Angelika Bikner-Ahsbahs

Jana Trgalova
email: jana.trgalova@univ-lyon1.fr

Andrea Maffia
email: andrea.maffia@unipv.it

Arthur Bakker
email: bikner@math.uni-bremen.de

Dorota Lembrér
email: dorota.lembrer@mau.se

Rhythmic instrumental orchestration: Joining two theoretical perspectives in designing an online summer school

Keywords: Design-based research, instrumental genesis, rhythm, online learning, summer school

Due to the pandemic situation in 2020 the ERME summer school (YESS10) was designed for an online format using a conference system. Our design choices were based on previous experiences with YESS and the use of the research pentagon as a tool to think about research. This paper elaborates theoretically and empirically the specification of the concept of instrumental orchestration of the pentagon use through its rhythmic implementation into the summer school. Research results indicate that this specification had two main effects: The students described their instrumentation of the pentagon as a structuring tool in various ways. The most relevant pentagon use for the students' experience of growing expertise was listening to and observing how the others used the pentagon.

Introduction and background

Due to the pandemic in spring 2020 the Young ERME Summer School (YESS) of the European Society for Research in Mathematics Education (ERME) switched to using a synchronic online conference system. Two out of seven thematic working groups (TWG7, TWG5) run in a virtual environment. As neither clear conceptions nor research were available for the design of the school with respect to distance learning of PhD-students via a conference system, the two experts and two brokers (previous PhD-students of YESS) decided to conduct a design-based research study involving the conference system Adobe Connect. Planning jointly these TWGs led the brokers to decide to act as assistances and to provide several Adobe tools to be shared in the TWGs. The leading experts made common design choices based on their experiences in the previous YESS-TWGs. As each student had to submit a short description of their research project, all the students were asked to read all the papers before the school and prepare a presentation of their project to be discussed during the school. Prior to the school, pairs of critical friends were built within both TWGs to make them engage deeper into the friend's paper and present their critical feedback in an explicit presentation. The main design choice concerned the research pentagon (Bikner-Ahsbahs, 2019) as a tool to think and reflect about research in terms of research aims, questions, objects, methods, and situations involving the situation of the project in the field and creating a specific situation for the research in the project (Figure 1,left). The design of how to use the pentagon took into account previous experiences with the pentagon as well as the fact that the students coming from different time zones, and simultaneously being involved in their families during the school, required a clear timetable so that they would be able to manage their pandemic situation and attend YESS10. This resulted in a sequence of five situations of using the pentagon (Figure 1, middle). This sequence was repeated in all the ten sessions in the same order in both TWGs, however with some variations (Figure 1, middle & right). Based on these design choices, we ask: How did the students experience the rhythmic orchestration of the pentagon use and the pentagon use itself as contributing to their learning about research?

Theoretical framework

Instrumental approach and instrumental orchestration

Considering the pentagon as a tool that participants need to appropriate in order to use it efficiently, instrumental approach (Vérillon & Rabardel, 1995;[START_REF] Rabardel | People and technology -a cognitive approach to contemporary instruments[END_REF] is one of two theoretical approaches in our research framework. The key idea is the distinction between an artefact -material or symbolic object available to a subject, and an instrument -a merging construct that consists of the artefact and the related mental schemes coming from the use of the artefact by the subject in a given context. The process of transformation of an artefact into an instrument, called instrumental genesis, consists of two interrelated processes: instrumentation leading to the constitution and the evolution of schemes of using the artefact in the subject, and instrumentalisation during which the subject adapts and personalizes the artefact according to her knowledge and beliefs. The development of schemes of use manifests itself in the subject's invariant organization of behavior in a given class of situations (Vergnaud, 1990). [START_REF] Trouche | Instrumentation in Mathematics Education[END_REF] suggests seeing instrumentation both as "an action (by which someone acquires an instrument)" and as "the influence of this action on a subject's activity and knowledge" (p. 307). Thus, using an artefact yields both pragmatic and epistemic outcomes.

In the context of students' learning, students' instrumental geneses need to be accompanied by a teacher. [START_REF] Trouche | Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students' Command Process through Instrumental Orchestrations[END_REF] introduced the notion of instrumental orchestration to describe how a teacher can plan and organize students' interactions with available artefacts in order to enhance individual and collective instrumental geneses. Referring to [START_REF] Trouche | Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students' Command Process through Instrumental Orchestrations[END_REF], Drijvers et al. (2010, p. 214) define an instrumental orchestration as "the teacher's intentional and systematic organization and use of the various artefacts available in a […] learning environment in a given […] task situation, in order to guide students' instrumental geneses". It consists of three elements:

1. A didactical configuration, which is "a configuration of the teaching setting and the artefacts involved in it" (Drijvers et al., 2010, p. 215);

2. An exploitation mode, which is "the way the teacher decides to exploit a didactical configuration for the benefit of his or her didactical intentions" and includes in particular the teacher's decisions "on the possible roles of the artefacts to be played" (p. 215); 3. A didactical performance, which involves "the ad hoc decisions taken while teaching on how to actually perform in the chosen didactic configuration and exploitation mode" (p. 215).

Design approach to orchestrate the pentagon use

The pentagon was implemented in the design of the sessions as a tool to structure and reflect on research (Bikner-Ahsbahs, 2019) individually and in interaction with others. Pairs of critical friends provided mutually more in-depth feedback in their presentation in addition to the spontaneous feedback coming from all the other peers. Whereas TWG7 introduced an article about the research pentagon before the school to be used for the preparation of the presentations, the expert of TWG5 introduced this tool in the first session. Besides this distinction, both experts used the tool permanently during the school as a reference object in their feedback and the discussion. We organized each of the 10 sessions in the TWGs by five situations of pentagon use (Figure 1, left) so that, in the course of these sessions, the students' roles as presenter, critical friend, observer and listener, feedback provider and receiver of peer and expert feedback varied regularly in time and space of the virtual environment. This organization aimed at enhancing participants' instrumental geneses to improve their research expertise.

The change from a face-to-face summer school to an online format required to theorize the meeting space. According to [START_REF] Lefebvre | The Production of Space[END_REF], any space is established with respect to three dimensions: (1) A space is represented materially (also technologically), (2) it is representational in that it represents some idea (here: switching YESS to an online school), and (3) it is (re-)produced as a social space by the people involved. All the three dimension apply to YESS10. The digital conference system Adobe Connect used as a technological space linked to the individual material home computer spaces, constituted a representation of the virtual space. It provided tools for sharing and distributing information among the students, hence, for creating the social space of acting and interacting of the participants supported by these tools, e.g. the breakout rooms, chat, and note taking possibilities were included into the didactic configuration. This notion of space developed structured in time with a frequency of three sessions per day and breaks of different lengths (session-short break-sessionlecture break-session-night break) repeated day after day.

In the summer school, the collective nature of teaching and learning is essential and this depends on the possibility of creating joint attention on the research presented. In face-to-face situations, this is established by building a reference space through deictic expressions, shown e.g., by linguistic means, through a projector, by gestures, gaze, body posture, moving [START_REF] Balantani | Joint attention and reference construction: The role of pointing and "so[END_REF][START_REF] Stukenbrock | Deixis in der face-to-face-Interaktion[END_REF]. However, most of these deictic expressions could not be shared in our virtual space. So, we designed the pentagon use in a way that it could facilitate creating a reference space because all the students could refer to this diagram sharing its structure, such as the vertices and their relations, with their peers, and simultaneously talking about their own research.

An implicit key idea in this design was rhythm as a way to structure time. Rhythm emerged as a critical design element when analyzing the data.

Rhythm as a design element [START_REF] Lefebvre | Rhythmanalysis. Space, Time and Everyday Life[END_REF] stressed that space, time, and energy are interrelated by rhythm, the latter being bodily grounded and thus always active. Rhythm as a structure of time is created in the course of sessions by an "ordered variation of changes" (Dewey, 1934, p. 160). Figure 1 (right) shows the linear rhythm as a repetition of sessions (three per day) with pentagon use and breaks. Within each session, the order of five situations of pentagon use is cyclically repeated involving regular and spontaneous variations by providing all students the opportunity to act in their way in all the roles and addressing various research topics and foci. This rhythmic characteristic of the didactical configuration was enriched by a rhythmic exploitation mode in which the experts regularly referred to the pentagon explaining while pointing to vertices and connections in varying research projects, and hence, inviting the students to use the pentagon as a reference space, too. Thus, the group could create joint attention on the research presented and discussed. As the experts have used the pentagon in previous summer schools, they related their feedback (didactical performance) to students' actual needs, highlighting blind spots, missing aspects, or a change of view, in using the pentagon as an organizing tool for research. So there is constructive interaction of various rhythms, so-called "eurhythmia" (Lefebvre, 2004, p. 16). Lefebvre emphasizes that rhythm allows measuring changes: "No rhythm without repetition in time and in space, without reprises, without returns, in short without measure [measure]" (p. 6). As the same situations of pentagon use are repeated the students had the opportunity to experience and compare these uses within eurhythmia, thus measure changes of own and others' pentagon uses related to research and hence, measure their own development of expertise on research.

Methodology

We use rhythmanalysis [START_REF] Lefebvre | Rhythmanalysis. Space, Time and Everyday Life[END_REF] to answer the research question focusing on how the students experienced the rhythmic instrumental orchestration, which instrumentation processes they reflect and how the instrumental geneses of the pentagon may have contributed to their learning about research. As we could not observe these processes in time, we shaped this research similar to action research. We drew on our experience of conducting the TWGs, our design choices and note taking. An external expert interviewed 11 volunteers of the 20 participants about the efforts they invested into the school, their experience of the pentagon use and of our design to develop their learning about research. These interviews were video recorded and transcribed verbatim (names were replaced with pseudonyms). Thus, we did not observe the rhythmic processes directly, but rather approached them through the students' individual experience of the (eu)rhythmic orchestration in their interviews. This experience begins with a first contact with the pentagon and ends with reflections on own research. Related to this frame, we conduct our analyses in three steps considering the five different roles (Figure 1, middle) in which each student acted and interacted. First, we identified quotes in the interviews that refer to these roles, and described how each student experienced his/her individual instrumental genesis of the pentagon distinguishing between instances of instrumentation and instrumentalisation within the orchestration (Figure 2, vertical analyses expressed as vertical arrows).

