Armand Hatchuel

Special Issue 'Expanding the boundaries of rationality: Towards new models of decision making for radical uncertainty' The quest for non-Bayesian decision-making heuristics : towards a logic of imagination

INTRODUCTION

In this special issue, papers explore different trends of non-Bayesian decision heuristics. They all address, to some extent, the challenges of managing in the unknown i.e., in contexts where surprises, discoveries and innovations are frequent events. After a brief reminding and synthesizing of four main conceptual assumptions of Bayesian decision making (Savage 1954[START_REF] Savage | The foundations of statistics[END_REF], we will develop a quick overview of non-Bayesian heuristics that we classify according to the Bayesian and Savagean assumptions they selectively reject. This approach hopefully captures the management issues and world views that each trend tries to address.

The development of Bayesian decision making in the 50's has been a major achievement both for theory and for practice (Wald 1950a, b;[START_REF] Savage | The theory of statistical decision[END_REF][START_REF] Savage | The foundations of statistics[END_REF][START_REF] Raiffa | Applied Statistical Decision Theory[END_REF] in deep interaction with management science [START_REF] Erickson | How Reason Almost Lost Its Mind -The Strange Career of Cold World Rationality[END_REF][START_REF] Giocoli | From Wald to Savage: Homo Economicus Becomes a Bayesian Statistician[END_REF]). In theory, utility functions for probabilistic lotteries allowed to compare decision alternatives facing uncertainties; in addition, the introduction of-so called subjective-prior probabilities of uncertain events made possible to compute, through Bayes's conditional probabilities theorem, the value of a new information about these events. Bayesian theory is still highly influential in several fields (statistics, medical science, finance, cognitive psychology, AI…) but also in industrial practice when investment policies face costly uncertainties like in pharma or oil and gas explorations.

However, Bayesian probability and logic have been also a highly controversial topic in statistical and management science (see in particular [START_REF] Shackle | Epistemics and Economics -a critique of economic doctrines[END_REF][START_REF] Shackle | The bounds of unknowledge[END_REF][START_REF] Lachman | Shackle's Place in the History of Subjectivist Thought[END_REF][START_REF] Hey | The possibility of possibilities[END_REF]. It was often argued (see for instance two recent special issues in European Management Review [START_REF] Elmquist | Innovation Theory and the (Re-)foundation of Management: Facing the Unknown[END_REF] and Academy of Management Review [START_REF] Alvarez | Imagination, Indeterminacy, and Managerial Choice at the Limit of Knowledge[END_REF] that the assumptions underlying the Savagean model were not adapted to major aspects of organizational life that are now commonplace for companies, states and consumers, namely: i) Contexts where fast and unpredictable changes occur and where unknown situations emerge and not only known-yet uncertain-events ii) the routinized mobilization of entrepreneurship, research, design and innovation which forces to recognize that unknown events are also proactively sought and provoked by organizations.

Actually, criticism of the Bayesian model also indicates directions of investigation and potential progress. As often in science, research tends to keep existing models as much as possible, while attempting to reject the most unrealistic assumptions [START_REF] Ehrig | Why we need normative theories of entrepreneurial learning that go beyond Bayesianism[END_REF][START_REF] Grandori | Unbounding bounded rationality: Heuristics as the logic of economic discovery[END_REF] and research has explored new families of heuristics that extends Bayesian decision framework [START_REF] Feduzi | Heuristic methods for updating small world representations in strategic situations of Knightian uncertainty[END_REF][START_REF] Ehrig | Theory-based learning and experimentation: How strategists can systematically generate knowledge at the edge between the known and the unknown[END_REF]. Such endeavors are expected to, hopefully, offer tractable alternatives to the standard approach.

THE MAIN CONCEPTUAL ASSUMPTIONS OF BAYESIAN DECISION MAKING.

Bayesian Decision theory [START_REF] Savage | The theory of statistical decision[END_REF][START_REF] Savage | The foundations of statistics[END_REF] was built by combining conceptual and technical axioms coming from rational choice theory and from Bayesian probability theory . These axioms can be synthesized through four conceptual major assumptions that we remind broadly below, avoiding unnecessary mathematical details.

Assumption 1: uncertainty in a closed and known world. The Bayesian decision model assumes i) a fixed list of decision alternatives D that are operational and actionable ; ii) a fixed set of events E that are all "uncertain" : meaning that the decision maker ignores which one will happen ; yet, it must be mentioned that the events that are elements of E are perfectly known, only their occurrence is uncertain. No lack of precision or incompleteness in the definitions of D and E are assumed [START_REF] Shackle | Epistemics and Economics -a critique of economic doctrines[END_REF][START_REF] Hey | The possibility of possibilities[END_REF][START_REF] Loch | Diagnosing Unforseeable Uncertainty in a New Venture[END_REF].

Assumption 2: measurability of decision utility with uncertain events. Coming from classic rational choice this assumption means that it is possible to measure (and compare) the utility (or costs, or consequences) of a decision alternative in D when any event in E occurs. Thanks to this measure, the decision maker can evaluate the variations of the utility of each alternative when the uncertainty of events in E is changed. For, instance when some new information about E is obtained. It is worth mentioning that this assumption does not mean that the measure is "objective". Both probabilities and utilities are subjective evaluations of the decision maker. Yet, once these values are accepted the Savagian decision making axioms require that the decision maker adopts all rational and consistent consequences of these values [START_REF] Savage | The theory of statistical decision[END_REF]. Thus measurability demands an "objective" continuity in conduct that fits with the assumptions of the model -This assumpation was discussed by Maurice Allais, Economics Nobel prize winner, and this discussion is famously reported in Savage's following books [START_REF] Savage | The foundations of statistics[END_REF].

Assumption 3: the uncertainty of events E can be measured by Bayesian probabilities.

This assumption is technically demanding but it has important consequences on the learning model that is central in Bayesian decision making. It allows to calculate the Bayesian utility of a new information that may change the so called "prior" or "subjective" probabilities of events E. In practice, the possibility of such calculations determined the wide adoption of Bayesian decision making in several domains.

Assumption 4: the decision maker cannot change E.

This assumption means that decisions do not influence the set E but only the probabilities of the elements of E. Simply put, the model takes for granted some events that are beyond the action of the decision maker. Technically, this means that decision alternatives and events are independent variables.

NON-BAYESIAN WORLDS : DISTINGUISHING BETWEEN THE UNCERTAIN AND THE UNKNOWN

All these assumptions correspond to several mathematical structures and properties [START_REF] Savage | The theory of statistical decision[END_REF](Savage , 1954) that we cannot describe here. However, the four assumptions portray the type of managerial and economic world view that fits with the model : classical investments with commercial uncertainties, financial policies in stable markets, diversification and make or buy strategies,…In such contexts, uncertainty may be important, but the stability of techniques and assets and the measurability of consequences are sufficient to justify the acceptance of the four schemes of assumptions.

Contrasting with this first world, we all recognize contexts where technology changes occur with accelerated pace, start-ups can grow or die very fast, human agency is key, social and environmental issues disrupt performance and responsibility and so on. In such contexts the assumptions of the Bayesian model are clearly non valid.

In economics, Frank Knight (see Nicolai Foss' 2023, this SI) is well known for having introduced the idea of non-probabilistic situations, called "Knightian uncertainty" where "judgment" is necessary for decision making. Recently, [START_REF] Hatchuel | A century old and still visionary. Fayol's innovative theory of Management[END_REF] remarked that, in 1916, before Knight, Henri Fayol, the founder of administrative science acknowledged unpredictable futures and introduced "the unknown" as a key element of any management "program" [START_REF] Fayol | Administration industrielle et générale[END_REF]). Actually, Fayol was both a manager and an accomplished scientist. Discoveries made in his laboratory contributed to the 1920 Nobel Prize in physics of Charles-Edouard Guillaume and to the business success of his company [START_REF] Hatchuel | A century old and still visionary. Fayol's innovative theory of Management[END_REF]. Thus, one of the first modern management thinkers was already facing a non-Bayesian world.

Are Knightian uncertainty or the popular "radical uncertainty" or "the Fayolian unknown" different expressions for the same notion ? It is clear that speaking of "uncertainty" when the situation includes future "surprises" or "discoveries" requires additional images and periphrasis. On the contrary, these traits are common sense attributes of the "unknown" simply understood as aknowledged ignorance. Scientific reasoning needs a sharp distinction between uncertain and unknown: the age of the sun is known with uncertainties (that can be high); the nature of "dark matter", if it exists, is unknown; a universal and quick cure of all cancers is unknown and uncertainty has no meaning about these entities. This remark has direct consequences for non-Bayesian heuristics. It allows to distinguish among Savagian assumptions those that describe a closed world (ie. all actions and events are known even if they are uncertain) from those which describe beliefs or probabilities (or any other quantities) associated to the realization of these known events. Let's summarize these notions through an example: one acknowledges that she may be affected by a well-known disease: this is an uncertain event. Yet, she can reject any probability value of that event: the uncertainty of the event is seen as not measurable.

Unfortunately the treatment for such well known disease may be unknown to day. Research is active but no one knows if some good news may be heard in the future.

To date, it seems that a non-Bayesian heuristics for rational decision making that would reject all four assumptions is not explored in the literature. For sure, authors, including those that appear in this issue may in principle reject all these assumptions. But when they elaborate a heuristic, researchers tend to prioritize the rejection of some of these assumptions according to some decisional context. Thus in the following, and for the sake of clarity, we describe some trends and emerging propositions of non-Bayesian heuristics by focusing on the assumptions that are explicitly rejected in the heuristics. Note that the same heuristic may appear in more than one trend .

Rejecting the measure of utility and its improvement (Assumption 2).

