In memory of Dr Hiroyuki Sorimachi, a pivotal researcher in the calpain field

This review was written in memory of our late friend, Dr Hiroyuki Sorimachi, who, following the steps of his mentor Koichi Suzuki, a pioneer in calpain research, has made tremendous contributions to the field. During his career, Hiro also wrote several reviews on calpain, the last of which, published in 2016, was comprehensive [START_REF] Ono | Calpain research for drug discovery: challenges and potential[END_REF]. In this manuscript, we decided to put together a review with the basic information a novice may need to know about calpains. We also tried to avoid similarities with previous reviews and reported the most significant new findings, at the same time highlighting Hiro's contributions to the field. The review will cover a short history of calpain discovery, the presentation of the family, the life of calpain from transcription to activity, the human diseases caused by calpain mutations and their therapeutic perspectives. In addition, the readers can find in Table 1 the listing of all the most relevant reviews in calpain research according to different subjects as well as the link of two online resources created by Dr. Sorimachi.

Subject Reference Online resources

CaMPDB: a resource for calpain and modulatory proteolysis http://.calpain. org/ [START_REF] Duverle | CaMPDB: a resource for calpain and modulatory proteolysis[END_REF] Calpain Research Portal, http://calpain.net/index.html Whole calpain family [START_REF] Ono | Calpain research for drug discovery: challenges and potential[END_REF][START_REF] Sorimachi | Structure and physiological functions of ubiquitous and tissuespecific calpain species. Muscle-specific calpain, p94, interacts with connectin/titin[END_REF][START_REF] Murachi | Calpain and calpastatin[END_REF][START_REF] Suzuki | Calpain: novel family members, activation, and physiologic function[END_REF][START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF][START_REF] Ono | Calpains: an elaborate proteolytic system[END_REF][START_REF] Sorimachi | Impact of genetic insights into calpain biology[END_REF][START_REF] Mellgren | Calcium-dependent proteases: an enzyme system active at cellular membranes?[END_REF][START_REF] Croall | Calcium-activated neutral protease (calpain) system: structure, function, and regulation[END_REF][START_REF] Goll | The calpain system[END_REF] Physiological system [START_REF] Baudry | Calpain-1 and Calpain-2 in the Brain: Dr. Jekill and Mr Hyde?[END_REF] (Calpain-1 and -2 in the brain); [START_REF] Litosh | Calpain-14 and its association with eosinophilic esophagitis[END_REF] (Calpain-14 oesophagus) Individual calpains [START_REF] Ono | An eccentric calpain, CAPN3/p94/calpain-3[END_REF][START_REF] Duguez | Calpain 3: a key regulator of the sarcomere?[END_REF]; [START_REF] Beckmann | Calpain 3, the "gatekeeper" of proper sarcomere assembly, turnover and maintenance[END_REF] (Calpain-3) Modulators of calpain activity [START_REF] Ono | Calpain research for drug discovery: challenges and potential[END_REF][START_REF] Dokus | Modulators of calpain activity: inhibitors and activators as potential drugs[END_REF] Inhibitors [START_REF] Donkor | An updated patent review of calpain inhibitors (2012 -2014)[END_REF] (patent review); [START_REF] Dokus | Modulators of calpain activity: inhibitors and activators as potential drugs[END_REF][START_REF] Hernandez | Recent advances in the synthesis, design and selection of cysteine protease inhibitors[END_REF] (cysteine protease inhibitors) Crystal structures and domains/calpain activation [START_REF] Sorimachi | The structure of calpain[END_REF][START_REF] Suzuki | Structure, activation, and biology of calpain[END_REF][START_REF] Khorchid | How calpain is activated by calcium[END_REF][START_REF] Campbell | Structure-function relationships in calpains[END_REF] Pathological mutations in calpains [START_REF] Branca | Calpain-related diseases[END_REF] (CAPN3, CAPN10); [START_REF] Sorimachi | Regulation and physiological roles of the calpain system in muscular disorders[END_REF] Table 1: Calpain review articles THE HISTORY OF CALPAIN FAMILY DISCOVERY.

