biological systems naturally present in various organisms enabled the development of synthetic sensing devices [START_REF] Van Der Meer | Where microbiology meets microengineering: design and applications of reporter bacteria[END_REF] giving rapid and inexpensive point of care measurement of various molecules of interest levels in a broad range of samples with applications in health [2], environment [3], industries and fundamental research [4]. The potential of these types of devices lies in the fact that they are able to deliver information on the presence and levels of certain chemicals without the need for expensive reagents, trained operators and large size facilities. Transcription factor based biosensors are a subset of those sensing devices repurposing transcription factor and inducible promoter for the quick detection of molecules of interest.

Cell-free biosensors emerged as an alternative for traditional whole-cell biosensors, solving several issues carried by these systems [5]. Cell-free biosensors are efficient, generating big signal over noise ratios [START_REF] Voyvodic | Plug-and-play metabolic transducers expand the chemical detection space of cell-free biosensors[END_REF]. They are suitable for the detection of molecules toxic for bacterial growth or molecules that are not able to cross the cell membrane. Their ability to be freeze dried on paper enables long term storage at room temperature without loss of activity pushing their use as point-of-care devices, especially in low resource communities [6]. The absence of living organisms in it is also facilitating their industrial developments with reduced regulatory issues and low biosafety concerns. Finally the ability to express each gene from distinct DNA fragments (plasmids or linear fragments) [7] and to fine tune their expression levels by varying their respective concentrations enable a fast development and optimisation of new biosensor candidates or more complex devices relying on them. This central property of straightforward tuning of gene expression is necessary for the development of the complex information processing systems that will be described in this chapter.

The main issue encountered in the development of transcription factor (TF) based biosensors is the limited number of molecules for which an interacting transcription factor has been described. The list of these metabolites known to trigger transcriptional response, either in natural systems or in the context of synthetic biosensors, have been described in various databases [8] [9] and compiled in a dataset of small molecules triggering transcriptional and translational cellular responses [10].

In order to detect molecules for which associated transcription factors are not available, new methodologies have been developed relying on the use of metabolic pathways. It has indeed been shown that enzymes could be used to convert non-detectable molecules into molecules known to regulate a characterized transcription factor [11]. This framework has been formalized and showcased as the Sensing-Enabling Metabolic Pathway (SEMP) concept [START_REF] Libis | Expanding Biosensing Abilities through Computer-Aided Design of Metabolic Pathways[END_REF]. This shifted the problem from finding a TF binding the molecule of interest to finding any potential enzymatic route between the molecule of interest and the known set of detectable molecules. This was addressed by repurposing bioinformatic tools traditionally used for retrosynthesis [START_REF] Delépine | RetroPath2.0: A retrosynthesis workflow for metabolic engineers[END_REF] into a platform able to find potential SEMP for any molecule of interest. This web-service called SENSIPATH is a CAD program predicting pathways of up to 2 steps allowing detection for any query compounds [START_REF] Delépine | SensiPath: computer-aided design of sensing-enabling metabolic pathways[END_REF]. This approach was then used for the development of a plug and play cell-free workflow where enzymes formalized as metabolic transducers were used as interchangeable modules converting various chemicals into detectable molecules called effectors [START_REF] Voyvodic | Plug-and-play metabolic transducers expand the chemical detection space of cell-free biosensors[END_REF]. Such biosensors were found to be highly efficient, the cell-free system potentialities allowing to reach optimal response for each biosensor by a systematic tuning of each component involved (reporter, transcription factor and enzyme encoding DNA). They were also applied for the commercially relevant sensing of molecules of interest in complex real-world samples (preservative in beverage, disease biomarker and drug in clinical samples).

In addition to its biosensing potential, this framework has also then been applied to the field of information processing. Indeed, the constructed devices sensing various molecules can also be conceptualised as information processing systems, converting an input signal (the concentration of detected molecules) into an output one (the level of protein synthesised by the reporter plasmid). This processing device uses the continuous concentration of input metabolites as analog signals able to be processed by the metabolic or the genetic layer featured in the system. Multiple molecules can be continuously converted into the same effector at the same time allowing the construction of multi-input devices derived from analog adders.

