Martin Skrodzki

Eric Zimmermann
email: eric.zimmermann@fu-berlin.de

Investigations of structures in the parameter space of three-dimensional Turing-like patterns *

In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found two-fold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.

Introduction

In his 1952 paper "The chemical basis of morphogenesis", Alan M. Turing presented a model for the formation of skin patterns [START_REF] Mathison | The chemical basis of morphogenesis[END_REF]. While it took several decades, the model has been validated by finding corresponding natural phenomena, e.g., in the skin pattern formation of zebrafish [START_REF] Asai | Zebrafish leopard gene as a component of the putative reactiondiffusion system[END_REF][START_REF] Scoones | A dot-stripe turing model of joint patterning in the tetrapod limb[END_REF]. More surprising, seemingly unrelated pattern formations can also be studied via the model, like e.g., the formation of plant patches around termite hills [START_REF] Pringle | Spatial self-organization of ecosystems: integrating multiple mechanisms of regular-pattern formation[END_REF]. Furthermore, three-dimensional Turing patterns have been observed in microemulsions [START_REF] Bánsági | Tomography of reaction-diffusion microemulsions reveals three-dimensional turing patterns[END_REF] as well as in physical simulations [START_REF] Teemu Leppänen | A new dimension to turing patterns[END_REF].

In 1984, David A. Young proposed a discretization of Turing's model, reducing it to an activator/inhibitor process on a discrete domain [START_REF] Young | A local activator-inhibitor model of vertebrate skin patterns[END_REF]. This model can be easily extended to the concept of three-dimensional Turing-like patterns [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF]. While the parameter space for two-dimensional Turing patterns has been investigated in the context of predator-prey models [START_REF] Vagner Weide Rodrigues | Pattern formation and bistability in a generalist predator-prey model[END_REF] or Ising models [START_REF] Merle | Turing-like patterns in an asymmetric dynamic ising model[END_REF], no such investigation is available for the three-dimensional Turing-like patterns resulting from the extension of Young's cellular automaton. This exploratory paper presents several findings and conjectures on the parameter space of these automata.

2 Three-dimensional Turing-like pattern from CA Based on previous models for patterns in the visual cortex of the brain ( [START_REF] Swindale | A model for the formation of ocular dominance stripes[END_REF]), Young presented his own two-morphogen model, see [START_REF] Young | A local activator-inhibitor model of vertebrate skin patterns[END_REF]. He assumes two types of cells: differentiated (pigmented) cells (DCs) and undifferentiated cells (UCs). Then, each DC produces an activator morphogen M 1 which differentiates nearby UCs and an inhibitor morphogen M 2 which causes nearby DCs to become undifferentiated. From this model, Turing-like patterns are obtained by starting with a rectangular grid of cells. We will model a cell at position (x, y) in the grid by its state s t (x, y) ∈ {0, 1} at time t, indicating a DC (s t (x, y) = 1) or UC (s t (x, y) = 0).

Initially, each cell is chosen to be a DC according to some random process. To determine the state of a cell (x, y) in the next time step, compute

s t+1 (x, y) =      1 ˜ > 0, s t (x, y) ˜ = 0, 0 ˜ < 0, , (1)
with ˜ = (x ,y )∈B R 2 (x,y) ω t,(x,y) (x , y ), where B R 2 (x, y) is the ball of radius R 2 around (x, y) and

ω t,(x,y) (x , y ) =      0 (x -x ) 2 + (y -y ) 2 > R 2 2 , w 1 • s t (x , y ) (x -x ) 2 + (y -y ) 2 < R 2 1 , w 2 • s t (x , y ) otherwise. (2)
That is, for any grid cell (x, y), all DCs within the circular region of radius R 2 are taken into account. Those that lie within the smaller circular region of radius R 1 contribute weight w 1 while those in the annulus between R 1 and R 2 contribute weight w 2 . If the sum of these weights is positive, the cell becomes differentiated; if the sum is zero, the cell does not change its state; if the sum is negative, the cell becomes undifferentiated. Note that the domain is assumed to be toroidal, i.e., two opposite borders are identified with each other.

In order to lift this model to a three-dimensional domain, the single necessary addition is a term (z -z ) 2 in the right-hand side of Equation (2). To further simplify Young's approach, it can be reduced to choosing initial DCs in the domain uniformly randomly with some probability ρ ∈ [0, 1] and furthermore fixing w 1 = 1, w 2 = -1, see [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF]. Given a domain size, these simplifications reduce the parameter space of Turing-like patterns to choosing a probability ρ ∈ [0, 1] as well as two radii R 1 and R 2 . It is this threedimensional parameter space that we aim to investigate in the following.

Fixing a finite parameter space

In order to experimentally investigate the parameter space spanned by R 1 , R 2 , and ρ, we first fix the size of the cubical three-torus to work on. To be able to fully classify the entire parameter space, we chose a grid size of 70 × 70 × 70. Choosing too large values for the radii R 1 and R 2 in a comparably small domain would obscure the patterns formed. Therefore, we reduced the choice for the two radii to be from {1, . . . , 40} ⊂ N. The value ρ determines the probability of initial activity of each cell in our cubical grid. To discretize the probability, we chose a domain I := {0, . . . , n ρ } ⊂ N with n ρ = 120. As the probabilities ρ = 0 and ρ = 1 only provide completely undifferentiated or completely differentiated1 domains respectively, we chose the lowest probability to be investigated to be at exp(-min ρ ) with min ρ = 12, i.e., the lowest probability as exp(-12) ≈ 6 • 10 -6 . As [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF] found their zero-dimensional behavior for extremely large values of ρ, we decided to discretize this dimension of the parameter space non-linearly. Therefore, the domain is mapped to a probability value via the following sigmoid σ :

I → [0, 1], x → exp(2•minρ •n -1 ρ •x-minρ) 1+exp(2•minρ •n -1 ρ •x-minρ) , (3)
which provides good resolutions at very low and very high probabilities as well as a decent resolution of the behavior between these extremes. In the following, we will assume that some x ∈ I was chosen and simply write ρ := σ(x) for brevity. The three parameters ρ, R 1 , and R 2 span a 121 × 40 × 40 grid where each included triple gives rise to a Turing-like pattern. Considering the types of structures found in previous work [START_REF] Teemu Leppänen | A new dimension to turing patterns[END_REF][START_REF] Bánsági | Tomography of reaction-diffusion microemulsions reveals three-dimensional turing patterns[END_REF][START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF], we expect to find seven different types of structures that can form in the three-dimensional domain. Two of these are the trivial patterns of only differentiated or only undifferentiated cells. The remaining five structures are displayed in Figure 1. Note that for the visualization of three-dimensional patterns, a different approach is necessary than for two-dimensional patterns, in order to gain insights into the presented structures. Thus, following [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF], the renderings in Figure 1 show only the isosurface between regions of DCs (green) and UCs (gray). Shrinking the structures by a factor of approximately R 1 , we reduce them as follows: spheres become 0-dimensional points, tubes become 1-dimensional strands, and areas remain 2-dimensional. In the following, we refer to the patterns by these "dimension" values.

Observe, that except for the 2-dimensional case, we obtain two possible scenarios for 0-dimensional and 1-dimensional structures, which come from the distinction into interior and exterior cell types. The following experimental part of the project now consists of partitioning the discretized parameter space into these seven cases, i.e., identifying for each parameter triple (ρ, R 1 , R 2 ) which type of Turing-like pattern arises.

