Amel Ennaceur

Zied Elouedi

Eric Lefevre

Belief AHP method: AHP method with the belief function framework

Keywords: Multi-criteria decision making, AHP, Belief function theory, Belief pair-wise comparison, Uncertain preferences

In this paper, the Analytic Hierarchy Process (AHP) method is extended to an uncertain environment where the uncertainty is represented by belief functions as interpreted in the Transferable Belief Model (TBM). Our proposed approach, called belief AHP, is developed to help the decision maker to determine what the best alternatives are, considering multiple conflicting criteria where both alternatives and criteria may be soiled with imperfection. The Belief AHP method aims at comparing subsets of criteria and groups of alternatives in order to reduce the pair-wise comparisons number. Furthermore, to handle uncertainty that may appear in the comparison procedure, we use basic belief assignments instead of exact ratios to elicitate expert preferences. Finally, to illustrate the feasibility of our approach and to judge its performances, we have applied our proposed method on a real application problem: we have considered the PVC life cycle assessment especially the end of life phase.

Introduction

In real life decision making situation, the decision maker faces with many cases in which he has to make a decision among different alternatives. However, the most preferable one is not always easily selected. So, the expert often needs to make judgments on alternatives that are evaluated on the basis of different criteria. [START_REF] Stewart | A critical survey on the status of multiple criteria decision making theory and practice[END_REF] [START_REF] Zeleny | Multiple Criteria Decision Making McGraw-Hill Book Company[END_REF] Within this framework, many methods have been proposed and each one has its own characteristics. In the following, we will focus on the Analytic Hierarchy Process (AHP), [START_REF] Saaty | How to make a decision: The Analytic Hierarchy Process[END_REF] [10] [START_REF] Saaty | The Analytic Hierarchy Process[END_REF] since it is amongst one of the most well known methods. [START_REF] Rao Tummala | A Note on the Computation of the Mean Random Consistency Index of the Analytic Hierarchy Process AHP[END_REF] The strength of this approach is that it is easier to understand and it can effectively handle both 1 qualitative and quantitative data. [START_REF] Liao | A fuzzy approach to business travel airline selection using an integrated AHP-TOPSIS-MSGP methodology[END_REF] [START_REF] Turskis | A model based on Aras-G and AHP methods for multiple criteria prioritizing of heritage value[END_REF] However, the AHP method is often criticized [START_REF] Joaquin | Some Comments on the Analytic Hierarchy Process[END_REF] [3] [START_REF] Wanga | An approach to avoiding rank reversal in AHP[END_REF] due to the exponentially increase of the number of pair-wise comparisons. In reality, the elicitation of preferences may be rather difficult when the number of alternatives and criteria is large. If the number of alternatives (criteria) in the hierarchy increases then, more comparisons are needed to be made. Another weakness of the AHP is its inability to handle the inherent uncertainty of the pair-wise comparison process. Typically, the scale is bounded and it is a verbal description of the intensity of preference and the Saaty scale is used, although any other scale could be used in this method. [START_REF] Beynon | An analysis of distributions of priority values from alternative comparison scales within AHP[END_REF] In some cases, expert would not be able to efficiently express any kind of preference degree between the available alternatives and criteria. This problem of judgment inconsistency has also be handled by some existing works. [START_REF] Kou | A cosine maximization method for the priority vector derivation in AHP[END_REF] [START_REF] Kou | Enhancing Data Consistency in Decision Matrix: Adapting Hadamard Model to Mitigate Judgment Contradiction[END_REF] In order to overcome these difficulties and to extend the AHP on a more realistic elicitation procedure, several AHP methods are combined within uncertain theories. One of these extensions is the Fuzzy AHP, [START_REF] Van Laarhoven | A fuzzy extension of Saaty's priority theory[END_REF] which utilized triangular fuzzy numbers to model the pair-wise comparisons. Since then, several fuzzy AHP developments have been proposed. [START_REF] Lootsma | Fuzzy Logic for planning and Decision-Making[END_REF] Besides, probabilistic AHP methods are introduced in, [START_REF] Basak | Probabilistic judgments specified partially in the Analytic Hierarchy Process[END_REF] handling pairwise comparison matrices based on probability theory, where each element of which is the prior probability. There are also other ways to solve AHP problems. [START_REF] Schoner | Ambiguous criteria weights in AHP: consequences and solutions[END_REF] [START_REF] Benítez | An approach to AHP decision in a dynamic context[END_REF] [36] [START_REF] Qattawi | Incorporating quality function deployment and analytical hierarchy process in a knowledge-based system for automotive production line design[END_REF] In particular in the belief function framework, Beynon et al. have proposed a method called the DS/AHP method [START_REF] Beynon | The Dempster-Shafer theory of evidence: An alternative approach to multicriteria decision modelling[END_REF] [START_REF] Beynon | DS/AHP method: A mathematical analysis, including an understanding of uncertainty[END_REF] comparing not only single alternatives, but also groups of alternatives. Besides, several works have been proposed by Utkin. [START_REF] Utkin | A new ranking procedure by incomplete pair-wise comparisons using preference subsets[END_REF] [START_REF] Utkin | Multi-criteria decision making by incomplete preferences[END_REF] The main feature of his approach is that it allows the expert to deal with comparisons of arbitrary subsets of alternatives and criteria. [START_REF] Utkin | The DS/AHP method under partial information about criteria and alternatives by several levels of criteria[END_REF] Additionally, Smarandache et al. [START_REF] Dezert | Multi-Criteria Decision Making based on DSmT-AHP[END_REF] have developed the DSmT/AHP which is based on the Dezert-Smarandache theory. [START_REF] Smarandache | Advances and Applications of DSmT for Information Fusion Collected works[END_REF] This method aimed at performing a similar purpose as DS/AHP that is to compare groups of alternatives. All these extensions are very useful and important. However, it requires introducing additional assumptions concerning the corresponding membership functions. Moreover, most elicitation procedures in this case are based on standard pair-wise comparisons which have some limits. Therefore, our proposed solution avoids the standard pair-wise comparisons and proposes a new elicitation technique based on the belief function framework. The expert has then the ability to express his assessment freely, because in some cases, he is unable to model his preferences due to absence of knowledge, lack experience, etc. The objective is then to extend the approach presented in [START_REF] Ennaceur | Handling Partial Preferences in the Belief AHP Method: Application to Life Cycle Assessment[END_REF] under a new comparison technique. In [START_REF] Ennaceur | Handling Partial Preferences in the Belief AHP Method: Application to Life Cycle Assessment[END_REF], we construct groups of alternatives and criteria, but we use the Saaty scale to compare each element of the hierarchy.

In this paper, we present what we call a belief AHP approach, a multicriteria decision making (MCDM) method adapted to imprecise and incomplete preferences, where the uncertainty is represented by belief functions as defined in the Transferable Belief Model (TBM). The choice of the TBM seems appropriate as it provides a convenient framework for dealing with incomplete and uncertain information, notably those given by experts. This theory is chosen because it has a powerful evidence combination rule, and it represents properly partial and total ignorance; it assigns beliefs to individual elements of the hypothesis set as well as their subsets. Our aim through this work is to propose an uncertain AHP approach. On one hand, our suggested solution offers a formalism for reducing the number of pairwise comparisons. It will be able to compare groups of alternatives instead of comparing only single alternatives between each other. To judge the importance of criteria, our approach offers tools allowing the expert to express his ranking even over subgroups of criteria. [START_REF] Ennaceur | Handling Partial Preferences in the Belief AHP Method: Application to Life Cycle Assessment[END_REF] Our method is then able to compare groups of criteria at the criterion level and groups of alternatives at the alternative level. Moreover, we are interested in treating the uncertainty that may appear in the comparison procedure. Hence, our solution moves from a scaled level of preference to response based on "yes" or "no". Then, the expert responses are described by a distributed assessment using a belief structure. The main objective of introducing uncertainty in the comparison scale is to model both precise data and subjective judgments with uncertainty can be consistently modeled under this framework. Rather than forcing the expert to provide exact representations of imprecise perceptions, we suggest using an imprecise representation instead. Also, using the belief function framework, we have the possibility to express our total ignorance, which was impossible under standard AHP.

Finally, to illustrate the feasibility of our approach, we have applied our proposed method on a real application problem: we have considered the PVC life cycle especially the end of life phase (the disposal problem). The challenge facing an expert here is the choice of the country where the environmental impact is the least important for the destruction of a kilogram of PVC.

In what follows, we first present an overview of the basic concepts of respectively the AHP and the belief function theory. Then, in the main body of the paper, we present our new approach: the belief AHP which is based on the belief function theory. Finally, our method will be illustrated by a real application problem in order to understand its real unfolding and to show the applicability of the developed approach.

Basic concepts

In this section, we briefly introduce a number of the key concepts in order to better understand our contribution.

Analytic Hierarchy Process

The AHP approach is a decision-making technique developed by Saaty [10] [11] to solve complex problems of choice and prioritization. The problem is structured hierarchically at different levels. The purpose of constructing this hierarchy is to evaluate the influence of the criteria on the alternatives to attain objectives. So, an AHP hierarchy has at least three levels: The highest level consists of a unique element that is the overall objective. Then, each level of the hierarchy contains criteria or sub-criteria that influence the decision. Alternative elements are put at the lowest level.