This statement not only means that indicators of utility are controversial or weak, it accounts also for the belief that utility evaluations cannot be made better by some additional information, i.e. learning itself is rejected. This direction of work is explored by the paper by Frey, Osterloh and Rust (2023, this SI) on the rationality of qualified lotteries . Such extreme heuristics target situations which may appear in human resources management or when there is the need to select important executives while acknowledging that, after a first shortlisting, judgment about the future behavior of remaining candidates is difficult and controversial. Authors revisit the old tradition of randomly choosing among qualified candidates and argue that such apparently "blind" decision is "locally" the best possible choice. They also claim that such heuristic contributes to avoid political coalitions and biases during the final step of selection. Theoretically, this heuristic stands at the frontier between non-Bayesian and Bayesian decision making. It still can be interpreted as a Bayesian decision making with uniform prior probability on the future performance of selected candidates, yet without any possible learning. It is assumed that no additional information or expertise would significantly modify the uniform prior probability. Hence it is a border line situation where Bayesian inference is possible but has no decisional value.

Rejecting the assumption that uncertainty can be measured by probabilities (Assumption 3)

It is worth mentioning that this line of criticism goes beyond the traditional epistemological debate about objective and subjective (prior) probabilities that has flourished in statistical theory and philosophy of science (Taper and Lele 2004). In management and connected areas, since the seminal work of [START_REF] Kahneman | Judgment under Uncertainty: Heuristics and Biases[END_REF], authors reject the idea that human cognition and beliefs obey to the consistency axioms that are requested by probability theory. The notion of belief is not rejected per se; but the way people build or changes their beliefs is not seen as universal or consistent. Thus, authors look for heuristics that adjust the notion of "belief" to the type of consequences and risks involved by the situation.

Bearing on the notion of "evidential weight" that has been early proposed by Keynes, Derbyshire, Feduzi and Runde (2023, this SI) suggest a heuristic that focuses on the parameters with the weakest "evidential weight" and the biggest impact. These two factors may be identified, for example, through the building of different scenarios for a complex investment. Scenarios are not prediction of the future, they only allow to enact the evidential weight/impact evaluation. These parameters should guide the decision maker towards a strategic monitoring process. Yet no Bayesian learning is assumed or warranted for the evolution of "evidential weights ". With this trend of research non-Bayesian decision making contributes to the class of risk management heuristics which traditionally relied on the assumption that one can evaluate the probability of critical events.

Rejecting the closed world assumption (Assumption 1 ).

This trend also often includes the rejection of assumption 4 by introducing the capacity of the decision maker to transform both E and D. Actually, this trend opens a wide range of research possibilities. For two reasons :

 it embeds the Bayesian logic in a larger process where the sets D and E are no more fixed and can evolve in surprising and diverse ways. This means acknowledging the existence of an expandable "unknown" world ( the "large world" in Foss' introductory note, 2023, this issue).  Surprising events or disruptions can be imposed by this outside world or can be voluntarily provoked by the decision maker (entrepreneuship, design, research, innovation..).. The latter case requires "expandable rationality" [START_REF] Hatchuel | Towards Design Theory and Expandable Rationality: The Unfinished Program of Herbert Simon[END_REF]) that fits with the unknown world.

Facing such unknown world, Knightian "judgment" is still helpful. The "evidential weight" can be seen as a type of "judgment" ; while "qualified lotteries" deal with a situation where judgment leads to stop seeking increased information. Yet, judgment has also to evolve reflexively with the dynamics of events that reveals parts of the unknown world and may generate also new unknowns. And when judgment is about some imagined and desirable entities new heuristics are proposed .

Heuristics for reflexive and dynamic judgment.

In management and entrepreneurship literature, "adaptive management", "strategic flexibility", "agility", "effectuation" [START_REF] Sarasvathy | Entrepreneurship as a science of the artificial[END_REF] are well known topics that also aim to capture dynamic decision making. However, in spite of their attractivity, such notions also need well-grounded and rational heuristic development [START_REF] Gigerenzer | Heuristic Decision Making[END_REF]. Obviously, not all agility or flexibility are efficient or relevant per se and the requested scope of adaptation depends on the departure situation. Hence, one should expect from a class of non-Bayesian heuristics that aim to revise D and E that they channel the generation of new D and new E towards an improvement of the value and relevance for the decision maker. Any such heuristic would offer an operational process for flexible or agile management and entrepreneurship. Scenario building for instance (see Derbyshire et al 2023, this SI) can be used not only to predict but also to help making robust and consistent judgments when new events appear. Likewise a scientific model is used to detect anomalies and organize the minimal revision that fits with the surprising data. Thus scientific heuristics (hypothesis, experimentation, modelling, ) that aim at detecting new causes and anomalies (see Grandori 2023, this SI) are possible guides for reflexive judgment and for navigating in the unknown.

Heuristics for imaginative exploration

Judgment may also require "imagination" (see also [START_REF] Foss | Knightian uncertainty and the limitations of the Savage heuristic[END_REF]Grandori 2023, this SI ). This idea was already in Simon [START_REF] Simon | Rationality as a Process Ans as a Product of Thought[END_REF]) when he discussed "creative" problem solving and design. But what type of rational heuristics include "imagination" safely, without creating nonsense?

The "logic" of imagination is a strange one. If one looks for a rational use of imagination, several thought models are possible. In Science, imagination can help building new hypothesis and new models, yet they have to be testable and controlled by experiments. These imaginative models may enrich a Bayesian exploration by extending causalities and events.

In mathematics, imagination plays a different role, it is used to elaborate a controlled extension (like in the design of complex numbers or non-euclidean geometries) where an imaginary entity is designed with no possibility to test its existence. Rather it is expected that it may help discovering previously unknown mathematical worlds. Yet, this design could create nonsense and contradictions. Therefore, it is submitted to tight controls of consistency and value of the new worlds. Can we find similar imaginative heuristics for decision making ? [START_REF] Hatchuel | Towards an ontology of design: lessons from C-K design theory and Forcing[END_REF] showed the correspondence between a formal logic of design and the theory of mathematical extensions, called Forcing. Forcing describes how one can create new sets, new numbers or new families of sets with a controlled heuristic.

Thus imagination can be safely used, not only to anticipate real observations like in science but also to "force" the generation of new worlds. Hence, there exists a class of non-Bayesian heuristics that can guide "creative" decision-making when required.

In this special issue, the paper on Chimera heuristics (Gilain et al 2023, this issue) follows Le [START_REF] Masson | Designing decisions in the unknown: a generative model[END_REF] and uses a design logic -namely, concept-knowledge theory [START_REF] Hatchuel | Towards Design Theory and Expandable Rationality: The Unfinished Program of Herbert Simon[END_REF]Weil 2003, 2007;Sharif Ullah et al. 2010 ) -to build a controlled non-Bayesian extension of D and E. This heuristic introduces imaginary decision alternatives or events (d* or e*) that have the form of desirable "chimeras" : a mixture of alternatives and events that does not exist, but would have a greater value than existing alternatives : for instance, a new robot that would perform better than all known robots . These chimeras have no warranted existence at present or in the future and need no notion of evidence or probability. The final aim of this heuristic is not the realization of these chimeras (that may never occur) but to systematically provoke a knowledge expansion in both D and E until some satisfactory and improved stage (D', E') has been reached, even if (d*, e*) is not part of it. Such heuristic does not aim at transforming "what is" (D, E) into what "ought to be" (d*, e*), like in classic design projects (Le [START_REF] Masson | The Interplay Between Creativity issues and Design Theories: a new perspective for Design Management Studies?[END_REF]), but to reach a new, yet unpredictable "is" (D', E') where new decision alternatives and new events have a higher value. Thus, in this heuristic learning does not only change the uncertainty of some events like in Bayesian logic. Learning reveals the value of ignored unknowns. At the end of the process, chimeras may remain chimeras, but the impact for the decision maker could be real and worth the journey. The decision maker has jumped "out of the box" of Bayesian thinking. It is interesting to remark that chimeras may be generated either by imaginative judgment, or by creating a chimeral scenario departing (ie. "jumping out") from a classic scenario building or from a standard Savagean model. What counts is controlling the process of generating new knowledge and new decision alternatives.

CONCLUSION : THE RATIONALITY OF NON-BAYESIAN WORLDS

Managerial worlds are not all alike. Some look Bayesian, others no. Thus, the rationality of non-Bayesian decision heuristics do not deny the rationality of Bayesian assumptions, it only reminds that their validity has to be restricted to some stable contexts of action and uncertainty. In such contexts Bayesian decision making is highly relevant and will remain so.

Non-Bayesian heuristics are necessary when the world becomes dominated by processes that generate unpredictability and unknowness. Their rationality is different : it still has boundaries, however it is an expandable rationality [START_REF] Hatchuel | Towards Design Theory and Expandable Rationality: The Unfinished Program of Herbert Simon[END_REF] : it calls for reflexive judgments, scientific inquiry and expanding the decision world rather than optimize it.

Interestingly, the papers appearing in this special issue suggest, although through different heuristics, that rationality in the unknown cannot be limited to exploration and reactive agility. There are cases where exploration has no sense and where random choices are better. Agility needs reflexive judgment on critical unknowns in non-probabilistic worlds. Such reflexivity may become transformative by introducing chimeras that may "force" knowledge expansion and extend the scope of decision alternatives. Hence, even if the class of non-Bayesian heuristics is still open to further research, the papers of this issue offer operational advances and food for thought on what rationality means in highly unknown situations.


Aurélien Barbotin

Armand Lablaine

Cyrille Billaudeau

Erdinc Sezgin

Rut Carballido-López

Measuring membrane fluidity in Bacillus subtilis using TIR-FCS

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Measuring membrane fluidity in Bacillus subtilis using TIR-FCS Aurélien Barbotin 1 , Armand Lablaine 1 , Cyrille Billaudeau 1 , Erdinc Sezgin 2 , Rut Carballido-López 1 1 MICALIS, INRAE, AgroParisTech, Université Paris-Saclay, Jouy-en-Josas, France 2 Science for Life Laboratory, Department of Women's and Children's Health, Karolinska Institutet, 17165 Solna, Sweden 180 DOPC SLBs TIR-FCS: FCS on a TIRF microscope Automated curves sorting TIR-FCS can generate hundreds to thousands of FCS curves per acquisition. As in any FCS experiments, these curves can be subject to artefacts, caused for instance by molecular clusters entering the observation area. Several methods exist to detect those, for instance using a quality metric based on the deviation of the curve to the fit. Unfortunately these also measure varying signal levels. Here we instead penalise fitting bias: a high penalty is given to fitting residuals with several adjacent points of the same sign (either positive or negative).

