Eric Vivier

Marie-Elise Truchetet

Thomas Pradeu
email: thomas.pradeu@u-bordeaux.fr

Re-thinking our understanding of immunity: Robustness in the tissue reconstruction system Seminars in Immunology (2018) Issue on "Redundancy and Robustness", guest edited by

Keywords: Tissue repair, tissue regeneration, robustness, redundancy, plasticity, surveillance

Robustness, understood as the maintenance of specific functionalities of a given system against internal and external perturbations, is pervasive in today's biology. Yet precise applications of this notion to the immune system have been scarce. Here we show that the concept of robustness sheds light on tissue repair, and particularly on the crucial role the immune system plays in this process. We describe the specific mechanisms, including plasticity and redundancy, by which robustness is achieved in the tissue reconstruction system (TRS). In turn, tissue repair offers a very important test case for assessing the usefulness of the concept of robustness, and identifying different varieties of robustness.

Introduction

Robustness can be defined as the maintenance of specific functionalities of a given system against internal and external perturbations [START_REF] Csete | Reverse engineering of biological complexity[END_REF][START_REF] Kitano | Biological robustness[END_REF]. The term, routinely used in engineering (e.g. [START_REF] Maynard | Architectural elements of language engineering robustness[END_REF]), is now pervasive in the life sciences [START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]. Systems and processes as diverse as bacterial chemotaxis, biochemical networks, cells, organisms, and ecosystems, among many others, have been described as robust [START_REF] Stelling | Robustness of cellular functions[END_REF][START_REF] Barkai | Robustness in simple biochemical networks[END_REF][START_REF] Alon | Robustness in bacterial chemotaxis[END_REF][START_REF] Wilmers | Understanding ecosystem robustness[END_REF][START_REF] Wagner | Robustness and evolvability in living systems[END_REF]. For example, a plane is robust when it continues to fly despite severe turbulence (for example thanks to the flexibility of its wings), and a bacterial cell is robust to modifications in genetic regulation when it tolerates a high number of these modifications [START_REF] Isalan | Evolvability and hierarchy in rewired bacterial gene networks[END_REF].

The notion of robustness, however, is very broad, and often elusive. To make it more precise, it has long been emphasized (e.g., [START_REF] Lesne | Robustness: confronting lessons from physics and biology[END_REF]) that two crucial questions must systematically be addressed when talking about robustness: first, what is robust, and second to what is it robust? In other words, a system is not robust in general; rather, it is robust to a certain kind of perturbations that can occur at a given level (or at a limited number of levels). The most stirring applications of the concept of robustness are those where talking about robustness seems directly operative, that is, sheds a new and important light on a given phenomenon, as illustrated by several cases including bacterial chemotaxis [START_REF] Alon | Robustness in bacterial chemotaxis[END_REF].

The aim of the present paper is to ask whether the concept of robustness can illuminate the processes of tissue repair and tissue regeneration, and whether, in turn, tissue repair and tissue regeneration offer a promising basis to better define the notion of robustness applied to biological phenomena. We are therefore interested in robustness at a particular level, namely that of tissues, and against a particular set of perturbations, namely damages made on tissues (physical or chemical aggressions, infectious agents, or "internal" stresses). Our focus on repair and regeneration at the tissue level is justified by the recent wealth of data on this issue [START_REF] Eming | Inflammation and metabolism in tissue repair and regeneration[END_REF], and by the obvious clinical interest of this topic, especially in the age of regenerative medicine [START_REF] Pang | An overview of the therapeutic potential of regenerative medicine in cutaneous wound healing[END_REF], but it is important to keep in mind that repair occurs also at other levels (including genetic [START_REF] David | Base-excision repair of oxidative DNA damage[END_REF] and cellular [START_REF] Tang | Self-repairing cells: How single cells heal membrane ruptures and restore lost structures[END_REF] level) in the organism. The idea that repairing oneself is fundamental to the organism's unity and individuality has been suggested at least since the 19 th century, particularly by physiologist Claude Bernard [START_REF] Bernard | Lectures on the phenomena of life common to animals and plants[END_REF].

More recently, the concept of robustness has been commonly associated with repair and regeneration [START_REF] Ninov | Current advances in tissue repair and regeneration: the future is bright[END_REF][START_REF] Galliot | Trends in tissue repair and regeneration[END_REF][START_REF] Bateson | Plasticity, Robustness, Development and Evolution[END_REF][START_REF] Laurent | Immune-Mediated Repair: A Matter of Plasticity[END_REF]. Much remains to be said, however, about how robustness and tissue repair can shed light one on the other. Tissue repair and regeneration involve a horde of components and pathways, including structural (e.g., fibroblasts, ECM, etc.) and immunological (e.g., neutrophils, macrophages, etc.) ones [START_REF] Eming | Inflammation and metabolism in tissue repair and regeneration[END_REF][START_REF] Laurent | Immune-Mediated Repair: A Matter of Plasticity[END_REF][START_REF] Gurtner | Wound repair and regeneration[END_REF][START_REF] Eming | Evolution of immune pathways in regeneration and repair: Recent concepts and translational perspectives[END_REF]. For this reason, we propose the concept of the "tissue reconstruction system" (TRS) to embrace all the different aspects of this phenomenon (see Figure 1). Repair is essential for the survival and maintenance of the body [START_REF] Bernard | Lectures on the phenomena of life common to animals and plants[END_REF][START_REF] Gurtner | Wound repair and regeneration[END_REF]. Failures in the repair process can lead to various pathological conditions, including fibrotic diseases, ulcers, hypertrophic and keloid scars, as well as cancers [START_REF] White | Inflammation, wound repair, and fibrosis: reassessing the spectrum of tissue injury and resolution[END_REF][START_REF] Menke | Impaired wound healing[END_REF][START_REF] Serra | From Inflammation to Current and Alternative Therapies Involved in Wound Healing[END_REF]. Repair is continuously occurring, to some degree, in organisms (e.g., skin renewal), in response to their constant exposure to damages of different types (physical, chemical, radiological, etc.). Even though there exists to a large extent a continuum between repair and regeneration [START_REF] Eming | Wound repair and regeneration: mechanisms, signaling, and translation[END_REF], the two phenomena can be considered distinct in several respects. Regeneration describes the capacity to regrow complex organs entirely, generally with the implication of several cell types [START_REF] Galliot | Trends in tissue repair and regeneration[END_REF][START_REF] Brockes | Comparative aspects of animal regeneration[END_REF][START_REF] Alvarado | Bridging the regeneration gap: genetic insights from diverse animal models[END_REF][START_REF] Birnbaum | Slicing across Kingdoms: Regeneration in Plants and Animals[END_REF]. In mammals, for example, the renewal of the epidermis is a form of repair, because it involves a single cell type (keratinocytes), whereas for the liver one can talk about regeneration as it involves several cell types (hepatocytes, sinusoidal endothelial cells, stellate cells, Kupffer cells, etc.) [START_REF] Michalopoulos | Liver Regeneration[END_REF]. Many repair mechanisms have been conserved across different taxa, including Drosophila, zebrafish, chick, and mammals [START_REF] Eming | Evolution of immune pathways in regeneration and repair: Recent concepts and translational perspectives[END_REF][START_REF] Eming | Wound repair and regeneration: mechanisms, signaling, and translation[END_REF]. The capacity to regenerate many complex organs such as limbs, however, is found only in a subset of living things [START_REF] Eming | Wound repair and regeneration: mechanisms, signaling, and translation[END_REF][START_REF] Brockes | Comparative aspects of animal regeneration[END_REF]. One important aim of this paper is to better clarify the similarities and differences between repair and regeneration, thanks to the concept of robustness.

We explain here how robustness can help better characterize the process of tissue reconstruction, through a description of the specific mechanisms, including plasticity and redundancy, by which robustness is achieved in the TRS. We also demonstrate that different repair-associated disorders (such as fibrosis, ulcers, and cancers) can be understood as the result of deregulated robustness. In turn, we show that the TRS offers a remarkable test case to defining the notion of robustness in a more precise and operational way, and more specifically to distinguishing different forms of robustness (structural vs. functional; preventive vs. corrective; partial vs. complete; dysfunctional vs. as a dysfunction).

What is robustness?

With the increasing attention paid recently to systems biology and complex systems, many living processes or systems have been described as "robust" [START_REF] Csete | Reverse engineering of biological complexity[END_REF][START_REF] Kitano | Biological robustness[END_REF][START_REF] Kitano | Towards a theory of biological robustness[END_REF]. The exact meaning of the word "robustness" often remains, however, elusive. The term originated in physics [START_REF] Lesne | Robustness: confronting lessons from physics and biology[END_REF],

and engineering [START_REF] Alon | Biological networks: the tinkerer as an engineer[END_REF] (though the engineering-related meaning is itself rooted in the physiology of the 19 th and 20 th century, including the work of Claude Bernard [START_REF] Kourilsky | The natural defense system and the normative self model[END_REF]). (On the relationship between biology and engineering, see [START_REF] Calcott | Engineering and Biology: Counsel for a Continued Relationship[END_REF]). In general, robustness is defined as the maintenance of specific functionalities of the system against internal and external perturbations. Two major requirements for any claim about biological robustness are to determine what exactly the robust system is, and against which type(s) of perturbations it is said to be robust. Importantly, robustness does not amount to conservation or absence of change. Robustness allows changes in the structure and components of the system owing to perturbations, but the key idea is that robustness leads to the maintenance of specific functions. It is likely that robustness is an evolved trait [START_REF] Wagner | Robustness and evolvability in living systems[END_REF][START_REF] Wagner | Robustness, evolvability, and neutrality[END_REF][START_REF] Wagner | Robustness and evolvability: a paradox resolved[END_REF]. Moreover, there are often trade-offs between robustness and other traits. In particular, systems that are evolved to be robust against certain perturbations can be extremely fragile to unexpected perturbations (see, e.g., [START_REF] Kitano | Biological robustness[END_REF][START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]).