For each role, we secondly compared these descriptions across all the interviews to situate the individual experience into the collective (Figure 2, horizontal analyses expressed by horizontal arrows). In the third step, we identified the students' individual reflections addressing their individual experience on learning about research in the interviews, interpreted them related to their instrumental genesis and through the lens of rhythmanalysis, and compared these across all the students.

Results

Based on our framework, we extracted quotes from the interview data, which express the students' experience in their reflections immediately after the summer school. Therefore, these results indicate what the students declare to have taken from the summer school rather than the learning processes themselves. The experiences of expertise are regarded as relevant outcomes for the PhD students who after the summer school may then continue their research with a refreshed expertise.

Step

1: The pentagon becoming an individual instrument

From the case of Emma we learned that some students' instrumental geneses started before YESS as Emma had read the paper and used the pentagon before the school already. The rhythmic organization of the situations in which she used the pentagon enhanced it. In the interview, she expresses a variety of instrumentation processes developed in different situations. She developed an instrument for analyzing own research and checking its coherence ("Also, just to see that you are making progress or making your project more coherent because I think it's very useful to talk about; if there's some parts of your research project that is not coherent.") and for analyzing other's research projects ("I used the pentagon when I read the papers, for instance when I read through my critical friend's paper and whether I found some indications of the research aim and the research object, research questions, the method and…"). She also developed instruments related to feedback, for example for structuring feedback for a critical friend ("when I did the presentation for my critical friend … I could give feedback on that so what I thought was making a lot of sense") and for structuring the feedback presentation ("it can also be quite hard to understand where they [the peers] are exactly in the process and what they would need the feedback on to make them come further because. I think the research pentagon helped in that matter because I could use it to structure my presentation of my feedback"). We identified similar instrumentation processes in other interviews, too.

The case of Nordy made us aware that students in TWG5 came to know the pentagon use in the first session when the expert introduced it for the first time. As Nordy was the first presenter, he did not have time ("I need some more time to read the whole text and to incorporate it, to think about this research pentagon") for appropriating the pentagon although he perceived its potential ("I really think that it could be a great thing to organize your work [with the pentagon]"). Compared with the case of Emma, this shows the lack of and need for instrumentation. However, his instrumental genesis was about to start. Other students from TWG5 showed similar experiences.

Nordy's beginning instrumentation process was based on the expert's didactical performance, when listening to the expert's systematically and repeatedly referring to the pentagon. Nordy says: "as long as our research topic group was going on, then her [the expert's] feedback was more and more referred to it". He relates his individual to the collective instrumental genesis, when stressing "we should do our feedback depending on the research pentagon to our critical friend, I try to do that, I think we handled it quite good".

Through the case of Kira we learned that the first parts the students grasp are vertices of the pentagon as an instrumentalisation in the sense that students select the artefact features they are going to use. The links between the vertices are more complex to make sense of and need perhaps a specific orchestration (e.g. a 'technical demo' by the expert). For example, Kira used the pentagon as a feedback-structuring instrument identifying missing vertices in a peer's paper ("I could more clearly identify what was missing for me and what was also good"). Then she points to the need to fill the links between two vertices, the research question, and the methods ("filling the link to the research question would mean to ask myself, or for him to ask himself the question 'How can I measure this?' What method can I use? …"). Kira also expresses another instance of instrumentalisation in the idea to take the pentagon home for her peers: "I will take the research pentagon into a small group discussion at the university." Thus, within one week, the instrumental genesis entailed instrumentation as well as instances of instrumentalisation for Kira as well as other students.

Our vertical analyses show that the individual processes of instrumental genesis led to various instruments where the vertices are the primary foci while the links between them are more difficult to understand. Instances of instrumentalisation appeared already as the students decided which vertices they take up. Thus, the pentagon provides instrumentation and instrumentalisation possibilities from the beginning.

Step 2: Rhythm and listening as essential parts for contributing to learning Kira connected the individual and the common use of the pentagon when she pointed to it as a diagram expressing a common reference space and resource in the didactical configuration of the virtual space ("what appeared were the edges and also a definition of what can be found, what can be identified as a research aim … the research pentagon in the middle on the whiteboard and then it was easy to just point somewhere"). The configuration of the virtual space where the pentagon was used as a diagram in screen sharing allowed sharing different research projects in the group ("This [the vertices] is very very different concerning what kind of research your project is about."). The repeated use of the pentagon emerged as a rhythmic exploitation mode. It helped the students to develop their expertise by listening to and observing others as well as practicing themselves to use the pentagon repeatedly. For example, Peter points to listening to and observing its repeated use by others ("I had the ability to see the pentagon in action for ten times."). Dan highlights practicing its repeated use themselves ("doing it like a few times to other colleagues, each one in a different subject area, and hearing the other teammates, how they raise questions, was really really valuable. I mean, I learnt from it a lot."). Therefore, the rhythmic configuration and exploitation mode seem to have the potential to support developing expertise on research. Mirka is a bit more precise as she expressed how the repeated use of the pentagon supports growing in expertise. It can happen during one session in the rhythm of different situations ("I think there were some students who couldn't find… who couldn't use it in a specific way, but during the discussions of their feedback, they realized that they have to be more specific.

[…]"), as well as through the repetition of the same situational use in the course of the school during the cyclic rhythm ("once you see how other students used it, I think it's a very nice way to use it [the pentagon] your own way, personally, everyone used it; they didn't use it the same way.")

Step 3: Summarizing students' reflections on learning about research

The students highlighted the relevant role of rhythm, linear as well as cyclic, for the different uses of the pentagon and their learning about research. The most important use, as shown in our horizontal analysis, is observing and listening to how others use the pentagon, hence, they benefit from the rhythm in the didactical performance of the experts as well as from observing and listening to the processes of instrumental genesis of peers to learn how to improve their own research. Lefebvre elaborates on rhythm related to measure change with respect to a reference and hence, enables to explain the phenomenon of experience of change in learning. For example, a student may identify that the research object is still a blind spot in the own research project. Through repeatedly listening to others talking about their research objects, he or she can compare and thus measure the change of view on the own object and the experience of learning on own research. For example, Surgeryfish reflects: "what I see from others and fit into my work. … A kind of learning, becoming aware."

Discussion on the theoretical specification of instrumental orchestration

The YESS10-study is deeply rooted in the common experience of an ontological change in the pandemic situation. This change raised the necessity to enable the participants of YESS10 to align their home situations with the summer school in space and time via a conference system. As this study shows, we have achieved this by a rhythmic synchronizing (cf. [START_REF] Akkerman | Relevance of Educational Research: An Ontological Conceptualization[END_REF] of the students' instrumental geneses of the research pentagon in our virtual space and time with the goal to advance the students' projects and by that improve their expertise on research. As a result, a rhythmic instrumental orchestration emerged as a specification of instrumental orchestration, theorized with the help of rhythmanalysis of the implemented rhythmic structure. This rhythmic instrumental orchestration involves a didactical configuration that provides a rhythmic organization of the various situations and kinds of tool use. Within this configuration, a rhythmic exploitation mode emerged, which involves the expert as well as the students allowing them to use the tool in various ways as well as listening to and observing how their peers use the tool. Within this mode, the expert synchronizes her didactical performance on the tool with the students' needs. In sum, the design choices in the rhythmic orchestration led to eurhythmia, which allowed the students to develop their expertise in a communal way by learning from their peers, particularly through listening.

Figure 1 :

 1 Figure 1: The research pentagon (left), sequence of pentagon use in a session (middle) and the sequence of sessions S1-S10 (right)

Figure 2 :

 2 Figure 2: Rhythmanalysis (Stn: Student No. n)

Acknowledgement

We thank Sanne Akkerman for her insightful comments on rhythmanalysis.


L Aquilina'
email: l.aquilina@brgm.fr

P Rose'

L Vaute'

M Brach'

S Gentier'

R Jeannot'

E Jacquot'

P Audigane4

T Tran-Vie

R Jung'

J Baumgktner'

R Baria'

A Gcrard'

A TRACER TEST AT THE SOULTZ-SOUS-FORF~TS HOT DRY ROCK GEOTHERMAL SITE

Soultz-sous-For& is the French site of the European Hot Dry Rock (HDR) project of which the objective is the creation of a heat exchanger at depth in the granite that constitutes the basement of the Rhine Graben. Geochemical studies permitted the characterization of the fluid circulation, which occur naturally in the Rhine Graben and are man-induced at the Soultz site. After the development of the heat exchanger, long-term circulation and tracer tests during the summer of 1997 have allowed for an improved characterization of the reservoir.

INTRODUCTION

Since 1987, Soultz-sous-Fori% has been the location of the European HDR project. The objective of this project is to create a heat exchanger at a depth of 4 km in the granitic basement. Two wells (GPK-1 and EPS-1) were drilled in 1987 and 1990 to depths of 2200 m. During this first phase, several geochemical tasks were carried out to characterize the chemistry of the fluids sampled in the granite [START_REF] Pauwels | Water-rock Interactions during Experiments within the Geothermal Hot Dry Rock Borehole GPKl at Soultz-sous-For&ts[END_REF][START_REF] Pauwels | Chemistry and Isotopes of Deep Geothermal Saline Fluids in the Upper Rhine Graben: Origin of Compounds and Water-Rock Interactions[END_REF]. The project then began a second phase wherein the GPK-1 bore hole was first deepened to 3600 m and a third well (GPK-2) was drilled to a depth of 3800 m (Aquilina & Brach, 1995; Aquilina et al., 1997). During 1995, a circulation test was carried out to initiate the heat exchanger and, during 1997, a long-term circulation test was successfully completed.