The calpain system (Clan CA, family C02; EC 3.4.22.17) is recognized as the third proteolytic system of the cell besides the lysosomal and proteasomal systems. In the following paragraph, we provide an historical perspective on calpain discovery. This would help a novice in the field understand how the research on this class of proteins developed during the years, and what underlies the complexity of defining this superfamily. The first Ca 2+ -dependent cysteine protease discovered is the one known today as calpain-2. In 1964, Guroff partially isolated calpain-2 from rat brain as a novel ubiquitous protease able to hydrolyse casein in a Ca 2+dependent manner [START_REF] Guroff | A Neutral, Calcium-Activated Proteinase from the Soluble Fraction of Rat Brain[END_REF]. The same year, Meyer, Fisher and Krebs identified the same protein as a factor that activates phosphorylase kinase, and named it KAF (kinase activating factor). In 1968, KAF was then described as a Ca 2+ -dependent proteolytic enzyme independently by Huston & Krebs and Drummond & Duncan [START_REF] Meyer | Activation of Skeletal Muscle Phosphorylase B Kinase by Ca[END_REF][START_REF] Huston | Activation of skeletal muscle phosphorylase kinase by Ca2+. II. Identification of the kinase activating factor as a proteolytic enzyme[END_REF][START_REF] Drummond | On the mechanism of activation of phosphorylase b kinase by calcium[END_REF]. In 1972, Goll's group re-identified (for the third time) a protein also corresponding to the current calpain-2 as Ca 2+ -activated sarcoplasmic factor (CaSF) able to hydrolyse Z-lines [START_REF] Busch | Ca 2+ -specific removal of Z lines from rabbit skeletal muscle[END_REF]. Finally, in 1978, Imahori and colleagues purified this 80kD protease to homogeneity for the first time from chicken skeletal muscle and called it Ca 2+ -activated neutral protease or CANP [START_REF] Ishiura | Studies of a calcium-activated neutral protease from chicken skeletal muscle. I. Purification and characterization[END_REF]. The same year, Murachi's group reported the occurrence of an inhibitor for CANP in rat liver known today as calpastatin, and partially purified it [START_REF] Nishiura | The occurrence of an inhibitor of Ca2+-dependent neutral protease in rat liver[END_REF]. During this time, it was also demonstrated by protein purification and Ca 2+ -activation assays that at least two different isoforms of Ca 2+ -dependent proteases were present, with different sensitivities to Ca 2+ concentrations [START_REF] Mellgren | Canine cardiac calcium-dependent proteases: Resolution of two forms with different requirements for calcium[END_REF]. In 1981, Murachi and colleagues reviewed all the literature of that time on this class of proteins and proposed that this new group of proteases, which requires an SH-reducing agent and the presence of Ca 2+ for full activity, should be called calpain. "Cal" stands for calcium, like in calcitonin or calmodulin, whereas "-pain" stands for the resemblance with well-known thiol proteases such as papain and clostripain [START_REF] Murachi | Intracellular Ca2+-dependent protease (calpain) and its highmolecular-weight endogenous inhibitor (calpastatin)[END_REF]. During this period, it was also discovered that both these proteases exist as the larger part of a heterodimer in association with a smaller subunit of 30kD (known today as calpain subunit 1 or CAPNS1) [START_REF] Imajoh | The amino-terminal hydrophobic region of the small subunit of calcium-activated neutral protease (CANP) is essential for its activation by phosphatidylinositol[END_REF]. The determination that the active site of calpain contained a cysteine only arrived in 1983 through protein sequencing [START_REF] Suzuki | Amino acid sequence around the active site cysteine residue of calcium-activated neutral protease (CANP)[END_REF]. One year after, these authors also published the first nucleotide sequence of a calpain, following cloning of the cDNA of what was named a calcium-dependent protease or CANP (probably the chicken CAPN11, that seems to be ubiquitous in birds) [START_REF] Ohno | Evolutionary origin of a calcium-dependent protease by fusion of genes for a thiol protease and a calcium-binding protein?[END_REF][START_REF] Sorimachi | Identification of a third ubiquitous calpain species--chicken muscle expresses four distinct calpains[END_REF]. This finding was the first to show that calpains contain domains sharing marked similarity to papain-like and calmodulin-like proteins [START_REF] Ohno | Evolutionary origin of a calcium-dependent protease by fusion of genes for a thiol protease and a calcium-binding protein?[END_REF].

As mentioned above, two different isoforms of these proteases were found, both ubiquitous, but with different sensitivities to Ca 2+ [START_REF] Yoshimura | Two distinct Ca2+ proteases (calpain I and calpain II) purified concurrently by the same method from rat kidney[END_REF][START_REF] Inomata | Comparison of low and high calcium requiring forms of the calciumactivated neutral protease (CANP) from rabbit skeletal muscle[END_REF]. Interestingly, the two main research groups working on this new class of proteases kept using different names, creating confusion in the literature.

Imahori's group named them µ-CANP and m-CANP based on the Ca 2+ concentration necessary for their full activity (micro-and milli-molar, respectively) [START_REF] Inomata | Comparison of low and high calcium requiring forms of the calciumactivated neutral protease (CANP) from rabbit skeletal muscle[END_REF]. On the other side, Murachi's group used the names calpain-1 for the former and calpain-2 for the later [START_REF] Yoshimura | Two distinct Ca2+ proteases (calpain I and calpain II) purified concurrently by the same method from rat kidney[END_REF].

The year 1989 changed the calpain field forever, when Sorimachi and colleagues, by screening human and rat muscle cDNA libraries, discovered and cloned a non-ubiquitous calpain for the first time, whose expression was mostly restricted to skeletal muscle, and called it p94 (known today as calpain-3) in reference to its molecular weight [START_REF] Sorimachi | Molecular cloning of a novel mammalian calcium-dependent protease distinct from both m-and mu-types. Specific expression of the mRNA in skeletal muscle[END_REF]. This finding led to the discovery of another tissue-specific calpain (known today as calpain-8) with expression restricted mainly in the stomach, suggesting the existence of a protease family with several members [START_REF] Sorimachi | A novel tissue-specific calpain species expressed predominantly in the stomach comprises two alternative splicing products with and without Ca(2+)-binding domain[END_REF]. For this reason, Suzuki's group proposed a new nomenclature: µ-CL and m-CL for µ-and m-Calpain large subunits respectively, 30K for the small subunit and the analogical name nCL-x for the newly discovered members (where nCL stands for novel Calpain Large subunits and x for the associated number) [START_REF] Sorimachi | Structure and physiological functions of ubiquitous and tissuespecific calpain species. Muscle-specific calpain, p94, interacts with connectin/titin[END_REF][START_REF] Sorimachi | A novel tissue-specific calpain species expressed predominantly in the stomach comprises two alternative splicing products with and without Ca(2+)-binding domain[END_REF]. Calpain-3 was then named nCL-1 and the future calpain-8 nCL-2. However, this nomenclature was never adopted outside of their group and this again created confusion in the literature. Since these first findings, many other calpain members have been discovered. Their first appearance in the literature is listed in Table 2.

Today, the nomenclature uses "CAPN" for the genes and "calpain" for the protein products, followed by an analogical number to identify the specific member of the protein family (eg.