Despite the fact that, historically, the concept of robustness took root to some extent in the concept of homeostasis, the two notions are different. Homeostasis is about maintaining constant (or almost constant, within a certain range) a value (e.g., body temperature in homeothermic animals) [START_REF] Cannon | Organization for Physiological Homeostasis[END_REF][START_REF] Kotas | Homeostasis, Inflammation, and Disease Susceptibility[END_REF]. Robustness, in contrast, is about maintaining a given function F against given types of perturbations (P1, P2, etc.).

Examples of robust processes or systems in biology abound [START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]. These include chemotaxis in bacteria [START_REF] Barkai | Robustness in simple biochemical networks[END_REF][START_REF] Alon | Robustness in bacterial chemotaxis[END_REF], cell cycle in budding yeast [START_REF] Li | The yeast cell-cycle network is robustly designed[END_REF], reliable development despite noise and environmental variations [START_REF] Félix | Robustness and evolution: concepts, insights and challenges from a developmental model system[END_REF], ecosystem reconstruction after a catastrophic event [START_REF] Wilmers | Understanding ecosystem robustness[END_REF], among many others.

As shown by Kitano [START_REF] Kitano | Biological robustness[END_REF], the four main mechanisms that ensure robustness are: system control, alternative mechanisms, modularity, and decoupling. System control consists in negative and positive feedbacks that enable the system to reach robustness against some perturbations. An example is bacterial chemotaxis, in which negative feedback plays a major role [START_REF] Yi | Robust perfect adaptation in bacterial chemotaxis through integral feedback control[END_REF]. Robustness can also be realized by alternative (or "fail-safe") mechanisms, that is, multiple routes to achieve a given function, which is to say that the failure of one of these routes can be compensated by another. This includes redundancy (where identical or nearly identical components can realize a given function) and diversity (where heterogeneous components can realize a given function). There are now many examples of these phenomena in the immune system (e.g., [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF]). Modularity is another important dimension of robustness: robustness is often achieved by modules, that is, flexible sets of components that collectively realize a given function, rather than by individual components [START_REF] Hartwell | From molecular to modular cell biology[END_REF]. Finally, decoupling is the prevention of undesired connection between low-level variations and highlevel functionalities. An example is the buffer mechanisms that decouple genetic variations from phenotypic expression, e.g., HSP chaperones [START_REF] Rutherford | Between genotype and phenotype: protein chaperones and evolvability[END_REF].

Here we focus on how the concept of robustness can be applied to the immune system and the TRS across the living world. Robustness has not been widely mentioned in immunology, though some exceptions exist (e.g., [START_REF] Kourilsky | The natural defense system and the normative self model[END_REF][START_REF] Feinerman | Variability and Robustness in T Cell Activation from Regulated Heterogeneity in Protein Levels[END_REF][START_REF] Jonjic | Functional plasticity and robustness are essential characteristics of biological systems: lessons learned from KLRG1-deficient mice[END_REF][START_REF] Mantovani | The chemokine system: redundancy for robust outputs[END_REF]). In particular, Mantovani [START_REF] Mantovani | The chemokine system: redundancy for robust outputs[END_REF] proposed that robustness provides a conceptual framework to understand intriguing aspects of the chemokine system, most prominently its redundancy (see also Mantovani, this special issue). Germain, Altan-Bonnet, and colleagues have explored theoretically and experimentally the mechanisms through which T cells can be both robust and adaptable to variations in protein expression [START_REF] Feinerman | Variability and Robustness in T Cell Activation from Regulated Heterogeneity in Protein Levels[END_REF]. Kourilsky has proposed to understand the immune system as conferring robustness to the whole organism via its capacity to systematically detect and respond to internal as well as external perturbations [START_REF] Kourilsky | The natural defense system and the normative self model[END_REF]. The question raised here is different and complementary, in so far as robustness is examined at the tissue level, and we ask which exact roles the immune system plays in this tissue-level robustness.

In what follows, we detail how the TRS works, mainly via five processes, namely plasticity, functional redundancy, constant surveillance, restraint, and dynamic adjustment.

We then show how pathologies associated with dysfunctions in tissue repair (e.g, fibrosis, ulcers, and cancer) can be understood as resulting from a deregulation of one or several of these five processes. We propose that the TRS offers a remarkable test case to define the notion robustness in a more precise and operational way, and more specifically to distinguishing different forms of robustness (structural vs. functional; preventive vs. corrective; partial vs. complete; dysfunctional vs. as a dysfunction). Importantly, we will consider both "repair" (defined as the partial reconstruction of an organ or tissue) and "regeneration" (defined as the complete reconstruction of a complex organ or tissue) examples, and explain how the concept of robustness helps clarify the differences between repair and regeneration.

The mechanisms that mediate tissue reconstruction

Tissue reconstruction is a complex and dynamic process, comprising overlapping, highly orchestrated stages -namely inflammation, tissue formation, and tissue remodeling [START_REF] Gurtner | Wound repair and regeneration[END_REF].

Tissue reconstruction involves many molecular and cellular components, which tightly interact. Understanding the interactions between these components and how they are regulated both spatially and temporally is a major aim for anyone interested in tissue repair, regeneration, and repair-associated pathologies. We show here that the TRS exhibits five key features that participate in robustness, and which are shared by many actors involved in the TRS: the TRS is plastic, redundant, under constant surveillance, restrained, and continuously dynamic.

Plasticity in the TRS

First, a major feature of the TRS is the plasticity of the cells involved in tissue reconstruction.

The word "plasticity" is used with different and sometimes confusing meanings in the scientific literature. Here we understand cell plasticity in two different and important senses [START_REF] Laurent | Immune-Mediated Repair: A Matter of Plasticity[END_REF]. The first sense is intra-lineage cell plasticity, that is, changes in cell function and phenotype within a given cell lineage -for example, M1 macrophages turning into M2 macrophages. This is sometimes called "functional plasticity" [START_REF] Galli | Phenotypic and functional plasticity of cells of innate immunity: macrophages, mast cells and neutrophils[END_REF]. The second sense is translineage cell plasticity, that is, the switch from one lineage to another -e.g., from macrophages to fibroblasts [START_REF] Chang-Panesso | Cellular plasticity in kidney injury and repair[END_REF]. This can also be called plasticity by "transdifferentiation" [START_REF] Das | Monocyte and macrophage plasticity in tissue repair and regeneration[END_REF] or by "reprogramming" -a phenomenon now known to occur in some non-immune cells [START_REF] Plikus | Regeneration of fat cells from myofibroblasts during wound healing[END_REF]. Actors of plasticity in tissue reconstruction are diverse, from immune to nonimmune cells. In what follows, we describe the main cellular actors in the repair process, with a particular emphasis on how they illustrate the phenomenon of plasticity. We show that this plasticity is central to the functioning of the TRS.

Far from being "one-shot" weapons, long-living neutrophils -which are central players in tissue reconstruction -are remarkably plastic. Indeed, neutrophils can differentially switch phenotypes, and display distinct subpopulations under different microenvironments [START_REF] Yang | The Diverse Biological Functions of Neutrophils, Beyond the Defense Against Infections[END_REF]. At the inflammatory stage of the repair process, neutrophils can play either a pro-resolving or an anti-resolving role. In addition to this intra-lineage plasticity, repair-associated neutrophils are capable of trans-lineage plasticity (plasticity by transdifferentiation) [START_REF] Balta | Qualitative and quantitative analysis of PMN/T-cell interactions by InFlow and super-resolution microscopy[END_REF][START_REF] Takashima | Neutrophil plasticity: acquisition of phenotype and functionality of antigen-presenting cell[END_REF][START_REF] Matsushima | Neutrophil differentiation into a unique hybrid population exhibiting dual phenotype and functionality of neutrophils and dendritic cells[END_REF][START_REF] Hampton | The lymph node neutrophil[END_REF].

Type 1 macrophages (M1) drive the early inflammatory responses that lead to tissue destruction, whereas type 2 macrophages ("M2" or "alternatively activated reparative macrophages") exert a central role in wound healing [START_REF] Biswas | Macrophage plasticity and interaction with lymphocyte subsets: cancer as a paradigm[END_REF][START_REF] Mantovani | Macrophage plasticity and polarization in tissue repair and remodelling[END_REF][START_REF] Nahrendorf | The healing myocardium sequentially mobilizes two monocyte subsets with divergent and complementary functions[END_REF][START_REF] Wynn | Macrophages in Tissue Repair, Regeneration, and Fibrosis[END_REF][START_REF] Wynn | Quantitative assessment of macrophage functions in repair and fibrosis[END_REF][START_REF] Mills | Anatomy of a Discovery: M1 and M2 Macrophages[END_REF]. Generation of a pro-type 2 microenvironment gradually leads to the switch from inflammatory to pro-repair macrophages. These cells promote tissue repair by producing pro-reparative cytokines and participate in a pro-type 2 microenvironment. A wide range of macrophage subtypes exists [START_REF] Das | Monocyte and macrophage plasticity in tissue repair and regeneration[END_REF][START_REF] Mantovani | Macrophage plasticity and polarization in tissue repair and remodelling[END_REF][START_REF] Jenkins | Local macrophage proliferation, rather than recruitment from the blood, is a signature of TH2 inflammation[END_REF][START_REF] Chávez-Galán | Much More than M1 and M2 Macrophages, There are also CD169+ and TCR+ Macrophages[END_REF]. Efficient tissue repair requires inflammatory macrophages, tissue repair macrophages, and resolving macrophages (producers of resolvins, IL-10 and TGF-b) [START_REF] Das | Monocyte and macrophage plasticity in tissue repair and regeneration[END_REF][START_REF] Wynn | Macrophages in Tissue Repair, Regeneration, and Fibrosis[END_REF][START_REF] Wang | Molecular mechanisms that influence the macrophage m1m2 polarization balance[END_REF]. Beyond intra-lineage plasticity, macrophages might participate actively in the tissue-remodeling phase of repair process by transdifferentiation into other cell types, notably endothelial cells [START_REF] Yan | Vascular endothelial growth factor modified macrophages transdifferentiate into endothelial-like cells and decrease foam cell formation., Vascular endothelial growth factor modified macrophages transdifferentiate into endothelial-like cells and decrease foam cell formation[END_REF].