FLUID-ROCK INTERACTIONS

Deepening of the GPK-1 borehole to 3600 m has shown that fluid circulation can take place at least down to 3500 m. Fluids sampled at this depth have the same compositional range as those previously sampled in the upper part of the granite and the sedimentary cover, which indicates that all these fluids have a common origin. However, it is also clear that they evolved differently.

Comparison of these deep fluids with shallower fluids sampled at Soultz and at other locations in the Rhine Graben gave information on the reactions occurring during the interaction of the fluids with both Triassic sandstone and granite within the Rhine Graben. Two clear trends can be observed between the fluids sampled in the granite and those collected in the Buntsandstein (see Fig. 1). The 87Sr/86Sr ratios vs. the Rb/Sr ratios of fluids and minerals indicate that mica dissolution occurs in the Buntsandstein. In the granite, the fluid composition is related to the dissolution of plagioclase and probably the illitic phase in the fractures. Thus, although the two fluids have a common origin, they have experienced separate evolutionary processes after their separation.

Several differences appear between the fluids at 3500 m and those sampled at 1800-2000 m. This is true not only for the organic-matter and metal-trace content, but especially for the Si02 and 6'*0 (SO,). These differences indicate that, although they have a common origin, the fluids in the deeper part took a distinct evolutionary pathway once they invaded the granite. The geothermometers at 3500 m all give values close to 230°C. This gives a clear demonstration of the existence of a hot (about 230°C) reservoir in the Rhine Graben (Aquilina et al., 1997). The discrepancy between the geothermometer calculations at 2000 and 3500 m implies that the fracture zone at 3500 m is closely connected to the reservoir. This is assumed to be due to intra-granite fluid circulation, extending from 2 to 5 km depth in the granite as stated by Benderitter and Elsass (1995).

FLUID CIRCULATION MODEL

The first phase of the Hot Dry Rock project, before completion of a heat exchanger, has led to intensive research that improved the understanding of natural fluid circulation in the Rhine Graben and at the Soultz In particular, the geothermometry indicates the proximity of a hot reservoir and suggests that circulation may take place in the shape of convection cells that function below the 3-km depth in the granite (Benderitter et al, 1995).

CIRCULATION TESTING

The borehole GPK-2 was drilled at the end of 1994 to a depth of 3880 m. It is located in the outer part of the seismic cloud recorded during the 1993 injections in GPK-I . At depth, the two wells are located about 450 m apart, which is a very large distance when compared to other HDR systems where this distance is generally less than 150 m. During the summer of 1995 two circulation tests of 18 and 20 days respectively were carried out and geochemically monitored to establish the commu-nication between the two wells. A short pulse of 10 kg of fluorescein was injected into GPK-2 but was never detected in GPK-I. This result indicated that no short circuit was present and that the reservoir was larger than 100,000 m3. An evolution of the chemistry of the brines produced was noticed during the test, which indicated that the connection was changing during the test.

During the summer of 1997, a long-term circulation test was carried out, this time with injection into GPK-1 and production via GPK-2. The aim of this test was (1) to characterize the deep reservoir, especially through tracer experiments, and (2) to test the hydraulic connections and the capability of circulating during a 4-month period.

Circulation was initiated in the middle of July and continued until the middle of November with essentially no interruptions. Injection and production flow rates varied slightly between 22 and 24 Us. No losses were observed during the production and no surface fluids were added to the loop. Although some degassing was observed, no corrosion was noticed. The temperature of the produced fluids increased continuously during the test and reached the value of 142°C at the end of the test.

Hydraulic stimulation of the wells GPK-1 and GPK-2 previous to circulation test resulted in the creation of several sets of fractures as determined by spinner-log flow testing (Baumgktner et al., 1996(Baumgktner et al., , 1998)) I 29 I 10/25 I 120 * Due to detection limit effects, fist detection time differs from the first arrival time which can be deduced from the curves and does not show major differences between the tracers which travel time is close to 72 +/-10 hrs.

The return curves for the tracers injected during the first tracer test (benzoic acid, fluorescein, and deuterium) are presented in Figure 2. The fluorescein curve is issued from several analytical sets which have been carried out in order to remove strong matrix effects. The confirmation of the values is under process. Fluorescein and benzoic acid show a relatively similar pattern with peak maxima occurring between about 6 and 10 days. Both return curves have the long flat tails that are characteristic of tracer in hydrothermal systems. Rhodamine and amino G, although the restitution is short, show different curves with a less defined peak at the begining of the restitution, amino G even showing a secondary peak clearly higher than the first increase.

The return curve for deuterium shows a completely different pattern, due partly to the fact that this tracer was injected only at depth. The curve shows two distinct increases. First, the curve rises simultaneously with fluorescein and benzoic acid to form a subtle peak. It can be concluded that a common pathway is present for the tracers injected from the surface and the one injected at depth. A second peak, much more significant than the previous one, is observed about 18 days after injection, whereas no increase is noticed in the other tracers at this time. It is assumed that this peak is related to the flow of fluids into the fractured zone at 3500 m into which deuterium was injected. However, the lack of increase for benzoic acid and fluorescein correlated to the deuterium increase and the deuterium-curve evolution are still not clearly understood.

A convenient method of plotting and comparing these returns is through the residence-time distribution function, E(t) (Danckwerts, 1953)

:
where Q is the fluid mass flow rate, F(t) is the tracer mass fraction (concentration) as a function of time and mp is the mass of the tracer pulse. What makes the E(t) function particularly useful is that it allows for the normalization of each tracer relative to the amount injected. Thus, for conservative tracers that neither adsorb nor decay, plots of E(t) should coincide as long as no changes have occurred within the reservoir to create different flow patterns. However, the fact that injection conditions (dilution of tracer in a different volume [6#2-2.5 m3]) differed from the first as regards the two other tests has to be taken into account.

Figure 3 shows plots of E(t) vs. time for benzoic acid, fluorescein for the first test and rhodamine WT and amino G for the subsequent two tests. The time scales for the rhodamine WT and amino G traces have been shifted in order to create common injection times with fluorescein and benzoic acid.

It is evident from the reduced relative return that rhodamine WT adsorbed strongly onto the reservoir rock. Whereas rhodamine WT is known to adsorb in ground-water tracing applications, it has been successfully used without adsorbing in a hydrothermal system at Steamboat Hills, Nevada, just 20°C warmer than the reservoir fluids at Soultz [START_REF] Rose | The Application of Rhodamine WT as a Geothermal Tracer[END_REF]. It is possible that differences in the mineral composition between the formations at Steamboat Hills and at Soultz have accounted for this enchanced adsorption of rhodamine WT in the Soultz reservoir.

The initial part of the fluorescein return curve is preserved by the small initial amino G curve. A large amino G peak then follows that overshoots the comparable portion of the fluorescein return curve. However, due to the different injection duration time, one could expect higher normalized curves for rhodamine and amino G than for benzoic acid and fluorescein. It can be concluded from this results and from the shape of the curves in Figure 2 that the first test shows a different tracer-behavior. This difference can be either attributed to (1) a strong adsoorption effect for both rhodamine and amino G, or (2) to the changes in the formation which have occurred between the two tracer tests, the most signifificant of changes being a redistribution of flow just below the casing shoe in GPK-1 at 2860 m, as revealed by the spinner logs.

production fluid temperature increased gradually, reaching a maximum value of 142°C at the end of the test. An abrupt drop in the injection-fluid pressure Integration of E(t) allows for a determination of R(t), the fractional recovery of each tracer at a given time: R(t) = j E(t)dt 0 Application of equation 2 over the period of time that each tracer was produced results in recoveries of 35%, 21%, 8%, and 2% for benzoic acid, fluorescein, amino G, and rhodamine WT, respectively. However recycling of the tracers has not been taken into account in this preliminary computation.

VOLUME

There exist two fundamental approaches for defining the fluid volume of a hydrothermal system based upon tracer-return data. The modal volume is the volume of the hydraulically connected fractures. The overall fluid volume is ususally much larger and consists of not only the volume of the hydraulically connected fractures but also the volume of the relatively stagnant regions (dead-end fractures, vugs, and pores) that are accessed primarily by diffusion.

The overall reservoir-fluid volume can be determined from an analysis of the long tailing portion of the return curve. After a conservative tracer has been produced and reinjected over a long period of time in a closed reservoir, the tail of the return curve attains a constant value (Ito and coworkers, 1977). The volume is estimated through the expression:

Mf =mt. Ft
where Mf is the overall fluid mass, m, is the mass of tracer injected and F, is the tracer mass fraction measured after a sufficiently long time for the tracer to be evenly mixed throughout the reservoir and for the return curve to attain a constant value.

Rose and coworkers showed that equation 3 can also be used to estimate the overall fluid mass for an open reservoir (Rose et al., 1995; Rose & Apperson, 1997). In such reservoirs, there is aquifer cross-flow and the long tailing portion of the return curve attains a constantly sloping profile after a sufficiently long time. An extrapolation of the linear portion of the return curve to its interception with the y axis yields a value for F, . Application of equation 3 then results in a value for Mf , which can be converted to an equivalent fluid volume.

Figure 2 shows that the benzoic acid and fluorescein return curve attained a rather constantly sloping profile after about 80 days. Extrapolation of this straight portion of the curve allows to define an intersect of the y axis and thus a concentration. Solution of equation 3 results, at the present state of the tracer processing, in a range of the reservoir-fluid volume of several millions of m3. This computation does not takes into account the tracer recycling and a possible extrapolation of the curves for an increased restitution time. It constitutes thus a minimum value.