Starting from this basis, we have implemented a machine learning architecture that allows integration of AI computations in cell-free systems [START_REF] Pandi | Metabolic perceptrons for neural computing in biological systems[END_REF]. One of the simplest architectures for signal processing is the perceptron [START_REF] Rosenblatt | The Perceptron, a Perceiving and Recognizing Automaton Project Para[END_REF]. A single-layer neural network transforms an input vector into an output value through the application of a weighted sum of the input followed by a thresholding function on the obtained result. Input data can be integrated as categorical or numerical (discrete or continuous). This signal-processing unit has been implemented in a cell-free system to enhance its multiplex sensing abilities [START_REF] Pandi | Metabolic perceptrons for neural computing in biological systems[END_REF]. This system is indeed capable of classifying complex samples in two categories (above or below a threshold of the reporter gene expression) based on the recognition of certain patterns of molecules present in those samples. A theoretical perceptron was first constructed and trained "in silico" to compute the optimal set of weights required to solve a defined problem of classification. The perceptron was then implemented in cell-free using a molecular model, giving concentrations of enzymes coding DNA that correspond to the weight of each input (see figure 1). The thresholding applied through the effector biosensor dose response curve.

[[ Figure 1 should be placed near here. ]] Different elements of the computing unit have a defined terminology: e.g. transducer, adder, actuator. A transducer, in cell-free terms, means one or more enzymes transforming one molecule into another. An adder consists of several reactions having different substrates but yielding the same compound. An actuator is simply a way for the cell-free system to report its activity, through the expression of a reporter gene, for example. Theoretically, many more parameters of a cell-free reaction could be implemented in such signal-processing manner.

The diversity of elements can be increased, as well as the number of layers in the perceptron, and the number of detectable molecules.

Here we describe a straightforward methodology that details how to use CAD tools to identify potential new biosensors for a chosen molecule, how to build and test these biosensor candidates and finally how to repurpose these cell-free biosensors into signal processing devices to implement neural computing in biological systems. This method to this date has been applied for the construction of a single layer 4-input perceptron but is easily scalable for more inputs and adaptable for more complex designs including multi-layer perceptron and other types of computational architectures. An example of potential implementation of a multi-layer perceptron using this system can be found in the supplementary figure 14 of the original metabolic perceptron paper [START_REF] Pandi | Metabolic perceptrons for neural computing in biological systems[END_REF].

Materials

Preparation of cell-free extract and buffer

Extract preparation equipment

In addition to common lab equipment, material for the preparation of Cell extract include large volume centrifugation equipment (for 1 L bottles) and a French Press.

Extract preparation media and buffers

Cells are grown in 2YTP media (31 g/L 2xYT, 40 mM potassium phosphate dibasic, 22 mM potassium phosphate monobasic) S30A and S30B buffers are also required for the preparation of the Cell Extract They are prepared with the following components:

 S30A = (

in silico materials

The R programming language (version 3.2.3) was used for fitting experimental data from actuator and transducers, as well as for defining and solving classification tasks. One can perform calculations described in section 3.6. with regular computational power, computational clusters are not required. Other programming languages can be used, but we recommend to use R in order to take advantage of our git repository containing all files from Pandi et al. work [START_REF] Pandi | Metabolic perceptrons for neural computing in biological systems[END_REF]. The repository is freely available here: https://github.com/brsynth/metabolic_perceptrons/tree/master/cell_free 3. Methods From this point using cross-references between databases or identifiers such as EC number, you can find enzyme candidates in UNIPROT with described catalysed reactions matching your expectations. You can also use the computational tool Selenzyme (http://selenzyme.synbiochem.co.uk/ ) with SENSIPATH provided references to find potential hits for other enzymes candidates (Note 3) 9. Retrieve DNA sequences for the identified parts. TF and Enzymes sequences can be codon optimised using any of the available tools. Promoter sequence is often defined as the 200 nucleotides before the start codon of a regulated gene. For Troubleshooting purposes, various size promoters can be synthesised to test for their response to the TF and their transcription initiation ability. (Note 4)