CAPN1 and calpain-1) [START_REF] Ono | Calpains: an elaborate proteolytic system[END_REF]. As is clear from Table 2, the analogical number in the gene/protein name does not correspond to the order of discovery. This results from the fact that some members of the family (CAPN8, 9, 15) were named differently than "calpain" at the time of their discovery while the Human Genome Nomenclature Committee was including consecutive numbers for other calpain members (Table 2 andFigure 1). Another discrepancy is due to the fact that CAPNS1 was incorrectly named CAPN4/Calpain-4 before the discovery of calpain-5.

Consequently, the gene list of the 15 family members lacks a "CAPN4", jumping from 3 to 5 (Figure 1). Of note, a second small subunit (CAPNS2) was identified in humans, homolog to CAPNS1, with a still unknown physiological role [START_REF] Schad | A novel human small subunit of calpains[END_REF].

A CURRENT DESCRIPTION OF CALPAIN FAMILY MEMBERS

The accepted definition of a calpain is a protein that has an amino-acid sequence significantly similar to the protease domain of conventional calpains, irrespective of proteolytic ability [START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF].

In mammals, the calpain family is composed of 15 gene products (Figure 2, Table 2) while additional calpains have been recognized in lower ontological classes [START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF][START_REF] Ono | Calpains: an elaborate proteolytic system[END_REF]. Because of their extensive sequence similarity, history and in the wake of protein kinase C nomenclature, the first two calpains discovered (-1 and -2) used to be called "conventional" [START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF]. The remaining family members are further divided into classical and non-classical calpains depending on the resemblance of their domain organisation to those of the conventional calpains (Figure 2).

Moreover, based on their expression profile and physiological role, nine calpains are classified as ubiquitous and six as tissue-specific (Figure 2).

Over the years, even if complicated by the instability of these proteins, a number of crystal structures of different calpains and calpain domains have been solved. These structural studies have provided useful information into tertiary and quaternary structure organizations, as well as giving insight on the activation of both classical and non-classical calpains (Table 3). It is worth mentioning that the structure of calpain-2 was the first calpain crystal structure ever solved, in 1999, and to date the only one that has been solved in its entirety. Campbell and Davies, inclusive of all calpain crystal structures known at that time [START_REF] Campbell | Structure-function relationships in calpains[END_REF], calpain-3 (PDB ID: 6BDT and 4OKH) and calpain-5 (PDB ID: 6P3Q) crystal structures have been resolved.

Classical calpains

The classical group is comprised of nine calpains, of which calpain-1, -2, -13 and -14 are expressed ubiquitously while calpain-3, -8, -9, -11 and -12 expression is restricted to particular tissues (Figure 2). By definition, they all share the same structural composition. This organisation was clarified following determination of the tridimensional structure of calpain-2

in 1999 [START_REF] Hosfield | Crystal structure of calpain reveals the structural basis for Ca(2+)-dependent protease activity and a novel mode of enzyme activation[END_REF] as well as the subsequent calpains (Table 3). The classical calpains consist of: i)

an N-terminal anchor helix region of variable length; ii) a catalytic domain named calpain-type cysteine protease conserved (CysPc), subdivided into two globular core domains called protease core 1 and 2 (PC1 and PC2 or DI and DII); iii) a calpain-type beta-sandwich (CBSW or DIII) domain and; iv) a penta-EF-hand domain (PEF or DIV). The N-terminal anchor is involved in the maintenance of the inactive state, as well as in Ca 2+ -dependant activation. In CysPc, the catalytic activity is possible thanks to a triad of amino acids composed of a cysteine, a histidine and an asparagine, as determined via site directed mutagenesis [START_REF] Arthur | Active site residues in m-calpain: identification by site-directed mutagenesis[END_REF]. From a tridimensional point of view, the active site resides in the catalytic cleft between the two PC domains, with the cysteine located in PC1 and the histidine and asparagine in PC2. The CBSW domain is also called C2-like (C2L) for the similarity with a common structural domain called C2, very often involved in Ca 2+ -dependant membrane targeting [START_REF] Zhang | Identification of novel families and classification of the C2 domain superfamily elucidate the origin and evolution of membrane targeting activities in eukaryotes[END_REF]. The PEF domain is characterized by five repetitive EF-hand motifs and a helix-loop-helix structure involved in Ca 2+ binding. The fifth EF-hand of the domain acts as a dimerization module to generate both homodimers and heterodimers [START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF].

Mammalian calpain-1 and -2 are the most studied and therefore well-known calpains, both are ubiquitously expressed with similarities in substrate specificities and mode of inhibition. As mentioned above, the functional form of calpain-1 and -2 exists as a heterodimer, where the gene product of CAPN1 or CAPN2 is the large catalytic subunit and the small 30 KDa protein acts as regulatory subunit (calpain subunit 1, or CAPNS1) [START_REF] Hosfield | Crystal structure of calpain reveals the structural basis for Ca(2+)-dependent protease activity and a novel mode of enzyme activation[END_REF][START_REF] Strobl | The crystal structure of calcium-free human m-calpain suggests an electrostatic switch mechanism for activation by calcium[END_REF]. Interestingly, the crystal structure of CAPNS1 was solved even before any calpain members. The subunit is composed of an N-terminal Glycine rich region (GR) and a PEF domain. The dimerization with the large subunit occurs through the fifth EF hands of their respective PEF domains [START_REF] Blanchard | Structure of a calpain Ca(2+)-binding domain reveals a novel EFhand and Ca(2+)-induced conformational changes[END_REF]. CAPN1 and CAPN2, are the only calpains where association with CAPNS1 has been demonstrated to be required in vivo. In fact, disruption of CAPNS1 in mice results in destabilization of both calpains [START_REF] Arthur | Disruption of the murine calpain small subunit gene, Capn4: calpain is essential for embryonic development but not for cell growth and division[END_REF][START_REF] Zimmerman | The calpain small subunit gene is essential: its inactivation results in embryonic lethality[END_REF].