CONCLUSIONS

, Three tracer tests were conducted during the circulation test, with breakthrough measured for five tracers including benzoic acid, fluorescein, amino G, rhodamine WT and deuterium. An analysis of the return-curve data indicated that rhodamine WT adsorbed strongly onto the reservoir rock, whereas the remaining four tracers behaved more conservatively. Deuterium was injected downhole in order to trace the flow processes of brine through a set of fractures intersected by the injection well at the 3500-m depth.

A comparison of the return curve for amino G, which was injected late in the circulation test, with that of benzoic acid and fluorescein, which were injected early, seems to show that formation changes occurred during the test, resulted in a modified interwell flow pattern but it can also be interpreted as an impact of the mass of tracer injected on the appearent adsoption coefficient. An analysis of the long tailing portion of the fluorescein return curve indicated an overall reservoir-fluid volume in the range of several millions of m3.

  site. Chemical and isotopic analyses of fluids collected in the graben and from surface springs indicate the following phases of evolution: (1) Dilution of Triassic primary-brines by meteoric waters and migration throughout the whole Buntsandstein aquifer during the Eocene continental period. (2) Burial of the Triassic reservoir during graben formation (Oligocene) and sealing of the formations by Tertiary deposits. (3) Water-rock interaction in the sedimentary aquifer and equilibration with temperature. (4) Circulation of the brines from Buntsandstein into the granite. The Buntsandstein formation almost acts as a confined aquifer within which the fluids are trapped. At present, fluid circulation in the graben is thought to occur through three different systems: (1) On each side of the graben, mixing of the basin fluids with surface water infiltrating from the Vosges and the Black Forest. (2) Strong migration from the lower part of the graben to the upper part of the granitic basement. This migration may occur through several convection cells. (3) The possibility of a third level of circulation within the granite due to the difference between fluids at 1800-2000 m and those at 3500 m.

  . The most significant set of fractures in GPK-1 occured just below the casing shoe between 2850 m and 2960 m. The fractures in this section received about 60% of the fluid entering the reservoir. Two additional sets of fractures at 3230 m and 3490 m received about 22% and 15% of the fluid, respectively. In GPK-2, there is 1 a nearly continuous set of fractures between about 1 I 3200 and 3330 m and an additional set between 3460 and 3560 m.IA sharp decrease of the injection pressure at a constant injection flow rate was observed after about 60 days into the test. This change was related, as observed on flow logs carried out just before and after this event, to a change in the distribution of the flow in GPK-1. The fraction of brine flowing into the reservoir via a large set of fractures between 2850 and 2960 m remained unchanged with about 60% of the total flow. However, the flow into one large fracture at 2860 m (approximately 10 m below the casing shoe) increased from about 24% to about 44% of the total flow into the reservoir. The relationship between the changes in the spinner-log data and the pressure drop in GPK-1 is not evident. Three tracer tests were carried out during the circulation test, which are summarized in Table1. In the first test, fluorescein and benzoic acid were mixed with 6 m3 of water in a tank. The contents of the tank were then injected into well GPK-I at a rate of 78 l/min. Deuterium was injected downhole using a dump bailer just above the large fracture zone at 3500 m. During the two remaining tests, amino G and rhodamine WT were mixed with 2 to 2.5 m3 of water and were injected in the same manner as fluorescein and benzoic acid.

Figure

  Figure

Table 1 : Main Aspects of the Tracer Tests

 1

	Tracer	Mass Date	1st detection
		0%)
	Deuterium	8.6	07/30	44
	Fluorescein	150 08/01	70-75
	Renmic acid RhodamineWT I 22 I 10109 I 292 08/01	72 84
	Amino G


A Perrin

P Herbelin

F Jorand

S Skali-Lami

L Mathieu

Effects of hydrodynamic conditions on the bacterial and amoebic biomass in freshwater biofilms

Naturally present in river water supplied cooling circuits, bacteria and free-living amoebae (FLA) can encounter conditions favourable to their multiplication in freshwater biofilms, especially since the water warms up. The presence of freshwater biofilms onto surface of industrial cooling circuits can reduce the thermal performance and represents a potential public health risk. Several factors play a role on the behavior of freshwater biofilm. Among those factors and to better manage those industrial problems, it is essential to understand how hydrodynamic conditions influence the dynamics of microorganisms into biofilms and identify the areas of the cooling system where it could occur.

and maturation detachment, erosion and dispersion (Simoes et al., 2012 ;Picioreanu et al., 2001;Rochex et al., 2008) Boundary layer

Fluid

EPS EPS

In most of the pipes of cooling circuits, the flow is turbulent with a Reynolds number (Re), ratio of inertial forces to viscous forces in the fluid, ranging from 4.10 4 to 2.10 7 .

The viscous sublayer (next to the surface), with a very small thickness, is characterized by large velocity gradients. Flow inside this viscous sublayer is considered to be essentially laminar, with a characterized shear rate (γ).

Hydrodynamic conditions is one of the main parameters playing a role on the biofilm formation and organization. Especially this shear rate influences the transport of nutrients and cells to the surface and it regulates adhesion of microbial cells, biofilm  Rapid colonization and stabilization in 7 days at about 10 6 cells cm-².  Bacterial densities do not differ statistically according the wide range of shear rates comprised between 1.000 to 151,000 s -1 .

Recovery rate and spatial organization were determined by cell nuclear DAPI staining and epifluorescence microscopic observation (Olympus BX50) (100-500x magnification). Ten microscopic views were captured on each coupon (Nikon DS-Fi2), which were analysed with the software MATLAB® in order to produce the recovery rate according to the script of Lewandowski and Beyenal (2013).

Each collected coupon was gently washed by immersion in phosphate buffer saline to remove deposits loosely attached to the substrata. Then deposits of biofilm were dispersed by sonication (15 min, ultrasonic batch) (Goudot et al., 2012). Bacterial cells nucleic acid were stained with SYBR® Green I. Cells were recovered by filtration through a 0.2 µm pore-size polycarbonate membrane and counted with an epifluorescence microscopy (Olympus BX50) (Noble et Fuhrman, 1998 ;Patel et al., 2007). Quantitative PCR with 16S rRNA-gene-targeted specific primers (Eub338F et Eub518R) have been also done to quantify bacterial cells (Ferrand et al., 2014) (data not shown, similar results as microscopy). Thermophilic free-living amoebae were cultured on non-nutrient agar plates overlaid with E. coli. Amoebic migration fronts were observed on an inverted microscope (80-160x magnification). Amoebic clades were determined according to morphologic criteria (size and shape of cyst and trophozoite stages).

 Establishment of viable eukaryotic microorganisms within biofilms under high shear rates, in opposition to theorical concepts.  Rapid colonization after 1-2 days  Amoebic concentration ranging from 10 1 to 10 3 amoebic cells cm -² and values are a hydrodynamic-independant phenomena Densities do not differ statistically.  However, amoebic density would trend to increase as shear rate decreases (tests required to confirm). The hypothesis are :

• A low shear rate would be favourable to amoebic-cell sedimentation. • A high shear rate could be responsible for erosion of biofilm and cell detachment would occur. • Amoebic density might depend on the bacteria density.

2

 Amoebic diversity is hydrodynamic-independant.  One of the two clades, Tubulinea clade, has been mostly identified on stainless steel. Belonging to the Tetramitia clade, Naegleria genus is also frequently observed in balance with Tubulinea.

31 000 55 000 13 000

DENSITY DIVERSITY

The designed experimental process has enabled to highlight that FLA colonization of freshwater biofilms is possible at shear rate up to 151,000 s -1 , even if convection mechanism doesn't technically provide it. These results confirm that other mechanisms, such as turbulence, locomotion or surface properties have allowed the establishment of FLA in biofilms.

Finally, hydrodynamic conditions have shaped the spatial organization of microbial cells on the surface : higher shear stresses have reduced the recovery rates and number of clusters. However no significantly difference in the bacterial and amoebic densities has appeared between the hydrodynamics conditions despite of the wide range of shear rates (high shear rates tending to reduce FLA concentration into biofilm needs to be confirmed).

How the recovery rate and the number of clusters could be affected by the shear stresses without modify microbial charge ? Given the relevance of hydraulics on the biofilm growth, a study of the biofilm cohesive strength and the density, porosity and thickness of biofilms will be helpful to better understand the structure of freshwater biofilms into cooling circuits, considering different materials (concrete, PVC) and microbial factors (microbial diversity and amoebic morphology).

growth and finally the erosion and detachment of biofilm (see figure above).

The shear rate in cooling circuits ranges from 34,000 s -1 to 72,000 s -1 (Perrin, 2019).

DISCUSSION AND PROSPECTS DIVERSITY

(optimized and adapted to Abe et al., 2012;Mathieu et al., 2014;Pellieux et al., 2012)



  et Gersten, 2017 Shear rate (γ) is a function of the rotational speed () of the disk and the radial distance (r) from the axis of rotation. The reactor consists of a 60L, PMMA reaction vessel, and contains sixty stainless steel coupons. It is supplied with freshwater. The main tank is maintained at warm temperatures (42 degrees Celsius). Four radius positions on the disk corresponds to four different hydrodynamic conditions. (Schlichting et al., 2017) Rotating disk reactor  coupons Cell recovery rate on the coupons decreases as shear rate increases.Only one layer appears on the surface.  The spatial organization differs according to shear rates. Microbial clusters on surface under low shear rates are larger and in greater numbers than at high shear rates.


Alain Kuzniak
email: alain.kuzniak@u-paris.fr

Assia Nechache
email: assia.nechache@cyu.fr

Understanding mathematical work and mathematical thinking through individuals' actions analyses: a networking approach

Keywords: Mathematical Working Space, mathematical work and mathematical thinking, action and knowledge, methodology

The theory of Mathematical Working Space (ThMWS) provides theoretical and methodological tools to describe and characterize mathematical work. A particular methodology has been developed to analyse actions produced by subjects when they perform mathematical tasks. In this paper, this methodology is introduced by an example of a geometrical task. Then, we show how two theories on knowledge construction based on actions were combined with ThMWS. Finally, we present the contributions drawn from these networking cases which provide insight into mathematical work as well as the thinking of subjects by articulating their knowledge and actions.