A noisy FCS curve exhibits high residuals (χ) but still represents a valid measurement. Adding a penalisation term for fitting bias (χ nl ) allows to discriminate artefactual curves from valid but noisy curves.

We tested this metric experimentally using a supported lipid bilayer (SLB). TIR-FCS in a DOPC SLB generates artefactual curves (bottom right) that can bias overall diffusion measurement. Filtering these curves based on their χ nl (vertical dotted line) allows for unbiased diffusion coefficient measurement.

Cold Shock: A sudden drop in temperature (cold shock) induces a decrase of membrane fluidity. To compensate for this, B. Subtilis changes the fatty acid composition of its membrane. We could observe the effect of this remodelling on the diffusivity of Nile Red occuring after cells grown at 37°C were transfered at 20°C. The diffusivity immediately decreases after transfer from 20°C

Upon starvation, bacteria of the Bacillus genus are able to differentiate as resilient spores. During its development, the spore is isolated from the environnment by a mother cell. Here we compared diffusion of Nile Red in the membranes of vegetative cells and in the membranes of the mother cell at a late stage of sporulation. We found that Nile Red slows down considerably in the mother cell. It appears also that there are 2 components of diffusion present during sporulation, perhaps caused by a membrane heterogeneity or interaction with an unknown element in the cytoplasm.

Membrane

FCS curve

Green: FCS curve. Black: fit with known model Bacteria need to adapt their membrane fluidity in response to a wide range of stimuli, whether physical or chemical. For instance, the bacterium Bacillus subtilis changes the fatty acid composition of its membrane in response to a change in temperature. Changes in membrane fluidity were either estimated from biochemical methods like fatty acids analysis or indirectly assessed using environment-sensitive probes, but never directly measured. We show here that Fluorescence Correlation Spectroscopy (FCS) used on a TIRF microscope (TIR-FCS) can be used to measure the diffusion of a well-known membrane marker, Nile Red. Because relative changes in diffusion coefficients are directly linked to a change in membrane fluidity, we can quantify changes in membrane fluidity that could so far only be qualitatively assessed. To facilitate the analysis of the large amount of data generated by TIR-FCS, we developed a simple method to automatically exclude artefactual FCS curves. As a result, we could perform high-throughput FCS measurements revealing the impact of changes in temperature and of sporulation on membrane fluidity.

  fluidity and temperature Membrane remodeling during the sporulation process Conclusion D [µm²/s] 37°C, steady-state 20°C, 180 mins Incomplete recovery of membrane fluidity: at t=0 and T=37°C, diffusivity of Nile Red is maximum (5 +/-1 µm²/s) in the membrane of B. Subtilis. Immediately after cold shock, membrane diffusivity is strongly reduced and starts to recover. After 3h at 20°C, diffusivity of Nile Red has increased (3.7 +/-0.5 µm²/s) but is still lower than at 37°C. Biophysical Society meeting San Diego, California -February 2023 FCS_BACSUB FCS allows, among others ,measurements of diffusion speeds. In FCS, fluorescent molecules diffuse in and out of an observation volume, inducing intensity fluctuations that can be autocorrelated to measure diffusion speed. FCS experiments are usually performed on a confocal microscope. They can also be done on a TIRF microscope, to benefit from increased axial selectivity, to observe spatially-varying diffusion coefficients and to parallelise FCS acquisitions. pipeline for imFCS Outcome of a TIR-FCS acquisition on exponentially-growing B. Subtilis cells, at 37°C, labeled with Nile Red We use FCS to measure the diffusion coefficient of the fluorescent membrane marker Nile Red in the membrane of the rodshaped bacterium Bacillus subtilis. Faster or slower diffusion speeds can be linked to higher or lower membrane fluidity. Bright-field image (top) and diffusion maps (bottom) of Nile Red in B. subtilis at 37°C (left), immediately after cold shock (middle) and 3h after cold shock (right) High noise, no bias χ = 0.37 χ nl = 3.2x10 -3 χ = 0.37 χ nl = 9.1x10 -2 No noise, high bias Simulations Representative FCS curves acquired in vegetative cells, at 37°C and after 5 mins or 3h at 20°C Recovery of membrane fluidity after a cold shock in B. subtilis. Merging of 3 different experiments: FCS performed at 37°C, 6-30 mins after cold shock or 180 mins after cold shock. Left: Bright-field images of a vegetative and sporulating cell. Right: average of FCS curves acquired in these cells. Residuals y i : value of FCS curve at point i, ŷ i : value of the fit, n adj : number of adjacent points on the FCS curve with residuals of same sign. non-linear Residuals

  (y i -ŷ i ) 2 χ nl = Σ n adj,i (y i -ŷ i ) 2


domain: sdv info

text:
Yoann Nicolas

Hélène Bret

Elda Cannavo

Petr Cejka
email: petr.cejka@irb.usi.ch

Valerie Borde
email: valerie.borde@curie.fr

Raphael Guerois
email: raphael.guerois@cea.fr

Molecular insights into the activation of Mre11-Rad50 endonuclease activity by Sae2/CtIP

Keywords: DNA repair, Double-strand breaks, Nuclease, Meiosis, Homologous Recombination, AlphaFold2

In S. cerevisiae, the Mre11-Rad50-Xrs2 (MRX)-Sae2 nuclease activity is required for the resection of DNA breaks with secondary structures or protein blocks, while in humans, the MRE11-RAD50-NBS1 (MRN) homologue with CtIP is needed to initiate DNA end resection of all breaks. Phosphorylated Sae2/CtIP stimulates the endonuclease activity of MRX/N. Structural insights into the activation of the Mre11 nuclease are available only for organisms lacking Sae2/CtIP, so little is known how Sae2/CtIP activates the nuclease ensemble. Here, we uncover the mechanism of Mre11 activation by Sae2 using a combination of AlphaFold2 structural modeling, biochemical and genetic assays. We show that Sae2 stabilizes the Mre11 nuclease in a conformation poised to cleave substrate DNA. Several designs of compensatory mutations establish how Sae2 activates MRX in vitro and in vivo, validating the structural model. Last, our study uncovers how human CtIP, despite considerable sequence divergence, employs a similar mechanism to activate MRN.

Introduction

DNA double-strand breaks (DSBs) represent a threat to cellular survival and genome stability. Cells possess two main DSB repair pathways: template-independent and potentially mutagenetic nonhomologous end-joining (NHEJ) and template-directed, hence more accurate, homologous recombination (HR). HR is initiated by a process termed DNA end resection. In resection, the 5'terminated DNA strand at a DSB site is nucleolytically degraded, resulting in the formation of a 3'terminated single-stranded DNA (ssDNA) overhang. The overhang is then bound by the RAD51 strand exchange protein to form a nucleoprotein filament with the capacity to identify and pair with homologous DNA, which serves as a repair template. Once the DNA strand exchange takes place, the invaded 3'-terminated end can prime DNA synthesis. There are multiple HR sub-pathways leading to diverse genetic outcomes, however they all share the initial DNA end resection process [START_REF] Cejka | DNA End Resection: Mechanism and Control[END_REF][START_REF] Kawale | Mechanism and significance of chromosome damage repair by homologous recombination[END_REF] .

There are at least three conserved nucleases or nuclease complexes that catalyze DNA end resection in eukaryotic cells. Their usage and relative importance are dependent on the nature of the DNA break, cell cycle stage and physiological context. Resection pathways are generally divided into initial shortrange and downstream long-range resection steps [START_REF] Cejka | DNA End Resection: Mechanism and Control[END_REF] . Short-range resection is catalyzed by the Mre11/MRE11 nuclease, which functions within the Mre11-Rad50-Xrs2 (MRX) complex in the budding yeast model Saccharomyces cerevisiae, or MRE11-RAD50-NBS1 (MRN) in human cells [START_REF] Mimitou | Sae2, Exo1 and Sgs1 collaborate in DNA double-strand break processing[END_REF][START_REF] Paull | Nbs1 potentiates ATP-driven DNA unwinding and endonuclease cleavage by the Mre11/Rad50 complex[END_REF] . Further downstream, either EXO1 or DNA2 nuclease extend the resection tracks to perform long-range resection [START_REF] Mimitou | Sae2, Exo1 and Sgs1 collaborate in DNA double-strand break processing[END_REF][START_REF] Zhu | Sgs1 helicase and two nucleases Dna2 and Exo1 resect DNA double-strand break ends[END_REF][START_REF] Gravel | DNA helicases Sgs1 and BLM promote DNA double-strand break resection[END_REF] . Beyond the initiation of DNA end resection, the MRX/N proteins also have nucleaseindependent functions to promote NHEJ, or to activate DNA damage checkpoint through the control of the apical Tel1/ATM kinase [START_REF] Oh | Xrs2 Dependent and Independent Functions of the Mre11-Rad50 Complex[END_REF] .