Innate Lymphoid Cells (ILCs) are a recently discovered family of immune cells that includes three subsets: ILC1, ILC2 and ILC3 [START_REF] Eberl | The brave new world of innate lymphoid cells[END_REF][START_REF] Spits | Innate lymphoid cells--a proposal for uniform nomenclature[END_REF][START_REF] Vivier | The evolution of innate lymphoid cells[END_REF]. ILC2-secreted amphiregulin, a protein shown to orchestrate tissue repair [START_REF] Zaiss | Emerging functions of amphiregulin in orchestrating immunity, inflammation, and tissue repair[END_REF], promotes wound healing by acting directly on fibroblasts, leading to ECM deposit. ILC responses to different stimuli allow intra-lineage plasticity between the different subsets [START_REF] Ohne | IL-1 is a critical regulator of group 2 innate lymphoid cell function and plasticity[END_REF][START_REF] Silver | Inflammatory triggers associated with exacerbations of COPD orchestrate plasticity of group 2 innate lymphoid cells in the lungs[END_REF]. This plasticity between different ILC subtypes might allow for rapid innate immune responsiveness in repair [START_REF] Almeida | Innate lymphoid cells: models of plasticity for immune homeostasis and rapid responsiveness in protection[END_REF][START_REF] Zhang | Cutting Edge: Notch Signaling Promotes the Plasticity of Group-2 Innate Lymphoid Cells[END_REF].

Overall, cell plasticity is a pivotal process by which tissue reconstruction is achieved. This is confirmed by the fact that, as detailed below, inappropriate realizations of cellular plasticity (excess or insufficiency) may lead to various disorders.

Functional redundancy in the TRS

Functional redundancy is another important feature of the TRS. Functional redundancy describes a situation in which different elements have similar functions or similar effects on a trait [START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]. Though some forms of functional redundancy occur in every organism as part of normal functioning, this phenomenon has often been observed in pathological contexts, where it appears that an organism deficient in one cell type can "compensate" this deficiency thanks to other cell types or other molecules or pathways [START_REF] Edelman | Degeneracy and complexity in biological systems[END_REF]. The TRS often displays "degeneracy", which refers to the existence of structurally diverse but functionally similar components [START_REF] Edelman | Degeneracy and complexity in biological systems[END_REF]. Overall, the TRS is characterized by a high level of redundancy, even though some components and pathways seem to be pivotal in the reconstruction process.

ILCs have potent immunological functions in experimental conditions, but their contributions to immunity in natural conditions are unclear. It has been shown that SCID patients with IL2RG and JAK3 mutations and ILC-deficient had no particular susceptibility to disease [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF]. Thus, ILCs appear to be dispensable in humans who have a functional adaptive immune system, at least in the context of modern medicine and hygiene conditions [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF][START_REF] Rankin | Complementarity and redundancy of IL-22-producing innate lymphoid cells[END_REF].

Functional redundancy allows the evocation of an overall type 1 or 2 immune response rather than talking more restrictively about type 1 or 2 neutrophils/macrophages/T cells. Those cells often produce the same types of molecules (albeit sometimes with different temporal patterns). This redundancy is not only important to maintain robustness against perturbations; it also creates feedback loops (and thereby a virtuous or vicious circle, depending on the situation), participating in the establishment of a local microenvironment that displays particular features.

Besides immunological redundancy, immune cells participate in the secretion of structural molecules such as matrix metalloproteinases (MMP) altogether with fibroblasts, pericytes, and endothelial cells. The relative importance of macrophage and other immune cell contribution to tissue reconstruction compared to the aforementioned structural cells might depend on the nature of the tissue and the injury.

Constant surveillance

Tissue reconstruction is an active process where some actors are on constant standby. It is of major importance at the level of DNA repair, as DNA lesions occurring during reprogramming are monitored by a surveillance mechanism called the zygotic checkpoint [START_REF] Ladstätter | A Surveillance Mechanism Ensures Repair of DNA Lesions during Zygotic Reprogramming[END_REF]. At the tissue level, some cells, including various types of immune cells [START_REF] Fan | Hallmarks of Tissue-Resident Lymphocytes[END_REF], are highly specialized in the surveillance of damages. Of crucial importance are tissue-resident sentinel cells, as they are present and on standby before any damages.

ILCs are found preferentially on epithelial barrier surfaces such as the skin, lungs, and gut, where they protect against infection and maintain the integrity of the barriers. ILCs are tissue-resident sentinels enriched at mucosal surfaces. They exert a constant surveillance on epithelia, and have a complex crosstalk with their microenvironment. They are highly involved in tissue repair through their sentinel position and the cytokines they produce [START_REF] Klose | Innate lymphoid cells as regulators of immunity, inflammation and tissue homeostasis[END_REF][START_REF] Rak | IL-33-Dependent Group 2 Innate Lymphoid Cells Promote Cutaneous Wound Healing[END_REF].

Different tissues often have their preferential sentinels, such as NK cells in the liver, or Langerhans cells in the skin. Cells of the innate but also adaptive immune system are involved in this surveillance. In particular, tissue resident memory T cells (TRM) -which reside in tissues without recirculating through the blood or lymph, and constitute a transcriptionally and phenotypically unique T cell lineage -have been shown to be key guardians against viral infections [START_REF] Mueller | Tissue-resident memory T cells: local specialists in immune defence[END_REF].

Cells traditionally seen as non-immune such as epithelial cells (ECs) play an important role in this collaborative surveillance process. They line body surface tissues and provide a physicochemical barrier to the external environment. This barrier is not a mere passive mechanical protection. Frequent microbial and non-microbial challenges cause activation of ECs, with release of cytokines and chemokines as well as alterations in the expression of cell-surface ligands. Epithelial stress is rapidly sensed by tissue-resident immune cells, which can directly interact with self-moieties on ECs and initiate both local and systemic immune responses. ECs are thus key drivers of immune surveillance at body surface tissues [START_REF] Dalessandri | Beneficial Autoimmunity at Body Surfaces -Immune Surveillance and Rapid Type 2 Immunity Regulate Tissue Homeostasis and Cancer[END_REF].

Restraint of the TRS

Detecting and responding to damages is so central for the organism's survival that the TRS is always on alert, ready to be triggered. But at the same time this system also constitutes a potential threat for the organism (inflammation, tissue formation, and tissue remodeling can all go awry, with potentially dramatic consequences), and must therefore be constantly kept under control. Numerous cells restrain the TRS through negative feedback, active production of pro-resolving molecules, and other dynamic mechanisms. These cells are important at all stages but they are particularly crucial for the pro-resolving phase after inflammation.

Pro-resolving neutrophils demonstrate the ability to: (i) produce several pro-resolving mediators (as lipoxins), (ii) form NETs and aggregated NETs, according to a cell-density dependent sensing mechanism, which dismantles the pro-inflammatory gradient by degrading the inflammatory cytokines and chemokines, (iii) store and release the proresolving protein annexin A1 [START_REF] Jones | The role of neutrophils in inflammation resolution[END_REF].

Inflammation resolution is partly mediated by the clearance of apoptotic neutrophils by macrophages through efferocytosis [START_REF] Stark | Phagocytosis of apoptotic neutrophils regulates granulopoiesis via IL-23 and IL-17[END_REF]. Non-apoptotic neutrophils can leave the injury site by reverse transmigration. Recently described resolving macrophages (producers of resolvins, IL-10 and TGF-b) are important actors of repair regulation.

In mice, some regulatory T cells (Tregs) are able to produce amphiregulin, favoring the resolving phase of the inflammation process [START_REF] Burzyn | A special population of regulatory T cells potentiates muscle repair[END_REF]. Depletion of muscle Tregs has profound impact on muscle regeneration with loss of regenerative fibers, collagen deposition and fibrosis, leading to a disorganized tissue structure. In the absence of Tregs, effector T cell infiltrate increases in the injured muscle and the switch from inflammatory to anti-inflammatory macrophage diminishes.

Dynamic adjustment of the TRS

TRS is a highly dynamic process implying a large recruitment of various cells, with movements in a tri-dimensional matrix, and with many back and forth between different steps that are not fixed and can often overlap. The dynamic character of the TRS is visible at the level of the recruited cells, but also of the resident cells.

Standby periods are not to be considered totally at rest. Resident cells are never completely motionless. Moreover, cells are constantly replaced in a dynamic process.

Tissues are continuously exposed to potentially hazardous environmental challenges in the form of inert material and microbes. In the epidermis, for example, Langerhans cells (LC) form a dense network of cells capable of capturing antigens and migrating to the lymph node after crossing the basement membrane into the dermis, and they are able to promote tolerance or immune responses [START_REF] Seneschal | Human epidermal Langerhans cells maintain immune homeostasis in skin by activating skin resident regulatory T cells[END_REF]. Velocity of migration is partly regulated by the microenvironment, and skin Tregs display a much slower migration compared to effector CD4+ T cells, although acute inflammation results in a rapid increase in their motility [START_REF] Mueller | Tissue-resident T cells: dynamic players in skin immunity[END_REF].

Gradients of chemokines largely participate in cell recruitment when damages occur.

CD14+ monocytes and neutrophils are very mobile cells, highly and promptly recruited in case of injury [START_REF] Achachi | UV Radiation Induces the Epidermal Recruitment of Dendritic Cells that Compensate for the Depletion of Langerhans Cells in Human Skin[END_REF]. The recruitment of neutrophils during the inflammatory phase is linked to a sharply regulated communication system based on the CXC chemokine/CXC receptors balance [START_REF] Nair | Study Investigators, Safety and efficacy of a CXCR2 antagonist in patients with severe asthma and sputum neutrophils: a randomized, placebo-controlled clinical trial[END_REF]. The injury triggers the production of G-CSF that converts the CXCR4 dominant signaling to that of CXCR2 in the bone marrow microenvironment, leading to the release of more mature neutrophils into the peripheral blood stream [START_REF] Silvestre-Roig | Neutrophil heterogeneity: implications for homeostasis and pathogenesis[END_REF]. Functional aberrancy in these systems leads to impaired wound healing [START_REF] Su | Chemokine Regulation of Neutrophil Infiltration of Skin Wounds[END_REF]. In a collaborative pathway, the release of chemoattractant factors by neutrophils, such as lactoferrin, attracts monocytes and activates macrophages [START_REF] Frangogiannis | Regulation of the inflammatory response in cardiac repair[END_REF]. A counterpart to recruitment is obviously needed to ensure robustness of a tissue, or else an overabundance of cells could lead to tissue destruction.