Exploration of the parameter space

Our exploration of the parameter space consisted of picking a parameter triple and starting the CA of [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF] with it. After the automaton had either converged or after 20 iterations/seconds had passed, the current state was rendered using the Marching Cubes Algorithm [START_REF] Lorensen | Marching cubes: A high resolution 3d surface construction algorithm[END_REF]. The experimentalist then classified the visualized pattern according to their closest resemblance with the images in Figure 1 or (if the pattern clearly had not converged yet) continued the generation for several more iterations. Separating isosurfaces of volumetric distributed triples "UC all" (behind white), "DC spheres" (between white and beige), "DC pipes" (between beige and green), and "DC all" (remaining volume on top). Bottom right: The respective triple groups from the parameter set, giving rise to the separating isosurfaces. All figures follow the color coding according to Figure 1.

The distribution of structures in our segment of the parameter space is shown in a histogram in Figure 2. The dominating structure is "DC all" followed by "UC all", which enclose all other structures in the parameter space. This encapsulating behavior can be seen in the two-dimensional layers of the parameter space shown in the left images of Figure 2, where the "DC all" states reside on top and the "UC all" states on the opposing side of the parameter cube. Note that in the left of Figure 2 and in the upcoming illustrations, we decided to neglect the "DC all" triples for visual reasons: Rendering these would hide the distribution of the other structures within the parameter space. Going back to the histogram in Figure 2, the remaining non-trivial structures follow with decreasing amount, with "UC pipes" and "UC spheres" having the rarest occurrences.

One first trivial observation regarding the distribution of structures can be drawn from the behavior of the activator and inhibitor radii: If R 1 ≥ R 2 , all cells are differentiated (given that the random process guided by ρ > 0 creates at least one active cell), because the inhibitor range does not have any impact.

When considering the special case of the volume of the activator ball and the inhibitor annulus around it being equal, we obtain:

4 3 πR 3 1 = 4 3 πR 3 2 - 4 3 πR 3 1 ⇒ R 1 = R 2 3 √ 2 .
Thus, we can separate the entire three-dimensional parameter space by a separating hyperplane, see Figure 2, top left. Above this separation, the triples are more likely to produce "DC all" structures and below it, they tend to form "UC all" structures. However, as we are dealing with a discretization-i.e., the volume of kernels is not 4 3 πR 3 i , i = 1, 2 but the number of discrete cells within-the obtained structures in our experiments deviate from this theoretical assertion. All such structures ("areas", "UC pipes", and "spheres") occur in the vicinity of this linear separation, see Figure 2, top left.

Within our discretization of the parameter space, we find that those triples form volumetric groups that give rise to the following four structures: "UC all", "DC all", "DC pipes", and "DC spheres". On the lower left of Figure 2, we draw isosurfaces in white, beige, and dark green that separate the respective triple groups, which are shown in the bottom right. From the dark green up to the beige isosurface, we find a slender wedge consisting of "DC pipes". The volume between the beige isosurface and the highly non-planar white surface is filled with "DC spheres" while the rest of the parameter space consists of "UC all". The other structures ("areas", "UC pipes", and "UC spheres") do not obtain volumetric extent in our discretization and with the chosen parameter bounds. They all lie close to the linear separation plane shown in the top left of Figure 2. These observations motivate two conjectures.

Conjecture 1

The wedge, formed in the parameter space by the "DC pipes" triples, given as volume between the beige and dark green isosurface, Figure 2, will grow larger and wider with increasing values for R 1 and R 2 .

Conjecture 2

The collections of parameter triplets associated to the structures: "areas", "UC pipes", and "UC spheres" will gain volumetric extent in the parameter space when increasing the maximal values of R 1 and R 2 .

From the above discussion, it is clear that we were able to find all structures displayed in Figure 1. Thereby, we positively answer a corresponding conjecture, posed in [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF]. Our experiments lead us to the following conjecture.

Conjecture 3 A Turing-like pattern of dimension d (cf. 1) exhibits structures of all dimensions from 0 to d -1. Furthermore, each structure of dimension 0, . . . , d -2 occurs twice as "UC" and "DC" version, while structures of dimension d -1 appear only once.

Towards automatic structure classification

The classification in our experiments where visually guided and depended to a certain extent on the experimentalist. This section is devoted to investigating an automatic classification of the found structures, based on the obtained data. Two entities of interest are the volume and surface area of the structures. The scatterplots shown in Figure 3 suggest that the structures are characterized to a large extend by their volume. Indeed, a corresponding histogram and a violin plot of the volume distribution confirm this first suspicion, see Figure 4.

Based on these preliminary analyses, we aim at building a predictor for the structure type, based on the volume found. We use a total of 2, 214 hand-classified structures to train the predictor. Note that this data is not uniformly selected, but rather chosen along the border of larger patches as shown in Figure 2. For cross-correlation, we separate the data set into five folds, with the distribution of structures as shown in Table 1. To train our predictor, we perform the following operation on all folds. For two neighboring structure types, we use the training data (all data points not in the fold) to find a certain volume-threshold to optimally distinguish these. Thus, we obtain four volume-thresholds for each fold, indicating five intervals that are associated with the respective structures. Then, we use these thresholds to classify the testing data from the fold, where we report an error if the hand-classified label does not agree with the threshold label of the structure, see Table 1. Overall, the prediction of the structure by the volume admits an average classification error of 8.39%. While this indicates the volume to be a very strong predictor, the results differ drastically over the different structures. While "DC spheres" exhibit an extremely low classification error of 3.47% across all folds and "DC pipes" are within the average with an error of 8.05%, all other three structures have way larger classification error, with 42.86% of "UC spheres" being wrongly classified, see Table 1. This confirms the impression from Figure 4 that "UC pipes" and "UC spheres" are not well separated solely by their volume. The scatterplots in Figure 3 suggest that the surface area of the structure does not carry a lot of information regarding the type of the structure, but that the activator radius, in conjunction with the volume, could provide a better classifier. On the same folds as used above, we train a k-nearest neighbor predictor. The used neighborhood sizes were k ∈ {1, . . . , 20}. As the two features, volume and activator radius, exhibit very different sizes, we normalize these two dimensions to exclude any artifacts. For our data set and the number of chosen folds, we obtain a minimal classification error for k = 1, where the mean classification error over all folds is 7.35%. Thus, most of the information on structures is already captured by the volume and the activator radius only provides a minimal gain in information.

Considering that the data is randomly generated and hand-classified, the predictors presented here already obtain a high level of accuracy. It is clear from our findings that the volume of a structure is the best predictor for its type. Further research is necessary to provide more precise classifications.

Conclusion

In this paper, we have presented a visually-guided investigation of structures in three-dimensional Turing-like patterns. Based on a discretization, we gave a complete partition of a finite portion of the parameter space and found five different non-trivial structures, see Figure 1. Thereby, we confirmed a conjecture of [START_REF] Skrodzki | Turing-like patterns revisited: A peek into the third dimension[END_REF], who proposed the existence of two-dimensional structures in three-dimensional Turing-like patterns. Based on our experiments, we furthermore provided statistical insight into the distribution of the different structures within the parameter space. From these data, we presented several conjectures both regarding three-and higher-dimensional Turing-like patterns.