MRN is generally required to initiate DNA end resection in human cells under all conditions, while MRX may be dispensable in budding yeast for the processing of chemically "clean" DSBs without protein adducts or secondary structures, such as hairpins [START_REF] Llorente | The Mre11 nuclease is not required for 5' to 3' resection at multiple HO-induced double-strand breaks[END_REF][START_REF] Shibata | DNA double-strand break repair pathway choice is directed by distinct MRE11 nuclease activities[END_REF][START_REF] Lobachev | The Mre11 complex is required for repair of hairpin-capped double-strand breaks and prevention of chromosome rearrangements[END_REF] . The unique feature of MRX/N is the ability to resect protein-blocked DSBs [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF] . Both yeast and human MRX/N help remove the NHEJ factor Ku from DNA ends, and in this way to channel non-productive NHEJ complexes into HR [START_REF] Chanut | Coordinated nuclease activities counteract Ku at single-ended DNA double-strand breaks[END_REF][START_REF] Reginato | Physiological protein blocks direct the Mre11-Rad50-Xrs2 and Sae2 nuclease complex to initiate DNA end resection[END_REF][START_REF] Wang | Plasticity of the Mre11-Rad50-Xrs2-Sae2 nuclease ensemble in the processing of DNA-bound obstacles[END_REF][START_REF] Deshpande | DNA-dependent protein kinase promotes DNA end processing by MRN and CtIP[END_REF] . MRX/N is also necessary to process meiotic DNA breaks, catalyzed in a programmed manner by Spo11 during the prophase of the first meiotic division [START_REF] Keeney | Meiosis-specific DNA double-strand breaks are catalyzed by Spo11, a member of a widely conserved protein family[END_REF][START_REF] Borde | The multiple roles of the Mre11 complex for meiotic recombination[END_REF][START_REF] Neale | Endonucleolytic processing of covalent protein-linked DNA double-strand breaks[END_REF] . Spo11 remains covalently attached to the DNA end postcleavage and needs to be removed by MRX before further processing and repair. The MRX/N complex employs a unique strategy, as it cuts the 5'-terminated DNA strand endonucleolytically some distance away from the DSB [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF][START_REF] Neale | Endonucleolytic processing of covalent protein-linked DNA double-strand breaks[END_REF] . The incision site is thus located on the internal side of the protein block, and can serve as an entry site for the long-range resection nucleases acting downstream. The endonuclease activity of the MRX/N complex is strongly stimulated by a phosphorylated co-factor, Sae2 in yeast and

Results

Alphafold2 predicts two conformations for ScMre11 and ScRad50 complex consistent with an autoinhibited and an activated state.

Using AlphaFold2-multimer (AF2) algorithm, we predicted the structure of the (ScMre11)2-(ScRad50)2 assembly (M2R2) adapting the length of ScRad50 coiled-coil to match the size limits imposed by AF2.

The models converged into two distinct classes of structures differing by a major conformational change at the interface between the Mre11 catalytic domains and the Rad50 dimer (Figure 1A and1B).

The class 1 model is similar to the auto-inhibited form of the nuclease as observed in the experimental structures of archaeal, bacterial and eukaryotic homologs (Figure 1A, Figure S1D) [START_REF] Rotheneder | Cryo-EM structure of the Mre11-Rad50-Nbs1 complex reveals the molecular mechanism of scaffolding functions[END_REF][START_REF] Lim | Crystal structure of the Mre11-Rad50-ATPgammaS complex: understanding the interplay between Mre11 and Rad50[END_REF][START_REF] Mockel | ATP driven structural changes of the bacterial Mre11:Rad50 catalytic head complex[END_REF] . In the autoinhibited state, the Rad50 ATPase domain obstructs access to the Mre11 active site. Class 2 model places the catalytic domain of Mre11 in an entirely different orientation (Figure 1B, Figure S1E), which is highly asymmetric, and more similar to the arrangement experimentally observed with the active state of the E. coli EcMre11 homologs in the context of a DNA bound to the EcRad50 groove (Figure 1C) [START_REF] Kashammer | Mechanism of DNA End Sensing and Processing by the Mre11-Rad50 Complex[END_REF] .

In the E. coli active state MR complex bound to DNA, the coiled-coils of EcRad50 are wrapped around the DNA in a closed state, which brings the EcMre11 active site into the proper geometry to cleave dsDNA substrates. The closed conformation of the ScRad50 coiled-coils was only generated by AF2 for models integrating longer lengths of the Rad50 coiled-coil. Using a longer construct of (ScRad50)2 alone, embedding up to 711 residues out of 1312, and without adding ScMre11, we could generate a symmetrical structural model of ScRad50 coiled-coil forming a long rod using AF2 (Figure 1D, Figure S1A andS1C). This closed conformation with contact points regularly distributed along the coiled-coil sequence is similar to the rod structure observed in the low resolution cryoEM map of the Escherichia coli MR homolog [START_REF] Kashammer | Mechanism of DNA End Sensing and Processing by the Mre11-Rad50 Complex[END_REF] and in the recently reported structure of the Chaetomium thermophilum MRN complex [START_REF] Rotheneder | Cryo-EM structure of the Mre11-Rad50-Nbs1 complex reveals the molecular mechanism of scaffolding functions[END_REF] (Figure S1B). Thus, despite the different means of achieving MR nuclease activation between organisms, in particular with or without a Sae2/CtIP homolog, our multiple sequence alignment led AF2 models to converge on distinct classes of states likely to represent key evolutionarily encoded functional steps.

Addition of ScSae2 to ScMre11 and ScRad50 favors the activated state conformation.

In most eukaryotes, stimulation of the endonuclease activity of the MR complex requires the binding of a phosphorylated co-factor, Sae2 in S. cerevisiae or CtIP in H. sapiens [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF][START_REF] Anand | Phosphorylated CtIP Functions as a Cofactor of the MRE11-RAD50-NBS1 Endonuclease in DNA End Resection[END_REF] . To probe the mechanisms underlying this process, we predicted the structure of the complex formed between ScMre11, ScRad50 and ScSae2 using AF2. For simplicity, we omitted ScXrs2, because it is largely dispensable for DNA resection in vitro and in vivo [START_REF] Oh | Xrs2 Dependent and Independent Functions of the Mre11-Rad50 Complex[END_REF] . We also focused on the C-terminal domain of Sae2, which was previously shown to bind Rad50 23 . We did not include the N-terminal domain of Sae2 that is involved in tetramerization [START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF][START_REF] Andres | Tetrameric Ctp1 coordinates DNA binding and DNA bridging in DNA double-strand-break repair[END_REF][START_REF] Forment | When two is not enough: a CtIP tetramer is required for DNA repair by Homologous Recombination[END_REF] . Different constructs and lengths were explored ranging from a large (ScMre11)2-(ScRad50)2-ScSae2(180-345) (M2R2S) assembly to a smaller one as shown in Figure 2A (see Methods). In contrast to the (M2R2) assembly in which two classes of conformations were generated, the presence of ScSae2 generated only the class 2 (Figure 2A), resembling the activated form of the E. coli MR nuclease (Figure 1B and 1C), while the class 1, corresponding to the auto-inhibited form, was no longer generated. The stimulatory function of scSae2 may thus be explained by shifting the equilibrium of the MR structure towards the active conformation.

A ternary complex forms between ScSae2, ScMre11 and ScRad50.

The structure of ScSae2 in its state bound to M2R2 (Figure 2A and Figure S2A) is distinct from any known fold in the PDB as established by the DALI server [START_REF] Holm | Dali server: structural unification of protein families[END_REF] (Table S1). In the model, the C-terminal domain of ScSae2 is predicted to bind one Rad50 monomer on its outer β sheet (Figure 2A). It also contacts the subunit of ScMre11 positioned to cleave dsDNA in the activated form, tightening the asymmetric arrangement of the Mre11 dimer on that of Rad50 (Figure 1C). Therefore, ScSae2 is predicted to create a trimeric interface connecting the outer β-sheet of ScRad50 and ScMre11 (Figure 2A and Figure S2D).

The region of ScMre11 in contact with ScSae2 and ScRad50 involves a short loop protruding from the catalytic domain and folded as a helix (Figure 2B). This ScMre11 helix makes only a few direct contacts with ScRad50 such as between residues E113 ScRad50 and R181 ScMre11 /R184 ScMre11 (Figure 2C, Figure S2B).

In prokaryotes, a similar but larger helix, assigned as the fastener loop [START_REF] Kashammer | Mechanism of DNA End Sensing and Processing by the Mre11-Rad50 Complex[END_REF] (Figure 1C), was found in the EcMre11 homologue, and was shown to play an important role to connect EcMre11 and EcRad50 by binding the outer β sheet of Rad50 through a large network of polar and ionic interactions. Compared to the structure of the E. coli homologs, the interface of the eukaryotic ScMre11-ScRad50 subunits involving the helix exhibits a much smaller interaction surface (Figure 1B), suggesting that a part of the fastener function in eukaryotes was taken up by Sae2 (Figure S2B) [START_REF] Gut | Structural mechanism of endonucleolytic processing of blocked DNA ends and hairpins by Mre11-Rad50[END_REF] . Indeed, the AF2 structural model of ScSae2 sheds light on this set of interactions, and Sae2 may strengthen the interaction of the small interface between ScMre11 and ScRad50 as will be seen below.

Anchoring of ScSae2 to the ScRad50 outer β-sheet

Focus on the predicted interface between ScSae2 and ScRad50 identified an extensive network of interactions involving side-chain and backbone atoms of ScSae2 (Figure 2, Figure S2B). The most central residue involved in these interactions is S267 ScSae2 , which specifically needs to be phosphorylated for optimal stimulation of the MRX endonuclease [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF][START_REF] Huertas | CDK targets Sae2 to control DNA-end resection and homologous recombination[END_REF] . The non phosphorylatable S267A ScSae2 mutant was shown to be inefficient in MR stimulation whereas phospho-mimicking S267E ScSae2 mutant partially retained the activity of phosphorylated wild-type Sae2. In our model, S267 ScSae2 is positioned near R77 ScRad50 enabling the formation of a salt bridge between both residues when S267 ScSae2 is phosphorylated (Figure 2B). Downstream of S267 ScSae2 , another salt bridge between D275 ScSae2 and R125 ScRad50 was also predicted (Figure 2B). The K81 ScRad50 ammonium moiety can form up to three hydrogen bonds with the carbonyl backbone atoms of the triad R273 ScSae2 -D275 ScSae2 . The involvement of K81 ScRad50 is consistent with the original report of absence of meiotic DNA break processing, sporulation defects and spore lethality in cells bearing the K81I (termed rad50S) mutation [START_REF] Alani | Analysis of wild-type and rad50 mutants of yeast suggests an intimate relationship between meiotic chromosome synapsis and recombination[END_REF] (Figure 2B). This was later explained by the inability to remove Spo11 from break ends [START_REF] Neale | Endonucleolytic processing of covalent protein-linked DNA double-strand breaks[END_REF] . In biochemical assays, the K81I mutation in ScRad50 was found to impair ScMre11 endonuclease activity by disrupting the physical interaction between phosphorylated ScSae2 C-terminus and ScRad50 [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF][START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF] .