Constructing candidate biosensors plasmids

1. Synthesise the previously isolated sequence of your Transcription Factor, its regulated promoter and any enzyme required to convert the molecule you want to detect into the TF binding effector. This mix is then typically incubated in a thermocycler applying the following program: 30 s at 98 °C then 35 cycles with (10 s at 98 °C, 30 s at Tm +3 °C, 30 s/kb at 72 °C) then 2 minutes at 72 °C. Tm beings the lowest melting temperatures of the 2 primers and kb being the size of the amplicon in kilo bases. Use primer pairs FBP/RBP on the template plasmid 114598 to reamplify the linearized reporter backbone and the primers FBC/RBC on the template plasmid 114597 to reamplify the linearized backbone for enzyme or TF. Use the newly designed primers from step 2 on the synthesised parts from step 1 to reamplify the inserts. 4. Run an electrophoresis on the PCR product on a 1% Agarose gel stained with SYBR Safe using an appropriate DNA ladder to be able to discriminate your amplicon by its size. After an approximate time of 30 minutes at 100 V identify and cut the band of the expected size by imaging on a blue light transilluminator. 11. Finally spread 100µL of the final mix on LB Agar + Ampicillin (100 µL/mL) plates and incubate overnight at 37 °C.

12. Small scale culture for screening 1. Select 4 colonies per assembly and inoculate them in 3mL LB medium + Ampicillin (100 µL/mL) overnight.

2. Purify the plasmids from 2mL the bacterial cultures using a plasmid miniprep kit and use the primers seqF and seqR to check the integrity of each cassette by sanger sequencing.

3. Save one correct clone per construction by freezing the rest of the liquid culture at -80 ºC after addition of Glycerol to final concentration of 25%.

13. Large scale culture for plasmid production. You will require big quantities (>100 µg) of each plasmid to run the cell-free reactions for the characterisation and optimisation of each biosensor candidate (Note 7). To do so you have to realize large scale culture and plasmid extraction for each construction.

1. Inoculate 300 mL of LB + Ampicillin (100 µL/mL) from the -80 °C glycerol stock and grow the cells overnight.

2. Pellet the cells by spinning them at 6,000 g for 15 min at 4 °C.

3. Use the Maxiprep kit to recover plasmid DNA from the pellet. 4. After the last step of your purification, resuspend the precipitated DNA in 200 µL pure water in order to have a final solution at high concentration.

5. Measure the final concentration using a nanodrop and adjust it at 1 µM by either diluting it with water either concentrating it using a speedvac machine.

Preparing in house Cell-Free Extract and Buffer

The method briefly described here is adapted from a widely used protocol [START_REF] Sun | Protocols for Implementing an Escherichia coli Based TX-TL Cell-Free Expression System for Synthetic Biology[END_REF]of 3-PGA powered cell-free mix with minor modifications mostly concerning the lysis method and the starting strain.

2. Buffer preparation 1. Starting from individual solutions of chemicals dissolved in pure water (described in detail in the original paper [START_REF] Sun | Protocols for Implementing an Escherichia coli Based TX-TL Cell-Free Expression System for Synthetic Biology[END_REF]) prepare an Amino Acid (at 4X concentration) and an Energy solution mix (at 14X concentration) that will be used for buffer preparation. The Amino Acid mix has to be prepared by mixing all the 20 canonical Amino Acids at a final concentration of 6 mM except for leucine at 5 mM. Prepare Energy solution with HEPES pH 8 700 mM, ATP 21 mM, GTP 21 mM, CTP 12.6 mM, UTP 12.6 mM, tRNA 2.8 mg/ml, CoA 3.64 mM, NAD 4.62 mM, cAMP 10.5 mM, folinic Acid 0.95 mM, Spermidine 14 mM, 3-PGA 420 mM. Store each Mix at -80 ºC.

2. Using previously prepared extract, Amino acid Mix, and the Energy Solution, you will have to evaluate in this order the best concentration of: (i) Mgglutamate, (ii) K-Glutamate, (iii) DTT, and (iv) PEG8000 to add to your buffer to optimize the protein production of your extract. You will run 4 consecutives calibration experiments of 8h cell-free reactions. For each calibration, prepare a master mix for 12 reactions by mixing 88µl extract, 66µl Amino Acid mix, 18.86µl energy solution and 13.2 µL of the Addgene plasmid #40019 concentrated at 200 nM. Add the remaining three components that you are not calibrating for from the following four (Mg-glutamate, K-glutamate, PEG8000

and DTT) at either a starting concentration (if the optimum has not been determined yet), or the optimal concentration determined in a previous step. The concentration that leads to the highest GFP signal at 8h is identified as the optimal one to be used for future calibrations and run.