Calpain-3 shares the same domains of the conventional calpains, but has additional specific sequences: NS at the N-terminal, the insertion sequence IS1 within the PC2 domain and the insertion sequence IS2 between the CBSW and PEF domains. The insertion sequence IS1 is primarily responsible for the uniquely rapid autolysis of calpain-3 that underlies enzymatic activation. The IS2 region contains a nuclear localization signal and calpain-3 has been observed in the nucleus and cytoplasm [START_REF] Herasse | Expression and functional characteristics of calpain 3 isoforms generated through tissue-specific transcriptional and posttranscriptional events[END_REF]. Interestingly, it was suggested that calpain-3 may have a non-proteolytic function with a role in Ca 2+ release from the sarcoplasmic reticulum [START_REF] Ojima | Non-proteolytic functions of calpain-3 in sarcoplasmic reticulum in skeletal muscles[END_REF]. The resolution of the crystal structure of the PEF domain of calpain-3 demonstrated that this calpain is able to form a homodimer through its PEF domain [START_REF] Ravulapalli | Homodimerization of calpain 3 penta-EF-hand domain[END_REF]. In these homodimers, the calpain-3 monomers are oriented in such a way that allows the catalytic sites to be positioned at opposite ends [START_REF] Partha | Crystal structure of calpain-3 penta-EF-hand (PEF) domain -a homodimerized PEF family member with calcium bound at the fifth EF-hand[END_REF]. Further investigation into the quaternary structure of calpain-3 revealed it can also form a homotrimer, the first demonstration of trimerisation of any calpain family member or PEF containing proteins [START_REF] Hata | A muscle-specific calpain, CAPN3, forms a homotrimer[END_REF].

Calpain-8 and -9 are mostly expressed in the gastrointestinal tract. Interestingly, it was shown that they are able to form a heterodimer with each other. The N-terminal anchor helix and Cterminal EF-hands of calpain-8 interact with the C-terminal EF-hands of calpain-9, forming a complex called gastric calpain (G-calpain) [START_REF] Hata | Calpain 8/nCL-2 and calpain 9/nCL-4 constitute an active protease complex, G-calpain, involved in gastric mucosal defense[END_REF][START_REF] Hata | A Gastrointestinal Calpain Complex, G-calpain, Is a Heterodimer of CAPN8 and CAPN9 Calpain Isoforms, Which Play Catalytic and Regulatory Roles, Respectively[END_REF]. The G-calpain is a protease complex where it has been shown that the catalytic activity is exerted by calpain-8, while calpain-9 functions as a chaperone-like regulatory subunit [START_REF] Hata | A Gastrointestinal Calpain Complex, G-calpain, Is a Heterodimer of CAPN8 and CAPN9 Calpain Isoforms, Which Play Catalytic and Regulatory Roles, Respectively[END_REF]. Besides G-calpain, calpain-8 is also able to homodimerize and calpain-9 to form a complex with CAPNS1 [START_REF] Hata | Calpain 8/nCL-2 and calpain 9/nCL-4 constitute an active protease complex, G-calpain, involved in gastric mucosal defense[END_REF][START_REF] Hata | A Gastrointestinal Calpain Complex, G-calpain, Is a Heterodimer of CAPN8 and CAPN9 Calpain Isoforms, Which Play Catalytic and Regulatory Roles, Respectively[END_REF].

Little is known about the remaining four classical calpains, apart from their organ distribution.

Calpain-13 is ubiquitous, while calpain-11 is expressed in testis and calpain-12 expression is restricted to hair follicle cells [START_REF] Ono | Calpains: an elaborate proteolytic system[END_REF]. There is no evidence that they heterodimerise with CAPNS1, except for the avian form of CAPN11 that was shown to be associated with the small subunit [START_REF] Macqueen | A newly classified vertebrate calpain protease, directly ancestral to CAPN1 and 2, episodically evolved a restricted physiological function in placental mammals[END_REF]. As for Calpain-14 initially thought to be ubiquitous, it is expressed at the highest level in the oesophagus where it constitutes a marker for tissue identity [START_REF] Fagerberg | Analysis of the human tissue-specific expression by genome-wide integration of transcriptomics and antibody-based proteomics[END_REF].

Non-classical calpains

Non-classical calpains present sequence homology with the classical calpains in the CysPc region but do not have complete similarity in domain organisation. In particular, they lack the PEF domain and for some the CBSW. According to the classification of Sorimachi [START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF], they are divided into three groups depending on their structure and additional domains: the PalB subfamily, which includes CAPN5, CAPN6, CAPN7 and CAPN10, is the most evolutionarily conserved group, the Sol subfamily, which comprises only one calpain (CAPN15) and the single group of demi-calpain (CAPN16), a very divergent calpain. They are all ubiquitous apart from CAPN6, which was found to predominantly be expressed in embryonic muscles and placenta [START_REF] Dear | A new subfamily of vertebrate calpains lacking a calmodulin-like domain: implications for calpain regulation and evolution[END_REF][START_REF] Dear | Diverse mRNA expression patterns of the mouse calpain genes Capn5, Capn6 and Capn11 during development[END_REF].