Last, the ScSae2-ScRad50 interface is potentially further stabilized by the interaction of the highly conserved F271 ScSae2 and F276 ScSae2 aromatic rings packing against the carbon moieties of aforementioned R77 ScRad50 and K81 ScRad50 sidechains, respectively (Figure 2B).

Inspection of the interface between ScSae2 and ScMre11 in the M2R2S ternary complex.

Several contacts can be distinguished between ScSae2 and ScMre11, which likely favor the cooperative binding of ScSae2 and ScRad50 and the assembly with ScMre11 in a suitable geometry for nuclease activation. First, the phosphorylated form of S267 ScSae2 can form a salt-bridge with R187 ScMre11 (Figure 2B). Three residues upstream in ScSae2, a second salt-bridge can be formed between R264 ScSae2 and E183 ScMre11 located in the short helix protruding from ScMre11 (Figure 2C). The model also predicts the formation of a small hydrophobic cluster involving L261 ScSae2 and F186 ScMre11 with a potential contribution of surrounding side-chains from F229 ScMre11 and Y254 ScSae2 (Figure 2D), which likely further stabilizes the complex. Altogether, a larger network of interactions was found in the two interfaces involving ScSae2 with ScRad50 and ScMre11, while direct contacts between the Rad50 and Mre11 subunits appear rather limited compared to the complex formed in the bacterial homologs [START_REF] Kashammer | Mechanism of DNA End Sensing and Processing by the Mre11-Rad50 Complex[END_REF][START_REF] Gut | Structural mechanism of endonucleolytic processing of blocked DNA ends and hairpins by Mre11-Rad50[END_REF] . This property may explain the need for a central role of phosphorylated ScSae2 in stabilizing a functional and active nuclease. The global integration of the models shown in Figures 1B,1D and 2A with the positioning of the DNA shown in Figure 1C could be achieved without any major structural rearrangement and converged on a coherent composite model (Figure 2E) representing the global architecture of the ScRad50-ScMre11-ScSae2 assembly and the DNA in the activated form of the complex.

Assessment of the S. cerevisiae M2R2S structural model by targeted mutagenesis in biochemical and in vivo assays

Our structural model of the activated form of the ScM2R2-Sae2 predicts that phosphorylation of S267 ScSae2 stabilizes a conformation of ScSae2 that promotes the positioning of ScMre11 on ScRad50 in its active state. To verify the predictions of this active structural arrangement, we designed 5 sets of mutants to probe the various interfaces predicted by AF2 (Figure 2F). We tested each mutation alone or in combination, to test for potential compensatory effects. To provide a simple quantitative readout for the compensatory effect of the mutations, we calculated the theoretical impact of the mutations assuming they were independent (i.e. on separate interfaces). Dividing the observed efficacy by the theoretical value gave us a rescue coefficient (R). A R value >1 suggests that the mutations compensate each other, and hence likely involve spatially close positions, while values ≈ 1 suggest no interaction.

We expressed and purified the wild-type and mutant recombinant proteins and protein complexes and tested for their capacity to endonucleolytically cleave protein-blocked DNA in reconstituted biochemical assays (Figure 3A and 3B and Figure S3). Additionally, we mutagenized S. cerevisiae cells and determined the effects of the individual and combined mutations in vivo in processes previously established to rely on the endonuclease activity of the MRX complex in conjunction with Sae2. To this point, we first employed the assay developed by Lobachev et al., which measures the recombination rate of DNA breaks terminated by hairpins formed by inverted repeated sequences, which require MRX-Sae2 activity for initial hairpin processing (Figure 3C) [START_REF] Lobachev | The Mre11 complex is required for repair of hairpin-capped double-strand breaks and prevention of chromosome rearrangements[END_REF] . Second, we probed for meiotic phenotypes. In S. cerevisiae, the Mre11 and Rad50 proteins are essential to promote DSB formation by the Spo11 transesterase, independently of the MRX nuclease activity and thus likely by a structural role (reviewed in [START_REF] Borde | The multiple roles of the Mre11 complex for meiotic recombination[END_REF] ). A complete loss of Mre11 or Rad50 therefore results in meiotic cells capable of sporulation, but forming only unviable spores, due to the absence of meiotic DSBs (see graph Figure 3D). However, the Mre11 nuclease activity is absolutely required for the removal of Spo11 oligonucleotides from meiotic DSB ends to allow for DSB resection and subsequent repair by HR, producing crossovers (Figure 3D). Mutants where the MRX is present but deficient in the Mre11 nuclease activity, such as rad50 K81I (called rad50S [START_REF] Alani | Analysis of wild-type and rad50 mutants of yeast suggests an intimate relationship between meiotic chromosome synapsis and recombination[END_REF][START_REF] De Jager | Genome instability and Rad50(S): subtle yet severe[END_REF][START_REF] Nairz | mre11S--a yeast mutation that blocks double-strand-break processing and permits nonhomologous synapsis in meiosis[END_REF] ) or sae2∆, therefore allow meiotic DSB formation but exhibit greatly impaired sporulation efficiency and spore viability, reduced or absent Spo11 oligonucleotides and a strong reduction in meiotic crossovers. Set 1 of mutants was designed to probe the predicted small interface between ScRad50 and ScMre11 as suggested by the model in Figure 2C. In this interface, R181 ScMre11 and R184 ScMre11 make a contact with E113 ScRad50 (Figure 2C). We replaced the positively charged arginines R181 ScMre11 and R184 ScMre11 with negatively charged glutamic acid residues, creating R181E ScMre11 and R184E ScMre11 mutants.

Reciprocally, on the Rad50 side, we generated the E113R ScRad50 charge reversal variant. In vitro, the E113R ScRad50 mutation alone strongly reduced the endonuclease activity (Figure 3A and3B). The negative effects of the E113R ScRad50 mutation could be partially rescued by either R181E ScMre11 or R184E ScMre11 , which had a moderate inhibitory effect in isolation (with ScRad50), with a rescue coefficient of 3.1 and 4.2, respectively. The observed rescue effect confirmed that the mutants impact the same interface, validating thus the structural model. In contrast, no rescue was observed in combination with R187E ScMre11 , which is positioned at the other end of the scMre11 helix containing R181 ScMre11 and R184 ScMre11 , and is thus less likely to interact with E113 ScRad50 (rescue coefficient of 1.2), indicating no interaction.

The hairpin cleavage genetic assays exhibited a much higher dynamic range (Figure 3C). The single mutations E113R ScRad50 , R181E ScMre11 and R184E ScMre11 reduced the recombination rates 560-, 120-and 660-fold with respect to the wild-type strain, respectively. The recombination rates of the double mutants between E113R ScRad50 and R181E ScMre11 or R184E ScMre11 were no worse than those of the single mutant strains, suggesting an interaction (rescue coefficients of 561 and 293, respectively). Finally, the meiotic phenotype, as assessed by the sporulation efficiency, confirmed these results, with the three single mutants strongly affected, to levels similar to the sae2∆ mutation, whereas the combination of E113R ScRad50 with R181E ScMre11 or R184E ScMre11 revealed rescue coefficients of 44 and 9, respectively, validating the relevance of the interaction also in the context of meiotic recombination (Figure 3D).

Our biochemical and genetic experiments thus support the AF2 model predicting contacts between E113R ScRad50 and R181E ScMre11 /R184E ScMre11 . Set 2 of mutants was designed to probe the ScRad50 and ScSae2 interface by evaluating the contribution of a potential salt bridge connecting R125 ScRad50 and D275 ScSae2 (Figure 2B). As above, we generated charge reversal mutations and evaluated the D275R ScSae2 and R125D ScRad50 mutants individually and combined. The rescue coefficient in biochemical experiments was rather modest (R=1.7) (Figure 3E), while in the genetic hairpin cleavage assays much more apparent (R=1751) (Figure 3F). Indeed, while the R125D ScRad50 mutation decreased recombination rate 760-fold in otherwise wildtype background, it increased recombination in the D275R ScSae2 cells, indicating a strong genetic interaction. Similarly, in meiosis, while both single mutants showed strongly decreased sporulation, similar to sae2∆, the R125D ScRad50 mutation rescued the defect of D275R ScSae2 (R=58), confirming the strong genetic interaction (Figure 3G). These in vivo results support the structural AF2 model for the interface between ScRad50 and ScSae2.

R77E substitution in Rad50 partially bypasses the requirement for co-activation of MRX by Sae2

Phosphorylation of the conserved S267 residue in Sae2 by cyclin-dependent kinase (CDK) strongly stimulates the endonuclease activity of MRX [START_REF] Huertas | CDK targets Sae2 to control DNA-end resection and homologous recombination[END_REF][START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF] . While a non-phosphorylatable S267A ScSae2 substitution mutant is strongly impaired, the phosphomimetic S267E ScSae2 retains some activity and partially bypasses the requirement for CDK in resection [START_REF] Huertas | CDK targets Sae2 to control DNA-end resection and homologous recombination[END_REF] . The AF2 model predicted that phosphorylated S267 ScSae2 forms salt bridges with neighboring R187 ScMre11 and R77 ScRad50 (Figure 2B). In this way, phosphorylated S267 ScSae2 promotes the positioning of the Mre11 and Rad50 subunits to specifically stabilize the nuclease in its active conformation. The third set of our mutants was designed to probe specifically this crucial trimeric interface. We designed S267R ScSae2 ; the positive charge of this normally negatively charged residue (in its phosphorylated form) was anticipated to confer a strong resection defect. We then set out to test this mutant in combination with charge reversal substitutions R187E ScMre11 and R77E ScRad50 . As anticipated, the S267R ScSae2 mutant conferred a strongly impaired endonuclease activity to MRX in vitro (Figure 4A). A combination of S267R ScSae2 and R77E ScRad50 rescued resection in vitro (R=1.7) (Figure 4A). Likewise, the R187E ScMre11 and S267R ScSae2 mutations compensated each other in vitro (R=1.9). Hairpin cleavage assays verified these interactions in a cellular setting: the S267R ScSae2 substitution was as defective as the sae2Δ strain, and the R77E ScRad50 mutation increased recombination in S267R ScSae2 cells (R=1330) (Figure 4B). Likewise, a combination with R187E ScMre11 partially rescued recombination in S267R ScSae2 (R=552). Corroborating this effect on hairpin cleavage, in meiosis, the S267R ScSae2 mutant was as defective as the sae2∆, and this defect was compensated by the R77E ScRad50 mutation, increasing sporulation efficiency (R=16) (Figure 4C), although not to a level sufficient to ensure spore viability (Figure S4). It is important to note that Spo11 catalyzes 150-200 DSBs in each meiotic S. cerevisiae cell. In order to achieve a full spore viability, all of these breaks have to be resected and repaired. Therefore, even a substantial rescue of DNA end resection, with only a few unrepaired breaks remaining, is not expected to result in spore viability. R187E ScMre11 also compensated the single S267R ScSae2 , to a more modest extent, as measured by sporulation efficacy (R=9) (Figure 4C). These results converge towards S267 ScSae2 closely and functionally interacting with R77 ScRad50 and R187 ScMre11 .