Once the hierarchy is built, the decision maker starts the prioritization procedure to determine the relative importance of the elements on each level of the hierarchy. Elements of each level are paired (with respect to their upper level decision elements) and then compared. This method elicits preferences through pair-wise comparisons based on a nine-point scale, [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF] which translates the preferences of a decision maker into crisp numbers (see Table 1). After filling all the pair-wise comparison matrices, the local priority weights are determined by using the eigenvalue method. The objective is then to find the weight of each criterion, or the score of each alternative by calculating the eigenvalue vector. With these values, the AHP method permits to compute a consistency ratio to check if the matrix is consistent or not. The last step aggregates all local priorities from the decision table by a simple weighted sum. The global priorities thus obtained are used for final ranking of the alternatives and the selection of the best one.

Belief function theory

In this subsection, we briefly review the main concepts underlying the belief function theory as interpreted by the TBM. Details can be found in [START_REF] Ph | The combination of evidence in the Transferable Belief Model[END_REF] [START_REF] Ph | The Transferable Belief Model[END_REF].

Basic concepts

Let Θ be the frame of discernment representing a finite set of elementary hypotheses related to a problem domain. We denote by 2 Θ the set of all the subsets of Θ.

The impact of a piece of evidence on the different subsets of the frame of discernment Θ is represented by the so-called basic belief assignment (bba), called initially basic probability assignment. [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] It quantifies the impact of a piece of evidence on the different subsets of the frame of discernment. [START_REF] Ph | The Transferable Belief Model[END_REF] A bba is a function denoted by m that assigns a value in [0, 1] to every subset A of Θ such that:

A⊆Θ m(A) = 1 (1)
A basic belief mass (bbm) m(A) quantifies the portion of belief that supports A without supporting any strict subset of A, and that could be transferred to subsets of A if further information justifies it.

Combination

The belief function theory offers interesting tools for combining beliefs induced by distinct pieces of evidence. Let m 1 and m 2 be two bba's defined on the same frame of discernment and are issued from distinct sources. One of the combination rules is the conjunctive rule: [START_REF] Ph | The combination of evidence in the Transferable Belief Model[END_REF] (m

1 ∩ m 2 )(A) = B,C⊆Θ,B∩C=A m 1 (B)m 2 (C) (2)

Discounting

The technique of discounting allows us to take in consideration the reliability of the information source that generates the bba m. Let β = 1 -α be the degree of reliability (α ∈ [0, 1]) assigned to a particular belief function. If the source is not fully reliable, the bba it generates is "discounted" into a new less informative bba denoted m α : [24]

m α (A) = (1 -α)m(A), ∀A ⊂ Θ (3) m α (Θ) = α + (1 -α)m(Θ) (4)

Coarsening and refinement

Sometimes, beliefs are induced by information sources with different but compatible frames of discernment. The coarsening and refinement operations allow us to establish relationships between these different frames in order to express beliefs on anyone of them. Let Ω and Θ be two finite sets, the idea behind the refinement consists in obtaining one frame of discernment Ω from the set Θ by splitting some or all of its events. [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] Conversely, the coarsening consists in forming a frame Θ by grouping together the events of the frame of discernment Ω (see Fig. 1).

Figure 1: Illustration of a coarsening Θ of Ω associated with a refining ρ of Θ.

Let us define a mapping ρ:

2 Θ → 2 Ω such that: [4] ρ({θ}) = ∅ ∀θ ∈ Θ (5) ρ({θ}) ∩ ρ({θ }) = ∅ if θ = θ (6) θ∈Θ ρ({θ}) = Ω (7)
So, given a disjoint partition ρ({θ}) one may set: [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]

ρ(A) = θ∈A ρ({θ}) (8)
With this notion of refining, we can define the vacuous extension. This operation allows one to convey a mass function m Θ , expressing a state of belief on Θ, to a finer frame Ω , a refinement of Θ. Stemming from the least committed principle, this operation is denoted with an arrow pointing up, and is defined by:

m Θ↑Ω (ρ(A)) = m Θ (A) ∀A ⊆ Θ (9)

Decision making

The TBM considers that holding beliefs and making decision are two distinct processes. Hence, it proposes a two level model:

• The credal level where beliefs are entertained and represented by belief functions.

• The pignistic level where beliefs are used to make decisions and represented by probability functions called the pignistic probabilities, denoted BetP : [5]

BetP (A) = B⊆Θ |A ∩ B| |B| m(B) (1 -m(∅))
, ∀A ∈ Θ (10)

Belief AHP approach

This section is dedicated to the presentation of the basic steps needed to ensure the ranking of alternatives in an uncertain environment. In this context, we introduce our belief AHP approach which is a combination of AHP method and the belief function theory.

Identification of the candidate criteria

Our set of criteria is given by Ω = {c 1 , ..., c m } where Ω is the frame of discernment involving all the possible criteria related to the MCDM problem. Denote the set of all subsets of criteria by 2 Ω , and let C k be a subset of Ω. To identify the candidate criteria, our main idea is to allow the expert to express his opinions on groups of criteria instead of single one. The decision maker chooses these subsets by assuming that criteria having the same degree of preference are grouped together. If an expert identifies a group of criteria, then we could suppose that all of its elements have the same importance. So, by comparing two subsets of criteria C k and C j , an expert makes a choice between C k and C j . Since we are not performing pair-wise comparisons of criteria but relating to groups of criteria, these sets of criteria should not consider a criterion in common, because if one criterion is included in two groups, then each group will give a different level of favorability. By generalization, these subsets can be defined as:

C k is preferred to C j , ∀ k, j|C k , C j ∈ 2 Ω , C k ∩ C j = ∅ (11)

Identification of the candidate alternatives

In many complex problems, experts are able to compare only subsets of alternatives and cannot evaluate separate ones. To solve this problem, that means to reduce the number of alternatives which consequently decreases the number of comparisons, our method suggests not necessarily to consider all of them but just to choose groups of those alternatives. One of the possible solutions of this task is to use the DS/AHP method. [17] [18] Like the criterion level, we assume that there is a set of all the possible alternatives Θ = {a 1 , ..., a n } consisting of n elements. Denote the set of all subsets of Θ by 2 Θ , and let A k be a subset. At this level, we apply the same hypotheses assumed in DS/AHP to identify the subsets of alternatives. Besides, the decision maker expresses his preferences by comparing groups of alternatives (except the set Θ which represents the total ignorance on alternatives). For example, A k is preferred to A j , that means that A k is more preferable than A j . The main aim behind this hypothesis was explained in. [START_REF] Beynon | The Dempster-Shafer theory of evidence: An alternative approach to multicriteria decision modelling[END_REF] [18]

Computing the weight of considered criteria

After identifying the candidate sets of criteria, what is left is "how to calculate the weight of these criteria?". At this point, pair-wise comparisons process is extended to an uncertain environment. We consider the same basic assumptions of AHP method, that all the elements in the same hierarchy are totally independent. In fact, preferences expressed on one criterion do not influence the ones expressed on another criterion.

Under this approach, our proposed scale may be different from the traditional one. In this study, we assume a new set of choices. Thus to express his preferences, the decision maker responses to the following question: "Is the subset of criteria C j important?". Besides, he only selects the related linguistic variable. He indicated whether a criterion was more or less important to its partner by "yes" or "no". In other terms:

Ω C j = {yes, no} (12)
where Ω C j is the set of possible choices.

After identifying the frame of discernment Ω C j corresponding to each subset of criteria, C j , our approach suggests to use a belief pair-wise comparison matrix. In order to express his opinions, we accept that the expert may define uncertain or even unknown assessments. Indeed, we assume that each subset of criteria is described by a basic belief assignment (m Ω C j ) defined on the possible responses. In other words, to quantify the subjective judgments with uncertainty, a preference degree may be assigned to each decision maker's response. For instance, in a problem of purchasing a car, the following type of uncertain subjective judgments was frequently used: "the comfort criterion is evaluated to be more important than style with a confidence degree of 0.8". In fact, the decision maker responses to the question "is comfort criterion important regarding the style criterion?". Thus, the answer was: comfort criterion is more preferable than style criterion and 0.8 is referred to the degree of belief. Finally, the same process is repeated for each pair of subsets.

In some cases, the expert is not able to give any information about alternatives and/or criteria. Thus, the obtained bba will be a vacuous basic belief assignment, and we will get a single subset m

Ω C i j (Ω C i ) = 1.
This case is referred to as total ignorance. Therefore, the decision maker can express his preferences more freely and all the elicitated data can be modeled.

To model the pair-wise comparison matrix, some priorities must be respected. We consider X, the pair-wise comparison matrix, is an k ×k matrix in which k is the number of groups of criteria being compared. It has the following characteristics:

1. The first step is to model the pair-wise comparison matrix. Let d ij is the entry from the i th column of pair-wise comparison matrix (d ij represents the different bbm's of the identified bba).

m

Ω C j i (a) = m Ω C i j (a) = d ij , ∀a ⊂ Ω C i ( 13
)
where m

Ω C i j
represents the importance of C i with respect to the subset of criteria C j (for simplicity we denote the subset of criteria by j instead of C j ), i = j, and m

Ω C j i is the negation of m Ω C i j .
As regarding the previous example, if we have "the comfort criterion (C) is evaluated to be more important than style criterion (S) with a confidence degree of 0.8", that is m Ω C S ({yes}) = 0.8, then we can say that "the style criterion is evaluated to be less important than comfort criterion with a confidence degree of 0.8": m Ω S C ({no}) = 0.8.

2.

Like the traditional AHP method, where the principal diagonal contains entries of 1, we set:

m

Ω C i i (Ω C i ) = 1 ( 14
)
Once the pair-wise comparison matrix is complete, our objective is then to obtain the priority of each subset of criteria. In fact, within the belief comparison matrix, our problem is "what is the appropriate function to use in order to obtain a single representation value of these different bbas to get the priority vector". The idea is to combine the obtained bba using the conjunctive rule of combination. Indeed, this function is chosen since we can regard each subset of criteria as a distinct source of information which provides distinct pieces of evidence.