Future research has to be directed at finding a thorough mathematical description of the different structures described in this work, e.g., based on the number of connected components or the Euler characteristic. We have shown that the volume is a strong predictor to classify the "DC" structures, but it does not work well on the "UC" structures. Furthermore, a larger portion of the parameter space has to be investigated to better understand the zero-, one-, and two-dimensional structures that arise. The experiments presented in this paper can help narrow down a corresponding portion of the parameter space and thus help focus on the relevant parts.

Figure 1 :

 1 Figure 1: Five of the seven possible Turing-like patterns in a three-torus, with the trivial cases of all DCs or all UCs not shown. The images show the isosurface separating DC (green) and UC (gray) cells. Patterns include sphere-like structures, (a) and (e), pipe-like structures, (b) and (d), and area-spanning structures (c)).

Figure 2 : 2 - 1

 221 Figure 2: Top and bottom left: Plot of the three-dimensional parameter space consisting of activator radius R 1 , inhibitor radius R 2 , and activation (x as argument for ρ = σ(x)). The parameter triples are colored by found structures. In the top left image, the shown hyperplane with increase 3 √ 2 -1 indicates equal volume covered by the R 1 ball and the R 2 annulus. Note that the color for all-DC triples is black, and we disabled their rendering for enhanced visibility of the other structures. Note further that the x-axis is not a linear depiction of the activation probability, but the argument for σ(x), see Equation (3). Top right: Histogram over the found structures. Note that the y-axis in the histogram follows a logarithmic scale. Bottom left:Separating isosurfaces of volumetric distributed triples "UC all" (behind white), "DC spheres" (between white and beige), "DC pipes" (between beige and green), and "DC all" (remaining volume on top). Bottom right: The respective triple groups from the parameter set, giving rise to the separating isosurfaces. All figures follow the color coding according to Figure1.

Figure 3 :

 3 Figure 3: Two plots indicating the relation between volume and surface area (left) as well as volume and activator radius (right) of the structures.

Figure 4 :

 4 Figure4: Left: Histogram over 2,214 data points from the parameter space, selected from the border of two structure areas. The bins on the x-axis represent the percentage of activated cells in the final state (excluding "UC all" and "DC all" at the far ends). The stacked bars are colored following the color scheme of Figure1. Bars are printed up to 85 elements and cut from there, where the actual values are given on top. Right: Violin plot over the different types of structures, indicating the distribution of the volume that is taken by each structure. Note the box-plot overlay indicating the mean as well as the first and third quartile. Outliers are shown in red, most notably for the "areas". Furthermore, "UC pipes" and "UC spheres" do not show a clear separation by their volume.

Table 1 :

 1 Distribution of the different structures over the folds of the data set used for cross-correlation. The errors given indicate classification errors of the structures solely based on the volume of the structure.

		Fold1 Fold2 Fold3 Fold4 Fold5 Error (%)
	"DC spheres"	116	115	115	115	115	3.47
	"DC pipes"	77	77	77	77	77	8.05
	"areas"	29	30	29	30	29	14.97
	"UC pipes"	20	20	20	20	21	18.81
	"UC spheres"	6	6	5	6	5	42.86
	Error (%)	7.26	9.27	8.06	9.68	7.66

Note that complete DCs can turn to complete UCs within one step, e.g., if R2 R1.

* This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Illustrating Mathematics program; This research was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -455095046; Furthermore, this research was supported by the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics", RIKEN iTHEMS, and the German National Academic Foundation.

† corresponding author


BACKGROUND AND RATIONALE

Food systems have been recognised to play a major role in climate change and it was estimated that they contributed to 34% of total anthropogenic greenhouse gas emissions (GHGe) in 2015 [START_REF] Crippa | Food systems are responsible for a third of global anthropogenic GHG emissions[END_REF]. A French study evaluated that food accounted for 24% of the carbon footprint of households [START_REF] Barbier | L'empreinte énergétique et carbone de l'alimentation en France[END_REF]. The largest contributor of these emissions is agriculture since it represents more than two-thirds of the food system's GHGe [START_REF] Barbier | L'empreinte énergétique et carbone de l'alimentation en France[END_REF]. The remaining third of the GHGe is due to transportation, consumption, industrial processes and packaging. A recent study showed that even if we were to stop fossil fuel emissions, the GHGe from food systems would not allow to stay under the targeted 1.5°C increase in global temperatures [START_REF] Clark | Global food system emissions could preclude achieving the 1.5° and 2°C climate change targets[END_REF] highlighting the importance of cutting down agricultural GHGe. It has also been shown that the structure of our diets can be modified to reduce their carbon footprint and remain within a "safe operating space for humanity" as modelled by planetary boundaries [START_REF] Willett | Food in the Anthropocene : The EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF]. In particular, the reduction of meat and dairy consumption and the increase of plant-based food consumption are major levers to reduce the environmental impact of our diets [START_REF] Hedenus | The importance of reduced meat and dairy consumption for meeting stringent climate change targets[END_REF][START_REF] Scarborough | Dietary greenhouse gas emissions of meat-eaters, fish-eaters, vegetarians and vegans in the UK[END_REF].

In developed countries particularly, where meat and dairy consumption is high, a shift towards more plant-based diets has been advocated by scientists for its beneficial impacts on health and the environment [START_REF] Willett | Food in the Anthropocene : The EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF]. However, this shift may not be easily achieved by consumers [START_REF] Hartmann | Consumer perception and behaviour regarding sustainable protein consumption : A systematic review[END_REF]. This can be explained by the fact that eating meat is associated with pleasure and has strong cultural, social and personal values in our societies [START_REF] Macdiarmid | Eating like there's no tomorrow : Public awareness of the environmental impact of food and reluctance to eat less meat as part of a sustainable diet[END_REF]. The reluctance to reduce meat consumption may also be associated with the lack of knowledge of the environmental footprint of meat [START_REF] Hartmann | Consumer perception and behaviour regarding sustainable protein consumption : A systematic review[END_REF][START_REF] Macdiarmid | Eating like there's no tomorrow : Public awareness of the environmental impact of food and reluctance to eat less meat as part of a sustainable diet[END_REF]. The underestimation of the GHGe of different types of foods, especially animal products, has been previously highlighted and was considered as a potential area for interventions [START_REF] Camilleri | Consumers underestimate the emissions associated with food but are aided by labels[END_REF]. In line with this idea, studies have shown that consumers with a higher knowledge regarding the impact of human behaviours on climate change declared higher intentions to adopt environmentally-friendly behaviours [START_REF] Truelove | Perceptions of behaviors that cause and mitigate global warming and intentions to perform these behaviors[END_REF].

Environmental labels on food products have been described as potential drivers of the choice of environmentally-friendly products, as they may increase consumers' knowledge about the environmental impact of the food products allowing consumers to make more informed decisions when choosing food products [START_REF] Camilleri | Consumers underestimate the emissions associated with food but are aided by labels[END_REF]. In addition, consumers wish to be informed about the environmental footprint of the products that they consume [START_REF] Gadema | The use and usefulness of carbon labelling food : A policy perspective from a survey of UK supermarket shoppers[END_REF]. However, when presented with those labels, many consumers found them hard to understand and to use [START_REF] Gadema | The use and usefulness of carbon labelling food : A policy perspective from a survey of UK supermarket shoppers[END_REF]). Yet, the use of an environmental label is directly linked to its understanding [START_REF] Grunert | Sustainability labels on food products : Consumer motivation, understanding and use[END_REF]. Thus, there is a need of a clear environmental label that conveys key information regarding the environmental impact of foods and that is easily understood by consumers.