Biochemical assays revealed that the R77E ScRad50 mutant was able to partially support resection without Sae2, more than any other variant constructed in this study (Figure 4D). Additionally, R77E ScRad50 promoted resection with pSae2 variants non-phosphorylatable at S267, but phosphorylated at other Sae2 sites, which is necessary to prevent protein aggregation (Figure 4E) [START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF] . In sae2Δ cells, the recombination rate in the presence of R77E ScRad50 increased notably (R=2489), showing that R77E ScRad50 can partially bypass the requirement for Sae2 even in a cellular setting (Figure 4F). This was confirmed in meiosis, where R77E ScRad50 strongly compensated the sae2∆ mutant for sporulation (R=13) (Figure 4G). However, the functional rescue was likely not sufficient to ensure the repair of all meiotic DSBs, because spores remained unviable in the double mutant (Figure S4).

According to the AF2 model, the negative charge of R77E ScRad50 may place it in a favorable conformation with respect to positively charged R187 ScMre11 , and thus partially bypass the need for phosphorylated S267 ScSae2 normally sandwiched in between. To our knowledge, R77E ScRad50 is the first reported mutation in the MRX complex that partially bypasses the requirement for Sae2 or its phosphorylation at the S267 site to co-activate its endonuclease activity in vitro and in vivo.

Assessing the existence of a physical contact between ScSae2 and ScMre11.

The AlphaFold2 model suggests that the ScSae2, ScMre11 and ScRad50 ternary assembly could be additionally cooperatively stabilized by direct contact between ScSae2 and ScMre11. Three residues upstream of S267 ScSae2 , we identified a salt bridge between the positively charged R264 ScSae2 and negatively charged E183 ScMre11 (Figure 2C) that could contribute to this tripartite stabilization. We set out to test a charge reversal mutation R264E ScSae2 in combination with E183R ScMre11 . Both point mutants exhibited defects in the reconstituted endonuclease assay in vitro. A notable rescue effect was observed with combination of R264E ScSae2 and E183R ScMre11 in vitro (R= 1.8) (Figure 5A). In the hairpin cleavage recombination assay, the compensatory effect was quite dramatic (R= 2670, the strongest of any mutant combination in this study) (Figure 5B). Confirming these compelling in vitro and in vivo results, the meiotic sporulation efficiency defect of the E183R ScMre11 was almost entirely rescued by the R264E ScSae2 mutation, reaching nearly wild-type levels (R=10) (Figure 5C). This high rescue of sporulation prompted us to investigate if other readouts of meiotic recombination were also rescued. Indeed, this was the case, since spore viability of the double mutant was also nearly at a wild-type level, suggesting that most meiotic DSBs are repaired to ensure meiotic progression and viable spores (Figure 5D). Consistent with these dramatic compensatory effects, the amount of Spo11 oligonucleotides was nearly wild-type in the double mutant (Figure 5E). The combination of the biochemical and cellular assays validates the structural model that identified a close and specific interaction between the R264E ScSae2 and E183R ScMre11 residues, promoting the Mre11 nuclease.

The small hydrophobic cluster between Sae2 and Mre11 favors the Mre11 nuclease

Finally, we tested the functional relevance of a hydrophobic cluster our model identified at the interface between Mre11 and Sae2, by mutating the L261 ScSae2 residue, conserved in Saccharomycetaceae species (mutant 5, Figure 2F). In both the biochemical and in vivo meiotic assays, this mutant was defective, showing decreased in vitro resection (Figure S5A), as well a reduced sporulation (Figure S5B) and spore viability (Figure S5C). These results indicate that this hydrophobic patch supports optimal Mre11 conformation and nuclease activity.

Discussion

In all eukaryotes, the nuclease activity of the MR machinery is strongly dependent on a conserved partner from the Sae2/CtIP family, whose sequences have diverged dramatically during evolution. The physical interaction between MR and Sae2 was reported to be weak and transient [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF][START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF] , likely due to the necessity to keep the MR endonuclease under a tight control. Nevertheless, the AlphaFold2 generated models reported here predict a ternary complex formation that was validated by mutagenesis in a combination of biochemical experiments in vitro and genetic assays in both vegetative and meiotic yeast cells.

Our structural, biochemical and genetics study in S. cerevisiae revealed that the transition from the inactive state of Mre11 to its active state, is triggered by the establishment of a cooperative set of interactions centered on the phosphorylated status of S267 ScSae2 (Figure 6). The addition of a negative charge on S267 ScSae2 by CDK-mediated phosphorylation switches the balance between unfavorable and favorable interactions that can form at the outer β-sheet of Rad50 with Mre11. In ScSae2, the conserved residues L261 ScSae2 , R264 ScSae2 and D275 ScSae2 provide key complementary contributions to the stabilization of the tripartite complex with ScMre11 and ScRad50. The validated structural model provides further insights on how Mre11 could be released from its auto-inhibitory conformation (Figure 1A) upon S267 ScSae2 phosphorylation. In the Sae2 bound state, the backbones of residues R320-Q322 (in helix α9) (Figure S6A) are predicted to induce a steric clash with the capping domain of ScMre11 in its inactive state, thus favoring its relocation toward the active state. Interestingly, the telomeric protein Rif2 was proposed to antagonize this active conformation to specifically inhibit the endonuclease activity of MRX at telomeres [START_REF] Marsella | Sae2 and Rif2 regulate MRX endonuclease activity at DNA double-strand breaks in opposite manners[END_REF][START_REF] Khayat | Inhibition of MRN activity by a telomere protein motif[END_REF][START_REF] Roisne-Hamelin | Mechanism of MRX inhibition by Rif2 at telomeres[END_REF] .

Sae2 and CtIP homologs are evolutionarily related although their sequence diverged drastically during evolution. Intriguingly, the residues L261 ScSae2 , R264 ScSae2 and D275 ScSae2 whose importance for the formation of the tripartite complex with Mre11 and Rad50 have been shown, are essentially conserved in the small subset of Saccharomycetaceae species (Figure 7A and Figures S2C andS7D) questioning the generality of the unraveled mechanisms. In ScSae2, the C-terminal domain adopts a combination of two small β-strands packing against α2 and α7-α9 helices (Figure S2A) that seems completely absent in human CtIP (Figure 7A). On another hand, in most eukaryotes CtIP contains a highly conserved "CxxC…RHR" motif (termed C-rich-H domain) that is completely absent in ScSae2 and in related species (Figure 7A). Interestingly, in Yarrowia lipolytica, a yeast belonging to the same subphylum as S. cerevisiae (Ascomycotina), the features specific to either ScSae2 (2 β-strands domain) or HsCtIP (Crich-H domain) appear to coexist in the Sae2/CtIP hybrid ortholog, noted YlCtIP (Figure 7A).

A model of the structure of the Y. lipolytica Rad50-Mre11-CtIP complex generated with AF2 provides key insights to understand how the C-rich-H domain could contribute to the activation of the MR nuclease. The C-rich-H domain is predicted to fold into a small two-helix domain, whose geometry is constrained by the highly conserved cysteine and histidine residues forming a cluster (Figure 7B and Figures S7A andS7B). The structural arrangement of the YlMR-CtIP ternary complex is very similar to that obtained for the ScMR-Sae2 complex. Of utmost interest, as in the structure of ScMR-Sae2, an apolar cluster is also formed between YlCtIP and YlMre11 involving the short fastener helix of YlMre11 (Figure 7C). In the case of Y. lipolytica, one of the helices of the C-rich-H domain mediates this interaction by exposing conserved apolar residues that pack against aromatic residues of the YlMre11 fastener helix, as in the MR-Sae2 complex (purple triangles in Figure 7A). In S. cerevisiae, deprived of C-rich-H domain, this interaction was in part mediated by L261 scSae2 as measured by 'mutant set 5' assays (Figure 2D and Figure S5). Thus, the model of the Y. lipolytica assembly suggests that a role of the highly conserved C-rich-H domain is to create an anchor between Mre11 and CtIP to stimulate the active conformation of the nuclease. This orientation would additionally place conserved basic residues in favorable position to interact with DNA.