To better understand, we consider X, as defined above, the pair-wise comparison matrix. For each row of the matrix, we apply the conjunctive rule. That means, for each subset of criteria (i = {1, . . . , k}), we will get the following bba:

m

Ω C i = ∩ k j=1 m Ω C i j (15)
Finally, these obtained bba m Ω C i is transformed into pignistic probabilities, denoted by BetP Ω C i using Equation 10.

The proposed solution addresses the problem of inconsistency. In fact, if the preferences are inconsistent, then we obtain m(∅) = 1. Thus, the expert may be guided to reformulate his preferences. Otherwise, any belief function will be found.

Computing the alternatives priorities

Following the same reasoning, belief pair-wise comparison matrices are also constructed, and evaluate each group of alternatives regarding each criterion. By applying Equation 15for each obtained pair-wise comparison matrix, we get a bba which quantifies the degree of belief assigned by the expert to each subset of alternatives. However, our purpose is to combine these obtained bba's to get a single belief function, but the problem here is that each subset of alternatives has its own frame of discernment. For instance, if we say that alternative A is preferred to B, and alternative C is preferred to B, this does not means that alternative A is indifferent to C. Consequently, each obtained bba is defined on a different frame of discernment.

The idea to use is then to standardize all the frames of discernment. Obviously, we propose to use the concept of refinement operations, [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] which allows us to establish relationships between different frames of discernment in order to express beliefs on anyone of them. The objective is then consists in obtaining one frame of discernment Θ from the set Θ A k by splitting some or all of its events. As mentioned above, each bba m Θ A k represents the belief over all possible answers (yes or no). However, at this stage, we want to know which alternative is the best (Θ = {A, B, C}). As a result, Θ A k is considered as a coarsening of Θ (see subsection 2.2.4), and we get the following relation:

m Θ A k ↑Θ (ρ k (ω)) = m Θ A k (ω) ∀ω ⊆ Θ A k ( 16
)
where the mapping ρ k from Θ A k to Θ is a refinement, and

ρ k ({yes}) = {A k } and ρ k ({no}) = {A k }.
Once we have standardized our frame of discernment Θ, we can now combine the resulting bba using the conjunctive rule in order to obtain a belief function reflecting the importance of alternatives to a given criterion:

m c k = ∩ l j=1 m Θ A i ↑Θ ( 17
)
where l is the number of subsets of alternatives. Finally, we obtain m c k representing the opinions-beliefs of the expert about his preferences regarding the set of alternatives.

Updating the alternatives priorities

To update the alternatives priorities with respect to the criterion weight, we have to define a rule for combining them. On the one hand, we have priorities concerning criteria and groups of criteria instead of single ones. On the other hand, the sets of alternatives are compared pair-wise with respect to a specific single criterion.

In order to overcome this difficulty, our approach proposes to regard each pignistic probability distribution of a specific set of criteria as a measure of reliability. This factor is used to measure most heavily the bba evaluated according to the most important criteria and conversely for the less important ones. If we have C k a subset of criteria, then we get β k its corresponding measure of reliability:

β k = BetP Ω C k ({yes}) (18)
As a result, two cases will be presented: First, if the reliability factor represents a single criterion c k , then the corresponding bba will be directly discounted:

m α k c k (A j ) = β k .m c k (A j ), ∀A j ⊂ Θ ( 19
)
m α k c k (Θ) = (1 -β k ) + β k .m c k (Θ) (20)
where m c k (A j ) the relative bbm for the subset A j (obtained in the previous step: Equation 17), β k its corresponding measure of reliability, and we denote

α k = 1 -β k .
Second, if this factor represents a group of criteria, their corresponding bba's must be combined, then it will be discounted by their corresponding measure of reliability. In this case, each element of a specific group of criteria has its own pair-wise matrix that the sets of alternatives with respect to this criterion. Therefore, we apply the conjunctive rule of combination to obtain a single representation value of these different bba's. As a result, the obtained bba compares the sets of alternatives according to this set of criteria.

Let C k a subset of criteria, and c i ∈ C k , then we apply the conjunctive rule of combination to obtain m C k :

m C k = ∩ h j=1 m c i ( 21
)
where h is the number of items of a specific group of criteria C k . Finally, these obtained bba's will be discounted by their corresponding measure of reliability. We apply the same process used in Equations 19 and 20 to get m α k C k .

Decision making

To this end, and after updating the priorities of the alternatives sets with respect to their set of criteria, we must combine the overall bba's in order to help the expert to make a decision:

m f inal = ∩ l j=1 m α k C k ( 22
)
where l is the number of subsets of criteria.

In this context, we choose to transform the final bba into pignistic probabilities using Equation 10. The decision maker will choose the alternative which has the highest value of pignistic probabilities.

PVC life cycle application

The main aim of this section is to show an applicable way of our proposed approach. In fact, the potential of the belief AHP method may be illustrated by considering a real application problem. Here, we consider "the PVC (Polyvinyl chloride) life cycle" especially "the end of life phase". The problem considered attempts to rank countries based on their environmental impact to the disposal of PVC product. Thus, our challenge is to use or not the PVC in general, but to know in which country the environmental impact is less important for the destruction of a kilogram of PVC?

By following the Belief AHP procedure described in the previous section, the decision maker has to identify the set of criteria and alternatives and he has to indicate his preferences. Then, the following can be done manually or automatically by our implemented software.

Identification of the candidate criteria and alternatives

Following discussion with the knowledgeable expert, it was decided to restrict the number of criteria to four areas:

1. Acidification (C 1 ): the gradual decrease in the pH of the oceans.

Human toxicity infinite (C 2 ): this category concerns effects of toxic

substances on the human environment.

3. Fresh water aquatic ecotoxicity infinite (C 3 ): this category indicator refers to the impact on fresh water ecosystems, as a result of emissions of toxic substances to air, water and soil.

4. Terrestrial ecotoxicity infinite (C 4 ): this category refers to impacts of toxic substances on terrestrial ecosystems.

Apart from the four criteria, the initial interview also identified six selected countries: "Switzerland (SW), France (FR), USA (US), England (ENG), Italy (I) and Spain (S)" which make the set of alternatives. We would generate a decision hierarchy by which it is possible to evaluate different alternatives. A figure showing the different levels of the hierarchy is presented (see Fig. 2).

Computing the weight of considered criteria

Given the necessary details of the criteria, the decision maker was asked to indicate his level of preference between them. Importantly, he was made aware that when assigning the same scale values to criteria, he was grouping them in the same subset.

To evaluate the responses of the pair-wise comparison question, the decision maker will be able to express his judgment with some degrees of uncertainty. That's, each element of the matrix will be labeled by a bba expressing a belief on judgment choice. So, the question is how to construct these bba's to obtain uncertain preferences. For collecting these expert beliefs, we have used a method for eliciting the expert opinions and we have generated associated belief functions (see Table 2).

Table 2: The weights preferences assigned to the criteria according to the expert's opinion

{C 1 } {C 4 } Ω 1 = {C 2 , C 3 } {C 1 } m Ω {C 1 } {C1} (Ω {C1} ) = 1 m Ω {C 1 } {C4} ({yes}) = 0.4 m Ω {C 1 } Ω1 ({yes}) = 0.9 m Ω {C 1 } {C4} (Ω {C1} ) = 0.6 m Ω {C 1 } Ω1 (Ω {C1} ) = 0.1 {C 4 } m Ω {C 4 } {C1} ({no}) = 0.4 m Ω {C 4 } {C4} (Ω {C4} ) = 1 m Ω {C 4 } Ω1 ({no}) = 0.3 m Ω {C 4 } {C1} (Ω {C4} ) = 0.6 m Ω {C 4 } Ω1 (Ω {C4} ) = 0.7 Ω 1 = {C 2 , C 3 } m ΩΩ 1 {C1} ({no}) = 0.9 m ΩΩ 1 {C4} ({yes}) = 0.3 m ΩΩ 1 Ω1 (Ω Ω1 ) = 1 m ΩΩ 1 {C1} (Ω Ω1 ) = 0.1 m ΩΩ 1 {C4} (Ω Ω1 ) = 0.7
From Table 2, we conclude that the decision maker has identified three subsets of criteria {C 1 }, {C 4 }, and {C 2 , C 3 }. For instance, the expert may say that {C 1 } is evaluated to be more important than {C 4 } with a confidence degree of 0.4. That means, 0.4 of beliefs are exactly committed to the criterion {C 1 } is more important than {C 4 }, whereas 0.6 is assigned to the whole frame of discernment (ignorance).

Then, the next step consists in combining the bba's corresponding to each criterion using the Equation 15. The obtained bba's are reported in Table 3.

Table 3: Belief pair-wise matrix: Partial combination

C 1 C 4 Ω 1 = {C 2 , C 3 } Weight m Ω {C 1 } ({yes}) = 0.94 m Ω {C 4 } ({no}) = 0.58 m Ω Ω 1 ({yes}) = 0.03 m Ω {C 1 } (Ω {C 1 } ) = 0.06 m Ω {C 4 } (Ω {C 4 } ) = 0.42 m Ω Ω 1 ({no}) = 0.63 m Ω Ω 1 (∅) = 0.27 m Ω Ω 1 (Ω Ω 1 ) = 0.07
After that, the resulting bba's are transformed into pignistic probabilities (see Table 4). Table 4: Belief pair-wise matrix: pignistic probabilities

{C 1 } {C 4 } Ω 1 = {C 2 , C 3 } BetP BetP Ω {C 1 } ({yes}) = 0.97 BetP Ω {C 4 } ({yes}) = 0.21 BetP Ω Ω 1 ({yes}) = 0.089 BetP Ω {C 1 } ({no}) = 0.03 BetP Ω {C 4 } ({no}) = 0.79 BetP Ω Ω 1 ({no}) = 0.911

Computing the alternatives priorities

Like the criterion level, the judgments between decision alternatives over different criteria are dealt within an identical manner. For example, to evaluate the alternatives according to the criterion C 1 (Acidification), the decision maker is asked to evaluate the following subsets of alternatives: {EN G, U S}, {F R, I, S} and {SW }, and we get the matrix presented in Table 5.