Efferocytocis, transmigration, and specific apoptosis allow recruited cells to be cleaned up after damages.

As the rest of the paper will show, two kinds of consequences follow from this analysis of the five key features of the TRS. First, it offers an important basis to re-think some tissue reconstruction-associated pathologies as dysfunctions of robustness. Second, it offers a test case to assess the usefulness of the notion of robustness in physiological and pathological conditions, and leads to distinguishing different forms of robustness.

Dysfunctions of the tissue reconstruction system

It has been suggested by Kitano and others that the concept of robustness can shed light on certain pathological processes [START_REF] Kitano | Biological robustness[END_REF]. Pathologies could result from robustness as a dysfunction (the process under consideration is robust, but this robustness is detrimental to the organism, as happens for example in AIDS or some cancers, where the robustness of a system is "hijacked" [START_REF] Kitano | Biological robustness[END_REF][START_REF] Kitano | Biological robustness in complex host-pathogen systems[END_REF][START_REF] Kitano | Cancer as a robust system: implications for anticancer therapy[END_REF]) or a dysfunctional robustness, which is to say a rupture of robustness (i.e., the process should be robust, but is not). This approach applies very well to the dysfunctions of the TRS.

Mechanisms that mediate tissue reconstruction to ensure robustness are constantly challenged. These mechanisms are sometimes overwhelmed, leading to various consequences depending on the situation, from the rupture of robustness to the promotion of the disease thanks to robustness-associated mechanisms, and to an excess of robustness.

The final consequence of each situation is a pathological process. Through concrete examples (ulcers, fibrosis, and cancers), we will illustrate these different threats to the TRS to ensure robustness, emphasizing in each case exactly which mechanisms are challenged (see Table 1, which present several additional examples).

Ulcers, or rupture of robustness

Pathological situations of insufficient repair such as ulcers underlie that the TRS mechanisms ensuring robustness are overwhelmed. Since the robustness of the tissue can be jeopardized, it is important to analyze the various components listed earlier. The value of a more detailed analysis of component robustness is dual. This makes it possible to precisely identify vulnerabilities, which vary depending on the clinical situation, but also to work out innovative therapeutic strategies. As we saw, cell plasticity is a crucial dimension of the TRS and it is especially true for neutrophils. This is confirmed by the fact that incapacity of neutrophils to switch plastically from one state to the other can contribute to ulcers, e.g., skin or gastric ulcers. Impossibility of tuning the response toward a pro-resolving phase by experimentally blocking neutrophils in a pro-inflammatory state directly leads to chronic inflammation and deregulation of the TRS [START_REF] Whitmore | Cutting Edge: Helicobacter pylori Induces Nuclear Hypersegmentation and Subtype Differentiation of Human Neutrophils In Vitro[END_REF], while the reintroduction of very plastic cells in damaged tissues can overcome this defect. Understanding that in this case the ulcer is a rupture of the robustness due to insufficient cellular plasticity allows to consider completely new therapeutic options. Several studies in animal models showed that adipose tissuederived stem cell sheet application to mucosal or skin wounds accelerates wound healing and decreases the degree of fibrosis [START_REF] Perrod | Cell Sheet Transplantation for Esophageal Stricture Prevention after Endoscopic Submucosal Dissection in a Porcine Model[END_REF][START_REF] Kato | Allogeneic Transplantation of an Adipose-Derived Stem Cell Sheet Combined With Artificial Skin Accelerates Wound Healing in a Rat Wound Model of Type 2 Diabetes and Obesity[END_REF].

While inflammation has to be regulated to ensure the completion of the TRS, a failure in that process can lead to a rupture of robustness. A deficiency of efferocytosis has been identified as a causative agent of sterile chronic granulomatous disease in mice [START_REF] Zeng | An efferocytosis-induced, IL-4-dependent macrophage-iNKT cell circuit suppresses sterile inflammation and is defective in murine CGD[END_REF]. In chronic ulcers, favoring the resolving phase (e.g., through efferocytosis, pro-Treg therapeutics, or resolving compounds) could be an innovative strategy [START_REF] Chan | Interleukin 2 Topical Cream for Treatment of Diabetic Foot Ulcer: Experiment Protocol[END_REF][START_REF] Reis | Lipoxin A4 encapsulated in PLGA microparticles accelerates wound healing of skin ulcers[END_REF][START_REF] Lohmann | Glycosaminoglycan-based hydrogels capture inflammatory chemokines and rescue defective wound healing in mice[END_REF]. Even though defects of inflammatory regulation are clearly involved in the deregulation of the TRS, therapeutic avenues to counteract these defects are still in their infancy, and mostly limited to animal models. A better understanding of the crucial place of that mechanism in the global robustness of the TRS will tend to raise the interest for therapeutics targeting regulation.

In ulcers, the loss of epithelial cells disturbs the ongoing surveillance of the TRS. In the eye, the inflammation of the cornea leads to damages of this protective barrier. Given that the cornea is an avascular tissue and contains few immune cells, corneal resident cells function as sentinel cells as well as immune modulators during corneal inflammation. They are able to sense bacterial infection through toll like receptor (TLR)-mediated detection. As a consequence, a loss of substance (i.e., a very significant injury) could lead to the disappearance of key first-line sentinel cells, normally responsible for the recruitment of other crucial cells in the repair process [START_REF] Allen | The Silent Undertakers: Macrophages Programmed for Efferocytosis[END_REF][START_REF] Fukuda | Corneal Fibroblasts as Sentinel Cells and Local Immune Modulators in Infectious Keratitis., Corneal Fibroblasts as Sentinel Cells and Local Immune Modulators in Infectious Keratitis[END_REF]. Other resident cells are involved in this mechanism (see Table 1). Targeting sentinels could constitute a new therapeutic avenue in the treatment of chronic ulcers.

Finally, due to the reduction of the dynamic flow to the damage site, new cells cannot come from the upstream and revitalize the system in an ulcer. Promoting the migration and proliferation of cells could accelerate wound healing [START_REF] Liu | CXCR4 antagonist delivery on decellularized skin scaffold facilitates impaired wound healing in diabetic mice by increasing expression of SDF-1 and enhancing migration of CXCR4-positive cells, Wound Repair Regen[END_REF][START_REF] Kai | Accelerated Wound Healing on Skin by Electrical Stimulation with a Bioelectric Plaster[END_REF].

Each of the five components of robustness can be compromised depending on the type of ulcer. For clinicians, thinking according to our classification and identifying which mechanism is deficient can therefore change very concretely their therapeutic management.

Fibrosis or excess of robustness

Keloid and hypertrophic scars can also be seen as the result of a dysfunction in the fundamental mechanisms of the TRS. One could consider fibrosis as a kind of hypertophic scar, and as such fibrosis could follow from a deregulated TRS as well. In a normal repair cycle, the resolution of damage-induced inflammation allows the system to rebuild itself efficiently. In contrast, the absence of resolution means the persistence of inflammation and also, especially in fibrosis, a disconnection between the levels of resolution and remodeling.

An excess of plasticity can also be pathological. For example, epithelial-mesenchymal transition (EMT) reflects a high level of cell plasticity essential during embryogenesis and wound healing, but EMT can be aberrantly regulated in fibrosis [START_REF] Skrypek | Epithelial-to-Mesenchymal Transition: Epigenetic Reprogramming Driving Cellular Plasticity[END_REF][START_REF] Gasparics | Alterations in SCAI Expression during Cell Plasticity, Fibrosis and Cancer[END_REF]. Cell plasticity could also be a hurdle for achieving some cell therapy. A pro-fibrotic microenvironment results in systematic M2 polarization even if macrophages of another type are injected. In contrast, the infusion of stabilized pro-resolving macrophages is associated with reduced kidney interstitial fibrosis and inflammation, as well as preservation of the phenotype and functions of macrophages [START_REF] Guiteras | Macrophage Overexpressing NGAL Ameliorated Kidney Fibrosis in the UUO Mice Model[END_REF]. Thus, a precise knowledge of the proper physiopathology of the studied condition is crucial to understanding whether a higher or a lower plasticity is needed.

The cause of fibrosis is sometimes attributed to the persistence of damage triggers such as chronic infection. Nevertheless, in hepatitis C, it is the inadequacy of the TRS response rather than the persistence of the infection that is at stake [START_REF] Rios | Chronic hepatitis C liver microenvironment: role of the Th17/Treg interplay related to fibrogenesis[END_REF]. Tregs or proresolving cells have also been suspected to be involved in more general fibrotic processes, such as systemic sclerosis (SSc) [START_REF] Ugor | Increased proportions of functionally impaired regulatory T cell subsets in systemic sclerosis[END_REF][START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF]. From this point of view, promoting the resolution of inflammation could be considered as a key aim to reverse fibrosis [START_REF] Brennan | Specialized pro-resolving mediators in renal fibrosis[END_REF][START_REF] Grabiec | The role of airway macrophages in apoptotic cell clearance following acute and chronic lung inflammation[END_REF].

A constant monitoring is an essential element of the TRS responsiveness. The fact that it is provided by resident cells guarantees this prompt response when damages occur. However, in some cases, including fibrosis, this surveillance can be over-stimulated and associated with an overly sustained response. As described before, innate immune signaling via TLRs is a key driver of persistent fibrotic response. Chronic signaling on resident mesenchymal cells underlies the switch from a self-limited repair response to non-resolving pathological fibrosis characteristic of systemic sclerosis. Limiting the responsiveness of resident cells to innate stimulation could be of interest to prevent fibrotic processes [START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF].