Introduction

Within the theory of Mathematical Working Space (ThMWS), the understanding of mathematical work is mainly supported by the analysis of the actions, which students and teachers perform in the school context. In this paper, we wonder how to better understand the cognitive dimension of mathematical work and we explore this issue through common analysis of actions with some cognitive approaches.

We first justify why actions can be used as a common key-point for discussion between ThMWS and cognitive theories. After giving some of the key-constructs of the ThMWS used in the paper, we provide an example to introduce the methodology we have implemented to account for the mathematical work of individuals performing mathematical tasks. We then consider how research using both ThMWS and Abstraction in Context (AiC), or APOS theory, has opened up other avenues to describe and characterize the articulations between actions of individuals and their knowledge. It is thus possible to better understand subjects' mathematical work and mathematical thinking.

Articulating mathematical work and mathematical thinking through subjects' actions analysis

The approach of teaching and learning issues using the notion of mathematical work is relatively new in the field of research on mathematics education. Indeed, the great majority of research on teaching and learning refers to the idea of mathematical thinking as CERME conferences indicate through the topics of the working groups on mathematical contents. These topics have long been expressed in the form of specific modes of thinking such as algebraic thinking, geometric thinking, and probabilistic thinking. The focus on "mathematical thinking" urges to attend "thinking" with the real risk of dissolving the mathematical specificity in all the fields, which are concerned with thinking. Indeed, the study of thinking supposes a multiplicity of approaches, which are often far apart from mathematical concerns.

Recently, cognitive sciences have also approached thinking through cognition with the objective of being "the science of thinking". The cognitivist current is itself a composite combination of disciplines such as neuroscience, artificial intelligence, the phenomenology of perception, etc. This current is probably at the origin of the collapse of the primacy of mental representations for the benefit of action in the field of mathematics education. In this view, action is prior to mental representation as the neurobiologist [START_REF] Berthoz | La simplexité[END_REF] endorses it, and thinking and consciousness relate directly to action and perception. This viewpoint is not new in the field and was already pointed out in his genetic epistemology by Piaget who considered that mathematical thinking is not "pre-formed but a creation constantly pursued by actual and new actions which open a new set of possibilities hitherto inconceivable" (Piaget, 1957, p. 34). The support of cognitive sciences gives this approach a new visibility. This change of perspective on mathematical thinking is valued from the point of view of ThMWS and more generally of studies that favour an entry through mathematical work. In this case, the focus on work naturally places action as a key theoretical and methodological tool in the study of mathematics teaching and learning in school contexts. However, considering mathematical thinking and mathematical work as complementary seems to be necessary to account for the complexity of school mathematics. In particular, some cognitive theories, such as AiC and APOS, focus on the construction of abstract mathematical knowledge and can both complement and benefit from approaches to mathematical work that focus more on the outcomes and processes of work.

Describing and characterizing mathematical work: the ThMWS approach

The theory of MWS: motivations and issues

In the theory of MWS [START_REF] Kuzniak | Mathematical Working Spaces in Schooling: an introduction[END_REF], mathematical work is considered as intellectual work that involves significant cognitive activity of which the orientation and purpose are defined and supported by mathematics. In this framework, mathematics is envisaged as a specific human work and therefore, mathematical work is seen in both its cognitive and epistemological dimensions.

The Mathematical Working Space (MWS)

The vertical planes of the MWS To grasp the specific activity of students solving problems in mathematics, the two epistemological and cognitive facets of mathematics are present and articulated in two planes: one of an epistemological nature in close relation to the mathematical content; the other of a cognitive nature, related to the visible action of the individual solving problems (Figure 1, left). Three components in interaction are introduced for the purpose of describing the work in its epistemological dimension: a set of concrete and tangible signs (R), a set of artefacts (A), a theoretical system of reference (T). The second level is centred in the subject; three cognitive components are introduced as follows: visualisation (V) related to deciphering and interpreting signs; construction (C) depending on the used artefacts and the associated techniques; proving (P) conveyed through discourses producing validations, and based on the theoretical system of reference.

Analysing mathematical work through ThMWS aims at tracking down the process of bridging the epistemological plane and the cognitive plane in accordance with three different geneses: An instrumental genesis (Ins) related to the use of artefacts, a discursive genesis (Dis) related to theoretical part of work, and finally a semiotic genesis (Sem) that expresses the importance of signs for mathematical work. The elaboration of mathematical work is perceived as the harmonious networking and intertwining of these geneses within the vertical planes of the diagram (Figure 1, right).

To achieve the description and characterization of mathematical work, studies performed within ThMWS theory are based on the analysis of the subject's various actions. Thus, the methodology of studying subjects' actions is central in the approach and is developed in the next section.

A cognitive analysis of subject's actions: The example of Alphonse's Parcel

Our starting point is that cognitive activities of a subject at work are not directly observable but they can be deduced from the visible actions of the subjects when they are performing mathematical tasks. The methodology we use must both consider mathematical content and give access to subjects' cognitive processes. We have chosen to introduce our methodology by an example, which is used along the text to show the combination of the different theories involved in the networking study: Alphonse's parcel [START_REF] Kuzniak | On forms of geometric work: a study with pre-service teachers based on the theory of Mathematical Working Spaces[END_REF]. The task statement is given in the form of a text to be read. Alphonse has just returned from a trip in Périgord where he saw a parcel of land in the shape of a quadrilateral that had interested his family. He would like to estimate its area. To do this, during his trip, he successively measured the four sides of the plot and found, approximately, 300 m, 900 m, 610 m, 440 m. He's having a hard time finding the area. Can you help him by showing him the method to be followed?

Data given in the task does not enable to find a unique solution to the problem because quadrilaterals with the same side lengths can have different forms and areas. However, the majority of students used the following very common Theorem in Act (TiA): figures with the same perimeter or the same side lengths have the same area. The main objective of the whole didactic situation explicitly destabilized the students and helped them to banish this false Theorem in Act.

To give a general overview of our methodology, we provide an analysis of Francis's solution of the problem. It is based on a corpus of data collected during the resolution of the mathematical task.

This corpus consists of his written and oral productions. We start dividing his activity into episodes. Each of the episodes is related to a sub-task self-prescribed by the student to achieve the global task: it includes a sequence of mathematical actions that lead to the realization of the sub-task. All the actions are analysed in detail by using ThMWS. The MWS diagrams are also used to visualize the circulation of work during the resolution of the task [START_REF] Kuzniak | On forms of geometric work: a study with pre-service teachers based on the theory of Mathematical Working Spaces[END_REF]. I drew a perpendicular to the large base and this line perpendicular to the large base was also perpendicular to the small base of 300m and as two lines are perpendicular to the same line, they are parallel so it makes a trapezoid. Then a bottom-up analysis gives a summary overview of the different episodes, and helps to deduce the logical organization of Francis' mathematical actions. The analysis seeks to identify how the work-generating processes interact within the components of MWS. From the data, we surmise that mathematical work produced by Francis is triggered by the referential and initiated in the [Sem-Ins] plane to construct a convex quadrilateral with the shape of a trapezoid using technological tools. These tools are used by the student as semiotic tools to collect data and make measurements. Construction of the quadrilateral is then justified with the use of theoretical tools characterizing the trapezoid. The formula for calculating the area of a trapezoid (technological tool) is applied to produce the result (in this case, the area of the land). To conclude the analysis, observations made during the two previous steps are used to characterize the mathematical work. In Francis' case, we may consider that his work circulates through the different geneses and planes of the MWS diagram. He uses drawing and measure tools to support his reasoning and argues about a particular figure, so his work is compliant with a geometrical paradigm based on measurement and drawing (Geometry I, see [START_REF] Kuzniak | On forms of geometric work: a study with pre-service teachers based on the theory of Mathematical Working Spaces[END_REF]. Finally, his mathematical work is not mathematically correct, as the result is false and based on a construction of an inappropriate figure and TiA theorem.

The description above provides the most accurate account possible of the student's choices and actions. However, it is sometimes difficult to be sure, of what really triggered and motivated these actions. Some of Francis' actions are material and easy to interpret, such as constructing the trapezium, but others are more challenging. Why did he choose to use a scale for the construction and then measure in the drawing? When did he focus on the trapezoid? Why didn't he finally see his mistake? To progress with the interpretation of subjects' actions, we explore other methods of action analysis based on theories, which are rather focused on mathematical thinking.

Combining ThMWS with cognitive theories

Several studies have combined ThMWS and cognitive theories. In this paper, we only refer on the studies with AiC and APOS. For each research, we first present its general purpose and introduce the method, which the authors used. Then, we apply this method to Alphonse's task and show the new contribution that can be drawn from the approaches to advance the analysis of Francis' actions and progress in its understanding.

A joint study between the AiC and MWS theories

In their study of the development of functional thinking of students working in four different MWS, [START_REF] Psycharis | A framework for analysing students' learning of function at upper secondary level: Connected Working Spaces and Abstraction in Context[END_REF] combined Abstraction in Context (AiC theory, [START_REF] Dreyfus | The nested epistemic actions model for Abstraction in Context: Theory as methodological tool and methodological tool as theory[END_REF] and ThMWS. Using a task on the optimal shape of a gutter, they analysed students' progress in their understanding of covariation by having them work successively on four gutter models in different environments (sheet of paper, DGS, etc.). Each of these models is associated with a different MWS. The connections between these MWS are described and explained thanks to the MWS geneses involved. This first analysis through the MWS theory is completed with the AiC theory to account for the processes of constructing abstract mathematical knowledge. In this theory, abstraction is defined as an activity of vertical reorganization of previous mathematical constructs into a new abstract construct. According to the authors, the process of abstraction passes through three stages of knowledge construction shaped by three epistemic actions: recognizing (R), building-with (B) and constructing (C). These actions are nested. They allow tracing the process of knowledge construction and are used as a methodological tool for analysis.