Table 5: Belief pair-wise matrix regarding C 1 criterion

C 1 {EN G, U S} {F R, I, S} {SW } {EN G, U S} m Θ {EN G,U S} {EN G,U S} (Θ {EN G,U S} ) = 1 m Θ {EN G,U S} {F R,I,S} ({no}) = 0.05 m Θ {EN G,U S} {SW } ({yes}) = 0.4 m Θ {EN G,U S} {F R,I,S} (Θ {EN G,U S} ) = 0.95 m Θ {EN G,U S} {SW } (Θ {EN G,U S} ) = 0.6 {F R, I, S} m Θ {F R,I,S} {EN G,U S} ({yes}) = 0.05 m Θ {F R,I,S} {F R,I,S} (Θ {F R,I,S} ) = 1 m Θ {F R,I,S} {SW } ({yes}) = 0.2 m Θ {F R,I,S} {EN G,U S} (Θ {F R,I,S} ) = 0.95 m Θ {F R,I,S} {SW } (Θ {F R,I,S} ) = 0.8 {SW } m Θ {SW } {EN G,U S} ({no}) = 0.4 m Θ {SW } {F R,I,S} ({no}) = 0.2 m Θ {SW } {SW } (Θ {SW } ) = 1 m Θ {SW } {EN G,U S} (Θ {SW } ) = 0.6 m Θ {SW } {F R,I,S} (Θ {SW } ) = 0.8
As in the criterion level, for the subset of alternatives {EN G, U S}, we use the Equation 15 in order to combine the obtained bba (see Table 6):

m Θ {EN G,U S} = m Θ {EN G,U S} {EN G,U S} ∩ m Θ {EN G,U S} {F R,I,S} ∩ m Θ {EN G,U S} {SW }
Then, a similar process is repeated for the rest of alternatives, and we get m Θ {F R,I,S} and m Θ {SW } . Table 6: Belief pair-wise matrix: Partial combination

C 1 {EN G, U S} {F R, I, S} {SW } Priority m Θ {EN G,U S} ({yes}) = 0.38 m Θ {F R,I,S} ({yes}) = 0.24 m Θ {SW } ({no}) = 0.52 m Θ {EN G,U S} ({no}) = 0.03 m Θ {F R,I,S} (Θ {F R,I,S} ) = 0.76 m Θ {SW } (Θ {SW } ) = 0.48 m Θ {EN G,U S} (∅) = 0.02 m Θ {EN G,U S} (Θ {EN G,U S} ) = 0.57
Subsequently, we proceed now with the standardization of our frame of discernment. By applying the Equation 16, in the subset of alternatives {EN G, U S} for example, we get the following bba:

m Θ {EN G,U S} ↑Θ ({EN G, U S}) = m Θ {EN G,U S} ({yes}) m Θ {EN G,U S} ↑Θ ({EN G, U S}) = m Θ {EN G,U S} ({no}) m Θ {EN G,U S} ↑Θ (Θ) = m Θ {EN G,U S} (Θ {EN G,U S} )
To simplify, we can note by m Θ {EN G,U S} the bba m Θ {EN G,U S} ↑Θ , and the similar process is repeated for the rest of alternatives. The obtained matrix is presented in Table 7.

Table 7: Belief pair-wise matrix: Refinement

C 1 {EN G, U S} {F R, I, S} {SW } Priority m Θ {EN G,U S} ({EN G, U S}) = 0.38 m Θ {F R,I,S} ({F R, I, S}) = 0.24 m Θ {SW } ({F R, I, S, U S, EN G}) = 0.52 m Θ {EN G,U S} ({F R, SW, I, S}) = 0.03 m Θ {F R,I,S} (Θ) = 0.76 m Θ {SW } (Θ) = 0.48 m Θ {EN G,U S} (∅) = 0.02 m Θ {EN G,U S} (Θ) = 0.57
Finally, the obtained bba's can be directly combined using the conjunctive rule of combination (see Table 8). Then, as shown in the previous step, the computation procedure is repeated for the rest of comparison matrices to get Table 9.

Updating the alternatives' priorities

After computing the belief functions for each set of alternatives with respect to each criterion, we must update this bba with their corresponding measure of reliability. Firstly, this step concerns the groups of criteria, that is the subset {C 2 , C 3 }. Therefore, by using the Equation 21, we propose to combine the bba relative to the C 2 and

C 3 criteria (m C 2 ,C 3 = m C 2 ∩ m C 3 ).
Then, these obtained bba's are discounted by the measure of reliability β C 2 ,C 3 = 0.089 (obtained from Table 4 and using the Equation 18). So, we use Equations 19 and 20 to get the Table 10. After that, the next step concerns the criterion {C 1 } and {C 4 }. The relative bba are directly discounting using Equations 19 and 20, where the reliability measures β C 1 = 0.97 and β C 4 = 0.21 (see Table 11).

Synthetic utility and decision making

The final step is then to choose the best alternatives. First, we apply the conjunctive rule of combination (Equation 22), that leads us to get a single bba denoted by m f inal . Then, we transform the obtained bba to a pignistic probability and we get this final result (see Table 12). The decision maker wants to know in which country the environmental impact is less important for the destruction of a kilogram of PVC.

The alternatives are now ranked according to their pignistic probabilities, as follows: {U S}, {EN G}, {S}, {I}, {F R} and {SW }. Consequently, "USA" is the recommended country since it has the highest values.

For the sake of comparison, we have assumed that the expert is able to express his preferences with certainty and we have used the standard AHP method. We have obtained the following ranking: {F R}, {I}, {EN G}, {S}, {U S} and {SW }. We can conclude that, when the expert freely expresses his preferences the obtained result may be affected. Thus, we can obtain a more accurate decision since we have tried to closely model the decision maker assessment.

This research proposes that the differences between classic AHP method and belief AHP approach are raised by two potential factors. One, belief AHP method can provide the decision maker to express his preferences with some uncertainty and imprecision rather than deterministic value options. This gives an advantage over classical AHP method in solving complex problems. Besides, by using our approach, we try to model all the elicitated information. Indeed, our method expects to find the most cautious solution. Second, by using belief AHP method the number of comparisons is reduced. So, the use of subsets of criteria and/or alternatives for pair-wise comparison is hence the second factor. In our case for example, if we have adopted the classical AHP, then there would be 15 comparisons per criterion between the decision alternatives level. That means, we will get 60 comparisons, and at the criterion level, we will have 6 comparisons. As a result, the number of pair-wise comparisons is then 66. However, by using the belief AHP, the number of comparisons decreases because instead of using single elements, we have used subsets. In fact, in our case the decision maker had made 3 pair-wise comparisons at the criterion level, and at the alternative level, he had made 10 pair-wise comparisons. In the worst case, the number of pair-wise comparisons is equal to the standard one, when singleton criterion is compared to singleton alternative.

Experimental study

In this Section, the performances of the proposed method, belief AHP, are compared. Firstly, the evaluation algorithm used in the comparisons is described. Secondly, the method is tested on randomly generated matrices. Then, we discuss the obtained results.

Evaluation algorithm

Despite the possible differences between the obtained results of two MCDM methods, we cannot conclude the superiority of one over another. Unless we have a solid basis, we compare the ranking results against the closeness of the rankings of each method to the actual rankings.

To do so, we need to compare each set of rankings provided by AHP and the proposed method with a ranking that has already been produced by an alternative, yet reliable ranking method. This alternative ranking will be considered as a basis, or actual ranking of the alternatives and will be used to measure the closeness of the rankings provided by AHP, belief AHP to reality. These evaluations are provided in the following Sections.

A method which is accurate in MCDM problems should also be accurate in single dimensional problems. Therefore, we use the Weighted Sum Model (WSM) method since in single-dimensional environment, [START_REF] Fishburn | Additive Utilities with Incomplete Product Set: Applications to Priorities and Assignment[END_REF] it yields the most reasonable results. Hence, some previous works compare the obtained results using WSM to those obtained by other MCDM method. [38] [39] This evaluation criterion has been applied in order to evaluate crisp and fuzzy MCDM methods. [38] [39] We therefore consider that at this level, there is no need to compare our method to other multi-criteria decision making approaches since we compare our solution to one of the most reliable method in a unidimensional context.

Besides, WSM is the simplest and still the widest used MCDM method. In this method, each criterion is given a weight, and the sum of all weights must be 1. Each alternative is assessed with regard to every criterion. The overall or composite performance score of an alternative is given by the equation:

P i = j v ij * ω j (23)
where P i is the priority of each alternative, ω j is the weight of each criterion and v ij is the score of each alternative regarding each criterion.

In order to overcome these issues, in the next Section, we test a simulation algorithm that compares the ranking results of belief AHP under different scenarios.

Simulation algorithm

To generate reliable data for a numerical analysis in AHP, simulation has been extensively used in prior research. [START_REF] Triantaphyllou | Development and Evaluation of Five Fuzzy Multiattribute Decision-Making Methods[END_REF] [38] The simulation algorithm is summarized as follows:

1. We generate a random matrix for the decision performance and another one to represent the weight of each decision criteria. Based on these two matrices, the overall scores and ranks of the decision alternatives are calculated. These steps are usual steps in the WSM method.