Resident cells themselves can also be responsible for the excessive stimulation without any clear external trigger [START_REF] Cheng | Guards at the gate: physiological and pathological roles of tissue-resident innate lymphoid cells in the lung[END_REF][START_REF] Hams | IL-25 and type 2 innate lymphoid cells induce pulmonary fibrosis[END_REF][START_REF] Li | Skin-Resident Effector Memory CD8+CD28-T Cells Exhibit a Profibrotic Phenotype in Patients with Systemic Sclerosis[END_REF].

A static TRS cannot result in normal repair. Indeed, different but more or less intricate phases must follow one another. Nonetheless, an excess of migration of pro-fibrotic cells into the tissue can be detrimental in a normal repair process. This happens, for example, in the lungs with fibrocytes. These cells enter the lungs in response to their chemoattractant CXCL12, and differentiate into fibroblasts or myofibroblasts, leading to excessive deposition of collagen-rich extracellular matrix. It has been shown that inhibiting the flow of fibrocytes to the lungs by a peptide called R1R2 attenuates pulmonary fibrosis by reducing the invasion of fibrocytes through basement membrane-like proteins [START_REF] Chiang | R1R2 peptide ameliorates pulmonary fibrosis in mice through fibrocyte migration and differentiation[END_REF].

Cancer or hijacking of robustness

Cancerous tumors have been related to deregulated repair by Dvorak, who describes them as "wounds that do not heal" [START_REF] Dvorak | Tumors: wounds that do not heal. Similarities between tumor stroma generation and wound healing[END_REF]. It is now well established that an inflammatory microenvironment promotes cancer [START_REF] Mantovani | Cancer-related inflammation[END_REF]. It has also been suggested that the formation and maintenance of a cancerous tumor could be seen as a robust process [START_REF] Kitano | The theory of biological robustness and its implication in cancer[END_REF]. Here we consider cancer as evolving from a damaged tissue, where the TRS could act to prevent the expansion of the injury and promote repair. The cancerous tumor, to drive its own development, hijacks some properties of the TRS that normally ensure robustness in physiological conditions.

Epithelial-mesenchymal transition (EMT) is an important process in embryonic development, fibrosis, but also in cancer metastasis. The activation of EMT in cancer allows cells to acquire migratory, invasive, and stem-like properties. SCAI is characterized as a tumor suppressor inhibiting metastasis in different human cancer cells, and which is thought to be reduced in some tumors. SCAI expression decreases in a model of endothelialmesenchymal transition, which suggests that it could be important for cell plasticity.

Nevertheless, its role in cancer remains to be further investigated, as its expression could be associated with high or low progression of the tumor depending on the type of cancer [START_REF] Gasparics | Alterations in SCAI Expression during Cell Plasticity, Fibrosis and Cancer[END_REF].

Macrophage polarization could influence immune checkpoint therapy resistance. The plasticity of macrophages is used by cancerous tumors, and targeting this plasticity could be of interest to increase the response to immunotherapy such as ipilimumab [START_REF] Soto-Pantoja | Unfolded protein response signaling impacts macrophage polarity to modulate breast cancer cell clearance and melanoma immune checkpoint therapy responsiveness[END_REF].

Redundancy may explain cancer resistance to certain treatments. Recently developed immunotherapies do not target the tumor cells as such; instead, they promote the local immune responses in the tumor microenvironment, which has some important consequences on key immunological actors of the TRS. Redundancy of the TRS becomes central in these conditions [START_REF] Lavi | Redundancy: a critical obstacle to improving cancer therapy[END_REF]. For example, IL-6 belongs to a family of cytokines with highly redundant functions, which use the glycoprotein 130 chain for signal transduction. It has an important role in the pathophysiology of multiple myeloma, where it supports the growth and survival of the malignant plasma cells in the bone marrow. Because of this redundancy, targeting IL-6 is highly difficult. Antibodies against glycoprotein 130 constitute a better option, as they can overcome this redundancy [START_REF] Burger | Due to interleukin-6 type cytokine redundancy only glycoprotein 130 receptor blockade efficiently inhibits myeloma growth[END_REF].

Insofar as cancer may be seen as both a cause and consequence of tissue damages, cancer cells could activate the TRS. In particular, cancerous tumors can use the above described restrain mechanisms of the TRS to induce a type of immune tolerance that will be at their own advantage. Tumor associated macrophages (TAMs) are one of the main actors of this phenomenon. Different therapeutic strategies have been proposed to address this problem, such as the suppression of TAM recruitment, their depletion, the switch of M2 TAMs into antitumor M1 macrophages, and the inhibition of TAM-associated molecules [START_REF] Santoni | Triple negative breast cancer: Key role of Tumor-Associated Macrophages in regulating the activity of anti-PD-1/PD-L1 agents[END_REF].

Immune surveillance could be considered as insufficient in cancer. The TRS lets cancer cells grow and develop as if it were incapable of "seeing" them. Resident memory CD8+T cells (TRMs) represent a recently described subset of long-lived memory T cells that remain in the tissues, do not recirculate, and are therefore very important actors in immunosurveillance. It has been shown that TRMs were present in human non-small cell lung tumor tissues, and their frequency was correlated with better overall survival than other infiltrating immune cells. In that case, the cancer misleads the immunosurveillance system, which suggests that strategies increasing the number of TRMs or activating them such as vaccines could be developed following this concept [START_REF] Granier | Tissue-resident memory T cells play a key role in the efficacy of cancer vaccines[END_REF].

Prior to metastatic cell arrival, a premetastatic niche in distant organs could be an important step in the metastatic cascade. This phenomenon suggesting highly dynamic process from cancer cells could be preceded by neutrophil migration and recruitment. As such the dynamic property of the TRS is used to prepare the basis for tumor cell engraftment in parenchyma [START_REF] Donati | Neutrophil-Derived Interleukin 16 in Premetastatic Lungs Promotes Breast Tumor Cell Seeding[END_REF].

Overall, the concept of robustness helps better understand TRS-associated pathologies, either as a deficiency in the fundamental processes by which robustness is normally realized (plasticity, etc.), or as an emerging, local form of robustness that is detrimental to the organism.

Conclusion: the virtues of thinking about tissue reconstruction in terms of robustness

In light of the various physiological and pathological examples examined in this paper, we propose that it is extremely fruitful to conceive of the tissue reconstruction system in terms of robustness, for three main reasons.

First, the recognition by Kitano and others [START_REF] Kitano | Biological robustness[END_REF] of different robustness-promoting mechanisms (system control, alternative mechanisms, modularity, decoupling) constitutes a useful conceptual framework to better describe the TRS and its dysfunctions in pathological situations. For example, plasticity and redundancy of immune components within the TRS have been described by scientists who were supportive of the concept of robustness [START_REF] Mantovani | The chemokine system: redundancy for robust outputs[END_REF][START_REF] Mantovani | Macrophage plasticity and polarization in tissue repair and remodelling[END_REF], and it seems likely that continuing to apply this concept will reveal even more plasticity and redundancy. Moreover, thinking in terms of robustness helps understanding that even a situation that could seem static, such as skin renewal for example, is in fact the outcome of a highly dynamic, continuously ongoing, process, and that it is pivotal to study in detail the mechanisms ensuring this process. It also suggests that the "default state" of the TRS is to be on alert, which means that tissue reconstruction is always active, though under constant restraint. As soon as the brake is lifted, the whole process of tissue repair (i.e., inflammation, tissue formation, and tissue remodeling) is triggered, which guarantees a higher capacity to react to various and often inevitable damages. Of course, this constant activation is energetically costly, but one should keep in mind that it is a low-level activation, and that it is probably essential for survival.

From a pathological point of view, the emphasis on the redundancy of the TRS, for example, is of the utmost importance. It shows that it is often entirely inadequate to hope for important benefits by intervening on just one actor or pathway. Indeed, in many cases, although some cells or pathways seemed crucial in a pathological process in vitro, their inhibition in vivo does not lead systematically to the pathological phenotype, because of the redundancy of some components and pathways. In other words, some pathological conditions reveal the role of certain cells or pathways, which are not normally indispensable, but become central when other components are missing. For example, alarmins IL-25, IL-33, and TSLP have a high level of redundancy, which makes anti-fibrotic treatments very difficult to develop. Blocking a single pathway is most often ineffective, so it is more promising to consider modulation of the response at a very early stage, or to identify common pathways that could be targeted. Moreover, acting on one of the mechanisms of robustness while others play a more significant role is not effective in achieving repair. This is probably what happens when one treats fibrosis with immunosuppressive therapy in cases where restoring cellular plasticity would in fact be more adequate. As a general rule, then, one should never draw hasty conclusions about whether or not some actors have an important role in the TRS before having tested them in real-life pathological conditions.

Second, the example of the TRS can, in turn, help us make some crucial conceptual distinctions about robustness. On the basis of the examples explored here, one can indeed distinguish functional vs. structural robustness, partial vs. complete robustness, and corrective vs. preventive robustness (Figure 2). Functional robustness in the case of the TRS means that tissue function (or, at least, one tissue function) is restored, but not tissue structure. For example, after significant skin injury, a scar will form, which will restore the protective function of the skin, but the initial structure of the skin will not be restored. In contrast, regeneration often leads to the restoration of both the structure and the function of the tissue. For example, adult zebrafish fins, including their complex skeleton, regenerate exactly to their original form within two weeks after an amputation. Importantly, some forms of complete tissue regeneration can also be observed in mammalian embryos, but this capacity is subsequently lost for most tissues (the most significant exception in humans is the liver, which can indeed regenerate, though it does not always recover its initial structure). Along similar lines, it is important to emphasize that robustness to a given challenge and at a certain level can be more or less effective. Robustness is partial when tissue function and/or structure is not completely restored, as illustrated by most cases of tissue repair in mammals, for example. Robustness is said to be complete when tissue function and/or structure is entirely restored, as illustrated by cases such as fin regeneration in zebrafish already mentioned, limb regeneration in many amphibians [START_REF] Brockes | Amphibian Limb Regeneration: Rebuilding a Complex Structure[END_REF], or tissue regeneration in many echinoderms [START_REF] Carnevali | Regeneration in Echinoderms: repair, regrowth, cloning[END_REF]. Furthermore, robustness can be corrective or preventive. It is corrective when it consists in the active restoration of a strongly disturbed state. For example, tissue repair or regeneration after significant damages is a form of corrective robustness, because it follows a major perturbation (damages), and it involves, as we saw, a complex, dynamic, and regulated interplay of many different components. In contrast, robustness is preventive when it occurs in the absence of a major perturbation while minimizing the risk of a major perturbation and its detrimental consequence. For example, epithelial repair occurs continuously in the body, which requires an extremely rich orchestration of events [START_REF] Peterson | Intestinal epithelial cells: regulators of barrier function and immune homeostasis[END_REF]. This preventive robustness helps insure that the skin is always sufficiently "sealed off" and at the same time sufficiently smooth to achieve its functions.