Using AiC for Alphonse's solution of the task, we first focus on recognizing. If we look again at episode 1 (figure construction), we surmise that the student recognized a construction problem as a previous meta-construct (R0) for which he knew mathematical knowledge involved in the figure construction. While constructing, using the given side lengths but ignoring the word "successive", he recognized (R1) a partially correct construct, a trapezoid, and with it its area formula, another previous construct (R2). Thus, he realized its shape by an approximation and calculated its area via measuring, the latter being a building-with (B) action built on the recognized constructs. The mathematical object "trapezoid" determines the follow-up of Francis' work and as it is only partially correct, he cannot construct the expected knowledge about the various areas of quadrilaterals having the same side lengths -construction of a new construct (C). In the example of the work carried out by Francis, the mathematical object named "trapezoid" is considered under different aspects during the completion of the task and referring to the three components of the epistemological plane: its drawing aspect (representamen), the material artefact aspect (ruler, compass, square) or symbolic artefact (area calculation formula) associated with the trapezoid, and its property and definition. All

A joint study between the APOS (Action, Process, Object, Schemas) and MWS theories

In her research, Camacho (2021) uses the APOS [START_REF] Arnon | APOS Theory-A framework for research and curriculum development in mathematics education[END_REF] and ThMWS together to study the linear algebraic concepts of eigenvalues and eigenvectors in relation to their geometrical representation in the plane. This study helps to further consider action by underlining the diversity of meanings of this term. Indeed, in the APOS theory, which assumes the heritage of Piaget, the term Action is one of the basic terms of the theory. The acronym APOS stands for Action, Process, Object and Schema. The action to which it refers is a mental structure that must be distinguished from "material action", which Dubinsky interpreted as actions that are performed by a subject but are external to the subject (Arnon et al., 2014, p. 7). From a set of actions on objects, the subject can derive a property arriving at a more general Action that allows passing from Action to Process through the mental mechanism of interiorization. The mechanism of encapsulation then transforms the Process into an Object as a cognitive entity. A Schema is then build by acting on the mathematical object. In her research methodology, Camacho applies an approach, specific to APOS theory, which consists of discovering the cognitive functioning of subjects according to their failure or success about an Object in some mathematics tasks. It is then a matter of recognizing, which of the components of the quadruplet (APOS) is operative or not. In Alphonse task, the starting point of the analysis is an initial reflection on the mathematical Object, with its Schemas, which is at stake in the situation. For us, it is the Area of a quadrilateral (Object) and this area is associated to a decomposition by triangulation (Schema) or more generally to the triangulation of any polygon, [START_REF] Kuzniak | On forms of geometric work: a study with pre-service teachers based on the theory of Mathematical Working Spaces[END_REF]. Francis however, activates another scheme acting on another mathematical object: he constructs a trapezoid by approximation and calculates its area with a formula via measuring. The actions as conducted do not lead to processes of building the area of a quadrilateral with the given side lengths because he elaborates the schema of an already existing mathematical object (trapezoid area), which is insufficient for the task.

In APOS, authors seek to elaborate the genetic decomposition of the Object of concern to be more precise in their analysis. Camacho elaborates the genetic decomposition of the Object "collinearity" within a digital environment and in terms of the different MWS geneses. When we analyse Francis' work with the MWS, we may notice that he relies on his old and well-established knowledge of the trapezium of which he knows both the different elements (signs, artefacts, properties) in relation to what we referred to as the Epistemological Entity associated with the trapezium. He also knows the cognitive processes related to the use of the corresponding cognitive unit. He knows well the Object "trapezoid", unfortunately, it is not the right Object, which should be used to perform this task about the area of a generic quadrilateral. The case of Francis shows that students do not really regulate their work once they have embarked on a solution that seems to be in line with their knowledge. An illusion of knowing-how that fosters certainty in action emerges. In the example, it was however possible to see that the order of the sides used in the construction was not the one indicated by the statement (semiotic control). It was also possible to see that different quadrilaterals fulfilled the same condition since Francis searched, by approximation, among these figures the one with parallel sides (semiotic and instrumental control). An internal control of the instruments (instrumental control) should have been based on another construction without approximation. Finally, it is interesting to see that his proof reasoning is biased to reach the desired result (discursive control).

Conclusion and perspectives

All of the studies we reviewed emphasise that ThMWS and its organization around the three geneses was an important structuring element in the analysis of actions and helped to describe and better understand the relationship between cognitive and epistemological aspects. Conversely, the triadic and prismatic geometric structure of the MWS generates theoretical notions that are not always easy to interpret as a tool for studying issues of mathematics teaching and learning. However, these joint studies on students' mathematical actions focussing on cognitive aspects enrich and flesh out a number of emerging notions in ThMWS. In particular, they have provided us with a better understanding of the importance and value of the following ideas:

1. The Epistemological Entities and Cognitive Units account for the importance of knowledge and cognitive functioning of individuals who perform a task using particular mathematical objects. To achieve description and characterization of mathematical work, which captures adequately interactions between cognitive and epistemological aspects, it is important to make visible the mathematical knowledge involved in the solution by identifying precisely the components of the Epistemological Entity and the Cognitive Unit associated with a mathematical object 2. These Cognitive units and Epistemological Entities depend on forms of work implemented, which can be described by different paradigms associated with the teaching contents. Based on AiC epistemic actions and on APOS contributions, it is thus possible to describe how the observed work conforms to particular modes of mathematical reasoning.

3. Finally, we have also made progress in understanding the regulation and control of outcomes through the importance of knowledge and cognitive repository as tools for regulating actions. However, we also met the attracting force of an "illusion of knowing" on acting.

The question of actions regulation and control is a research direction that we plan to develop further. Controls and regulation questions are not new in the field of mathematics education, it has existed since at least the first research on problem solving, seeking to make students more autonomous. Particularly [START_REF] Arzarello | Semiotic and theoretic control in argumentation and proof activities[END_REF], who also emphasise the variety of actions implemented by students and the teacher, have recently addressed this. They have analysed the way in which gestures, considered as a semiotic resource, intervene in the learning and teaching process of mathematics. This extension of research to gestures and the body raises the question of how to analyse and identify their real impact on students' actions and beyond on their mathematical work and thinking. We are now engaged in a new project (E-ESMEA) dealing with enacting in astronomy education. We are mainly interested in the mathematical part of the project: relations between proportionality and velocity. The project focuses on the concept of embodied action, which assigns a key role to perceptual and bodily processes in the formation of abstract concepts and knowledge. The sensory-motor engagement of the body carries these processes in situations out that constitute the functional system, on which actions are based. From this study, we expect that it will help us to identify the limits and the right levels of analysis of mathematical action that are really necessary to advance in the mastery of mathematical work and in the learning of mathematical knowledge. This is, in our opinion, a fundamental question to develop a didactic approach in phase with the school reality.

Figure 1 :

 1 Figure 1: The MWS diagrams

Episode 1 :

 1 Construction of the figure Episode 2: Justification of the nature of the figure I took the large base 900 m then from the two ends of the 900m with the compass, I made 440 on one side and 610 on each side and then with the ruler I tried to find the 300 with the two arcs of circle.

Action 1 :

 1 Choice of scale Action 2. Construction of the figure Action 3. Francis' explanation of his construction Action 4. Francis' description of the fact that it is a trapezium. Episode 3: Determining the area of the figure. Action 5: Measurement of the height h of the trapezoid Action 6: Calculation of the area from the formula (B+b). h/2

  these elements constitute what we call the Epistemological Entity associated to the trapezoid overlapping with what is called Context in AiC. The subject's visualization of the "trapezoid" object as being a quadrilateral with two parallel lines, his construction of a figure and the area with techniques associated with the artefacts to develop the trapezoid, and his justification of these constructions by citing the properties that characterize the trapezoid are all part of what we name the Cognitive Unit associated with the mathematical object "trapezoid". Identifying the existence and role of the Cognitive Unit and Epistemological Entity associated with the mathematical object "trapezoid" enables to characterize the mathematical work and understand mathematical thinking in terms of epistemic actions developed by AiC.

Table 1 . Francis' work division in three episodes each with two actions.

 1


Dandan Su

Yan Zhang

Sébastien Ulrich

Mihail Barboiu
email: mihail-dumitru.barboiu@umontpellier.fr

Constitutional Dynamic Inhibition/Activation of Carbonic Anhydrases

Keywords: Constitutional Dynamic Chemistry, Carbonic Anhydrase, Constitutional Frameworks

As one important member of metalloenzymes family, carbonic anhydrase (CA) is involved in the treatment of several diseases. Different approaches have therefore emerged to regulate the activity of CA, mostly acting on inner catalytic active site or outer microenvironment of the enzyme, leading to inhibition or activation of CA. In recent years, gradually increased attentions have been drawn into the adoption of constitutional dynamic chemistry (CDC) strategies for the screening and discovery of potent inhibitors or activators. The participation of reversible covalent bonds enabled the enzyme itself to select the optimal ligands obtained from diverse building blocks with comparatively higher degree of variety, resulting in the fittest recognition of enzyme ligands from complex dynamic systems. With the increasing implantation of CDC for enzyme target, it shows great potentials for the drug discovery-or environmental-oriented applications.

Introduction

Constitutional Dynamic Chemistry (CDC) and its application Dynamic Combinational Chemistry (DCC) [START_REF]Constitutional Dynamic Chemistry[END_REF] provide diverse opportunities for screening the chemical space at molecular and supramolecular levels for dedicated applications in chemistry, medicine and materials sciences. By using dynamic constitutional systems constructed by combination of reversible interacting components, it is possible to use various stimuli, such as light, heat, pH or ionic gradients, leading, in the presence of templating or molding agents, to self-selection or optimization of the fittest constituents.