2. From the performance matrix, we generated pair-wise comparison matrices of different alternatives that are compared to each criterion.

3. Each pair-wise matrix is transformed into belief pair-wise comparison. Indeed, the resulting bbas has only one focal element since we are in a certain context:

• If the actual preference value regarding the alternative a with bbm, m({yes}) = 1 (If 1 < p ≤ 9).

• If p = 1 then m(Θ) = 1 • If 1 9 ≤ p ≤ 1 2 then m({no}) = 1
4. We apply the suggested method to compute the overall priorities and to rank alternatives.

5. We compare the obtained result with the ranking of the WSM method.

In order to gain a deeper understanding, a computational study was undertaken. The data were random numbers from the interval [START_REF] Saaty | How to make a decision: The Analytic Hierarchy Process[END_REF][START_REF] Zeleny | Multiple Criteria Decision Making McGraw-Hill Book Company[END_REF] (in order to be compatible with the numerical properties of the Saaty scale). In these test problems, the number of alternatives was equal to the following values: 3, 4, 5, 6, 7, 8, 9, and 10. Similarly, the number of criteria was equal to 3, 4, 5, 6, 7, 8, 9, and 10. Psychological experiments have shown that individuals cannot simultaneously compare more than seven objects (plus or minus two). [START_REF] Miller | The Magical Number Seven Plus or Minus Two: Some Limits on Our Capacity for Processing Information[END_REF] Therefore, we choose that the number of criteria and alternatives in the analysis should not exceed 10. Thus, a total of (8 × 8) different cases were examined with 100 replications (in order to derive statistically significant results) per each case. Each random problem was solved using the original and belief AHP methods.

Simulation results

One of the difficulties in comparing many MCDM methods is that there is rarely any way to check the accuracy of the methods. At the same time, it must be understood that these methods have different relative advantages in dealing with different types of data such as ordinal and uncertain data. The intent is not to establish the dominance of any one method, but rather to compare the relative accuracy in the case of known outcome. We also notice, from Tables 13 and14, that there is a significant difference in the percentage of contradiction between AHP and belief AHP methods. Most importantly, when comparing the overall results, we can remark that problems with less alternatives yielded fewer ranking contradictions. As it can be seen, the number of criteria did not play a prime role. For instance, from Table 14, our method is compared regarding 5 criteria and different alternatives the percentage of contradiction in almost cases is under 10%.

Importantly, belief AHP method uses only the minimum information to model the decision maker preferences. However, more uncertainty can be handled within the scale of preferences.

Conclusion

In this paper, a new approach inspired of the AHP method was developed. Our proposed method, named belief AHP, is based on the belief function framework. Indeed, this approach is shown to model the uncertain decision maker's preferences and to reduce the number of pair-wise comparisons. Instead of comparing single alternatives regarding single criterion, our belief AHP compares the sets of decision alternatives with respect to certain groups of criteria. Moreover, our method has adopted a yes/no choice and for each response we assign a degree of belief. Consequently, our approach follows the same outline as AHP method in uncertain environment. Finally, to illustrate the feasibility of our approaches and to judge their performances, we have applied our proposed methods on a real application problem. Our aim was to choose the appropriate country, where its environmental impact is the least important for the destruction of a kilogram of PVC, on the basis of several criteria.

Nevertheless, the proposed work is still subject to improvements. It can be extended into different directions. In fact, our method can be improved by defining a new consistency ratio in a belief function framework. Thus, the proposed method will be more flexible, if it will be able to calculate the consistency of a belief pair-wise matrix. Further work may be suggested on the comparison between the belief AHP approach and other methods.

Figure 2 :

 2 Figure 2: Belief AHP hierarchy.

Table 1 : The Saaty rating scale

 1

	Saaty's scale	Definition
	1	Equal importance
	3	Somewhat more important
	5	Much more important
	7	Very much more important
	9	Absolutely more important
	2,4,6,8	Intermediate values

Table 8 :

 8 The bba m C 1

	∅	{F R, I, S} {F R, I, S, EN G, U S} {F R, SW, I, S} {EN G, U S}	Θ
	m C1 0.1113	0.1559	0.2252	0.0109	0.2887	0.2079

Table 9 :

 9 The bba's m C 2 , m C 3 and m C 4

		∅	{F R}	{SW } {EN G, U S, I, S}	Θ
	m C2	0.541	0.128	0.012	0.226	0.093
		{S, SW, U S}	Θ
	m C3	0.84	0.16
		∅	{EN G, U S, F R} {I, S}	{SW }	{SW, I, S}	Θ
	m C4	0.261	0.26	0.17	0.009	0.203	0.097

Table 10 :

 10 The bba m C 2 ,C 3 and m

	α C 2 ,C 3
	C 2 ,C 3

Table 11 :

 11 The bba m

				α C 1 C 1 and m	α C 4 C 4 after discounting
		∅	{F R, I, S}	{F R, I, S, EN G, U S} {F R, SW, I, S} {EN G, U S}	Θ
	m αC 1 C1	0.1080	0.1512	0.2184	0.0105	0.28	0.2319
		∅	{EN G, U S, F R}	{I, S}	{SW }	{SW, I, S}	Θ
	m αC 4 C4	0.0548	0.0546	0.0357	0.0019	0.0426	0.8104

Table 12 :

 12 The final result using the belief AHP approach

	Alternatives EN G	U S	S	F R	I	SW
	BetP	0.2328 0.2464 0.1676 0.1512 0.1569 0.0451

Table 13 :

 13 Percentage of contradiction (%) between the WSM and the AHP

		Number of alternatives
	Number of criteria 3 4 5	6	7	8	9 10
	3	7 7 9	9 10	11	13 12
	4	7 7 9	9 11	11	11 11
	5	7 7 9	9 12	11	11 11
	6	8 6 9 10 11	12	13 13
	7	8 6 10 10 11	12	13 13
	8	8 6 10 10 10	1	12 13
	9	8 6 10 10 10	12	12 14
	10	9 6 11 11 10	12	14 14

Table 14 :

 14 Percentage of contradiction (%) between the WSM and the belief AHP

		Number of alternatives
	Number of criteria 3 4 5	6	7	8	9 10
	3	5 5 5	6	9	9	9 10
	4	5 5 5	6	9	9	9 11
	5	7 7 7	6 10	9	10 11
	6	7 7 9	6 10	9	10 12
	7	8 8 9	6 10	10	10 12
	8	8 8 9 10 10	10	10 12
	9	8 8 10 10 10	10	10 12
	10	8 8 10 10 10	10	10 12


Wenye Lin

Ying Wang

Yoan Coudert

Daniel Kierzkowski
email: daniel.kierzkowski@umontreal.ca

Leaf Morphogenesis: Insights From the Moss Physcomitrium patens

Keywords: Physcomitrum patens, Physcomitrella patens, leaf, heteroblasty, bryophytes, development, organogenesis, cellular dynamics

Specialized photosynthetic organs have appeared several times independently during the evolution of land plants. Phyllids, the leaf-like organs of bryophytes such as mosses or leafy liverworts, display a simple morphology, with a small number of cells and cell types and lack typical vascular tissue which contrasts greatly with flowering plants. Despite this, the leaf structures of these two plant types share many morphological characteristics. In this review, we summarize the current understanding of leaf morphogenesis in the model moss Physcomitrium patens, focusing on the underlying cellular patterns and molecular regulatory mechanisms. We discuss this knowledge in an evolutionary context and identify parallels between moss and flowering plant leaf development. Finally, we propose potential research directions that may help to answer fundamental questions in plant development using moss leaves as a model system.

INTRODUCTION

Leaves are photosynthetic organs with mainly determinate growth that evolved several times independently during land plant diversification [START_REF] Tomescu | Megaphylls, microphylls and the evolution of leaf development[END_REF][START_REF] Nelissen | Leaf growth in dicots and monocots: so different yet so alike[END_REF]. These organs can be large with a complex structure as some flowering plant leaves, or small and anatomically simple as bryophyte phyllids (hereafter called leaves). However, they all display a predominantly flat shape as an adaptation to optimize light capture. The genetic basis of leaf development has been extensively studied in flowering plants ( [START_REF] Bar | Compound leaf development in model plant species[END_REF][START_REF] Du | Molecular Mechanisms of Leaf Morphogenesis[END_REF][START_REF] Maugarny-Calès | Getting leaves into shape: a molecular, cellular, environmental and evolutionary view[END_REF]. However, how genetic commands are coordinated between cells and translated into supracellular level organization, and the final leaf shape remains largely elusive. This is due to the complex, multilayer structure of flowering plant leaves with interweaving interactions between cells and tissues [START_REF] Malinowski | Understanding of Leaf Development-the Science of Complexity[END_REF]. The development of upright bryophyte gametophores, or leafy shoots, has contributed to the colonization of new environments by plants and helped mosses to thrive on land for hundreds of million years [START_REF] Mitchell | Cryptogamic ground covers as analogues for early terrestrial biospheres: Initiation and evolution of biologically mediated proto-soils[END_REF]. Moss leaves are lateral appendages attached to the stem of gametophores. In the model species P. patens (Physcomitrum patens, formerly known as Physcomitrella patens), leaves are small and composed of cells arranged principally in a single layer [START_REF] Courtice | Mutants of the moss Physcomitrella patens which produce leaves of altered morphology[END_REF]; Figures 1A-C). As the gametophore grows, leaves start to develop a midrib (a bundle of specialized conducting cells) and marginal serrations [START_REF] Sakakibara | Involvement of auxin and a homeodomain-leucine zipper I gene in rhizoid development of the moss Physcomitrella patens[END_REF][START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF][START_REF] Dennis | Quantitative methods in like-for-like comparative analyses of Aphanorrhegma (Physcomitrella) patens phyllid development[END_REF]; Figure 1E). At first glance, the lanceolate-shaped leaves of P. patens mirror the dominant leaf shape of flowering plants, but they have a much simpler structure and smaller size, and can be more easily imaged, which makes them an ideal system for studying leaf development. However, our understanding of moss leaf organogenesis is still limited. Here, we review current knowledge on P. patens leaf organogenesis, focusing on the cellular dynamics and molecular factors underlying leaf development.