When this process is interrupted, for example in ulcers, the organism is at a high risk of being damaged and invaded by pathogens or toxic substances. Of course, there will be a grey zone here, because it is not always clear whether a perturbation is major or not, and different tissues are likely to perceive perturbations differently. For example, the liver is constantly exposed to toxic chemicals that could endanger the rest of the body, and its regenerative capacities are certainly evolutionarily related to this particular exposition [START_REF] Michalopoulos | Liver Regeneration[END_REF]. An additional distinction seems important to better grasp the role of robustness in pathological contexts. Dysfunctions in robustness of the TRS can indeed be understood along two different lines. In some cases, the tissue fails to be robust, presumably because one or several important components or pathways of the TRS are not working properly. This is what we call a dysfunction of robustness. For example, we saw that the incapacity to realize cell plasticity can sometimes lead to the failure of tissue reconstruction. Yet, in other cases, the tissue is robust, but this robustness is, in this specific context, detrimental to the organism.

This is what we call robustness as a dysfunction. For example, a tumor can constitute a robust tissue, which is well vascularized, nourished, and constantly repaired, often via the co-optation of classical physiological mechanisms to the benefit of the tumor itself. Here again, this distinction between a dysfunction of robustness and dysfunctional robustness might prove useful in other contexts, beyond the case of the TRS.

A third and final consequence concerns the very understanding of immunity. Since the beginnings of immunology, immunity has been conceived primarily as a form of defense -most often against pathogens. Yet, if the perspective offered in this paper is correct, then immunity needs to be re-defined within a much wider context. Immune processes, we submit, concern not only defense, but also the construction (development) and reconstruction (constant repair; occasional repair after a significant damage; regeneration) of the organism,. Indeed, a typical immune system in nature is constantly busy surveying, renewing, and repairing the body,. This is not to say, obviously, that immune defense is not important, and has not been a major selective pressure in the evolution of immune systems.

Our suggestion is that immune systems have evolved under a multidimensional complex selective pressure, which includes a capacity to develop and repair as well as a capacity to defend against different sorts of threats. The way scientists traditionally delineate the immune system reflects an intellectual decision. This does not mean, of course, that the immune system is not "real", but rather that there exist many different ways to divide up living entities into different "systems". In the present paper, we have argued in favor of another intellectual decision by suggesting that it is more appropriate to focus on a functionally defined "system" of interest (namely the tissue reconstruction system) than on traditionally defined systems (such as the immune system). Repair and defense are probably just two sides of the same coin -a lesson that thinking immunity in terms of robustness might help us keep in mind. EMT and SCAI in renal fibrosis [START_REF] Gasparics | Alterations in SCAI Expression during Cell Plasticity, Fibrosis and Cancer[END_REF] -MET/EMT with the tumorinitiating ability required for metastatic colonization [START_REF] Yao | Mechanism of the mesenchymal-epithelial transition and its relationship with metastatic tumor formation[END_REF] -Plasticity between the epithelial and the mesenchymal states rather than a fixed phenotype [START_REF] Liao | Revisiting epithelial-mesenchymal transition in cancer metastasis: the connection between epithelial plasticity and stemness[END_REF] -UPR in macrophage polarization and plasticity with shift to M1-like profile [START_REF] Soto-Pantoja | Unfolded protein response signaling impacts macrophage polarity to modulate breast cancer cell clearance and melanoma immune checkpoint therapy responsiveness[END_REF] Functional redundancy ILC redundancy [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF] -IL-25, IL-33, and TSLP redundancy and fibrosis [START_REF] Vannella | Combinatorial targeting of TSLP, IL-25, and IL-33 in type 2 cytokine-driven inflammation and fibrosis[END_REF][START_REF] Gieseck | Type 2 immunity in tissue repair and fibrosis[END_REF] -Targeting porcupine in kidney fibrosis and Wnt O-acylation [START_REF] Madan | Experimental inhibition of porcupine-mediated Wnt O-acylation attenuates kidney fibrosis[END_REF] -IL-6 and glycoprotein 130 in the pathophysiology of multiple myeloma [START_REF] Burger | Due to interleukin-6 type cytokine redundancy only glycoprotein 130 receptor blockade efficiently inhibits myeloma growth[END_REF] Constant surveillance -Loss of substance (i.e., a very significant injury) and disappearance of sentinel cells in ulcers [103] -Langerhans cells and hypoxia [START_REF] Pierobon | Regulation of Langerhans cell functions in a hypoxic environment[END_REF] -Fibronectin-EDA and tenascin-C sensed by TLR4 on resident cells and fibrotic processes [START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF] -ILC2s in pulmonary fibrosis [START_REF] Cheng | Guards at the gate: physiological and pathological roles of tissue-resident innate lymphoid cells in the lung[END_REF][START_REF] Hams | IL-25 and type 2 innate lymphoid cells induce pulmonary fibrosis[END_REF]] -CD8+CD28-T cells and profibrotic cytokine IL-13 in the skin of systemic sclerosis (SSc) patients [START_REF] Li | Skin-Resident Effector Memory CD8+CD28-T Cells Exhibit a Profibrotic Phenotype in Patients with Systemic Sclerosis[END_REF] -TRMs in human non-small cell lung tumor tissue [START_REF] Granier | Tissue-resident memory T cells play a key role in the efficacy of cancer vaccines[END_REF] -Role of amphiregulin in orchestrating responses to tumors [START_REF] Zaiss | Emerging functions of amphiregulin in orchestrating immunity, inflammation, and tissue repair[END_REF] Restraint -Resolution deficiency and sterile chronic granulomatous disease [98-101] -Imbalance Treg/Th17 in pyoderma gangrenosum [START_REF] Caproni | The Treg/Th17 cell ratio is reduced in the skin lesions of patients with pyoderma gangrenosum[END_REF] -Chronic hepatitis C and hepatic fibrosis with the Th17/Treg balance Table 1. Main mechanisms involved in the robustness of the tissue reconstruction system (TRS), and its dysfunctions in major pathological situations (ulcer, fibrosis, and cancer). preventive robustness. The four cases presented here are merely illustrations, among others, showing how these distinctions can be applied to real-life cases. In severe skin injury, robustness is corrective, partial, and functional. In liver regeneration in mammals, robustness is corrective, almost complete, and functional. In the continuous renewal of epithelia, robustness is preventive, complete, and functional. Finally, in limb regeneration in salamander, robustness is corrective, complete, and functional.

Figure 2

 2 Figure 2 sumps up the three distinctions proposed here (structural vs. functional; partial vs.

  Figures and Tables

-

  [START_REF] Rios | Chronic hepatitis C liver microenvironment: role of the Th17/Treg interplay related to fibrogenesis[END_REF] -Role of TRegs in SSc[START_REF] Ugor | Increased proportions of functionally impaired regulatory T cell subsets in systemic sclerosis[END_REF] -SSc and TLRs with persistence of the response[START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF]] -Resolving inflammation against fibrosis and specialized pro-resolving lipid mediators [112] -Macrophages and efferocytosis [113] -TAMs recruitment in triple negative breast cancer [124] -Tregs in tumor progression [138] -Tregs and cancer cell clearance [139] -Tregs and cancer immunotherapies with IL-2 [140] -To target immune checkpoints such as CTLA4, PD1 or TIGIT to both interfere with Treg function and enhance effector responses at the same time [Cancer cells and use of the dynamic potential of neutrophils [126] -CCL26 in colorectal cancer cells invasion by inducing TAM infiltration [142] -Inhibitors of the receptor tyrosine kinase c-MET and impairment of the mobilization and recruitment of neutrophils into tumors [143]

Figure 1 .

 1 Figure1. Overview of the "tissue reconstruction system" (TRS). Many various components and pathways are involved in both tissue repair and tissue regeneration. Crucial components of the TRS include structural (e.g., fibroblasts, extracellular matrix, etc.) and immunological (e.g., neutrophils, macrophages, etc.) components. The concept of TRS is intended to embrace all the main entities and mechanisms responsible for tissue repair and tissue regeneration.

Figure 2 .

 2 Figure 2. The exploration of the tissue reconstruction system (TRS) leads to distinguishing different types of robustness, namely structural vs. functional robustness, partial vs. complete robustness, and corrective vs. preventive robustness. The four cases presented here are merely illustrations, among others, showing how these distinctions can be applied to real-life cases. In severe skin injury, robustness is corrective, partial, and functional. In liver regeneration in mammals, robustness is corrective, almost complete, and functional. In the continuous renewal of epithelia, robustness is preventive, complete, and functional. Finally, in limb regeneration in salamander, robustness is corrective, complete, and functional.

Acknowledgements

We would like to thank Cécile Contin-Bordes, Paôline Laurent, Alberto Mantovani, Jean-François Moreau, and Derek Skillings for discussions about tissue repair and robustness. Thomas Pradeu has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme -grant agreement n°637647 -IDEM.


Nicolas Gauvrit
email: ngauvrit@me.com

Jean-Charles Houillon

Jean-Paul Delahaye

Generalized Benford's Law as a Lie Detector

may come

IntroductIon

During the late 19th century, an intriguing phenomenon was discovered by [START_REF] Newcomb | note on the frequency of use of the different digits in natural numbers[END_REF]: The first significant digit (leftmost nonzero digit) of seemingly random numbers often fails to follow a flat distribution with an equal proportion of 1s, 2s, … , 9s, as one would expect, but instead follows a decreasing distribution, with more 1s than 2s, more 2s than 3s, and so forth. The same phenomenon was later rediscovered and detailed by [START_REF] Benford | the law of anomalous numbers[END_REF]. According to what is now referred to as Benford's law or Newcomb-Benford law (NBL), the distribution of the first significant digit X of a "random" number follows a logarithmic law given by P(X = d) = Log(1+1/d), where Log stands for the base 10 logarithm and d stands for a digit (in the range of 1-9; see Table 1).