In the last decades, DCC [START_REF]Constitutional Dynamic Chemistry[END_REF] dealing with the studies of dynamic chemical libraries built on reversible covalent interactions, showed bright prospects in diverse applications such as drug discovery, [START_REF] Ramström | [END_REF]3] materials science, [4,5] gene delivery, [6,7] catalyst screening, [8,9] etc. Among these, enzyme catalysis has attracted increasing attention from incipient development of the field. For example, DCC has been applied for highly selective enzyme-catalyzed asymmetric synthesis, resulting in identification of the optimal substrates fitting to the target enzyme active sites. [10][11][12][13] Moreover, the catalytic activity of enzymes can be tuned effectively with the participation of dynamic constitutional frameworks, [14] as well as discrete enzyme classification through pattern recognition methodology. [15] As a type of most commonly used metalloenzymes, carbonic anhydrase (CA, EC 4.2.1.1) has been selected and integrated with dynamic constitutional systems for the screening and identification of its inhibitors and activators. CAs are zinc-containing metalloenzymes, which are encoded by five gene families, including α-CAs, β-CAs, γ-CAs, δ-CAs and ζ-CAs, with diverse catalytic activity and tissue distribution. [START_REF] Supuran | Carbonic anhydrase inhibitors[END_REF] This superfamily plays a vital role in maintaining a stable body environment by catalyzing the carbon dioxide hydration reaction. [17,18] Therefore, the tuning of catalytic activity of CAs became a key breakthrough point for the treatment of several diseases. For example, the inhibitors of CAs can reduce the production of bicarbonate and lower the intraocular pressure, which is mainly used in the clinical treatment of glaucoma. [19][20][21][22] In addition, the treatment of tumors, epilepsy, acute mountain sickness and obesity are inseparable from the effects of carbonic anhydrase inhibitors (CAIs). [23][24][25][26][27] As for carbonic anhydrase activators (CAAs), they may improve memory and learning ability, paving the way for the treatment of Alzheimer's disease. [28,29] More specifically, the design of CAIs and CAAs usually involve the inner active site or the outer surrounding microenvironment of CA. It has emerged that CAIs can be mostly developed through direct coordination binding with the metal ion -zinc, which is playing the key role for catalysis inside the active site of the CA. [18] Meanwhile, the development of CAAs has been, comparatively, much slower, probably due to their complicated pharmacology and catalytic cycle. It has been suggested that the location of CAAs in the entrance of enzyme active site cavity is essential for activating enzyme activity, leading to acceleration of the proton-transfer step via intermediate water-wires. [18,30] Additionally, the external microenvironment of enzyme can also make a difference for enhancing the activity of CAs and need to be further studied for developing continuous processes for medical or environmental applications. [31] In both cases, DCC can provide efficient approaches for the screening of optimal molecular or supramolecular components from various dynamic constitutional systems, in order to inhibit or activate the catalytic activity of CAs.

Herein, we have summarized a set of CA inhibitors and activators which were identified through the DCC strategy. The aim is to highlight recent advances of dynamic constitutional systems with carbonic anhydrases as the target enzyme.

Representative examples will be illustrated below, hoping to inspire new ideas for the application of drug discovery or enzyme immobilization for environmental applications.

Enzyme inhibition

The active site of CA is mainly constructed around the Zn 2+ cation, which is located at the bottom of the active pocket and coordinated by His94, His96 and His119 and a water molecule. [18,24] During the general catalytic cycle (Fig. 1), the water molecule is first dissociated into proton and hydroxyl ions under neutral pH conditions prior to nucleophilic addition of the latter onto carbonic acid, generating bicarbonate, and proton-transfer of the former to regenerate the zinc-water coordination complex. This proton-transfer step is rate-limiting and therefore represent a leverage tool for designing CAI and CAA. [32,33] Regarding the design of CA inhibitors, not only the zinc-binding groups, but also other enzyme attaching moieties, such as the Zn 2+ coordinated water and the entrance to the active site, can be used as the action site. [34] Until now, five inhibition mechanisms of CA has been reported, 34 leading to demonstrated roles of CAI in numerous diseases, including glaucoma, epilepsy, obesity and kidney cancer. [35][36][37][38][39] Fig. 1 The catalytic mechanism of the carbon dioxide hydration reaction by CAs. [18] Lehn and Huc were the first group reporting a set of hydrophobic sulfonamide-based CAIs generated by reversible imine combinations of constitutional dynamic libraries composed of amine and aldehyde components. They have been demonstrated that a known aromatic sulfonamide CAI, is amplified from the combinatorial mixture of components, based on its favorable affinity toward bCAII, [40,41] via the tetrahedral coordination with the sulfonamide group to the Zn 2+ center. [18] Thereafter, other classes of sulfonamides CAIs and derivatives were exploited for the CA recognition efficiency through the binding with the enzyme active site. [42,43] Meanwhile, their X-ray crystallography were also disclosed since the first sulfonamides CAI was found through X-ray crystallography of the obtained enzyme-inhibitor adduct. [44] In this context, a series of CAIs have been obtained and extended from the DCC approach to target different CA isoforms. They have been illustrated below (Table 1), classified by different reversible bonds, including aminocarbonyl/imine interconversion, alkene cross metathesis and acyl hydrazones. Table 1. Representative reversible bonds for the dynamic generation of CAIs targeting carbonic anhydrases

Regulation

Reversible bonds Targeted CA Ref

Inhibition

Alkene cross metathesis bCAII [45] Acyl hydrazones bCAII [46] Amino-carbonyl/imine interconversion hCAI, hCAII, hCAIX [47-50] Amino-carbonyl/imine interconversion under thermodynamic control is one of the most popular dynamic covalent chemistry used for these purposes. Our group have used various aldehydes and amines to form stable Schiff base in a constitutional dynamic library (Fig. 2), for the purpose of corelating the inhibitory power of CAIs with their respective amplification is solution, useful for understanding the role of multivalent interactions with the enzyme toward the discovery of novel DCC CAIs. [47,49] Several active compounds from a pool of twenty components can be efficiently identified with the amplification of the most active CAIs, 1C, 1D, 2D (Fig.

2

) together with less active CAIs 1E or 4E which are probably forming via the outer side of the enzyme surface-directed reactions or via kinetic-rapid reactions in solution.

We have next showed that the flexible amines obtained by NaBH 3 CN reduction of imine adducts showed better performance on hCAI inhibition, while the rigid imine expressed higher selectivity towards hCAII over hCAI (Fig. 3). [48] At this point, it is important to highlight those reactions occurring in the active site are of outmost importance, but they may be related to the favourable diffusion and the oriented binding of the components within the active site. Parallel side reactions cand be favoured by respective binding of components on other regions situated out of active enzymatic centre. They are providing "parasite" reactions to be considered useful for the selection features of the combinatorial systems under considerations.

Supuran group has designed a series of sulfanilamide derivatives, while using imine Schiff bases chemistry to probe the structure-activity relationship of CA inhibitors, thus providing efficient screening of the most effective inhibitors for CA isoforms. [50][51][52][53] The precise position of the interactional groups on the molecular platform is highly important, substantially affecting the functional binding of the CAI within the active center. The impact of sulfonamide group position on the phenyl ring has been studied through a stopped-flow CO 2 hydrase assay, indicating that smaller K I values for hCAI and hCAII can be obtained from para position than meta-and orthopositions, under the condition of the same substituents on the isatin ring. They also illustrated the docking position in the active site of hCAIX with hydrogen bonds formed between Gln 92 and the imine group. In the end, compound a meta-substituted aromatic sulfonamide group presented the highest selectivity towards the tumor-associated isozyme hCAIX, compared to the cytosolic hCAI and hCAII. [50] Fig. 2 a) Constitutional dynamic library of amino, A-E and aldehyde, 1-4 components towards the discovery of isozyme-specific inhibitors; (b) Relative peak area expressing the amplification of the DCL function of inhibitory power (KI) against hCA II. [47] Considering the pharmacological importance for the discovery of isoform-selective CAIs, hCAI and hCA II, have been subjected to a parallel screening of the same CDL.

The use of parallel constitutional screening of CDL chemistry for the discovery of enzyme inhibitors is straightforward method that provides insights toward the generation of efficient classes of selective, high affinity inhibitors. These results by our group, [49] also point to the possibility of modulating the drug discovery methods by constitutional recomposition induced by a specific enzymatic target (Fig. 3). The structurally fittest CAIs have been found for each isozyme, illustrating that the response a CDC system may evolve differently to the presence of two different enzymatic effects. This example is demonstrating variable behaviours of CDC systems, showing precise adaptation under the pressure of variable external factors.

Fig. 3 Inhibition constants K I and the amplification of the constitutional dynamic library (CDL) against catalytically active human hCA I and hCA II cytosolic isosymes. [49] Aiming for the dual functionality, aromatic sulphonamideacyl-hydrazone can be combined with various aldehydes and used for the generation of a dynamic combinatorial library towards selection of bCAII inhibitors (Fig. 4a). [46] Amino acid derivatives were included into the dynamic system through hydrazone exchange reactions, to provide extra periphery recognition with the targeting enzyme. Then the screening of enzyme-ligand complexes was performed directly from the dynamic combinatorial library with Fourier transform ion cyclotron resonance mass spectrometry. As a result, the sulfonamide-amino acid conjugates with acyl-hydrazone linkages presented developable potential as CA inhibitors. Furthermore, Poulsen group reported a pre-equilibrated dynamic combinatorial library (pDCL) based on reversible alkene metathesis (Fig. 4b). [45] In this design, allyl ester benzene sulfonamide, which composed with an allyl substituent for terminal alkene metathesis and an aromatic sulfonamide moiety for bCAII recognition, was used to construct the dynamic scaffolds through the reversible reaction with ten additional building blocks respectively. Regarding to the fragment-based drug discovery to bCAII inhibitors, it was demonstrated that the cross-metathesis compound with a terminal acetate group, as well as homodimer and trans-isomers, exhibited more potent bCAII binding affinity with a lower equilibrium dissociation constant. Meanwhile, heterogeneous catalysis showed advantages on the convenience and benefits over homogeneous catalysis.