The PalB subfamily takes its name from an alternative term for the human CAPN7 (PALBH) through homology with the Aspergillus atypical calpain PalB [START_REF] Futai | Molecular cloning of PalBH, a mammalian homologue of the Aspergillus atypical calpain PalB[END_REF]. The members of this family all lack a PEF domain making it implausible that they are able to bind the small subunit. The PEF domain is replaced either by a domain that is similar in structure to the classical C2 domain (for calpain-5 and -6), or by a second CBSW (for calpain-7 and -10). Calpain-5, historically called hTRA3 for its homology to the TRA-3 sex determination gene of C. Elegans [START_REF] Mugita | Identification of a novel, tissue-specific calpain htra-3; a human homologue of the Caenorhabditis elegans sex determination gene[END_REF], has the structural feature of having 2 NLS that accounts for its dual localization in the nucleus and cytoplasm [START_REF] Singh | Calpain 5 is highly expressed in the central nervous system (CNS), carries dual nuclear localization signals, and is associated with nuclear promyelocytic leukemia protein bodies[END_REF]. Furthermore, calpain-5 has a unique conformational feature of three extended flexible loops in each PC domain. These loops are proposed to account for the high Ca 2+

concentrations required for optimal activity [START_REF] Velez | Structural Insights into the Unique Activation Mechanisms of a Nonclassical Calpain and Its Disease-Causing Variants[END_REF]. Calpain-6 is unique because it lacks a cysteine residue in the catalytic triad (substituted with Lys in humans) within the PC1 domain, suggesting that it may not possess proteolytic activity [START_REF] Dear | A new subfamily of vertebrate calpains lacking a calmodulin-like domain: implications for calpain regulation and evolution[END_REF]. In this calpain, the C2-like domain mediates the interaction with microtubules rather than phospholipids [START_REF] Tonami | Calpain 6 is involved in microtubule stabilization and cytoskeletal organization[END_REF][START_REF] Tonami | Calpain-6 deficiency promotes skeletal muscle development and regeneration[END_REF].

The SOL subfamily includes only CAPN15, also called SOLH in human, as the homologue of the small optic lobes (SOL) gene identified in D. Melanogaster [START_REF] Kamei | SOLH, a human homologue of the Drosophila melanogaster small optic lobes gene is a member of the calpain and zinc-finger gene families and maps to human chromosome 16p13.3 near CATM (cataract with microphthalmia)[END_REF]. Calpain-15 is a conserved ubiquitous protein that is highly expressed in the nervous system [START_REF] Zha | Biallelic variants in the small optic lobe calpain CAPN15 are associated with congenital eye anomalies, deafness and other neurodevelopmental deficits[END_REF] and has up to 6 Zn-finger repeats at the N-terminus, recently shown to be able to bind polyubiquitin [START_REF] Hastings | The zinc fingers of the small optic lobes calpain bind polyubiquitin[END_REF], a CysPC domain, and a SOL homology domain (SOH) at the C-terminal domain of unknown function [START_REF] Kamei | SOLH, a human homologue of the Drosophila melanogaster small optic lobes gene is a member of the calpain and zinc-finger gene families and maps to human chromosome 16p13.3 near CATM (cataract with microphthalmia)[END_REF].

The demi-calpain subfamily includes calpain-16, which consists of half the CysPC proteolytic domain and a calmodulin-binding motif (amino acid sequence motif IQ) in the C-terminal region [START_REF] Sorimachi | Calpain chronicle--an enzyme family under multidisciplinary characterization[END_REF]. The actual name of this protein is androglobin (ADGB) as registered in the HUGO database. Phylogenetic analyses indicate that this gene is probably not a descendant of an ancestral calpain, but rather a protein derived from the fusion of a calpain domain with a globulin in an exon-shuffling event [START_REF] Hoogewijs | Androglobin: a chimeric globin in metazoans that is preferentially expressed in Mammalian testes[END_REF]. Interestingly, the calpain domain of ADGB has retained only the cysteine of the typical triad but contains several His and Asn residues at nonstandard positions. It is not yet defined whether calpain-16 has proteolytic activity [START_REF] Hoogewijs | Androglobin: a chimeric globin in metazoans that is preferentially expressed in Mammalian testes[END_REF].

FROM TRANSCRIPTION TO ACTIVITY

Proteolytic events are irreversible post translation modifications. Therefore, they require strict control to occur at the right time, the right place and on the right targets in order to keep homeostasis and avoid cell self-destruction. Calpains cumulate all existing possibilities for controlling proteolysis, from expression to conformational changes. In this part of the review, we will present these mechanisms using the most relevant examples to illustrate how such control can be achieved.

Expression and alternative splicing

The strategies that the cell uses to regulate calpain function start by spatio-temporal modulation of the level or the form of transcription. Restricting the expression of certain calpains only to the required cells and time with the goal to avoid abnormal functions and deleterious effects is a simple but extremely efficient way of activity modulation. The most exquisite example of this is given by calpain-3. In adults, this calpain is expressed solely in the skeletal muscle.

Interestingly, gene transfer experiments showed that ectopic expression of calpain-3 in the heart leads to cardiotoxicity, demonstrating the absolute requirement to restrict the expression of the protein to the tissues that specifically require its activity [START_REF] Lostal | Titin splicing regulates cardiotoxicity associated with calpain 3 gene therapy for limb-girdle muscular dystrophy type 2A[END_REF]. The timing of expression is also an important regulation checkpoint. For example, during human and murine myogenesis, calpain-3 expression occurs only after innervation, a time when most of the other musclespecific proteins are already expressed [START_REF] Stockholm | Calpain 3 mRNA expression in mice after denervation and during muscle regeneration[END_REF]. Calpain-3 also provides a perfect example for alternative splicing as a control mechanism [START_REF] Spencer | Stable expression of calpain 3 from a muscle transgene in vivo: immature muscle in transgenic mice suggests a role for calpain 3 in muscle maturation[END_REF]. Alternative splicing of exons coding IS1 and IS2 were observed across tissues such as brain, smooth muscle, and embryonic lens or during development [START_REF] Herasse | Expression and functional characteristics of calpain 3 isoforms generated through tissue-specific transcriptional and posttranscriptional events[END_REF][START_REF] Fougerousse | Calpain3 expression during human cardiogenesis[END_REF]. These domains are important for full proteolytic capacity and their absence results in calpain-3 isoforms with reduced autolysis and/or lack of proteolytic activity, while potentially keeping the non-proteolytic function [START_REF] Stockholm | Calpain 3 mRNA expression in mice after denervation and during muscle regeneration[END_REF].