Many real-world datasets approximately conform to NBL (Hill, 1998;[START_REF] Nigrini | Benford's Law: Applications for forensic accounting, auditing, and fraud detection[END_REF]. For instance, the distance between earth and known stars (Alexopoulos & Leontsinis, 2014) or exoplanets [START_REF] Aron | crime-fighting maths law confirms planetary riches[END_REF], crime statistics (Hickman & Rice, 2010), the number of daily-recorded religious activities [START_REF] Mir | the Benford law behavior of the religious activity data[END_REF], earthquake depths [START_REF] Arroucau | Benford's law of first digits: From mathematical curiosity to change detector[END_REF], interventional radiology Dose-Area Product data [START_REF]the novel application of Benford's second order analysis for monitoring radiation output in interventional radiology[END_REF], financial variables [START_REF] Ausloos | Benford's law and theil transform of financial data[END_REF], and internet traffic data (Arshadi & Jahangir, 2014), were found to conform to NBL. In psychology, NBL was found relevant in the study of gambling behaviors [START_REF] Chou | Benford's law and number selection in fixed-odds numbers game[END_REF], brain activity recordings (Kreuzer et al., 2014), language [START_REF] Dehaene | cross-linguistic regularities in the frequency of number words[END_REF][START_REF] Delahaye | Culturomics: Le numérique et la culture [culturomics: the numerical and the culture[END_REF], or perception [START_REF] Beeli | Frequency correlates in grapheme-color synaesthesia[END_REF].

this is an open access article under the cc By-nc-nd license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Prop (%) 30.1 17.6 12.5 9.69 7.92 6.69 5.80 5.12 4.58

the SenSItIvIty and SpecIfIcIty of Benford analySIS

Human pseudorandom productions are in many ways different from true randomness [START_REF] Nickerson | the production and perception of randomness[END_REF]. For instance, participants' productions show an excess of alternations [START_REF]Bias and processing capacity in the generation of random time intervals[END_REF] or are overly uniform (Falk & Konold, 1997). As a consequence, fabricated data might fit NBL to a lesser extent than genuine data [START_REF] Banks | the apparent magnitude of number scaled by random production[END_REF]. Haferkorn ( 2013 These results support the so-called Benford analysis, which uses a measure of discrepancy from NBL to detect fraudulent or erroneous data [START_REF] Bolton | statistical fraud detection: A review[END_REF][START_REF] Kumar | data mining for detecting carelessness or mala fide intention[END_REF][START_REF] Nigrini | Benford's Law: Applications for forensic accounting, auditing, and fraud detection[END_REF]. It has been used to audit industrial and financial data [START_REF] Rauch | deficit versus social statistics: empirical evidence for the effectiveness of Benford's law[END_REF][START_REF] Rauch | liBor manipulation -empirical analysis of financial market benchmarks using Benford's law[END_REF] Hsü (1948), [START_REF] Kubovy | response availability and the apparent spontaneity of numerical choices[END_REF]Hill (1988) provided direct experimental evidence that human-produced data conform poorly to NBL. Another study went even further in formulating meaningful tasks by using the type of data known to exhibit a Benford distribution. [START_REF] Burns | sensitivity to statistical regularities: People (largely) follow Benford's law[END_REF] asked participants to guess real-world values, such as the US gross national debt or the peak summer electricity consumption in Melbourne. He found that although participants' first digit responses did not perfectly follow a logarithmic law, they conformed to the logarithmic distribution better than to the uniform distribution. Burns concluded that participants are not too bad at producing a distribution that conforms to NBL as soon as the task involves the type of realworld data that do follow NBL.

One limitation of Burns' ( 2009) study is that it only works at a population level. We cannot know from his data if a particular individual would succeed in producing a pseudorandom series conforming to NBL, since each participant produced a single value. Nevertheless, his and Diekmann's studies certainly suggest that using Benford's law to detect fraud is questionable in general since humans may be able to produce data confirming to NBL, in which case a Benford test will yield many undetected frauds, lacking sensitivity. As mentioned above, not all random variables or real-world datasets conform to NBL (and when they do, it is generally only in an approximate manner). Because many real-world datasets do not conform to NBL, a Benford test used to detect fraud not only may have low sensitivity but may also have low specificity.

GeneralIzed Benford'S law

Several researchers (e.g., [START_REF] Schmeiser | survival distributions satisfying Benford's law[END_REF] have studied conditions under which a distribution seems more likely to satisfy NBL. Fewster ( 2009) provided an intuitive explanation of why and when the law applies and concluded that any dataset that smoothly spans several orders of magnitude tends to conform to NBL. Data limited to one or two orders of magnitude would generally not conform to the law.

To pursue the question of why many data conform to NBL further, the conservative version of the NBL may be a better starting point than the mere first-digit analysis. Recall that in the conservative version, a

random variable X conforms to NBL if Frac(Log(X)) ~ U([0,1[).
In an attempt to show that the roots of NBL ubiquity should not be looked for in the specific properties of the logarithm, Gauvrit and Delahaye (2008,2009) defined a generalized Benford's law (GBL) associated with any function f as follows: A random variable X conforms to a GBL as-

sociated with function f if Frac(f(X)) ~ U([0,1[).
The classical NBL thus appears as a special case of GBL, associated with function Log.

Testing several mathematical and real-world datasets, Gauvrit and Delahaye (2011) found that several of them fit GBL better than NBL.

Of 12 datasets they studied, six conformed to the classical NBL, while

Study 1

In Study 1, we examined human pseudorandom productions in four different realistic settings, such as those where a classical Benford's law has been initially observed, and compared these responses with true sample values.

Participants

A sample of 169 adults (63 women) took part in this experiment.

Participants were recruited via social networks and e-mails. Ages ranged from 13 to 73 years (M Age = 40.9, SD = 11.6).

Method

Participants were randomly assigned to one of four groups: cities (n = 41), numbers (n = 44), stars (n = 36), or tuberculosis (n = 48). In each group, participants were informed that a series of 30 numbers had been selected at random from a dataset, and were instructed to produce what they thought would be a credible outcome-that is, to supply 30 plausible numbers. In the cities group, the list was the set of popula-

Measures

For each set, X of 30 values' (either fabricated or real samples) observed

distributions of the fractional parts of f(X), with f(X) = Log(X), f(X) = π × X 2 and f(X) = √(X)

Results

As expected, fabricated data were usually less consistent with GBL than real data (see Figure 1, Table 2). There is only one exception to this feature: GBL associated with square function in the case of numbers.

Depending on the context, different computational variants of GBL seemed more appropriate for segregating true values from fabricated ones. In the case of the largest US cities, for instance, human and real data did not significantly differ in terms of conformity to NBL or GBL with the square function, but they differed when calculated with a square-root function.

To analyze the specificity and sensitivity of a fraud detection tool based on GBL, we drew receiver operating characteristic (ROC) curves (see Figure 2) and computed the areas under the curves (AUCs). As shown in Table 3, different sets of data resulted in different patterns.

For the cities condition, classical NBL was barely efficient, whereas square root yielded better results. With the Plouffe database, all GBLs were relevant, although the one associated with function π × X 2 appeared to be the best one.

Discussion

Overall, NBL appeared to be a better means than the other tested variants of GBL for distinguishing between fabricated and real data. However, NBL is not always the best measure, as the cities condition showed. Even when NBL was an efficient measure, such as in the number condition, some other GBL may have been even better or at least as good as NBL, for example, GBL associated with square, for which the AUC was greater than the NBL AUC. Depending on the type of data one tests, different types of GBL could thus be adviced, either replacing or complementing classical Benford analysis.

A further argument in favor of the GBL analysis is that, with the growing popularity of the Benford analysis, potential swindlers might become aware of the necessity of conforming to the NBL. Alternative methods complementing the classical analysis (for another such method, see [START_REF] Miller | data diagnostics using secondorder tests of Benford's law[END_REF] could thus prove useful, especially in view of the fact that it would be particularly difficult to fabricate data conforming to a whole set of variants of GBL.

Study 2

One possible reason why [START_REF] Diekmann | not the first digit! Using Benford's law to detect fraudulent scientific data[END_REF] found that humans were able to produce accurate values (r) is that the notion was familiar to the participants (students in the social sciences), a feature that may have had an impact on the outcome. If this is true, our positive results in Study 1 might have been the result of too low a familiarity with the material at hand. In Study 2, we investigated the possible effects of familiarity, as well as that of cognitive effort, on the accuracy of the Benford analysis.

Participants

A sample of 124 first-year psychology students (103 women) from a distant learning program volunteered in the experiment. Ages ranged from 22 to 55 years (M Age = 38.27; SD = 9.08). Participants were recruited by e-mail and voluntarily accepted to participate. We chose distant learning students as participants because, contrary to ordinary students, they have various backgrounds and previous working experiences in a diversity of fields, warranting greater variation in the familiarity with the material.

Method

The experiment was performed online using a Google Form (https:// www.google.com/forms/about/). We used country population data, as this was believed to grant a somewhat larger variation in familiarity.

Participants were asked to produce series of only 20 data points to lower the risk of tiredness.

Participants were randomly assigned to one of two groups (no time pressure or time pressure condition). Each group included 62 participants. They were informed that they would have to supply a list of 20 values that could be the numbers of inhabitants of 20 randomly selected countries in the world. They were asked to try to guess what these populations could be. They were told that a true sampling would be performed for comparison with their answers. In the no time pres-sure condition, the instruction was to be "as accurate as possible, taking as much time as needed. " In the time pressure condition, the instruction was to be "as fast and accurate as possible. " The former condition is known to conduce to superior cognitive effort than the latter (e.g., [START_REF] Maule | the effects of time pressure on human judgment and decision making[END_REF].