Interventional studies using gold-standard design (i.e., randomised controlled trials) in realistic food purchasing environments are needed to demonstrate the effect of environmental labels on food choices [START_REF] Vyth | Methodological quality of front-of-pack labeling studies : A review plus identification of research challenges[END_REF]. Supermarkets are key environments to test the effect of food labelling interventions since they represented roughly 70% of the total food sales in 2019 in France (Insee, 2019). Studies testing the effect of environmental labels on food choices carried out in realistic food purchasing settings, such as online supermarkets [START_REF] Muller | Environmental Labelling and Consumption Changes : A Food Choice Experiment[END_REF], real supermarkets [START_REF] Pelletier | Informing Food Consumption Choices : Innovations in Measuring and Labelling[END_REF][START_REF] Vanclay | Customer Response to Carbon Labelling of Groceries[END_REF][START_REF] Vlaeminck | Food labeling and eco-friendly consumption : Experimental evidence from a Belgian supermarket[END_REF] or restaurants [START_REF] Brunner | Carbon Label at a University Restaurant -Label Implementation and Evaluation[END_REF] are scarce. To our knowledge, only one randomised controlled trial, consisting in an online food shopping task for soups, tested the effect of an environmental label on food choices [START_REF] Camilleri | Consumers underestimate the emissions associated with food but are aided by labels[END_REF].

We will conduct a 2-arm randomised controlled trial (with and without labels) to test the effects of an environmental label on food choices in a virtual supermarket. Participants will take part in two shopping tasks: 1/ selection of 3 products to prepare a home-made dish, and 2/ selection of a readyto-eat dish. These two tasks will be repeated for two scenarios: 1/ participants will be asked to select the foods as they would do for a usual day (everyday scenario), and 2/ participants will be asked to select the foods for an environmentally friendly meal (environmentally-friendly scenario). This experimental design will allow to compare food choices in the presence vs. the absence of an environmental label and to investigate whether the label is informative and likely to help individuals to choose more environmental-friendly food options when explicitly asked to do so. In line with previous results showing that participants were more likely to choose pulses for a meal at a restaurant compared to a home-cooking scenario [START_REF] Melendrez-Ruiz | A central place for meat, but what about pulses? Studying French consumers' representations of main dish structure, using an indirect approach[END_REF], we also hypothesise that it might be easier for a consumer to choose more environmentally friendly ready-to-eat dish than more environmentally friendly ingredients to cook it from scratch.

On top of the lack of knowledge regarding the link between food systems and climate change, it has been emphasized that food choice motivations are key barriers for a shift towards more plant-based diets [START_REF] Grunert | Sustainability labels on food products : Consumer motivation, understanding and use[END_REF]. Many criteria are considered when purchasing foods and differ among consumers. In the French population taste, price and geographical origin of products are the key declared criteria influencing food choice (European Commission. Directorate General for Health and Food Safety, 2020). Environmental considerations such as the carbon footprint were poorly rated among food choice motives by UK supermarket shoppers [START_REF] Gadema | The use and usefulness of carbon labelling food : A policy perspective from a survey of UK supermarket shoppers[END_REF]. Similarly, the environment was declared by only 15% of European respondents as an important driver influencing food choice (European Commission. Directorate General for Health and Food Safety. et al., 2020). An additional aim of the present study is to investigate the relationships between food choice motives and the effect of an environmental label on food choices.

OBJECTIVES, OUTCOMES MEASURES AND HYPOTHESES

OBJECTIVES OUTCOME MEASURES HYPOTHESES

Primary objectives 1.1

To study the effect of an environmental label on the environmental impact of ingredients and readyto-eat dish choice in a virtual supermarket.

Mean of PEF 1 score for 100g of each ingredient selected and PEF score for 100g of the ready-to-eat dish in the "everyday scenario".

▪ An environmental label will lower the environmental impact of the food choices made by the participants compared to no label. ▪ The effect of the environmental label will be greater when choosing a ready-to-eat dish compared to when choosing ingredients.

1.2

To investigate whether the environmental label is informative and likely to help individuals to choose environmentally friendly food options when they Mean PEF score for 100g of each ingredient selected and PEF score for 100g of the ready-to-eat dish in the "everyday scenario" and the "environmentally-friendly scenario".

▪ The reduction of the environmental impact of the foods chosen in the "environmentally-friendly scenario" compared to the "everyday scenario" will be greater for participants in the 1 PEF score is the "Product Environmental Footprint". Details regarding its calculation is described in the paragraph "Creation of an environmental score" (page 9).

are encouraged to make environmentally friendly food choices.

environmental label condition than in the no label condition.

Secondary objectives 2.1

To study the effect of an environmental label on the nutritional quality, liking, familiarity, level of process and price of ingredients and ready-toeat dish choices in a virtual supermarket.

The following variables will be calculated for the 3 ingredients (average) and for the ready-to-eat dish chosen in the "everyday scenario":

▪ FSA scores ▪ Liking ▪ Familiarity ▪ NOVA score (only for the average of the 3 ingredients)

▪ Price per calorie (continuous variables) ▪ The environmental label will have an impact on the nutritional quality (FSA score), liking, familiarity, NOVA score and price of the foods chosen by the participants.

2.2

To investigate the relationships between food motives and the effect of an environmental label on ingredients and ready-toeat dish selection in a virtual supermarket.

▪ Mean PEF score for 100g of each ingredient selected and PEF score for 100g of the ready-to-eat dish in the "everyday scenario". ▪ Level of ethical concern (questionnaire score)

▪ The environmental label will be more effective in reducing the environmental impact of food choices in participants with higher levels of ethical concern when making food choices.

EXPERIMENTAL DESIGN

This study will be a randomised controlled trial with two experimental arms: 1/ No label (no label condition) 2/ Environmental label (label condition)

A 2-block randomisation sequence (male and female) of 1:1 will be generated to allocate each participant to one of the experimental arm before recruitment using the Random Allocation software [START_REF] Saghaei | Random allocation software for parallel group randomized trials[END_REF].

Each participant will be asked to perform two hypothetical shopping tasks in a virtual shopping environment, first in an "everyday scenario": 1/ choice of 3 ingredients for an everyday home-cooked dish among 66 products, 2/ choice of a ready-to-eat dish for an everyday meal among 30 products. The order of presentation of the two tasks will be counterbalanced. Then, the participants will be asked to perform the same two tasks while choosing foods for a dish that is "good for the planet" ("environmentally-friendly scenario"). For a given participant, the order of presentation of the two tasks will remain the same as in the everyday scenario.

The experimental design is described Figure 1.

PARTICIPANTS AND RECRUITMENT Recruitment

Participants will be recruited thanks to the "PanelSens" database from the "Chemosens" platform. This database was declared to the relevant authority (Commission Nationale Informatique et Libertés;CNIL;n°1,148,039). Participants will be invited to take part in a lab study that aims to study food choices in a virtual supermarket. After completing the study, participants will receive a 10-euro voucher.

METHODS

Online shopping tasks

Shopping tasks

The experimental sessions will take place individually. After participants will have given their informed consent to take part in the experiment, they will be asked to sit down and will be equipped with a virtual reality headset. A qualified experimenter will be present to assist the participants.

The first step will consist in a test of the virtual environment so that participants can familiarise themselves with the virtual supermarket environment. Participants will be asked to navigate and test the functionalities of the virtual reality headset. This dummy supermarket will only include cosmetic products (shampoos, soaps, etc.).