We further compared the evolutionary properties of Sae2 and CtIP orthologs by performing AF2 structural predictions between human HsRad50, HsMre11 and HsCtIP (Figure 7D and Figure S7C). AF2 models of the human assembly lead to a conformation between the C-rich-H HsCtIP domain and HsMre11 that was not as compact as for the Y. lipolytica proteins, but the two domains come close to each other in a similar orientation. Conserved hydrophobic residues also protrude from the C-rich-H HsCtIP and could also favorably make contacts with the aromatic residues of the fastener HsMre11 with little conformational change required. The contribution of DNA or other unmodeled partners in the assembly may be necessary to reach the fully active geometry with the human proteins. Just downstream of the RHR motif, T847 HsCtIP , equivalent to S267 ScSae2 , is also phosphorylated by CDK [START_REF] Huertas | Human CtIP mediates cell cycle control of DNA end resection and double strand break repair[END_REF] . In this phosphorylated loop, the architecture of the state bound to the outer β sheet of HsRad50 is very similar to that modeled for the S. cerevisiae complex, with T847 HsCtIP contacting R79 HsRad50 , an arginine in the position equivalent to R77 ScRad50 (Figure 7E; compare with Figure 2B). The possibility of establishing a charged salt-bridge interaction upon phosphorylation may therefore explain the stabilization of the MR-CtIP complex as in ScSae2. In humans, phosphorylation on another residue, T859 HsCtIP , by ATM/ATR kinase was shown to also promote MR activity [START_REF] Wang | The interaction of CtIP and Nbs1 connects CDK and ATM to regulate HRmediated double-strand break repair[END_REF] . In the structural model, T859 HsCtIP contacts a basic residue, K6 HsRad50 , which may additionally support the interaction with Rad50 upon CtIP phosphorylation (Figure 7E). The variety of ways in which the physico-chemical complementarity is encoded at the interface between the homologs of Sae2/CtIP, Rad50 and Mre11 is also highlighted when looking at the case of SpCtp1, the ortholog of HsCtIP in Schizosaccharomyces pombe. A specific polypeptide containing the conserved 15 amino acids at the C terminus of SpCtp1 is sufficient to stimulate Mre11 endonuclease activity in the absence of phosphorylation by CDK [START_REF] Zdravkovic | A conserved Ctp1/CtIP C-terminal peptide stimulates Mre11 endonuclease activity[END_REF] . In SpCtp1, the position corresponding to the phosphorylated residues S267 ScSae2 and T847 HsCtIP is occupied by a hydrophobic amino-acid I283 SpCtp1 . On the other hand, in SpRad50, the position equivalent to the basic residues R77 ScRad50 and R79 HsRad50 is a L77 SpRad50 . This physico-chemical switch from an electrostatic to an apolar interaction could account for Ctp1 stabilizing the interaction with SpRad50 in a CDKindependent manner and promoting the formation of a tri-partite interface with SpMre11 to activate its endonuclease activity.

In conclusion, the comparative analysis of several interologous complexes has provided key insights into functional configurations and mechanisms for interpreting the role of the conserved motifs such as the C-rich-H domain in CtIP homologs. The CtIP and Sae2 families appear to have undergone considerable structural reorganization during evolution, while maintaining an ability to stabilize the active conformation of the Mre11 nuclease domain so that it can act on DNA clamped by Rad50. and 950-1312), in which the coiled-coils assembled into a rod-like structure. The regions in contact between two coiled-coils involve highly conserved glycines (Figure S1A) that are labeled in brown at different positions along the coiled-coils. The G1083 ScRad50 and G349 ScRad50 are reminiscent of the G849 EcRad50 in E. coli, whose mutation to alanine was shown to affect significantly the endonuclease activity of the bacterial enzyme [START_REF] Kashammer | Mechanism of DNA End Sensing and Processing by the Mre11-Rad50 Complex[END_REF] .

Figures Legends

Methods

Structural modeling

Sequences of S. cerevisiae, Y. lipolytica and H. sapiens for the Mre11, Rad50 and Sae2/CtIP were retrieved from UniProt database [START_REF] Uniprot | UniProt: the universal protein knowledgebase in 2021[END_REF] (Table S2). Mre11, Rad50 and CtIP from H. sapiens or Y. lipolytica were used as input of mmseqs2 homology search program [START_REF] Steinegger | MMseqs2 enables sensitive protein sequence searching for the analysis of massive data sets[END_REF] to generate a multiple sequence alignment (MSA) against the UniRef30 clustered database [START_REF] Mirdita | Uniclust databases of clustered and deeply annotated protein sequences and alignments[END_REF] . Homologs sharing less than 25% sequence identity with their respective query and less than 50% of coverage of the aligned region were not kept. For Sae2, we used HHblits [START_REF] Remmert | HHblits: lightning-fast iterative protein sequence searching by HMM-HMM alignment[END_REF] for its enhanced sensitivity with 8 iterations to retrieve remote homologs against the UniRef30_2020_06 database. In every MSA, in case several homologs belonged to the same species, only the one sharing highest sequence identity to the query was kept. Full-length sequences of the orthologs were retrieved and re-aligned with mafft 57 . To model the structures of the Mre11-Rad50-Sae2/CtIP complexes, the corresponding MSAs were concatenated. In the concatenated MSAs, when homologs of different subunits belonged to the same species, they were aligned in a paired manner, otherwise they were left unpaired [START_REF] Bryant | Improved prediction of protein-protein interactions using AlphaFold2[END_REF] . The number of paired and unpaired sequences in the final concatenated MSAs are reported in Table S2. Concatenated MSAs were used as inputs to generate 25 structural models (5 different seeds) for each of the conditions using a local version of the ColabFold v1.3 interface 59 running 6 iterations of the Alphafold2 v2.2.0 algorithm 60 trained on the multimer dataset [START_REF] Evans | Protein complex prediction with AlphaFold-Multimer[END_REF] on a local HPC equipped with NVIDIA Ampere A100 80Go GPU cards. Four scores were provided by AlphaFold2 to rate the quality of the models, the pLDDT, the pTMscore, the ipTMscore and the model confidence score (weighted combination of pTM and ipTM scores with a 20:80 ratio) [START_REF] Evans | Protein complex prediction with AlphaFold-Multimer[END_REF] . The scores obtained for all the generated models are reported in Table S3 for the model ranked first according to the confidence score. The models ranked first for each complex were relaxed using rosetta relax protocols to remove steric clashes [START_REF] Leman | Macromolecular modeling and design in Rosetta: recent methods and frameworks[END_REF] under constraints (std dev. of 2 Å for the interatomic distances) and were deposited in the ModelArchive database (https://modelarchive.org/) (Table S4). The global composite model of the (Mre11)2-(Rad50)2-Sae2 in complex with DNA was built by combining the S. cerevisiae models of (i) the long Rad50 dimer (Figure 1D, ModelArchive ID: ma-i9sv6), (ii) the Mre11 dimers in the active conformation with Rad50 (Figure 1B, ModelArchive ID: ma-jfoh4) and (iii) the trimeric assembly between Sae2, Mre11 and Rad50 (Figure 2A, ModelArchive ID: ma-qhmpy). Model (iii) was first superimposed on model (i) to recover the relative position of Sae2 and of one Mre11 nuclease domain in the context of the closed Rad50 coiled-coil, taking the outer βsheet of Rad50 binding to Sae2 as a reference. The Mre11 nuclease domain from (iii) was then taken as reference to model the relative position of the second Mre11 nuclease domain using model (ii). The orientation of Mre11 (1-412) regions were then kept fixed while Mre11 domains (429-533) binding to Rad50 coiled-coils as in model (ii) were modeled on the closed conformation of the Rad50 coiled-coil using model (i) as reference for superposition. The flexible linkers Mre11 (412-429) connecting the two Mre11 domains of each subunit were finally remodeled using RCD+ [START_REF] Lopez-Blanco | RCD+: Fast loop modeling server[END_REF] . The positions of DNA, nucleotides and Mn 2+ ions were modeled using the superposition of the composite global model with the bacterial assembly in PDB 6S85 taking the position of the Rad50 ATPase domains as a reference.

The resulting global model was relaxed using rosetta under the same conditions as described above.

The Molecular graphics and analyses were performed with UCSF ChimeraX [START_REF] Pettersen | UCSF ChimeraX: Structure visualization for researchers, educators, and developers[END_REF] . Multiple sequence alignments were represented using JalView [START_REF] Waterhouse | Jalview Version 2--a multiple sequence alignment editor and analysis workbench[END_REF] .

Yeast strains

S. cerevisiae strains used for all the meiotic experiments are diploids of the efficiently sporulating SK1 background [START_REF] Kane | Carbohydrate metabolism during ascospore development in yeast[END_REF] . Strains used for the hairpin-opening assays are derived from the haploid HS21 strain, of the CGL background [START_REF] Lobachev | The Mre11 complex is required for repair of hairpin-capped double-strand breaks and prevention of chromosome rearrangements[END_REF][START_REF] Lobachev | Inverted Alu repeats unstable in yeast are excluded from the human genome[END_REF] . Their genotypes are listed in Table S5. Cells were grown and all assays were performed at 30°C. Strains were transformed by the lithium acetate method. Spo11-His6-Flag3::NatMX allele was amplified by PCR. Haploid mre11∆, rad50∆ and sae2∆ strains for hairpin cleavage assays were provided by K. Lobachev. mre11∆ and sae2∆ in the SK1 diploid strains were obtained by replacing the entire coding sequence by the KanMX resistance marker [START_REF] Longtine | Additional modules for versatile and economical PCR-based gene deletion and modification in Saccharomyces cerevisiae[END_REF] . Introduction of all the mre11, rad50 or sae2 point mutations were done in two steps: first, the regions surrounding the aminoacids to mutate were replaced by the KanMX cassette by PCR with long flanking homology. The regions deleted were mre11∆113-291; rad50∆1-192 (which we also used as the "rad50∆" for the meiotic experiments, since it deletes the start codon and the first 192 residues of the protein); sae2∆193-345. Then, these strains were co-transformed with a NatMX selectable plasmid encoding both Cas9 and a guide RNA against the KanMX sequence and a mutagenized PCR fragment spanning the deletion as a healing fragment.

Transformants were selected on clonNAT, and all transformants were sequenced for correct introduction of the mutation and preservation of the remaining of the targeted gene.

Yeast hairpin cleavage assays

The rate of Lys+ recombinants was computed as the number of colonies growing on -Lys plates compared to the total number of colonies growing on YPD rich medium [START_REF] Lobachev | The Mre11 complex is required for repair of hairpin-capped double-strand breaks and prevention of chromosome rearrangements[END_REF] . The mean recombination rate was calculated from six different colonies of each strain using the Falcor website tool using the MSS-MLE [START_REF] Hall | Fluctuation analysis CalculatOR: a web tool for the determination of mutation rate using Luria-Delbruck fluctuation analysis[END_REF] and (https://lianglab.brocku.ca/FALCOR/index.html).