The starting concentration for each component are the following ones :(6mM for Mg-glu, 80mM for K-glu, 0 for DTT and 2% for PEG8000).

The tested concentration for each component are the following: Mg-glu : {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 mM} K-glu : {0, 20, 40, 60, 80, 100, 120, 140, 160, 180 mM} DTT : {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 mM} PEG8000 {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4%} 3. Prepare the final Buffer mix that will be used for running the cell-free reactions. The following quantities are for 1 tube of 650 µL (you need to scale up to have 2 tubes of buffer for 1 mL of final extract produced): per tube of buffer, add 128.97 µL of Amino Acid mix, 110.54 µL of Energy Solution, PEG8000, K-glu, Mg-glu and DTT according to the best determined concentration and pure water to adjust the volume to 650 µL.

Running Cell-Free reactions

All the cell-free experiments in the following parts should be run according to the same methodology. This method is for the preparation of N number of 20-µL cell free-reactions. All the following reactions have to be prepared on ice and have to be run in technical triplicate.

1. Thaw your cell free reagents (extract and buffer) on ice.

2. Pre-dilute your DNA and inducer stock: you will prepare 22-µL reactions each with 7.33 µL of extract, 9.17 µL of buffer and 5.5 µL of other components (DNA plasmids, inducer(s) and water to adjust). The easiest way to proceed if you have less than 5 different components to add per reaction is to make 20X solutions of the plasmids and inducers you want to use to add 1.1 µL of those to the reaction.

3. Prepare your cell-free Master Mix for (N+15%) reactions to compensate for the pipetting loss. Add (7.33*(N+15%)) µL of cell extract and (9.17*(N+15%)) µL of Buffer to a single 1.5-mL tube. 8. Pipette 20 µL from each tube to a well of a black 384-well plate pre-chilled. 9. Then cover the plate with a transparent sealing film.

10. Using a plate reader, monitor the green fluorescence (ex: 458 nm, em: 528 nm) at various gains during a 12-h kinetic run.

Cell-free biosensors characterizations and optimizations

In order to obtain the best possible biosensor response for our target we need to first develop an efficient biosensor for the TF binding effector, before optimizing the enzymatic conversion of our target molecule into our effector.

The first step is aimed to screen for any potential response of designed candidate biosensors and to answer 2 questions: first is there any interaction between the TF expression and the level of expression of the reporter and then is this interaction modulated in any way by the effector that we are trying to detect. To possibilities depending on the level of constraints you have with the input molecules. If you want to implement a computing device with a defined behavior but without any constraint on the nature of the inputs, you can reprogram the already developed benzoate based perceptron [START_REF] Pandi | Metabolic perceptrons for neural computing in biological systems[END_REF] to implement your desired computational function.

2. If you are interested in multiplexed sensing for defined molecules, you need to identify a common effector they can be converted into. You can use SENSIPATH with a reaction length of 2 to increase the chance to find a product accessible from the 2 or more substrates. Some central molecules (lactate, acetate, hydrogen peroxide, ammonium…) may be useful for that purpose but a specific attention has to be put on the development of biosensors with high dynamic range for these molecules with potential high noise issues related to their central positions on the metabolic networks.

  14 mM Mg glutamate, 60 mM K-glutamate, 50 mM Tris, pH 7.7) S30A is titrated using acetic acid, 2mM DTT are added just before use  S30B = (14 mM Mg glutamate, 60 mM K-glutamate, ~5 mM Tris, pH 8.2) S30B is titrated using 2 M Tris, 1 mM DTT are added just before use 3. Chemicals for the reaction buffer 1. Solutions of the 20 canonical amino acids, 19 of them concentrated at 168 mM, except leucine that is concentrated at 140 mM 2. Chemicals for the energy solution, including individual solutions of: HEPES pH 8, ATP, GTP CTP, UTP, tRNA, CoA, NAD, cAMP, folinic Acid, Spermidine and 3-PGA, all of them concentrated close to their limit of solubility.

3 .