Immature conformation of the protein

During protein formation, the association with a molecular chaperone to achieve the correct conformation appears to be necessary for at least some calpains. Such a requirement for chaperone intervention represents a means to prevent immediate activation of the protease, providing an additional point of modulation. A clear example for such a mechanism is obviously the conventional calpain small regulatory subunit (CAPNS1), which has a chaperone-like function on conventional calpains and is required for the stability of both calpain catalytic subunits. Only the formation of this heterodimeric structure allows the subsequent activation of conventional calpains [START_REF] Arthur | Disruption of the murine calpain small subunit gene, Capn4: calpain is essential for embryonic development but not for cell growth and division[END_REF]. Another example is G-calpain, the heterodimeric structure formed by calpain-8 and -9. The resulting protease complex functions with calpain-8

as the active proteolytic subunit and calpain-9 probably acting like a molecular chaperone, stabilizing the structure similarly to the action of CAPNS1 with calpain-2 [START_REF] Hata | Calpain 8/nCL-2 and calpain 9/nCL-4 constitute an active protease complex, G-calpain, involved in gastric mucosal defense[END_REF][START_REF] Hata | A Gastrointestinal Calpain Complex, G-calpain, Is a Heterodimer of CAPN8 and CAPN9 Calpain Isoforms, Which Play Catalytic and Regulatory Roles, Respectively[END_REF].

Interaction with inhibitors

One of the most intuitive regulatory mechanisms is the expression of modulatory proteins able to directly inhibit calpain by inducing conformational changes, which could impair proteolytic activity or substrate accessibility. Most of the data on modulators for these proteins have been generated from studies on conventional calpains and calpain-3, whereas there are very few examples of direct endogenous regulation of the other calpains. Calpastatin is the elite inhibitor of conventional calpains. It binds directly to all domains of the calpain molecule by occupying both sides of the active cleft, at the same time looping some of the adjacent amino acid residues to protect itself from proteolysis [START_REF] Moldoveanu | Concerted multi-pronged attack by calpastatin to occlude the catalytic cleft of heterodimeric calpains[END_REF][START_REF] Hanna | Calcium-bound structure of calpain and its mechanism of inhibition by calpastatin[END_REF][START_REF] Kiss | Local structural preferences of calpastatin, the intrinsically unstructured protein inhibitor of calpain[END_REF][START_REF] Emori | Endogenous inhibitor for calcium-dependent cysteine protease contains four internal repeats that could be responsible for its multiple reactive sites[END_REF][START_REF] Maki | Repetitive region of calpastatin is a functional unit of the proteinase inhibitor[END_REF]. It was also shown that G-calpain can be inhibited by calpastatin like for the conventional calpains [START_REF] Hata | A Gastrointestinal Calpain Complex, G-calpain, Is a Heterodimer of CAPN8 and CAPN9 Calpain Isoforms, Which Play Catalytic and Regulatory Roles, Respectively[END_REF]. For the calpains that do not bind calpastatin, other safeguard mechanisms exist. For example, calpain-3 activity in skeletal muscle is heavily controlled through interaction with a giant protein of the sarcomere, titin. This protein has calpain-3 binding sites at the N2A region (I-band) and in the C terminus (M-line).

The N2A titin is able to inhibit calpain-3 activity by protecting IS2 from autolysis [START_REF] Sorimachi | Skeletal muscle-specific calpain, p94, and connectin/titin: their physiological functions and relationship to limb-girdle muscular dystrophy type 2A[END_REF][100][101]. In addition, the association of calpain-3 with titin also forces the protease to a sarcomeric localization, providing another way to prevent aberrant activity via subcellular compartmentalization [START_REF] Lostal | Titin splicing regulates cardiotoxicity associated with calpain 3 gene therapy for limb-girdle muscular dystrophy type 2A[END_REF]102].

From an inactive state to an active state

The activation of calpain is a multi-step process starting with a protein locked in an inactive conformation. The question of how a calpain is maintained in an inactive state was answered when the crystal structure of Ca 2+ -free calpain-2 was solved. The structure showed that the geometric proximity of each residue of the catalytic triad would not allow the biochemical reactions to result in a nucleophilic and therefore active cysteine [START_REF] Hosfield | Crystal structure of calpain reveals the structural basis for Ca(2+)-dependent protease activity and a novel mode of enzyme activation[END_REF][START_REF] Strobl | The crystal structure of calcium-free human m-calpain suggests an electrostatic switch mechanism for activation by calcium[END_REF]. As the crystal structures were solved of other calpain family members in their inactive states, in the absence of Ca 2+ , the same arrangement of an unassembled catalytic triad was observed in calpain-1 and calpain-5 [START_REF] Pal | Crystal structure of a micro-like calpain reveals a partially activated conformation with low Ca2+ requirement[END_REF][START_REF] Velez | Structural Insights into the Unique Activation Mechanisms of a Nonclassical Calpain and Its Disease-Causing Variants[END_REF]. For calpain-9, an unassembled catalytic triad was observed even in the calcium-bound state [START_REF] Davis | The crystal structures of human calpains 1 and 9 imply diverse mechanisms of action and auto-inhibition[END_REF]. These observations suggests a general mechanism in accordance with the remarkable similarity in the 3D structure of the proteolytic cores of the different calpains.