LEAF INITIATION

In contrast to flowering plants where lateral organs are generated at the multicellular shoot apical meristem (Kuhlemeier, 2017), leaves in bryophytes are derived from a single shoot apical cell [START_REF] Gifford | Concept of Apical Cells in Bryophytes and Pteridophytes[END_REF][START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF]. This shoot apical cell is itself generated from a single shoot initial cell. Specification of the shoot initial cell requires both cytokinin and auxin [START_REF] Ashton | Analysis of Gametophytic Development in the Moss, Physcomitrella patens, Using Auxin and Cytokinin Resistant Mutants[END_REF][START_REF] Cove | Mosses as model systems for the study of metabolism and development[END_REF][START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF]. Factors including DEFECTIVE KERNEL 1 (DEK1), NO GAMETOPHORES 1 and 2 (NOG1 and 2) RECEPTOR-LIKE PROTEIN KINASE 2 (RPK2), and CLAVATA (CLV) function through APETALA2-type (AP2type) transcription factors to control the frequency of shoot initial cells [START_REF] Aoyama | AP2-type transcription factors determine stem cell identity in the moss Physcomitrella patens[END_REF][START_REF] Perroud | Defective Kernel 1 (DEK1) is required for threedimensional growth in Physcomitrella patens[END_REF][START_REF] Moody | Genetic Regulation of the 2D to 3D Growth Transition in the Moss Physcomitrella patens[END_REF][START_REF] Moody | NO GAMETOPHORES 2 Is a Novel Regulator of the 2D to 3D Growth Transition in the Moss Physcomitrella patens[END_REF][START_REF] Whitewoods | CLAVATA Was a Genetic Novelty for the Morphological Innovation of 3D Growth in Land Plants[END_REF][START_REF] Demko | Calpain DEK1 acts as a developmental switch gatekeeping cell fate transitions[END_REF][START_REF] Nemec Venza | PpRPK2 modulates auxin homeostasis and transport to specify stem cell identity and plant shape in the moss Physcomitrella[END_REF]. In P. patens, a shoot initial cell undergoes several rounds of stereotypic, oblique cell divisions that lead to the formation of a tetrahedral shoot apical cell, marking the transition from a so-called 2D to 3D growth mode (Figure 2A; [START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF]. These divisions are also regulated by DEK1, CLV, NOG1, and NOG2 genes [START_REF] Perroud | Defective Kernel 1 (DEK1) is required for threedimensional growth in Physcomitrella patens[END_REF][START_REF] Moody | Genetic Regulation of the 2D to 3D Growth Transition in the Moss Physcomitrella patens[END_REF][START_REF] Moody | NO GAMETOPHORES 2 Is a Novel Regulator of the 2D to 3D Growth Transition in the Moss Physcomitrella patens[END_REF][START_REF] Whitewoods | CLAVATA Was a Genetic Novelty for the Morphological Innovation of 3D Growth in Land Plants[END_REF][START_REF] Nemec Venza | PpRPK2 modulates auxin homeostasis and transport to specify stem cell identity and plant shape in the moss Physcomitrella[END_REF], and precisely fulfilled by mitotic spindle orientation regulators, including microtubule-associated protein TARGETING FACTOR FOR Xklp2 and SABRE [START_REF] Kosetsu | Cytoplasmic MTOCs control spindle orientation for asymmetric cell division in plants[END_REF][START_REF] Kozgunova | Spindle position dictates division site during asymmetric cell division in moss[END_REF][START_REF] Cheng | SABRE populates ER domains essential for cell plate maturation and cell expansion influencing cell and tissue patterning[END_REF]. Additionally, SOSEKI proteins might also be involved in apical cell identity specification and division [START_REF] Van Dop | DIX domain polymerization drives assembly of plant cell polarity complexes[END_REF].

The self-renewing activity of the shoot apical cell gives the gametophytic leafy shoot a capacity for indeterminate growth. Through successive asymmetric divisions, the apical cell maintains itself and gives rise to merophytes, which divide to generate leaf initials and cells that produce stem tissues (Figure 2A). This cell-autonomous capacity to rotate cell division planes in 3D initiates the growth of upright leafy gametophores and underlies its spiral phyllotaxy (Figure 2B; [START_REF] Kamamoto | Rotation angle of stem cell division plane controls spiral phyllotaxis in mosses[END_REF][START_REF] Véron | Phyllotaxis from a Single Apical Cell[END_REF]. Similar to flowering plant meristems, leaf initial outgrowth and shoot apical cell function in mosses involve auxin and PIN-FORMED (PIN) mediated auxin [START_REF] Viaene | Directional Auxin Transport Mechanisms in Early Diverging Land Plants[END_REF].

transport but the precise mechanism of their action is unclear [START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF]. SHORT-LEAF (SHLF), a bryophyte specific tandem direct repeat gene, is likely involved in the underlying mechanism, as SHLF expression is associated with auxin accumulation in the gametophore and the capacity of the shoot apical cell to generate leaves [START_REF] Mohanasundaram | The unique bryophyte-specific repeatcontaining protein SHORT-LEAF regulates gametophore development in moss[END_REF].

LEAF DEVELOPMENT

Leaf development in P. patens starts with the outgrowth of the leaf initial cell, which depends on auxin and cellulose biosynthesis [START_REF] Goss | A CELLULOSE SYNTHASE (CESA) gene essential for gametophore morphogenesis in the moss Physcomitrella patens[END_REF][START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF]. The leaf initial cell maintains meristematic potential and cleaves daughter cells basipetally [START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF]. Similar to the shoot apical cell, the divisions of the leaf initial cell seem to be controlled cell-autonomously but instead of rotating spirally, subsequent divisions alternate in the same plane and are almost perpendicular to each other (Figures 2C,D; [START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF][START_REF] Bascom | Longterm growth of moss in microfluidic devices enables subcellular studies in development[END_REF]. The orientation of the leaf initial cell division plane is likely controlled by microtubules as mutants lacking the cortical microtubule regulator TONNEAU1 develop thick multi-layered leaves [START_REF] Spinner | The function of TONNEAU1 in moss reveals ancient mechanisms of division plane specification and cell elongation in land plants[END_REF]. Eventually, the leaf apical cell stops dividing and becomes the pointed tip of the leaf. Daughter cells derived from the leaf apical cell divide further, first near the leaf base (Figure 2E). The proliferative activity of these daughter cells decreases gradually so that cells near the tip divide less frequently and give rise to smaller segments of the leaf (Figures 2C,D; [START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF]. Several rounds of longitudinal divisions within sectors derived from the leaf apical cell, especially in the outermost lateral portion of the leaf, lead to leaf broadening (Figure 2E). Additional transverse divisions also contribute to extending the daughter segments in the proximo-distal axis (Figure 2E; [START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF].

Quantitative analysis of the entire leaf growth at cellular resolution has not yet been performed in P. patens. However, the basipetal gradient of cell proliferation (Figure 2E; [START_REF] Harrison | Local Cues and Asymmetric Cell Divisions Underpin Body Plan Transitions in the Moss Physcomitrella patens[END_REF] and basipetal increase of cell sizes in mature leaves (Figure 2F; [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF][START_REF] Dennis | Quantitative methods in like-for-like comparative analyses of Aphanorrhegma (Physcomitrella) patens phyllid development[END_REF] indicate that cells near the leaf tip are the earliest to cease growth. Cells at the tip differentiate first as they become insensitive to exogenous cytokinin, while cell proliferation in more proximal leaf regions is stimulated by this hormone [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF]. Interestingly, basipetal gradients of growth, proliferation, and differentiation are key features of many flowering plant leaves and is controlled non-cell-autonomously by positional information [START_REF] Avery | STRUCTURE AND DEVELOPMENT OF THE TOBACCO LEAF[END_REF][START_REF] Andriankaja | Exit from proliferation during leaf development in Arabidopsis thaliana: a not-so-gradual process[END_REF][START_REF] Kuchen | Generation of Leaf Shape Through Early Patterns of Growth and Tissue Polarity[END_REF][START_REF] Fox | Spatiotemporal coordination of cell division and growth during organ morphogenesis[END_REF][START_REF] Kierzkowski | A Growth-Based Framework for Leaf Shape Development and Diversity[END_REF]. Thus, apart from the cell-autonomous behavior of the leaf apical cell, positional cues also likely play a role in controlling moss leaf growth.

Auxin is a fundamental player in plant organogenesis where it regulates cell proliferation, elongation, and differentiation in a positional and context dependent manner [START_REF] Vieten | Molecular and cellular aspects of auxin-transport-mediated development[END_REF][START_REF] Weijers | Auxin: small molecule, big impact[END_REF]. An essential common genetic toolbox involved in auxin biosynthesis, transport, and signaling is conserved between flowering plants and bryophytes [START_REF] Poli | Auxin regulation of axial growth in bryophyte sporophytes: its potential significance for the evolution of early land plants[END_REF][START_REF] Kato | Evolution of nuclear auxin signaling: lessons from genetic studies with basal land plants[END_REF][START_REF] Thelander | Auxin-mediated developmental control in the moss Physcomitrella patens[END_REF]. Auxin could provide positional information during moss leaf development. Although the signal of the GH3:GUS reporter has not been observed in the wild type leaves of P. patens, exogenous auxin treatments disrupt leaf growth [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF][START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF].