Fig. 4 Generation of the DCL with (a) acyl hydrazones exchange [46] and (b) alkene cross-metathesis reaction. [45]

Enzyme Activation

It has been reported that CA level may have influences on the memory therapy as well as spatial learning, thus, the tuning of CA activity can be important for the related treatments. [29,54] However, no CAAs have been exploited for clinical applications because of their complicated pharmacology and catalytic cycle. Considering the design of CAAs, it has been suggested that those CAAs ligands are usually located at the entrance of enzyme active site cavity. [30] For instance, spectroscopic evidence provided by X-ray crystallography showed that histamine, which is indeed located at the entrance of enzyme active site, exhibited activation of human CA isoform (hCAII) through the imidazole moiety which accelerated the proton-transfer rate-determining step of the CA catalytic cycle. [30,55] Furthermore, selective CAAs have been demonstrated with the structure of ureido/bis-ureido histamine derivatives, which showed activation effect towards hCAI. [56] Since the screening efficiency is relatively low by designing and synthesizing diverse ligands one by one, dynamic combinatorial approach was adopted to accelerate the process. Lehn and Ramström have reported an efficient screening method with Dynamic Deconvolution, [57,58] to select and amplify the active molecules for enzyme activations. During the procedure, a parallel screening among diverse building blocks was carried out, by removing the components one by one from the final dynamic combinatorial library and test the activity of the enzyme. If the removed component is active an important effect in the catalytic enzyme activity is observed.

With all these in mind, we combined agonistic inhibitor and activator components within different DCLs mixtures. Dynamic dominant behaviours in the expression of human of Carbonic Anhydrase hCA I activity are observed, most probably related to the differences in the long-distance binding behaviors of the of CAAs compared to the stronger direct binding of CAIs to Zn 2+ in the active enzyme pocket. The relative activity of the mixtures reveals the inhibitory effects dominate over the activating ones in DCLs containing both inhibitors and activators of the hCA I. [59] Apart from the active site, the external enzyme microenvironment is also worth to be exploited, since it is essential for the catalytic activity and selectivity of the enzymes. [60,61] In this respect, dynamic constitutional chemistry has also found its applications in screening optimal three-dimensional frameworks reversibly connecting at molecular and supramolecular levels for better enzyme activation effect.

Dynamic reversible connections between components can be transferred from

A-B binary type of combinations as previously described before toward 2D-3D Dynamic Constitutional Frameworks DCFs, constructed from bi(tri)functional components and core connectors via different covalent reversible chemistries. [60] For example, the direct addition of PEG-ylated dynamic constitutional frameworks linked with reversible imine bonds at equipped with H-bonding motifs, showed enzyme activation effect towards CA. [31,62,63] The mechanism can be that the in situ generation of dynamic framework via imine, bond formation simultaneously encapsulate CA through hydrogen-bonding and supplementary imino-bond formation with the amino groups on the enzyme surface. They are generating an adaptive protective layer around the enzyme, leading to more stabilized tertiary protein structures and they are providing proximal multivalent proton acceptor groups in the close proximity of the enzyme surface, assisting the proton shuttling during the catalytic cycle (Fig. 5).

Fig. 5 Concept of enzyme activation via dynamer encapsulation. [31] Our group has pioneered this field by reporting a straight-forward constitutional enzyme activation strategy, using dynamic constitutional frameworks as enzyme microenvironment activators (Fig. 6). [31] Fig. 6 Dynamic Enzyme Activation in Multivalent H-bonding DCFs composed of core centers, PEG connectors and (a) single-amide head, sDCF and b) multi-amide head mDCF. Enzymatic activity of bCA on hydrolysis reaction of p-nitrophenyl acetate, was determined by using UV absorbance at 400 nm recorded within time for different concentrations of DCFs relative to the concentration of the BTA core center [31] The DCFs were composed from benzenetrialdehyde, BTA core centers, water soluble polyethyleneglycol (PEG 1500 ) connectors and mono or dendrimeric amide-head components. The only former H-bonding components show a very low activity, while they became very active when they are multivalently assembled within DCFs. Interestingly the former PEGylated network, despite its strong interactions binding probably via imine-bond formation on the enzyme surface, did not enhance the CA activity. [31] Further results showed that the multi-amide substituted dendrimer (PAMAM) provided higher activation effect towards CA than single-amide substituted building blocks, emphasizing the importance of multivalent presentations of the active sites as proton sponges within DCFs. The enzyme catalytic activities were determined to be 1.7 mM sDCF and 0.62 mM MDCF respectively, presenting moderate activation effect compared to molecular activators, but attributed to the number of H-bonding interactions with enzyme surface and the facilitated proton shuttling.

Next, we changed the target enzyme from carbonic anhydrase to lipase, the catalytic enhancement effect on enzyme catalytic activity is still observed suggesting that observed effect is broad and effective potential applications with the DCFs-enzyme systems can be developed in the future. The shell-like encapsulation of the enzyme through the multivalent interactions between DCF and enzymes is most probably connected to the presence of the PEGylated matrix, contributing to the whole lipase interfacial activity enhancement. With such combined effects, the amide-PEG-DCFs accelerate lipase-catalyzed hydrolysis with high efficiencies. [START_REF] Zhang | for enzyme activity inhibition and activation are described by using active dynamic molecular libraries and dynamic molecular frameworks[END_REF] Fig. 7. Exponential activation of Carbonic Anhydrase by encapsulation in a) DCFs host-matrices with stereochemical and chiral discrimination. Activity change of bCA on the hydrolysis reaction of p-nitrophenyl acetate, with the addition of a dynamer 7 (concentrations relative to 1) followed by UV-vis spectroscopy. [62] Linear, cross-linked and optically active DCFs have been synthetized by including of diamino components, leading to an exponential activation of the bCA enzyme catalytic activity. While the DCFs-enzyme interaction is considered as shell-like matrix around the enzyme, the asymmetric DCFs may generate a chiral micro-environment for the enzyme encapsulation, leading to high exponential enhancement of the enzyme activity. [62] The nature and structural behaviors of functional groups used for the DCFs synthesis are highly important and their influence on CA activation has been further explored by using an efficient enzyme self-screening system, in order to obtain optimized microenvironment of CA (Fig. 8). In a given dynamic system, the interactions between CA and the functional DCFs were controlled by the competition between protein-DCF affinity and the direct DCFs formation via imine cross-linking reactions. Based imine bond formation with the amine groups and via non-covalent interactions on the enzyme iso-surface a re-equilibration can be achieved upon the addition of CA. It was found that lower incorporation of small amine components in DCFs exhibited higher activation effects, owing to the enzyme selection preferences.

The enzyme structure directed DCF formation kinetic studies were in good correlation with their activation effects, providing insights into the CA-substrate interactions upon addition of the DCFs. [63] This constitutional dynamic strategy offers a new opportunities for enzyme self-directed microenvironment optimizations and further tuning of enzyme activities. With Carbonic Anhydrase-CA as the target enzyme, both inhibitors and activators have been widely explored and the using dynamic constitutional chemistry may led to efficient easy discovery and rapid identification of highly efficient CAIs and CAAs candidates. It is worth noting that not only enzyme active site but also the surrounding microenvironment is essential for the regulation of enzyme activity.

The examples discussed here illustrated the developing advances with regarding to inhibitors and activators of CA, through the generation of dynamic adaptive libraries. Different enzyme-ligands complexes have been classified by the ligand architectures, as well as the type of reversible linkages. The inhibition capacity of small dynamic molecules towards CA were realized through binding at the inner enzyme active site. On the other hand, as for the design of activators of CA, dynamic constitutional frameworks composed of diverse building blocks were self-screened for the optimal outer microenvironment with the presence of CA, with activation effects further verified. It leaves the enzyme the freedom to select its fittest effector.

As an important disease-related enzyme target, CA has long attracted a lot attention from researchers, and the search of its inhibitors and activators has been continuously conducted. In this process, dynamic constitutional chemistry has showed high efficiency and demonstrated broader application potentials. However, there is still some space for further exploration. For example, ligands for several CA isoforms, including hCAI, hCAII, hCAIX and bCAII, have been screened, while others still remain to be studied. Operate the inner active pocket or outer enzyme/protein environment should constitute an important endeavor for future research activities in this field. Moreover, the exact activation and inhibition mechanism still need be explained through conformations of enzyme-dynamer structures. Therefore, further innovations can bring more exciting improvements to this field.

Conflicts of interest "There are no conflicts to declare".

Fig. 8

 8 Fig.8General concept of enzyme-driven microenvironment optimization by dynamic constitutional imine systems.63

63

 63 Fig.8General concept of enzyme-driven microenvironment optimization by dynamic constitutional imine systems.63

4 .

 4 Summary and outlookOver the past few decades, dynamic constitutional strategies have been emerged to generate enzyme-dynamic combinatorial systems that have been used in drug discovery-or environmental-oriented applications. The convergent combinatorial synthetic strategy uses simple synthetic building blocks with participation of reversible molecular and supramolecular linkages to provides a parallel and facile screening of complex mixtures of appropriate ligands or external networks from the adaptive constitutional polymers. It allows variability, versatility and easy screening for the best performances.

Acknowledgements : DS appreciates receiving a scholarship from China Scholarship Council as support at University of Montpellier, France. This work was supported by M-ERA NET 2019 grant to MB via Agence Nationale de la Recherche ANR-20-MERA-0001-01, SMARTMATTER. YZ thanks financial support from Natural Science Foundation of Jiangsu Province (BK20180625) and National First-class Discipline Program of Light Industry Technology and Engineering (LITE2018-20).