Then, participants will start the shopping tasks under two scenarios. If participants are allocated to the "label condition", the label will be presented and explained to the participants prior to the shopping tasks with a pop-up message.

1) For the first scenario, in the first task, we will ask the participants to navigate in the aisles of a virtual environment of 66 foods and to select 3 food items in order to prepare a home-cooked dish for an everyday meal. In the second task, the shopping environment will be composed of 30 foods and participants will be asked to select 1 ready-to-eat dish for an everyday meal. The order of these tasks will be balanced.

2) The second scenario will be broken down in the same two tasks. We will keep the same order for the two tasks as in the first scenario. This time we will ask the participants to choose foods for a meal that is "good for the planet".

Experimental virtual supermarket

This study will be undertaken in a virtual reality environment that mimics the aisles, shelve and position of food products of a real supermarket. This virtual environment will be developed by our partner STRATEGIR under Oculus Quest 2 (https://www.strategir.com/fr/). The food products available in the virtual supermarket will be real products that can be found in three French supermarkets (Intermarché, Carrefour and Picard). Participants will see the front of the products on the shelves in the same size as if they were in a real supermarket. With a hand controller, participants will be able to navigate in the virtual supermarket, come closer, seize and examine closely the products they would like to choose. Once a product is selected, they will have the option to either put the product in their shopping cart or to put it back on the shelf.

Choice of the foods available in the supermarket

Based on a recent study carried out in a virtual supermarket (Melendrez et al. working paper), four food groups will be included in our virtual supermarket: "Meat, egg, fish and substitutes", "legumes", "vegetables" and "starchy foods". The number of foods composing each category, for each task of the shopping scenarios ("everyday scenario" and "environmentally-friendly scenario"), is described in Table 2. The foods included in each category are either canned, dry, fresh, or frozen.

As participants will be asked to select foods for a main dish only, the items included are all savoury foods. They are typical foods that can be part of a main dish. We will focus on the main dish as it is the meal item where we found the greater variability in terms of environmental impact, as compared with starters and desserts. Food sub-categories were chosen with distinct environmental impacts to maximise the likelihood of substitutions between food products of contrasted environmental impact.

Availability and placement of the products in the virtual supermarket

The first rule we adopted to create our virtual environment was to keep the same types of shelves as in a real supermarket. The shelves are defined by the storage method of the products. 4 types of shelves have been identified:

• With canned products • With dry products • With fresh products • With frozen products
As in a real supermarket, within each shelf, we grouped products from the same food category (Table 3). To control for availability effect, we chose to allocate the same space to each product.

Intervention (environmental label)

As mentioned in the design of the experiment, an environmental label will be added to the front of the pack of the 96 food items included in the virtual supermarket.

Development of an environmental score:

In order to identify the environmental impact of the 96 foods included in the virtual supermarket, we matched each experimental food product with items in Agribalyse database2 . To create our environmental score, we used the PEF, Product Environmental Footprint, score recommended by the European Commission [START_REF] Pant | Suggestions for updating the organisation environmental footprint (OEF) method[END_REF]. This score has the aim to evaluate the environmental impact of a food product based on 16 indicators such as climate change (kg CO2 eq), ozone depletion (kg CFC-11 eq), marine eutrophication (kg N eq), land and water use or acidification (mol H + eq). These 16 indicators are calculated throughout the life cycle of the products. In order to obtain a unique PEF score, the indicators are weighed according to a method described by the European Commission [START_REF] Pant | Suggestions for updating the organisation environmental footprint (OEF) method[END_REF]. PEF scores for the foods present in the Agribalyse database are expressed in mPt for 1 kg of product. Pt means "eco-indicator point" and on a scale, 1 mPt represents the mean annual environmental load of an average European inhabitant.

In order to highlight PEF score variability across the different products presented in the virtual supermarket we created a segmentation of this score into five modalities. We defined the cut-offs across modalities as the quintile values from a larger selection of similar foods from Agribalyse database. For the first shopping task (choice of 3 ingredients to compose a home-cooked dish), we created quintiles for the PEF score of 1655 foods that correspond to the food groups included in the virtual supermarket (Table 4). We replicated this method for the second task (choice of a ready-to-eat dish among 30 products). The quintiles were therefore calculated for the 281 foods from Agribalyse database with corresponding food groups (Table 4).

The process for the creation of the environmental score, explained for the first shopping task, is summarised Figure 2. Since no environmental label design is currently being used in France for food products, we created a design considering previous findings regarding the most effective label designs.

< Q1 ≤ 0,17 0 < Q1 ≤ 0,20 0,17 < Q2 ≤ 0,35 0,20 < Q2 ≤ 0,36 0,35 < Q3 ≤ 0,62 0,36 < Q3 ≤ 0,46 0,62 < Q4 ≤ 1,36 0,46 < Q4 ≤ 0,84 1,36 < Q5 ≤ 6,09 0,84 < Q5 ≤ 3,23
The ideal label proposed by Carrero and collaborators is "traffic-light labels" [START_REF] Carrero | Designed to Be Noticed : A Reconceptualization of Carbon Food Labels as Warning Labels[END_REF]. It has been shown that green is often seen as a validation ("go") and a positive colour whereas red is associated with negative aspects and danger [START_REF] Schuldt | Does Green Mean Healthy? Nutrition Label Color Affects Perceptions of Healthfulness[END_REF]. This was put forward in research preceding the adoption of a nutritional front-of-pack label [START_REF] Ducrot | Effectiveness of Front-Of-Pack Nutrition Labels in French Adults : Results from the NutriNet-Santé Cohort Study[END_REF].

In September 2020, the French Agency for ecological transition (ADEME) published a design for a label assessing the environmental impact for the life cycle analysis of products or services (ADEME, 2020), see Figure 3. The colour was changed to obtain a "traffic light label" since it is considered as the best option to increase understanding and therefore behaviour change in consumers, see Figure 3. The label will be placed on the bottom part of the products.

Questionnaires after the shopping experience

After the shopping experience, participants will be asked to complete one questionnaire with the virtual reality headset and four questionnaires on a separate computer (Qualtrics). These questionnaires will be presented in the following order:

1) After the four shopping tasks, participants will be asked to evaluate their familiarity and liking of the eight products selected under the virtual reality headset. The eight products will successively appear under the headset and participants will give a familiarity score (frequency of consumption of a similar product, ranging from "never" to "very often") and a liking score (score ranging from 1 to 10) for each product.

2) An investigation questionnaire will then be presented to participants on a computer to identify aim-guessers and record participants' feedback about their experience in the virtual supermarket. We will use questions adapted from the presence questionnaire used in previous studies [START_REF] Schnack | Immersive virtual reality technology in a threedimensional virtual simulated store : Investigating telepresence and usability[END_REF][START_REF] Witmer | The Factor Structure of the Presence Questionnaire[END_REF]. We will ask participants to evaluate, for each scenario, the importance of the presumed price, taste, packaging size, nutritional quality and environmental impact when making their choices ("not influenced" to "very influenced" by each criteria). For participants in the "label condition", we will also investigate if they paid attention, understood and used the environmental label while selecting products. Participants in the "no label condition" will also be asked if they noticed the presence of an environmental label in the virtual supermarket (manipulation check) and if they understand the meaning of the environmental label (displayed in the questionnaire). We will also ask questions to investigate what participants understood when asked to choose an environmentally-friendly dish.