Auxin distribution within the developing leaf could be controlled by canonical auxin efflux carriers PIN-FORMED A and B (PpPINA and B) [START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF][START_REF] Viaene | Directional Auxin Transport Mechanisms in Early Diverging Land Plants[END_REF]. The expression of PpPINA is displaced along the proximo-distal axis of the leaf during growth and correlates with developmental gradients (Figure 2G). This correlation is also evident at the subcellular level, where PpPINA protein is localized bipolarly on both apical and basal cell sides close to the leaf tip, while it is distributed more uniformly in cell membranes near the leaf base (Figure 2G; [START_REF] Viaene | Directional Auxin Transport Mechanisms in Early Diverging Land Plants[END_REF]. Alternatively, auxin gradients in mosses might also be achieved by callosecontrolled plasmodesmata-mediated diffusion [START_REF] Coudert | Three ancient hormonal cues co-ordinate shoot branching in a moss[END_REF]. Gametophores of the shlf mutant produce shorter leaves with a decreased cell number, similar to plants grown with a high auxin concentration or overexpressing PpPINA. Auxin activity gradients can be detected in leaves of the shlf mutant where GH3:GUS signal is present in very young leaves and at the tip of more developed leaves [START_REF] Mohanasundaram | The unique bryophyte-specific repeatcontaining protein SHORT-LEAF regulates gametophore development in moss[END_REF]. As SHLF might regulate plasmodesmata frequency, it could control leaf morphogenesis through the regulation of auxin gradients [START_REF] Mohanasundaram | The unique bryophyte-specific repeatcontaining protein SHORT-LEAF regulates gametophore development in moss[END_REF].

Class III Homeodomain-Leucine Zipper (HD-ZIP III) transcription factors are key players during flowering plant development and are associated with auxin synthesis and transport [START_REF] Prigge | Evolution of the class III HD-Zip gene family in land plants[END_REF][START_REF] Ariel | The true story of the HD-Zip family[END_REF][START_REF] Turchi | Interplay of HD-Zip II and III transcription factors in auxin-regulated plant development[END_REF]. There are five HD-ZIP III homologs in the P. patens, namely PpC3HDZ1-5 [START_REF] Yip | Class III HD-Zip activity coordinates leaf development in Physcomitrella patens[END_REF]. PpC3HDZ expression colocalizes with actively developing regions in adult leaves. Strikingly, PpC3HDZ knock-down induces pronounced, multicellular protrusions along leaf margins that resemble the distal portion of wild-type leaves. It indicates that in PpC3HDZ knock-down plants, daughter cells derived from the leaf apical cell, whose normal divisions produce characteristic leaf sectors (Figures 2C,D), could recapitulate the cell-autonomous behavior of the apical cell itself. Additionally, in PpC3HDZ knock-down leaves, cell number is reduced, suggesting that HD-ZIP III genes are important for the establishment and/or the maintenance of the proliferative activity of the daughter cells [START_REF] Yip | Class III HD-Zip activity coordinates leaf development in Physcomitrella patens[END_REF]. PpC3HDZ expression domain mirrors PpPINA distribution, and both PpC3HDZ knock-down and PpPINA/PpPINB knock-out lines produce narrower leaves with a reduced cell number, indicating that HD-ZIP III function could be at least partially related to auxin-dependent positional information [START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF][START_REF] Viaene | Directional Auxin Transport Mechanisms in Early Diverging Land Plants[END_REF][START_REF] Yip | Class III HD-Zip activity coordinates leaf development in Physcomitrella patens[END_REF].

LEAF HETEROBLASTIC DEVELOPMENT

Although all leaves of P. patens initiate from single cells, their morphology changes gradually up the gametophore axis (Figure 1A; [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF][START_REF] Dennis | Quantitative methods in like-for-like comparative analyses of Aphanorrhegma (Physcomitrella) patens phyllid development[END_REF], a phenomenon called heteroblastic development [START_REF] Zotz | Heteroblasty-A Review[END_REF]. As in Arabidopsis, juvenile leaves in P. patens are much smaller than adult leaves (Figures 1B,C; [START_REF] Courtice | Mutants of the moss Physcomitrella patens which produce leaves of altered morphology[END_REF][START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF]. The width of the juvenile leaf is relatively constant except at the tapering tip, resulting in a roughly oblong shape (Figure 1C). By contrast, adult leaves have a lanceolate shape with a narrow base, a broader middle part, and a pointy tip (Figure 1B). The increase in size during heteroblastic development results from an increase in cell number but not cell size [START_REF] Dennis | Quantitative methods in like-for-like comparative analyses of Aphanorrhegma (Physcomitrella) patens phyllid development[END_REF]. In flowering plants, auxin and cytokinin regulate cell proliferation and differentiation in an opposite manner during leaf development [START_REF] Shani | Cytokinin regulates compound leaf development in tomato[END_REF][START_REF] Kierzkowski | A Growth-Based Framework for Leaf Shape Development and Diversity[END_REF][START_REF] Skalák | Multifaceted activity of cytokinin in leaf development shapes its size and structure in Arabidopsis[END_REF]. Exogenous treatments with auxin, or knock-out mutants in auxin efflux carriers PpPINA and PpPINB, cause a decrease in moss leaf size by reducing cell number [START_REF] Bennett | Plasma Membrane-Targeted PIN Proteins Drive Shoot Development in a Moss[END_REF][START_REF] Viaene | Directional Auxin Transport Mechanisms in Early Diverging Land Plants[END_REF]. By contrast, cytokinin treatment increases cell proliferation and enhances adult leaf characteristics [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF]. Altogether this indicates an important role of cytokinin-auxin cross-talk in moss leaf heteroblastic development.

The transition from juvenile to adult leaves in P. patens is also associated with morphological changes of the marginal cells that become long and narrow (Figure 1D) as well as the formation of a multilayered midrib (Figure 1E; [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF][START_REF] Dennis | Quantitative methods in like-for-like comparative analyses of Aphanorrhegma (Physcomitrella) patens phyllid development[END_REF]. Marginal cells tend to grow slightly outward at their distal end to form marginal serrations that usually appear on the apical half of the adult leaf [START_REF] Barker | Heteroblasty in the moss, Aphanoregma patens (Physcomitrella patens), results from progressive modulation of a single fundamental leaf developmental programme[END_REF]. The leaf margin in flowering plants has a distinct cell morphology and plays an important role during leaf development [START_REF] Bilsborough | Model for the regulation of <em>Arabidopsis thaliana</em> leaf margin development[END_REF][START_REF] Nakata | Roles of the Middle Domain-Specific WUSCHEL-RELATED HOMEOBOX Genes in Early Development of Leaves in Arabidopsis[END_REF][START_REF] Kalve | Leaf development: a cellular perspective[END_REF][START_REF] Alvarez | Active suppression of a leaf meristem orchestrates determinate leaf growth[END_REF][START_REF] Hayakawa | Flat leaf formation realized by cell-division control and mutual recessive gene regulation[END_REF]. Marginal cells in P. patens could play a role in controlling upper leaf size and shape by restricting or promoting growth around its circumference. The wrinkled leaves with undifferentiated marginal cells of crinkly4 mutants likely result from mechanical conflicts in lamina cells [START_REF] Demko | The phenotype of the CRINKLY4 deletion mutant of Physcomitrella patens suggests a broad role in developmental regulation in early land plants[END_REF]. The leaf margin could also work in tandem with the midrib in the bigger leaves and ensure proper leaf flattening by preventing blade twisting. However, the role of marginal cells in moss leaf morphogenesis remains elusive.

THE ROLE OF MIDRIB

Unlike vascular plants which conduct water through the xylem, mosses neither possess vessel elements or tracheids, nor have an interconnected network of veins. Instead, mosses mostly rely on external water conduction by capillary action to carry out water-dependent physiological functions [START_REF] Proctor | Structure and eco-physiological adaptation in bryophytes[END_REF]. However, the adult leaves conduct water through a multilayered tissue called the midrib that is reminiscent of a vascular bundle and arises through a series of periclinal and anticlinal cell divisions [START_REF] Ligrone | Conducting tissues and phyletic relationships of bryophytes[END_REF][START_REF] Sakakibara | Involvement of auxin and a homeodomain-leucine zipper I gene in rhizoid development of the moss Physcomitrella patens[END_REF][START_REF] Yip | Class III HD-Zip activity coordinates leaf development in Physcomitrella patens[END_REF]. These cell divisions give rise to different cell types within the leaf midrib, including thick-walled stereids and thin-walled, elongated hydroids (Figure 1E). Hydroids are the main water-conducting cells. They are initially alive, before undergoing programmed cell death, and have fully degenerate protoplasm upon maturation [START_REF] Xu | Contribution of NAC transcription factors to plant adaptation to land[END_REF]. Stereids also undergo programmed cell death but may mainly serve a supporting purpose. Therefore, hydroids resemble xylem vessel elements and tracheids, except that they lack a lignified secondary cell wall, while stereids resemble xylem fiber cells [START_REF] Ligrone | Conducting tissues and phyletic relationships of bryophytes[END_REF].