 3 Additional components for the reaction buffer including K-glutamate solution at 3000 mM, Mg-glutamate solution at 100 mM, PEG8000 solution at 40% w/w and DTT solution at 1M. 4. Reference plasmid pBEST-OR2-OR1-Pr-UTR1-deGFP-T500 (Addgene #40019) for reaction buffer calibration experiments 2.2. Cloning of DNA parts and production of plasmids Additional materials for the construction and production of plasmid encoding the elements for Biosensors include : 1. High fidelity PCR Material (Q5 High-Fidelity 2X Master Mix or equivalent) 2. Template plasmid for backbone amplifications (plasmids #114597 and #114598 from Addgene) and respective primers : FWD primer Backbone for CDS : FBC: cccGGTCTCtGCTTactttatctgagaatagtc REV primer Backbone for CDS: RBC: cccGGTCTCtCATCatatctcttcttaaagttaaac FWD primer Backbone for Promoter : FBP:cccGGTCTCtATGCgtaaaggcgaagagctgttc REV primer Backbone for Promoter RBP:cccGGTCTCtTAAGaatagtaatacaggatccgaatcgtttcag 3. Thermocycler 4. TAE buffer, 50X 5. 1% agarose gel with SYBR™ Safe DNA Gel Stain or equivalent 6. Monarch® DNA Gel Extraction Kit or equivalent. 7. Thermo Scientific NanoDrop 1000 for DNA concentration determination or equivalent.

3. 1 .

 1 Identifying sensing routes for a target molecule [[ Figure 2 should be placed near here. ]] 1. Choose a target molecule to sense. Check that the molecule you want to sense isn't one of the components of the Cell-Free Buffer you are using. (Note 1) 2. Identify the InChi identifier or your target. To do so you can either search it by name on the database Pubchem ( https://pubchem.ncbi.nlm.nih.gov/ ) and search for the InChi in the section Identifier. The other possibility is to go on Pubchem sketcher (https://pubchem.ncbi.nlm.nih.gov//edit3/index.html) and to draw the molecule to retrieve it's InChI by replacing SMILEs by StdInChI. 3. Go to http://sensipath.micalis.fr/ to access the online platform. Use the section "Query with a Standard InChI" and paste the InChI of your molecule. 4. Run the tool first for one step pathways. If the results are not satisfying, you can run a second attempt for two step pathways. This can be the case if no pathway to a detectable molecule ( in green on the graph view) is identified or if the identified detectable molecule is non suitable for cell-free biosensors ( components of the Cell-Free Buffer) (Note 2) 5. Identify the promising effector molecules from the "Pathways view" section. If the molecule you want to sense appears in green, it means it is directly detectable without need for enzymatic conversion. For developing a biosensor for it you can skip the enzyme related steps. 6. Download the database of detectable molecules from: https://github.com/brsynth/detectable_metabolites and isolate the lines corresponding to the identified effector. First identify Transcription factor that interacts with your component. You can use column E to retrieve the names of it or, if this information is not available, use the column B that contains literature reference of papers describing potential sensing mechanisms for the molecule.7. Once you find a TF of interest you have to find the promoters that are potentially regulated by it. To do so you can search for it in databases such as RegulonDB (http://regulondb.ccg.unam.mx/) for E. coli, Subtiwiki (http://subtiwiki.unigoettingen.de/) for B. subtilis. You can also try a naive bibliographic search for this regulator to identify features linked to it like regulated promoters but also mechanism of action and possible existing design of biosensors using it. 8. Identify the enzymes converting the molecule you want to sense into the effector for which you found a TF. By clicking on the edges from the SENSIPATH graph view you can retrieve the references associated with the enzymatic reaction of interest.

2 .

 2 Design and orders primers for golden gate cloning of the synthesised parts. Genes (TF and enzymes) requires overhangs in the format, FWD : ccGGTCTCtGATG…. REV : ccGGTCTCtAAGC…. promoters requires overhangs in the format, FWD : ccGGTCTCtCTTA…. REV : ccGGTCTCtGCAT…. 3. Run High fidelity PCRs to amplify with the correct Golden Gate overhangs the Vectors and the synthesised inserts. The reaction typically consists of pipetting 25 µL of 2X Q5® polymerase Master Mix, 2.5 µL of FW and RV primer at 10 µM, 1µL of template DNA (at concentration around 100ng/µL) and 19 µL of water.