3) From this will follow a food choice motivation questionnaire. We will use the adaptation of the Food Choice Questionnaire [START_REF] Steptoe | Development of a Measure of the Motives Underlying the Selection of Food : The « Food Choice Questionnaire[END_REF] for the French population [START_REF] Cottet | La compréhension des moteurs des comportements alimentaires : Une approche par le food choice questionnaire. 12ème Journée du Marketing Agroalimentaire[END_REF]. This version includes 24 items that are grouped in 9 subscales (health, convenience, sensory appeal, natural content, ethical concerns, weight control, mood, familiarity and price). In order to have a better understanding of the sustainability motives consumers have for selecting their food, we will also ask independently the "Environmental welfare" subscale from the recently published Sustainable Food Choice Questionnaire (SUS-FCQ) [START_REF] Verain | Sustainable food choice motives : The development and cross-country validation of the Sustainable Food Choice Questionnaire (SUS-FCQ)[END_REF]. This subscale is composed of 5 items and its reliability was considered, by its authors, high enough to be used independently. The 5 items were translated in French and back-translated to English. For each item, participants will be asked to give a score ranging from 1 to 4 (1= not at all important, 2= a little important, 3= moderately important and 4 = very important). We will calculate the average scores for the "ethical" subscale of the FCQ and for the "environmental welfare" subscale of the SUS-FCQ. We will use the score with the highest Cronbach's coefficient in the analyses. The Food Choice Questionnaire will be complemented with questions regarding potential changes in food consumption habits that will inform future research work and will not be analysed as part of the present study. This latter was studied by [START_REF] Gadema | The use and usefulness of carbon labelling food : A policy perspective from a survey of UK supermarket shoppers[END_REF] 3 . Will follow, questions evaluating participants' declared capacity to make food choices based on the environmental impact or the nutritional quality of products.

4) A socio-demographic questionnaire to collect data regarding the participants (age, gender, level of qualification, etc.) will be completed by the participants.

STUDY FLOW

RECRUITMENT (Chemosens platform)

▪ An email will be sent to individuals in the PanelSens database.

Inclusion criteria will be mentioned in the email. ▪ If a person wishes to participate in the experiment, a questionnaire to assess eligibility will be sent.

INFORMED CONSENT (On paper)

▪ Participants will read the information sheet ▪ Participants will sign the consent form

RANDOMISATION (Random Allocation Software)

▪ Participants will be randomly allocated to one of the two experimental conditions (1:1) No label (X)

Environmental label (E)

SHOPPING TASKS (virtual reality headset)

▪ Participants will navigate in a dummy virtual supermarket. ▪ Everyday scenario ( 1): Participants will be asked to go shopping in the supermarket twice to select foods for an everyday meal:

-Task 1: Selection of 3 items to compose a home-cooked dish among 66 products -Task 2: Selection of 1 ready-to-eat dish among 30 products (The order of these two tasks will be counterbalanced) ▪ Environmentally-friendly scenario (2): Participants will be asked to go shopping in the supermarket twice to select environmentally friendly ready-to-eat dishes:

-Task 1: Selection of 3 items to compose a home-cooked dish among 66 products -Task 2: Selection of 1 ready-to-eat dish among 30 products (The order of these two tasks will be the same as in the everyday scenario)

FAMILIARITY AND LIKING OF THE PRODUCTS SELECTED (virtual reality headset)

▪ Under the virtual reality headset, participants will be asked to evaluate their familiarity and liking of the 8 products they selected during the 4 shopping tasks.

INVESTIGATION QUESTIONNAIRE Qualtrics

▪ This questionnaire aims to gather information regarding the shopping tasks participants underwent under the virtual reality headset. We will identify potential aim-guessers and food motives during the online shopping experience. For participants in condition E, we will investigate if they paid attention, understood and used the environmental labelling.

FOOD CHOICE QUESTIONNAIRE Qualtrics

▪ Food choice motives will be assessed through an adaptation of the "Food Choice Questionnaire".

SOCIO-DEMOGRAPHIC CHARACTERISTICS Qualtrics

▪ Participants will complete a questionnaire on socio-demographic data.

The complete study flow is represented in Figure 4.

STATISTICAL ANALYSES

All statistical analyses will be performed using SAS version 9.4. The level of significance will be set at p < 0.05 for all pre-registered analyses.

Definition of the population for analysis

Participants who will be included in the statistical analyses are the ones having completed the shopping tasks and answered all of the questionnaires. Aim-guessers will be excluded in the sensitivity analysis.

Participant's characteristics

Participants' characteristics, overall and for each condition, will be displayed in a table. We will include gender, age, highest educational qualification, current employment, number of people living in the household, the perceived financial status of the household, dieting status and BMI. Continuous variables will be summarised using means and standard deviations. Categorical variables will be summarised using counts and percentages.

Overview of the statistical analysis

Objective Dependent variables (Y)

Independent variables (x)

Data used Statistical analysis

1.1.

To study the effect of an environmental label on the environmental

Mean of PEF scores for 100g of each ingredient selected and PEF score for 100g of the readyto-eat dish.

-Environmental label (0= no / 1= yes)

Only data from scenario 1 ("everyday scenario")

Mixed model analysis with random effect of participants. PEF = label (0/1) + type_of_food (I/RTEd) + label*type_of_food impact of ingredients and ready-toeat dish choice in a virtual supermarket.

-Type of food (I4 / RTEd5 )

If label*type_of_food interaction is not significant, it will be removed from the model.

1.2.

To

Description of variables

Dependent variables

Primary outcome

The primary outcome is the environmental impact of the food selected by the participants. The environmental impact of the food selection will be quantified using the PEF score in mPt for 100g of product throughout the life cycle of the products.

The environmental impact of the 3 ingredients (i) chosen during task 1 of each scenario is calculated by the mean of the PEF scores of each product (PEFi) 7 . 6 We will calculate the average for the ethical subscale of the FCQ (Cottet et al. translation) and for the "environmental welfare" subscale of the SUS-FCQ. We will use the score with the highest Cronbach's coefficient. 7 eCO2 stands for CO2 equivalent

𝑃𝐸𝐹 𝑗 = ∑ 𝑃𝐸𝐹 𝑖 𝑖=3 𝑖=1
3 PEFj is the environmental impact for 100g of the selection of the 3 ingredients (mPt/100g). PEFi is the environmental footprint for 100 g of each ingredient i.

The environmental impact of the ready-prepared dish chosen (task 2 of each scenario) is assessed by the PEF score for 100g of this product.

Secondary outcomes

▪ The nutritional quality of the dishes chosen by participants is assessed by the FSA score (the British Food Standards Agency nutrient profiling system) [START_REF] Rayner | The UK Ofcom Nutrient Profiling Model -defining 'healthy' and 'unhealthy' food and drinks for TV advertising to children[END_REF], as described for each product [START_REF] Julia | Nutri-Score : Evidence of the effectiveness of the French front-of-pack nutrition label[END_REF].

The nutritional quality of the home-cooked dish (j) chosen during task 1 of each scenario is calculated by the mean of the FSA scores of each of the 3 ingredients (i) composing the dish.

FSAi scores are calculated for 100g of the ingredient i.

The nutritional quality of the ready-prepared dish chosen (task 2 of each scenario) is assessed by the FSA score for 100g of this product.

▪ The liking score of the home-cooked dish (j) is the average of the 3 ingredients (i).