The Ca 2+ -induced activation mechanism of the proteases was described thanks to the observation of active versus inactive calpain-1 crystal structures [START_REF] Moldoveanu | A Ca(2+) switch aligns the active site of calpain[END_REF]103]. It appears that after binding of two Ca 2+ ions, one to PC1 and one to PC2, an unstructured loop region within PC2 undergoes a conformational change becoming a β-sheet. As a result of the new conformation, a side chain in PC2 is displaced from the active cleft, permitting the proper formation of the catalytic triad [START_REF] Khorchid | How calpain is activated by calcium[END_REF]. Assembly of the protease core to an active state through the cooperative binding of two Ca 2+ ions to the equivalent sites used in calpain-1 and -2 was also demonstrated for calpain-3 and -5 [START_REF] Ye | Structures of human calpain-3 protease core with and without bound inhibitor reveal mechanisms of calpain activation[END_REF][START_REF] Velez | Structural Insights into the Unique Activation Mechanisms of a Nonclassical Calpain and Its Disease-Causing Variants[END_REF]. As for calpain 9, the distance between the catalytic residues is still too great for the formation of the salt-bridge linking the two PC domains even after Ca 2+induced conformational changes. This prevents formation of the active triad and indicates additional and currently unknown auto-inhibition and activation mechanisms for calpain-9 [START_REF] Davis | The crystal structures of human calpains 1 and 9 imply diverse mechanisms of action and auto-inhibition[END_REF]. Ca 2+ -induced conformational changes to align the catalytic triad is only one layer of calpain activation. Proteolytic removal of N-terminal or internal peptides can be required for enzyme activation. Such cleavage can lead to additional structural changes to assemble the catalytic site, to free the active site sterically blocked by a peptide, or else change the interactions with other proteins including inhibitors. In particular, it has been demonstrated that the conventional calpains autolyse their first N-terminal domain rapidly after Ca 2+ binding [111]. Resolution of the crystal structure of calpain-2 showed that the N-terminal anchor of calpain-2 binds to CAPNS1, tethering together the two molecules [START_REF] Hosfield | Crystal structure of calpain reveals the structural basis for Ca(2+)-dependent protease activity and a novel mode of enzyme activation[END_REF]. The internal autolysis of the N-terminal segment is an intermolecular event. Release of the tether between the large and small calpain subunits allows more conformational flexibility between PC1 and PC2 and therefore may aid alignment of the catalytic triad. It is important to note, that while this autolysis occurs rapidly after Ca 2+ -induced activation it is not essential for enzyme activation [112]. N-terminal autolysis has been shown to reduce the Ca 2+ concentration required for activation [112]. Rapid autolysis of the N-terminal anchor has also been observed in calpain-1 [START_REF] Moldoveanu | A Ca(2+) switch aligns the active site of calpain[END_REF], calpain-3 [113] and calpain-9 [START_REF] Davis | The crystal structures of human calpains 1 and 9 imply diverse mechanisms of action and auto-inhibition[END_REF]. Calpain-3 is distinctive within the calpain family in that autolysis is so rapid that the full-length protein can barely be detected in vitro [START_REF] Ono | An eccentric calpain, CAPN3/p94/calpain-3[END_REF]. The underlying molecular mechanism is autoproteolysis of a further internal peptide, IS1, a sequence present at the interface of PC1 and PC2. The key event for calpain-3 activation is the intramolecular cleavage of IS1 that in turn allows for the opening of the cleft to calpain-3 substrates [START_REF] Ye | Structures of human calpain-3 protease core with and without bound inhibitor reveal mechanisms of calpain activation[END_REF]. IS1 was found to be disordered and to act as an internal inhibitory polypeptide whose cleavage is required for full activation [113,114]. This is supported by the crystal structure showing the N-terminus of

Substrate recognition

Those calpains that possess enzymatic activity exert an intracellular limited proteolytic activity on their target that modulatory rather than degradative. Calpains do not "erase" but instead "modulate" their target by generating new fragments of the protein and consequently alter its original function with changes to protein-protein interactions or subcellular distribution [START_REF] Ono | Calpains: an elaborate proteolytic system[END_REF][121][122][123].

The analysis of crystal structures and kinetics has elucidated the positions surrounding the cleavage site that are most influential for calpain cleavage [START_REF] Moldoveanu | Crystal structures of calpain-E64 and -leupeptin inhibitor complexes reveal mobile loops gating the active site[END_REF][START_REF] Cuerrier | Calpain inhibition by alpha-ketoamide and cyclic hemiacetal inhibitors revealed by X-ray crystallography[END_REF][START_REF] Cuerrier | Development of calpain-specific inactivators by screening of positional scanning epoxide libraries[END_REF][START_REF] Qian | Cocrystal structures of primed side-extending alpha-ketoamide inhibitors reveal novel calpain-inhibitor aromatic interactions[END_REF][START_REF] Moldoveanu | Concerted multi-pronged attack by calpastatin to occlude the catalytic cleft of heterodimeric calpains[END_REF][START_REF] Hanna | Calcium-bound structure of calpain and its mechanism of inhibition by calpastatin[END_REF]124]. Peptide libraries, machine learning and position-specific scoring matrix analysis have been used to reveal amino acid sequence preferences and develop calpain cleavage site predictors [START_REF] Cuerrier | Development of calpain-specific inactivators by screening of positional scanning epoxide libraries[END_REF][124][125][126][127][128][129][130][131]. Structural data showed the active cleft is flanked by two flexible loops that may confer interaction specificity to calpain substrates through variability in sequences [START_REF] Moldoveanu | Crystal structures of calpain-E64 and -leupeptin inhibitor complexes reveal mobile loops gating the active site[END_REF]. Calpain substrates identified thus far are involved in a range of cellular processes including signal transduction, membrane repair and structural sarcomeric proteins, many of which still need to be validated under physiological conditions [132-142].

ABNORMAL CALPAINS AND THERAPIES

What has been learnt from mutations in calpain genes?