5 .

 5 Recover the DNA from the Gel using a DNA Gel extraction kit, following the kit's instructions to purify your DNA fragment. Determine the titer of purified DNA using a nanodrop spectrophotometer. 6. Prepare a golden gate reaction to insert each fragment in its respective backbone (Note 5). To do so you need to calculate the molarity of your DNA. You can use the tool available at https://nebiocalculator.neb.com/#!/dsdnaamt using the nanodrop determined concentrations of each purified DNA fragment. For the golden gate reaction incubate 100 fmol of insert with 50 fmol of linearized backbone in a tube with 1 µL T4 DNA ligase, 1 µL BsaI enzyme, 2 µL T4 DNA ligase Buffer and water to adjust the volume to 20 µL. Incubate the mix at the 37 °C for 1 h and then 16 °C for 5 minutes. (Note 6) 7. Transform DH5α competent cells with the Golden Gate Reaction product. Incubate 5 µL of golden gate product with 50 µL chemically competent cells at 4° C for 30 min. 8. Heatshock at 42 °C for 45 s. 9. Incubate at 4 °C for 3 min. 10. Add 300 µL LB media and incubate at 37 °C for 1h.

Finally, add water

  to this mix to reach a final volume of 237.6 µl. Prepare each calibration reaction by mixing in PCR tubes 19.8 µl of the master mix with 2.2 µl of the tested component concentrated at 20X. Then, take 20 µl from each PCR tube and pipette them inside individual wells of a 384 wells microplate before incubating the plate for 8h while measuring GFP signal produced (ex: 458 nm, em: 528 nm).

4 .

 4 Mix by briefly vortexing. 5. Pipette 16.5 µL of that mix in N PCR tubes (in strips).6. Add the respective other inputs (plasmids, inducers, water…) from the 20X stocks, to each PCR tube up to 22 µL. 7. Close these tubes and mix them by briefly vortexing and bench centrifugation.

2 .

 2 answer those, you need to characterize the behavior of the reporter plasmid in a cellfree reaction in presence or absence of Transcription Factor and in presence or absence of Effector. If you have identified multiple candidates for the sensing of the same molecule this first experiment can also be used to decide what are the TFs and the promoters most promising for the nexts steps. 1. Run cell-free reactions varying the quantity of added plasmids and chemicals in the following possibilities: Reporter DNA concentration at 0 or 30 nM, TF DNA concentration at 0 or 30 nM and inducer concentration at 0, 100 µM or 1 mM. Use end point kinetics results at 8 hours to evaluate the potential mechanism of your TF (activation or repression) and a possible response of your system to the inducer. 3. Find the optimal plasmid concentration for the Reporter and TF that results in a maximal response for your biosensor. With 3 concentrations of inducer (0 µM, 100 µM and 1 mM), test a combination of concentrations gradients for the added plasmid DNA varying reporter and TF DNA concentration on a logarithmic scale (0, 0.1, 0.3, 1, 3, 10, 30 and 100 nM) 4. Using the previously determined best pair of plasmid concentration, evaluate the inducer dose response of the constructed sensor. Run a cell-free experiment with inducers concentration at : 0 nM, 0.5 nM, 1 nM, 2 nM, 5 nM, 10 nM, 20 nM, 50 nM, 100 nM, 200 nM, 500 nM, and 1,000 nM. Dose response curve is obtained by plotting (fluorescence at concentration x / fluorescence at concentration 0) for each datapoint. Use the final results obtained to assess the performance of the developed sensor and its conformity with your objective of sensitivity. (Note 10) 5. Once you have a satisfying sensor for your effector, you can start screening for enzymes candidates to convert your target into this effector. The first experiment to do consists of screening for any potential activity on each of the selected enzymes candidates. Prepare a cell-free experiment with the previously identified best concentration for reporter and TF DNA, varying Enzyme DNA concentration at 0 nM or 10 nM and the molecule of interest at 0 mM or 1 mM. 6. Select the best enzyme candidate and optimize the expression of the enzyme on a dose response curve (same concentrations as in step 3) by varying the DNA concentration coding for the enzyme on the scale of 0, 0.1, 0.3, 1, 3, 10, 30, 100 nM.