Self-reported data on level of expertise in the field of country populations were also collected using a 6-point Likert scale, from 0 = absolutely naïve about country populations to 5 = expert in country populations. This measure serves as an indication of the participants' familiarity with the material.

Sixty-two true samples of 20 country populations were selected from real-world data (found at http://data.okfn.org/data/core/ population#data) to be compared with participants' productions.

Measures

For each set X of 20 values (either fabricated or real samples), observed distributions of Frac(Log(X)) were computed. The discrepancy from uniformity was assessed in each case using the Kolmogorov-Smirnov statistic D.

Results

Reported expertise was rather low (M = 1.3; SD = .97; range of 0-4).

To assess the effect of expertise, we performed an analysis of variance (ANOVA), with the dependent variable D and the independent variable Level of Expertise (6). No significant effect was found, F(4, 119) = 0.64, p = .63. The same procedure yielded nonsignificant statistics for GBL associated with the square-root function, F(4, 119) = 1.29, p = .28, and with GBL associated with function π × X 2 , F(4, 119) = 0.20, p = .94. Correlation analysis showed nonsignificant links between level of expertise and D, for classical NBL, r(122) = -.11, p = .20 , GBL associated with a square-root function, r(122) = -.14, p = .10, and also with function πX 2 , r(122) = -.06, p = .53.

To assess the effect of cognitive effort, we performed an ANOVA with a dependent variable D and an independent variable Group (2).

There was a significant influence, F(2, 183) = 7.33, p < .001, but a Tukey HSD test showed that the two groups did not differ significantly from one another (p = .46), although they both differed from real data values.The same procedure was used to assess the effect of cognitive effort related to other variants of GBL. No effect was found for GBL associated with the square-root function, F(2, 183) = 1.53, p = .22, or with function πX 2 , F(2, 183) = 0.01, p = .99.

Discussion

Familiarity does not appear to influence the quality of participants' responses in terms of GBL. The experimental procedure used to assess the possible effect of time pressure or cognitive effort yielded no significant results either. These results thus suggested that producing fraudulent data that would remain undetected under the Benford analysis is not necessarily a matter of familiarity or cognitive effort.

It is, however, fair to note two limitations. First, time pressure, although widely used to increase cognitive effort, probably does not result in large differences, especially if a task requires little effort in general. Concerning familiarity with the material, no participants declared themselves experts (score of 5 on the scale) about country populations, most of the sample lying in the range of 0-3, so that we would not have detected a specific effect only appearing with true experts.

concludInG dIScuSSIon

We performed the first investigation of the generalized Benford analysis, an equivalent of the classical Benford analysis, but based on the broader GBL. Results from Study 1 rendered mild support for the generalized Benford analysis, including the classical Benford analysis.

They also draw attention to the fact that different types of data yielded different outcomes, suggesting that the best way of detecting fraud using GBL associated with some function f would be obtained either by finding the function f that best matches the particular data at hand or by combining different analyses. Although the classical Benford analysis was validated in our studies, it occasionally failed at detecting humanproduced data as efficiently as other generalized Benford analysis.

The present positive results could have been the result of our sample characteristic, in which participants, contrary to real swindlers, might have put little effort into the task since the stakes were low. Plus, the participants were not highly familiar with the material at hand. To rule out the possibility that our results resulted from such features and GBL would be inapplicable in real situations, Study 2 aimed at demonstrating that cognitive effort and familiarity with the material have little effect on the participants' responses. The data supported this view, although further studies (including higher levels of cognitive pressure and true experts) would be recommended.

With Benford analysis having become more common in fraud detection, new complementary analyses are needed [START_REF] Miller | data diagnostics using secondorder tests of Benford's law[END_REF]. The GBL analysis potentially provides a whole set of such fraud detection methods, which means making it more difficult, even for informed swindlers intentionally conforming to NBL, to remain undetected.

http://www.ac-psych.org 2017 • volume 13(2) • 121-127 127 Falk, r., & Konold, c. (1997). Making sense of randomness: implicit encoding as a basis for judgment. Psychological Review, 104, 301-318. doi: 10.1037/0033-295X.104.2.301 Fewster, r. M. (2009). A simple explanation of Benford's law. The American Statistician, 63, 27-32. doi: 10.1198Statistician, 63, 27-32. doi: 10. /tast.2009Statistician, 63, 27-32. doi: 10. .0005 gauvrit, n., & delahaye, J.-P. (2008)). Pourquoi la loi de Benford n'est pas mystérieuse [Why Benford's law is not mysterious]. Humaines, 182, 7-16. gauvrit, n., & delahaye, J.-P. (2009). loi de Benford général- Criminology, 26, 333-349. doi: 10.1007Criminology, 26, 333-349. doi: 10. /s10940-010-9094-6 hill, t. P. (1988)). random-number guessing and the first digit phenomenon. Psychological Reports, 6, 967-971. doi: 10.2466Reports, 6, 967-971. doi: 10. / pr0.1988Reports, 6, 967-971. doi: 10. .62.3.967 hill, t. P. (1998)). the first digit phenomenon: A century-old observation about an unexpected pattern in many numerical tables applies to the stock market, census statistics and accounting data. American Scientist, 86, 358-363. hsü, e. h. (1948). An experimental study on "mental numbers" and a new application. Journal of General Psychology, 38, 57-67. doi: 10.1080/00221309.1948.9711768 Kreuzer, M., Jordan, d., Antkowiak, B., drexler, B., Kochs, e. F., & schneider, g. (2014). Brain electrical activity obeys Benford's law. Anesthesia & Analgesia, 118, 183-191. doi: 10.1213/

Mathématiques et Sciences

  ) compared algorithm-based and human-based trade orders and concluded that algorithm-based orders approximated NBL better than human-based orders. Hales, Chakravorty, and Sridharan (2009) showed that NBL is efficient in detecting fraudulent data in an industrial supply-chain context.

  , to gauge the scientific publication process (de Vries & Murk, 2013), to separate natural from computer-generated images (Tong, Yang, & Xie, 2013), or to detect hidden messages in images' .jpeg files (Andriotis, Oikonomou, & Tryfonas, 2013).As a rule, the Benford analysis focuses on the distribution of the first digit and compares it to the normative logarithmic distribution.However, a more conservative version of Benford's law states that numerical values or a variable X should conform to the following property: Frac(Log(X)) should follow a uniform distribution in the range of[0,1[. Here, Frac(x) stands for x-Floor(x), Floor(x) being the largest integer inferior or equal to x. The logarithmic distribution of the first digit is a mathematical consequence of this version[START_REF]the first digit problem[END_REF].

  10 conformed to a GBL with function f(x) = π × x 2 , and nine with square-root function. On the other hand, none conformed to GBL with function Log o Log. These findings suggest that a GBL associated with the relevant function-depending on the context-might yield more specific or sensitive tests for detecting fraudulent or erroneous data.We addressed this question in two studies. In both studies, each participant produced a whole series of values, allowing analyzing the resulting distribution at an individual level. In Study 1, we examined three versions of GBL in four different situations in order to compare the sensitivity and specificity of different types of GBL analyses. Study 2 explored the potential effects of variations of familiarity with the material and of cognitive effort on the productions.

  tions of the 5,000 most populated US cities. Numbers corresponded to the dataset of numerical constants published by Simon Plouffe and described in the experiment as an extensive encyclopedia of mathematical constants. Participants assigned to the stars group were told that the dataset was the list of distances from earth to all known visible stars expressed in light-years. Last, the tuberculosis group dealt with the set of known country-wise incidences of tuberculosis as measured in 2012.Real samplings of 30 numbers from the corresponding databases were also performed. The set of populations of the biggest US cities came from an online dataset (http://factfinder2.census.gov/). Numbers were randomly selected from the Simon Plouffe database of numerical values (http://www.plouffe.fr). The distances to the stars were read from the HYG2.0 dataset (http://www.astronexus.com/hyg) and multiplied by 3.262 to render them in light-years instead of parsecs.Lastly, the tuberculosis dataset was downloaded from the World Health Organization's (WHO) website (http://www.who.int/tb/country/data/ download/en/).

  , were computed. The two last functions were selected on the basis of previous studies indicating that they led to satisfying fits with several numerical datasets(Gauvrit & Delahaye, 2009).The deviation from GBL was measured by the Kolmogorov-Smirnov statistic D-that is, the maximum difference between the cumulative distribution function of Frac(f(X)) and the cumulative distribution function of U([0,1[). D thus serves as a proximal measure of conformity to GBL. This statistic is a classical measure of distance between distributions that grounds the classical Kolmogorov-Smirnov test.

  isée [generalized Benford's law]. Mathématiques et Sciences Humaines, 186, 5-15. doi: 10.4000/msh.11034 gauvrit, n., & delahaye, J.-P. (2011). scatter and regularity implies Benford's law… and more. in h. Zenil (ed.), Randomness through computation (pp. 53-69). london, england: World scientific. gauvrit, n., & Morsanyi, K. (2014). the equiprobability bias from a mathematical and psychological perspective. Advances in Cognitive Psychology, 10, 119-130. doi: 10.5709/acp-0163-9 haferkorn, M. (2013). humans vs. algorithms -who follows newcomb-Benford's law better with their order volume? in F. A. rabhi & P. gomber (eds.), Enterprise Applications and Services in the Finance Industry (pp. 61-70). Berlin, germany: springer. hales, d. n., chakravorty, s. s., & sridharan, v. (2009). testing Benford's law for improving supply chain decision-making: A field experiment. International Journal of Production Economics, 122, 606-618. doi: 10.1016/j.ijpe.2009.06.017 hickman, M. J., & rice, s. K. (2010). digital analysis of crime statistics: does crime conform to Benford's law? Journal of Quantitative

tAble 1 .

 1 Proportion of 1s, 2s,…, 9s as First significant digit in a series conforming to nBl Note. NBL = Newcomb-Benford law.

	Digit	1	2	3	4	5	6	7	8	9

Author Note

 P., oikonomou, g., & tryfonas, t. (2013). JPeg steganography detection with Benford's law. Digital Investigation, 9, 246-257. doi: 10.1016/j.diin.2013.01.005