Yeast meiotic assays

Meiosis was induced synchronously by pre-growing cells in SPS pre-sporulation medium as previously described [START_REF] Murakami | Gel electrophoresis assays for analyzing DNA double-strand breaks in Saccharomyces cerevisiae at various spatial resolutions[END_REF] . Sporulation efficiency was measured after 24 h in sporulation medium as the percentage of cells that showed cells with two or more spores. At least two independent experiments were performed and at least 400 cells were counted for each genotype. For spore viability, cells were put on solid SPO plates for 48 h at 30°C. Spo11 oligonucleotide analysis was performed by immunoprecipitating Spo11-His6-Flag3 cells at t = 4hr in sporulation medium essentially as described in [START_REF] Neale | End-labeling and analysis of Spo11-oligonucleotide complexes in Saccharomyces cerevisiae[END_REF] , with the following modifications: Dynabeads Protein G beads (Thermo Fisher Scientific : 10003D) and mouse anti-FLAG M2 monoclonal antibody (Sigma-Aldrich : F3165) were used. Samples were run on an Invitrogen NuPAGE Novex Bis Tris gel 4-12% gradient SDS-PAGE gel and dried under vacuum before exposing to a Phosphorimager screen.

MRX and Sae2 purification

All proteins were expressed in Spodoptera frugiperda 9 (Sf9) cells. The Mre11 construct (vector pTP391 provided by T. Paull) contained a His-tag at the C-terminus, and was co-expressed and co-purified with Xrs2, tagged at the C-terminus with FLAG (pTP694 vector from T. Paull). The heterodimer was purified using affinity chromatography using NiNTA resin (Qiagen) followed by anti-FLAG M2 affinity resin (Sigma) [START_REF] Cannavo | Sae2 promotes dsDNA endonuclease activity within Mre11-Rad50-Xrs2 to resect DNA breaks[END_REF] . Rad50, FLAG-tagged at the C-terminus (vector pFB-Rad50-FLAG), was purified by using anti-FLAG M2 affinity resin (Sigma) [START_REF] Cannavo | Stepwise 5' DNA end-specific resection of DNA breaks by the Mre11-Rad50-Xrs2 and Sae2 nuclease ensemble[END_REF] . Phosphorylated Sae2 (pSae2), MBP-tagged at the N-terminus and his-tagged at the C-terminus, was prepared as described previously in the presence of phosphatase inhibitors [START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF] (vector pFB-MBP-Sae2-his). The MBP tag was cleaved and removed during protein purification. The λ phosphatase treatment was carried out as described previously [START_REF] Cannavo | Regulatory control of DNA end resection by Sae2 phosphorylation[END_REF] . Point mutagenesis of the expression vectors was carried out using the QuikChange site directed mutagenesis kit (Agilent) according to standard procedures. The mutant proteins were then expressed and purified in the same way as the wild type proteins.

Endonuclease assays

Nuclease assays (15 μl volume) were carried out with oligonucleotide-based DNA substrate blocked at all ends with streptavidin (1 nM molecules, oligonucleotides, 210:

(GTAAGTGCCGCGGTGCGGGTGCCAGGGCGTGCCCTTGGGCTCCCCGGGCGCGTACTCCACCTCATGCATC and 211: GATGCATGAGGTGGAGTACGCGCCCGGGGAGCCCAAGGGCACGCCCTGGCACCCGCACCGCGGCACTTAC) 11 , in a buffer containing 25 mM Tris-acetate, pH 7.5, 1 mM dithiothreitol, 5 mM magnesium acetate, 1 mM manganese acetate, 1 mM ATP, 80 U/ml pyruvate kinase (Sigma), 1 mM phosphoenolpyruvate, 0.25 mg/ml bovine serum albumin (New England Biolabs). 30 nM streptavidin (Sigma), was added to the substrate containing the reaction mixture and incubated for 5 min at room temperature.

Recombinant proteins were then added on ice and incubated for 30 min at 30°C. Reactions were terminated and the DNA products were separated by denaturing gel electrophoresis and detected by autoradiography [START_REF] Pinto | Methods to Study DNA End Resection II: Biochemical Reconstitution Assays[END_REF] . The data were quantitated using Fiji (Image J). The signal below the substrate band in no protein lane was removed from all lanes as a background. DNA end resection was then calculated as the fraction of the endonucleolytic DNA cleavage products in each lane. The data were plotted using GraphPad Prism (V8).

Figure S3

: Purified recombinant proteins used in this study. Related to Figure 3 Mre11 and Xrs2 proteins were co-expressed together and purified as a complex. Rad50 variants were purified separately. Phosphorylated Sae2 and variants (indicated as pSae2) were expressed and purified in the presence of phosphatase inhibitors, resulting in an electrophoretic mobility shift. Sae2 prepared without phosphatase inhibitors (right lane, Sae2) was included as a reference. The polyacrylamide gels were stained with Coomassie blue. 1.9 0.9 1.8 2.3 Table S1: Ten top hits from the DALI server analysis.

Table S2: Description of conditions used to generate models.

Table S3: Summary of evaluation scores for models generated by AlphaFold2.

Table S4: Links to the ModelArchive database containing model coordinates, multiple sequence alignments and evaluation scores.

Table S5: Yeast strains used in this study.

Table S1: Top ten hits retrieved from the DALI server analysis of PDB structures potentially similar to the C-terminal domain of ScSae2. The first hit with a Z-score of 3.1 aligned with a fraction of the fold (mainly the long α7 helix) without significant resemblance to the structure of the ScSae2 bound domain.

No: Chain Z rmsd lali nres %id PDB Description 1: 5lbm-C 3.1 3.9 72 90

Table S3: Summary of evaluation scores for the model generated with the highest confidence score for all the models generated by AlphaFold2.

Index Model name

Figure 1 :

 1 Figure 1: Structural modeling of S. cerevisiae Mre11-Rad50 assembly.

Figure 2 :

 2 Figure 2: Structural modeling of S. cerevisiae Sae2-Mre11-Rad50 assembly.

Figure 3 :

 3 Figure 3: Validation of the S. cerevisiae M2R2S structural model by targeted mutagenesis

Figure 4 :

 4 Figure 4: R77E substitution in Rad50 partially bypasses the requirement for co-activation of MRX by Sae2

Figure 5 :

 5 Figure 5: Validation of the interface between ScSae2 and ScMre11

Figure 6 :

 6 Figure 6: Schematic model for the molecular mechanism of activation of the nuclease complex.

Figure 7 :

 7 Figure 7: Evolution at the sequence and structural level of the interactions between Sae2/CtIP orthologs and the Mre11-Rad50 complex.

Figure S4 :Related to Figure 4

 S44 Figure S4: R77E substitution in Rad50 does not restore spore viability of sae2S267R or sae2∆.Related to Figure4

Figure S5 : 5 (

 S55 Figure S5: The small hydrophobic cluster between Sae2 and Mre11 favors Mre11 nuclease. Related to Figure 5

Figure S6 :

 S6 Figure S6: Models illustrating the molecular events occuring the transition between the inactive and active conformations. Related to Figure 6

Figure S7 :

 S7 Figure S7: Details on the structural modeling of Y. lipolytica and H. sapiens Mre11-Rad50-Sae2 assembly. Related to Figure 7

  Figure 3

  Figure 4

  Figure 5 A B

7 Supplementary Figures and 5

 5 Supplementary TablesLegends of the Supplementary Figures are in the main manuscript.

Figure S1 :

 S1 Figure S1: Details on the structural modeling of S. cerevisiae Mre11-Rad50 assembly. Related to Figure 1

Figure S2 : 2 Figure S3 :

 S22S3 Figure S2: Details on the structural modeling of S. cerevisiae Mre11-Rad50-Sae2 assembly. Related to Figure 2 Figure S3: Purified recombinant proteins used in this study. Related to Figure 3

Figure S4 :

 S4 Figure S4: R77E substitution in Rad50 does not restore spore viability of sae2S267R or sae2∆. Related to Figure 4

Figure S5 : 5 Figure S6 :

 S55S6 Figure S5:The small hydrophobic cluster between Sae2 and Mre11 favors Mre11 nuclease. Related to Figure5FigureS6: Models illustrating the molecular events occuring the transition between the inactive and active conformations. Related to Figure6

Figure S7 :

 S7 Figure S7: Details on the structural modeling of Y. lipolytica and H. sapiens Mre11-Rad50-Sae2 assembly. Related to Figure 7

  Figure S4

Acknowledgments

We thank Kirill Lobachev (Georgia Tech University) for the hairpin assays yeast strains, and Matt Neale (Sussex University) for advice on the Spo11 oligonucleotides protocol. This work was funded by Agence Nationale de la Recherche (ANR-21-CE44-0009) to R. G. and V. B. and Fondation pour la Recherche Médicale (Equipe FRM EQU201903007785) to V.B. The Swiss National Science Foundation (SNSF) (Grants 310030_207588 and 310030_205199) and the European Research Council (ERC) (Grant 101018257) support the research in the Cejka laboratory.

Declaration of interests

The authors declare no competing interests.

Supplementary material

Supplementary Figures legends

Supplementary Tables legends

Table S1: Ten top hits from the DALI server analysis. 3D VBD2341 Same as VBD2335 but mre11R184E/" Used in Figure 3D VBD2350 Same as VBD2335 but rad50E113R/" Used in Figure 3D VBD2362 Same as VBD2335 but mre11R181E/" rad50E113R/" Used in Figure 3D VBD2364 Same as VBD2335 but mre11R184E/" rad50E113R/" Used in Figure 3D VBD2347 Same as VBD2335 but sae2D75R/" Used in Figure 3G VBD2351 Same as VBD2335 but rad50R125D/" Used in Figure 3G VBD2369 Same as VBD2335 but rad50R125D/" sae2D75R/" Used in Figure 3G VBD2342 Same as VBD2335 but mre11R187E/" Used in Figure 4C VBD2346 Same as VBD2335 but sae2S267R/" Used in Figure 4C VBD2349 Same as VBD2335 but rad50R77E/" Used in Figure 4C