3. 6 .

 6 Design, Build and Test a perceptron 1. Define the architecture of the perceptron based on input molecules. There are 2

3 .

 3 Create and fit the cell-free model. It can be decomposed in 2 parts: actuators and transducers.4. An actuator is modelled with a modified Hill function; commonly used in biochemistry for simulating the binding of ligands to proteins according to the ligand concentration.

-

  total: concentration of input metabolite in µM -: concentration of input metabolite yielding half of the maximum induction of the system (also called IC50) -ℎ : Hill coefficient characterizing the cooperativity of the induction system -fc : stands for fold-change, corresponds to the dynamic range (in Arbitrary Units) of the system -basal : basal GFP fluorescence signal without input, i.e. the background noise of the system -lin : accounts for the linearity of the system when dealing with concentrations saturating the Hill transfer function

Acknowledgments

PS and JLF are supported by the ANR SynBioDiag grant number ANR-18-CE33-0015. LF is supported by the French National Research Institute for Agriculture, Food, and Environment (INRAE), through the "Métaprogramme BIOLPREDICT". MK acknowledges funding support from Ile-de-France region's DIM-RFSI, INRAE's MICA department and the University of Paris-Saclay.

We recommend at least some inspiration from our fitting process described in the notes section (Note 11) as it puts an emphasis on tackling the loss of signal of the actuator when the whole system is modelled then implemented.

Otherwise, one can adapt another fitting process for a particular project's needs.

6. Measure the model's performance with different metrics: Root Mean Square Deviation (RMSD); R²; Weighted R² ; Error percentage...

Ensuring high metrics (e.g. above 0.9 for scores between 0-1) on many experimental data points guarantees a robust model for predicting the weights schemes for each classification task.

7. Define your classification problem. Start by defining a set of tasks (here, classifications), which outputs either 0 (OFF) or 1 (ON); as well as a set of weights to be tested (equivalent to a range of possible enzyme concentrations in the cell-free experiment, for our work it was between 0.1-10nM).

8. Continue by sampling uniformly input values for your problem, that is to be resolved by the perceptron. For example, in the case of a binary classification of solutions composed of hippurate and cocaine, sample points in a given range, either for a "low" concentration or a "high" concentration for each of the compounds. Here, let us assume we can sample between 0 and 2 µM for low concentrations and between 80 and 100 µM for high concentrations.

9. For each of the two clusters, choose to sample either in the low or high range for each compound.

10. Then, two clusters have been produced and a binary classifier can be easily defined on these points: the perceptron set of weights and its corresponding fluorescence threshold. Please find a visual example of the definition of a classification task on figure 3. Further is detailed how to find the set of weights and the fluorescence "decision threshold".

[[ Figure 3 should be placed near here. ]]

11. Predict the best set of weights for solving this problem (aka "train" the perceptron on a classification task). Using the previously fitted (part 2) and benchmarked (part 3) model, simulate all possible input combinations with all possible sets of weights.

12. Screen for performant set of weights, i.e. those enabling a sharp threshold between the output states of the system ("ON" or "OFF"). Several thresholds can be tested for considering an output value as ON or OFF, for all possible simulations.

13. Manually select the best set of weights and corresponding threshold, i.e. those that show the highest and clearest difference between ON and OFF behaviors, and in most scenarios. Also, prefer those showing low enzyme concentrations (to avoid resource competition). One can also test several possibilities (several sets of weights and/or thresholds) in the following cell-free experiments.

14. Implement the designs predicted as best ones, to test it in a cell-free reaction.

Start by drawing a test set of chemical (input) combinations from the compositions used for the training of the in silico perceptron. Be cautious to evaluate enough points from the space of possibilities to capture the behavior of your system. If your perceptron is designed to solve the particular problem of binary inputs sample classifications, you can eventually build a complete test set with all the possible combinations of inputs (see figure 1D).

15. The designed test set can be prepared at 20X concentration in PCR tube strips to be evaluated on various perceptron implementations. We advise you to prepare a single master mix with the cell-extract, the buffer and the DNA coding for the reporter system and the various transducers at the desired concentration for each implementation of the perceptron that you designed. [START_REF] Pandi | Metabolic perceptrons for neural computing in biological systems[END_REF]. Use this master mix to evaluate the response of the system on the chemical test set previously designed. If the perceptron does not have the expected behaviour from in silico analysis, try another set of weights.