Several genes affecting midrib formation have been identified and characterized. For instance, among the seven moss genes that encode VND/NST/SND family transcription factors, six of them (PpVNS1-7, except PpVNS3) are expressed in the central region of newly formed leaves or in developing midribs. The triple mutant (ppvns1,6,7) forms morphologically normal midribs. However, stereid programmed cell death and hydroid proliferation are disrupted, and the water conducting capacity is greatly compromised [START_REF] Xu | Contribution of NAC transcription factors to plant adaptation to land[END_REF]. Interestingly, VNS orthologs are critical for xylem vessel element formation in angiosperms, suggesting that mosses and vascular plants may at least partly use the same molecular mechanism in shaping water-conducting tissue. The leaves of midrib-defective mutants tend to curl around the middle axis under low-humidity conditions, indicating a water transport deficiency. Nevertheless, compromised midrib formation does not seem to have a detrimental impact on overall moss growth, at least in laboratory conditions.

HD-ZIP III transcription factors also control midrib establishment. When their function is suppressed, midrib formation is abnormal and leaf shape becomes distorted [START_REF] Yip | Class III HD-Zip activity coordinates leaf development in Physcomitrella patens[END_REF]. Given that HD-ZIP III transcription factors ATHB8, ATHB15, and REVOLUTA are essential for procambial cell specification and vasculature development, the observation that HD-ZIP III proteins function in specifying moss waterconducting tissues further suggests that shared molecular mechanisms underpin conducting tissue development in convergent plant organs [START_REF] Kang | Cell cycling frequency and expression of the homeobox gene ATHB-8 during leaf vein development in Arabidopsis[END_REF][START_REF] Ohashi-Ito | Promotion of transcript accumulation of novel Zinnia immature xylem-specific HD-Zip III homeobox genes by brassinosteroids[END_REF][START_REF] Green | CORONA, a member of the class III homeodomain leucine zipper gene family in Arabidopsis, regulates stem cell specification and organogenesis[END_REF][START_REF] Prigge | Class III homeodomain-leucine zipper gene family members have overlapping, antagonistic, and distinct roles in Arabidopsis development[END_REF][START_REF] Donner | Regulation of preprocambial cell state acquisition by auxin signaling in Arabidopsis leaves[END_REF] and therefore that these mechanisms may have evolved before the divergence between bryophytes and vascular plants. Nonetheless, further research on midrib development is needed to uncover the regulatory circuits underlying the morphological differences between juvenile and adult leaves.

PERSPECTIVES

In this review, we summarized the current understanding of leaf development in the model species P. patens. Proper leaf morphogenesis in this moss seems to require coordination of cellautonomous and non-cell-autonomous developmental processes that are controlled via cross-talks between molecular regulators that are bryophyte-specific or shared by land plants. It is, however, still difficult to understand how these genetic and hormonal inputs are translated into cellular growth and division patterns and how they are coordinated in space and time within the mechanically connected tissue constituting the P. patens leaf.

In recent years, new research approaches combining genetics, quantitative live-imaging, biomechanics, and computational modeling have massively advanced our understanding of organ development in plants [START_REF] Hervieux | A Mechanical Feedback Restricts Sepal Growth and Shape in Arabidopsis[END_REF][START_REF] Solly | Regional Growth Rate Differences Specified by Apical Notch Activities Regulate Liverwort Thallus Shape[END_REF][START_REF] Fox | Spatiotemporal coordination of cell division and growth during organ morphogenesis[END_REF][START_REF] Hong | Heterogeneity and Robustness in Plant Morphogenesis: From Cells to Organs[END_REF][START_REF] Kierzkowski | A Growth-Based Framework for Leaf Shape Development and Diversity[END_REF][START_REF] Sapala | Mechanics, geometry and genetics of epidermal cell shape regulation: different pieces of the same puzzle[END_REF][START_REF] Wolny | Accurate and versatile 3D segmentation of plant tissues at cellular resolution[END_REF][START_REF] Whitewoods | Evolution of carnivorous traps from planar leaves through simple shifts in gene expression[END_REF][START_REF] Hernandez-Lagana | Organ geometry channels reproductive cell fate in the Arabidopsis ovule primordium[END_REF][START_REF] Vijayan | A digital 3D reference atlas reveals cellular growth patterns shaping the Arabidopsis ovule[END_REF]. The next step will be to apply such a multidisciplinary approach to study moss leaf development.

In comparison with the complex anatomy of flowering plant leaves, the single-cell-layered structure of P. patens leaf provides a unique opportunity to quantify the development of the entire organ in 3D. Such an approach, should provide a comprehensive picture of moss leaf developmental dynamics at cellular resolution that otherwise would be difficult to apprehend. In contrast with flowering plants, gene targeting by homologous recombination is very efficient in P. patens, and together with CRISPR/Cas9-mediated gene editing, permits to bypass genetic redundancy and rapidly generate highorder mutants [START_REF] Koshimizu | Physcomitrella MADS-box genes regulate water supply and sperm movement for fertilization[END_REF]. Combination of imaging and genetic approaches will, for example, help us to understand the precise role of molecular regulators in the control of cell-autonomous and non-cell-autonomous behaviors during growth and their role in juvenile to adult leaf transition.

Single-cell-layered leaves in P. patens will also be a huge asset to dissect the role of biomechanical signals regulating plant organogenesis. Largely eliminating complex interactions that occur between different tissue layers of developing organs, as in flowering plants, this moss model system should provide a better understanding of the mechanical interactions between individual cells and their role in the coordination of organ growth. As P. patens leaves are easy to manipulate and have relatively big cells, they should also enable precise measurements of cell mechanical properties using modern micro-indentation devices [START_REF] Routier-Kierzkowska | Cellular force microscopy for in vivo measurements of plant tissue mechanics[END_REF][START_REF] Robinson | An automated confocal micro-extensometer enables in vivo quantification of mechanical properties with cellular resolution[END_REF][START_REF] Majda | Cellular Force Microscopy to Measure Mechanical Forces in Plant Cells[END_REF] or turgor pressure manipulations [START_REF] Kierzkowski | Elastic Domains Regulate Growth and Organogenesis in the Plant Shoot Apical Meristem[END_REF][START_REF] Sapala | Osmotic Treatment for Quantifying Cell Wall Elasticity in the Sepal of Arabidopsis thaliana[END_REF]. All these experimental inputs, in combination with geometrically accurate templates extracted from confocal images, will enable the creation of biologically realistic simulations of the entire moss leaf. Such models will not only provide a comprehensive picture of P. patens leaf development, but also advance our general understanding of the mechanism governing plant organogenesis.

FIGURE 1 |

 1 FIGURE 1 | Leaf anatomy in P. patens. (A) A leafy shoot (or gametophore) with juvenile and adult leaves from the base to the top. (B) Adult (upper) leaf has a multicellular midrib and lanceolate shape. The leaf margin is magnified to show marginal serrations (red arrowhead) formed by cell tip outgrowths. (C) First juvenile leaf is composed of a single cell layer and has a rectangular shape. (D) Three cell shapes are commonly identified in adult leaves: long and narrow cells on the edge; smaller and more isodiametric cells close to the tip (top); long and broad cells near the base (based on Dennis et al., 2019). (E) TEM cross-section image of a midrib cell bundle with thick-walled stereids and thin-walled hydroids (marked with blue and red asterisk respectively). Scale bars: 200 µm in (A); 50 µm in (B,C); and 10 µm in (E).

FIGURE 2 |

 2 FIGURE 2 | Leaf growth in P. patens. (A) Schematic representation of the transition from 2D to 3D growth during leafy gametophore initiation. Gametophore shoot initial cell divides obliquely to generate the shoot apical cell (in green). Second and third divisions of the shoot initial cell are also oblique and generate a hair cell (in white) and the first leaf initial (L1, in red) consecutively. From this stage, the shoot apical initial cell becomes tetrahedral and generates subsequent leaf initials. (B) A single apical cell (marked with asterisk) produces leaves in a spiral phyllotactic pattern with leaf sequences indicated in consecutive numbers. (C) Schematic representation of the early development of the juvenile leaf. The leaf apical cell is shown in red and the cell recently cleaved from the apical cell is shown in blue. Red dotted lines indicate recent cell division (based on Harrison et al., 2009). (D) Schematic representation of clonal sectors arising from single cells shown in (C) at early stages of leaf development. (E) Schematic representation of cell division patterns within the segments generated by the leaf apical cell. Red lines represent new walls (based on Harrison et al., 2009). (F) Schematic representation of the distribution of cell sizes in the adult leaf. Bigger cells are located in the distal region of the leaf, smaller cells in proximal region (based on Dennis et al., 2019). (G) Schematic representation of the PpPINA protein localization (in green) in the adult leaf[START_REF] Viaene | Directional Auxin Transport Mechanisms in Early Diverging Land Plants[END_REF].

Frontiers in Plant Science | www.frontiersin.org

September 2021 | Volume 12 | Article 736212

ACKNOWLEDGMENTS

We thank Viraj Alimchandani and Andrea Gómez Felipe for critical reading of the manuscript. We also thank reviewers for their suggestions to improve the manuscript.

FUNDING This work was supported by the 2020-NC-267497 grant from Fond de Recherche du Québec Nature et Technologies and Discovery grant RGPIN-2018-04897 from the Natural Sciences and Engineering Research Council of Canada to DK. This work was also supported by the grant 31871245 from National Natural Science Foundation of China (NSFC) and the grant 2019YFA0903902 from National Key R&D Program of China (NKP) to YW.

AUTHOR CONTRIBUTIONS

WL, YW, YC, and DK wrote and edited the article. All authors contributed to the article and approved the submitted version.

Conflict of Interest:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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