𝐿𝑖𝑘𝑖𝑛𝑔 𝑗 = ∑ 𝑙𝑖𝑘𝑖𝑛𝑔 𝑖 3 𝑖=1
3 For the ready-to-eat dish, we will directly use the liking score given by participants. The liking score ranges from 1 to 10, 10 being the highest degree of liking.

▪ The familiarity score of the home-cooked dish is the average of the familiarity score of the 3 ingredients selected.

𝐹𝑎𝑚𝑖𝑙𝑖𝑎𝑟𝑖𝑡𝑦 𝑗 = ∑ 𝑓𝑎𝑚𝑖𝑙𝑖𝑎𝑟𝑖𝑡𝑦 𝑖 3 𝑖=1
3 The familiarity score of the ready-to-eat dish is directly used. Familiarity scores range from 1 to 5 (1=Never and 5=Very often).

▪ The NOVA score of the home-cooked dish is the average of the NOVA score of the 3 ingredients selected.

𝑁𝑂𝑉𝐴 𝑠𝑐𝑜𝑟𝑒 𝑗 = ∑ 𝑁𝑂𝑉𝐴 𝑠𝑐𝑜𝑟𝑒 𝑖 3 𝑖=1
3 The NOVA score of the ready-to-eat dish will not be analysed since, by definition, all of these foods will be categorised in category 4 (ultra-processed foods).

▪ The price per calorie of the home-cooked dish is the average of the price per calorie of the 3 ingredients selected. The price per calorie of the ready-to-eat dish is directly used.

𝑃𝑟𝑖𝑐𝑒 ( € 𝑘𝑐𝑎𝑙

) 𝑗 = ∑ 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑐𝑎𝑙𝑜𝑟𝑖𝑒 𝑖 3 𝑖=1 3 The price per calorie for one product will be calculated with the following equation:

Independent variables

Variables for the primary outcomes ▪ Labelling (presence (yes) vs. absence (no) of an environmental label) ▪ Scenario ("everyday scenario" vs. "environmentally friendly scenario") ▪ Type of food present in the virtual environment (Ingredients vs. Ready-to-eat dish)

Variables for the secondary outcomes ▪ Labelling (presence (yes) vs. absence (no) of an environmental label) ▪ Scenario ("everyday scenario" vs. "environmentally friendly scenario") ▪ Type of food present in the virtual environment (Ingredients vs. Ready-to-eat dish) ▪ Ethical concern scores will be calculated for: o the items of the "Environmental Welfare" subscale from the Sustainable Food Choice Questionnaire (SUS-FCQ). o the items of the "Ethical" subscale from the French translation of the Food Choice Questionnaire (SUS-FCQ). Each rating for individual items, such as "is produced in an environmentally friendly way", ranges from 1 to 4: 1 = Not at all important; 2 = A little important; 3 = Moderately important; 4 = Very important. Out of the two ethical concern scores described above, we will use the one with the highest Cronbach's alpha in the analyses. This score will be noted "Ethic_score" and will be regarded as a continuous variable.

Other variables

Socio-demographic characteristics ▪ Gender: 0=male, 1=female ▪ Age: continuous variable ▪ The level of education: This variable will be coded as the highest educational qualification from 1 to 4: 1= "< High school + 2-years diploma", 2= "high school + 2-years diploma", 3= "High school + 3 or + 4-year diploma" and 4 = "≥ High school + 5-year diploma". ▪ Employment status: The employment status of the participant will be coded from 1 to 6: 1 = "full or part-time", 2="student", 3="retired", 4="looking for a job", 5="looking after home" and 6="other". ▪ The number of people living in the household: this will be regarded as a continuous variable. ▪ Dieting status: 0 = no, 1 = yes. ▪ BMI: BMI will be calculated as weight (kg) / height (m 2 ). BMI data will be trimmed for implausible values excluding weight for less than 30 kg and more than 250 kg, height for less than 145 cm and more than 3m, BMI < 14, or BMI > 48 [START_REF] Hardy | Harmonised height, weight and BMI user guide[END_REF][START_REF] Miller | Recommendations for the truncation of Body Mass Index in population data[END_REF]. We will use the BMI as a continuous variable or use the categories defined by the WHO.

Missing data

We do not anticipate missing data for our analysis. It will be impossible for the participant to finish the shopping scenarios without having selected the 8 products. We have calculated the FSA-NPS scores and the PEF score for all of the products presented in our virtual supermarket.

We do not anticipate missing data for the dependent variables because the questionnaires will not allow missing answers. Data from participants who start but do not finish the study will not be included. Any a posteriori withdrawal will be reported and reasons for withdrawal will be documented (e.g., incorrect answers, technical problems).

Primary analyses MODEL 1.1: Mixed model analysis will be used to test the effect of labelling (categorical variable: yes or no), type_of_food (categorical variable: ingredients or ready-to-eat dish) and labelling* type_of_food interaction on PEF scores in the "everyday scenario", with random effect of participants to account for correlation between repeated measures. If the labelling*type_of_food interaction is not significant, it will be removed from the model.

MODEL 1.2:

• Mixed model analysis will be used to test the effect of labelling (categorical variable: yes or no), type_of_food (categorical variable: ingredients or ready-to-eat dish), scenario (categorical variable: "everyday scenario" or "environmentally friendly scenario"),labelling*scenario, labelling*type_of_food and scenario*type_of_food interactions on PEF scores, with random effect of participants to account for correlation between repeated measures. Non-significant interactions will be removed from the model. • If the labelling*type_of_food or the scenario*type_of_food interaction is statistically significant, two mixed model analysis will be carried out for the home-cooked dish (average of the 3 ingredients) and for the ready-to-eat dish separately. These models will test the effect of labelling (categorical variable: yes or no), scenario (categorical variable: "everyday scenario" or "environmentally friendly scenario") and labelling*scenario interaction on PEF scores, with random effect of participants to account for correlation between repeated measures.

Secondary analyses

▪ For the first secondary analysis (MODEL 2.1), mixed model analysis will be used to test the effect of labelling (categorical variable: yes or no), type_of_food (categorical variable: ingredients or ready-to-eat dish) and labelling*type_of_food interaction on 4 dependent variables, with random effect of participants to account for correlation between repeated measures. We will only look at the data from the first scenario ("everyday scenario"). If the labelling*type_of_food interaction is not significant, it will be removed from the models.

ANOVA model will be used to test the effect of labelling (categorical variable: yes or no) on NOVA score for task 1 (for the 3 ingredients chosen). We will only look at the data from the first scenario ("everyday scenario").

▪ For the second secondary analysis (MODEL 2.2), we will only use "everyday scenario" data.

o Mixed model analysis will be used to test the effect of labelling (categorical variable: yes or no), type_of_food (categorical variable: ingredients or ready-to-eat dish), labelling*type_of_food interaction, Ethic_score (continuous variable) and labelling* Ethic_score interaction on PEF scores, with random effect of participants to account for correlation between repeated measures. If the labelling*type_of_food interaction is not significant, it will be removed from the model. o If the interaction between the labelling condition and the type of food is statistically significant., two ANCOVA models will be carried out for the home-cooked dish (average of the 3 ingredients) and for the ready-to-eat dish separately. ANCOVA models will be used to test the effect of labelling (categorical variable: yes or no), Ethic_score (continuous variable) and labelling*Ethic_score interaction on the mean PEF score.