Notes

1. Most of the molecules present in the cell-free buffer are in high concentration masking any potential response for a biosensor designed to detect them. This list includes all the 20 amino acids, HEPES, ATP, GTP, UTP, CTP, tRNA, CoA, NAD, cAMP, folinate, spermidine, 3-PGA, Magnesium, Potassium and DTT.

2. If you cannot find satisfying results with the software SENSIPATH you can try to find other SEMP using the retrosynthesis workflow Retropath [START_REF] Delépine | RetroPath2.0: A retrosynthesis workflow for metabolic engineers[END_REF]. It has the advantage to allow prediction for pathways with more than 2 steps or pathways using promiscuous activity of enzymes to find potential new reactions. To run retropath for biosensor design, use the list of detectable molecules [10] as Sink, the molecule you want to detect as source and the reaction rules in the forward direction. Results coming from this workflow have to be taken with more care as it relies on less reliable predictions.

3. The Selenzyme tool is predicting potential enzymes catalysing a defined reaction based on similarity of sequences, reactions or other features existing between enzymes from a well-annotated database and enzymes potentially catalysing the query reaction.

The predictions should be manually checked in published literature/ datasets to evaluate if the identified enzyme can likely catalyse the given reaction.

4. The methodology described here is a standard to be used in the case where the identified promoter has limited annotation or features described. You are strongly encouraged to search for existing biosensing projects described in the literature that uses the same Transcription Factor with defined size promoters or synthetic ones (built by inserting TF binding sequences in another promoter) as these promoters may show a better response than the natural ones.

Golden Gate Assembly was chosen over other methods like Gibson Assembly as it

allows reusing the same primers to reamplify backbone for every new insert cloned.

6. This thermocycler protocol for Golden Gate is a variation of the fast Golden gate assembly protocol (1h 37 °C, 10 min 55 °C) that is adapted to backbone plasmid containing internal bsaI cut sites. Removing the last 55 °C step and adding one at 16

°C for 5 min avoid cutting the final assembly containing the BsaI site. First, fit the actuator experimental data (effector dose response curve obtained in part 3.5 section 3) to the Hill function model.

To do so, fit 100 times the actuator experimental values to the actuator model. Let all parameters be able to vary. Use ordinary least-square error or the R "optim" function (that uses the Limited-memory Broyden Fletcher Goldfarb Shanno algorithm) as the objective function for the data fitting. Keep the seed of the fitting process in memory so it ensures reproducibility. Retrieve fitted parameters for each fitting (a population of 100 sets of fitted parameters will be produced). From this population, save the mean, standard deviation, standard error and confidence interval; for each parameter.

To account for the decrease of signal in experimental actuator data when the whole system is implemented in cell-free, one needs to fit the transducer model in a specific way.

To fit the transducers models, we need to actually fit the whole model to the whole system's cell-free experimental data.

In order to have a coherent perceptron, we will actually constrain the actuator's parameters, then fit all transducers and the actuator together.

To do so, start by constraining each of the actuator's previously fitted parameters with bounds corresponding to the 95% confidence interval; or +/-one standard deviation from the mean, in the case of following parameters: the fold-change, termed "fc", and the baseline, termed "basal".

We call "pseudo-experimental" the data used to fit the whole system, but that originates from specific components' experimental data. These individual data were aggregated as described in the whole-system model: combine transducers data by simply adding their output and feeding the sum to the actuator. Note that in the cellfree experimental system, the same process happens: e.g. benzoate concentrations are added as a result of all transducers yielding benzoate, and the actuator takes this aggregated benzoate concentration as an input. As a result, we obtain "pseudoexperimental" data for the whole cell-free system, with experimental data only available for each component separately.

Once your whole system is modelled and you have constrained your actuator's parameters, you can fit your transducer parameters. Initialize the whole model by drawing actuator parameters values according to a Gaussian distribution centered on the mean of the parameters estimations, with a standard deviation equal to the standard error of this parameter estimation. Let all transducer parameters vary to fit the whole system to the previously described 'pseudo-experimental' data.