Sensitivity analyses

▪ Sensitivity analysis will be carried out to analyse if the results found for MODEL 1.1. remain the same when aim guessers are excluded. ▪ MODEL 1.1. will be adjusted for age, sex, level of education and BMI. ▪ MODEL 1.1. (raw and adjusted) will be replicated for the GHGe from the Agribalyse database instead of the PEF score.

Exploratory analyses

Descriptive analyses of the answers to the investigation questionnaire will be performed.

1/ We will calculate the scores of participants responding to the following questions depending on the label condition and the scenarios ("everyday" / "environmentally friendly"):

• How much did taste / nutritional quality / environmental impact / packaging size / presumed price influence participant's food choice in the virtual supermarket? (5 possible answers ranging from "strongly disagree" to "strongly agree")

2/ We will calculate the scores of participants responding to the following questions depending on the label condition:

• What participants understood when asked to make food choices that are "good for the planet" (8 possible answers).

• Participant's perception of their capacity to distinguish 2 foods depending on their environmental impact (5 possible answers ranging from "strongly disagree" to "strongly agree").

3/ We will calculate the scores of participants, in the labelling condition, responding to the following questions:

• If they noticed the label (0=they did not notice the presence of the label on the products chosen or thought that the label was present on less than half of the products chosen /1=the participant thought the label was present on all of the products chosen or on more than half of them).

• If they understood the environmental label presented on the foods in the virtual supermarket (0=the participant did not answer correctly / 1=the participant answered correctly).

• If they used the environmental label when making food choices (we will categorize the scale ranging from 0 to 100 into 4 parts).

Further exploratory analyses will be conducted to have a broader understanding of the effects of the environmental label on food choices and are summarised in Table 6. -Attention paid by the participant to the label (0=they did not notice the presence of the label on the products chosen or thought that the label was present on less than half of the products chosen /1=the participant thought the label was present on all of the products chosen or on more than half of them)

-Type of food (I / RTEd) -Scenario ("everyday scenario"/ "environmental-friendly scenario") -Use of the label by the participant (continuous variable ranging from 0 to 100) -Type of food (I / RTEd) -Scenario ("everyday scenario"/ "environmental-friendly scenario")

-Time spent to carry out each shopping task8 (continuous variable) -Environmental label (0= no / 1= yes) -Type of food (I / RTEd) -Score obtained for the question regarding the knowledge about the environmental impact of food systems (continuous variable).

-Environmental label (0= no / 1= yes) -Type of food (I / RTEd) (knowledge_score*labelling interaction will be interesting to analyse in order to test the moderating effect of knowledge on the environmental impact of food choices) Mean FSA scores -The average ratings for items in the health subscale of the FCQ -Environmental label (0= no / 1= yes) -Type of food (I / RTEd)

Sample size

We powered primary analyses in order to detect a d=0.50 effect size of labelling on PEF scores based the results of a previous randomised control trial that tested the effect of an environmental label on food choices [START_REF] Camilleri | Consumers underestimate the emissions associated with food but are aided by labels[END_REF]. A sample size of 122 participants will be required for 80% power at α = 0.05 (SAS 9.4). We will recruit a sample of 130 participants to account for potential data loss due to technical problems.

Figure 1 .

 1 Figure 1. Experimental design for the virtual reality tasks performed by participants

Figure 2 .

 2 Figure 2. Creation of an environmental score for the label

Figure 3 .

 3 Figure 3. Design of the environmental label inspired by the French environmental label of ADEME

Figure 4 .

 4 Figure 4. Complete study flow.

Table 2 .

 2 Foods included in the virtual supermarket

	Tasks for
	each shopping	Main Food Groups	Food Sub-groups	Number of products included
	scenario
		Starchy foods	-	12
		Vegetables	-	12
		Legumes	-	12
			Beef and lamb	6
	Task 1:
	Home-cooked dish		Other meat	6
		Meat, eggs, fish and substitutes	Fish	6
			Other animal products	6
			Plant-based meat substitutes	6
		Beef and lamb	/	6
		Other meat	/	6
	Task 2: Ready	Fish	/	6
	to eat dish	Vegetarian dishes (with animal	/	6
		products)
		Vegan dishes (without animal	/	6
		products)

Table 3 .

 3 Number of products composing each shelf of the virtual supermarket.

		Task 1: Home-cooked dish		Task 2: Ready to eat dish
	Food categories	Canned products	Dry products	Fresh products	Ready-to-eat dish category	Frozen products	Fresh products
		Dairy products	0	2	4	Dairy products	4	2
	Meat, egg,	Poultry and pork	0	0	6	Poultry and pork	3	3
	fish and substitutes	Cow or sheep meat	0	0	6	Cow or sheep meat	3	3
		Sea food	2	0	4	Sea food	4	2
		Plant-based substitutes	0	2	4	Plant-based substitutes	4	2
		Vegetables	5	0	7	Total	18	12
		Legumes	4	6	2
		Starchy foods	1	10	1
		Total	12	20	34

Table 4 .

 4 Food groups from the Agribalyse database that are common with the food items included in the virtual supermarket

Food groups from Agribalyse used to create a list of foods from which we created quintiles

	Task 1: Choice of 3 ingredients to compose a	Task 2: Choice of a ready-prepared dish among 30
	home-cooked dish among 66 products	products
	• Fruits, vegetables, legumes, nuts and
	seeds	• Starters and dishes
	• Cereal products
	• Milk and dairy products
	• Meat, egg and fish
	These food groups represent:
	N=1655 foods	N=281 foods

Table 5 .

 5 Limits of the quintiles created for the PEF scores of the different foods selected. The values in the following table are PEF scores expressed in (mPt/kg of product)

	Ingredients	Ready-to-eat dishes
	1655 foods	281 foods
	0

  The 5 mixed model analysis that will be carried out are:o FSA = labelling (categorical variable: yes or no), type_of_food (categorical variable: ingredients or ready-to-eat dish) and labelling* type_of_food interaction o Liking = labelling (categorical variable: yes or no), type_of_food (categorical variable: ingredients or ready-to-eat dish) and labelling* type_of_food

	interaction
	o Familiarity = labelling (categorical variable: yes or no), type_of_food
	(categorical variable: ingredients or ready-to-eat dish) and labelling*
	type_of_food interaction
	o Price per calories (€/kcal) = labelling (categorical variable: yes or no),
	type_of_food (categorical variable: ingredients or ready-to-eat dish) and
	labelling* type_of_food interaction

Table 6 .

 6 Summary of exploratory analysis.

	Data used of	Dependent variable	Independent variables	Statistical analysis
	the analysis
	Both labelling	Importance of food	-Environmental label (0= no / 1= yes)
	conditions and	choice motives for	-Scenario ("everyday scenario"/
	the 2 scenarios	each scenario	"environmental-friendly scenario")
	will be	(taste, nutritional
	considered	quality,
		environmental
		impact, presumed
		price and packaging
		size)
		Continuous variables
	Data for
	participants in
	the "label			Mixed model analysis
	condition" only
		Mean PEF scores

Agribalyse is a database with the environmental impact, calculated thanks to a life cycle analysis, of 2480 different foods created by the French Environmental and Energy Management Agency (ADEME). https://agribalyse.ademe.fr/app/aliments

These questions have the purpose to inform an additional research project.

Ingredient (3 ingredients are chosen by the participant)

RTEd: ready-to-eat dish

The time spent in the supermarket is calculated under the virtual reality headset. We can also identify the time spent between for each task performed by participants.