The necessity for the various systems adopted to control calpain activity, described in the previous paragraphs, is also stressed by the large number of publications describing the deleterious consequences of calpain overexpression and/or exacerbated activity. At the same time, the biological importance of calpain is highlighted by the several pathological conditions associated with calpain inactivation or absence. An extensive number of studies are present in the literature regarding this topic, but this part of the review will focus on the genetic disorders caused by mutations in human calpain genes (Table 2), and on what we have learnt from the most important findings on mammalian animal models.

The identification of inactivating mutations in CAPN3 as the genetic cause of limb girdle muscular dystrophy (LGMD) is a milestone that marked both the calpain and the muscular dystrophy fields [143]. The association of calpain deficiency with a pathology was in plain contrast with the initial concept of calpain hyperactivation as an aggravating agent in myopathies [144]. The discovery totally changed the perspective on the involvement of calpains in diseases, opening an entire new line of research. However, besides the calpain-3-related

LGMD (LGMD-R1), so far only three additional monogenic disorders have been identified as being caused by mutations in calpain family members, one for the classical calpains and 2 for the non-classical calpains, while associations of pathological conditions with high-risk genetic variations is more frequent (Table 4).

LGMD-R1 is the most frequent form of LGMD, caused by recessive mutations in CAPN3 and characterized by progressive skeletal muscle degeneration with no involvement of other tissues Despite the large number of studies, the first disease-causative human mutation in conventional calpains was identified more than 50 years after the intial discovery of calpain (Tables 2 and4). Finally, this year, several variants in CAPN15 have been associated with developmental eye disorders [START_REF] Zha | Biallelic variants in the small optic lobe calpain CAPN15 are associated with congenital eye anomalies, deafness and other neurodevelopmental deficits[END_REF]. Loss of function mutations were identified in patients from 4 different families, presenting with microphthalmia and/or coloboma (disruption of the optic fissure closure). Some of the affected patients also displayed developmental delays and hearing loss. In the same study, it has been shown that calpain-15 KO mice suffered a marked growth deficit and exhibit various eye defects, including anophtalmia or, microphtalmia and cataract, confirming a critical role for calpain-15 in the development of the ocular and neuromotor systems and comforting the implication of calpain 15 as a bona fide causative gene for the disease [START_REF] Zha | Biallelic variants in the small optic lobe calpain CAPN15 are associated with congenital eye anomalies, deafness and other neurodevelopmental deficits[END_REF].

In addition to these monogenic diseases, it has also been suggested that single nucleotide polymorphisms (SNPs) in CAPN8/9 might be linked with gastric ulcer susceptibility [START_REF] Hata | Calpain 8/nCL-2 and calpain 9/nCL-4 constitute an active protease complex, G-calpain, involved in gastric mucosal defense[END_REF]

Promising therapeutic approaches to cure a problem in calpain

According to the 2016 classification by Ono, Saido and Sorimachi, calpain-related diseases can be divided in three main categories: type 1, caused by exacerbated calpain activity; type 2, caused by parasites that use their or the host's calpain for their own survival; type 3, those genetic disorders caused by loss or gain of function mutations in one of the calpain genes (described in the previous chapters) [START_REF] Ono | Calpain research for drug discovery: challenges and potential[END_REF]. Accordingly, a therapeutic strategy would involve either the inhibition of calpain activity (type 1, type 2 and gain of function type 3), or the rescue of calpain function (for loss of function type 3).

A comprehensive list of calpain inhibitors has been provided in a previous review by Sorimachi in 2016 [START_REF] Ono | Calpain research for drug discovery: challenges and potential[END_REF]. We also refer the readers to a review by Dókus and colleagues about the known modulators of calpain activity and their possible application as potential drugs [START_REF] Dokus | Modulators of calpain activity: inhibitors and activators as potential drugs[END_REF]. For now, only one inhibitor has been used in clinical trials [174][175][176]. Unfortunately, the mere inhibition of calpain might not be sufficient to resolve the pathology. Sometimes it may even have a series of undesired effects because of the lack of specificity, or the need for more fine-tuned regulation. Hyperactive mutations were identified in CAPN5. Interestingly, it was shown that calpain-5 KO mice do not present an apparent phenotype, suggesting that inhibition of calpain-5 might be a good candidate for a safe therapeutic strategy [177]. In addition, the availability of the crystal structure of calpain-5 in the inactive conformation provides useful structural information that can potentially guide effective inhibitor design [START_REF] Velez | Structural Insights into the Unique Activation Mechanisms of a Nonclassical Calpain and Its Disease-Causing Variants[END_REF].

For calpain loss-of-function mutations, the most intuitive strategy would be replacing the necessary level of expression via gene therapy. This may not be straightforward for a congenital form with developmental defects like in calpain-15 related diseases, but it might be for recessive mutations in LGMD-R1. To date, a gene transfer strategy mediated by vectors derived from the adeno-associated virus (AAV) is under development at Genethon for LGMD-R1. A first preclinical proof of concept was obtained in 2006, with a study showing the efficacy of an AAV1 vector expressing calpain-3 under control of a muscle-specific promoter to restore the phenotype of mice deficient in calpain-3 after local administration [START_REF] Duguez | Calpain 3: a key regulator of the sarcomere?[END_REF]. The data showed transgene expression with correct protein localization at the sarcomere level and restoration of proteolytic activity, with correction of muscle atrophy and improvement in strength. However, a systemic administration of the same vector caused cardiotoxicity related to the ectopic expression of calpain-3 in the heart. To overcome the cardiotoxic adverse effect without compromising the therapeutic effect in skeletal muscle, modified vectors were developed [178].

A second generation vector was designed by introducing target sequences of a cardiac musclespecific microRNA (miR208a) in the transgene expression cassette, or by the use of promoter sequences derived from endogenous promoters. In order to confirm the tolerance of this new type of vector, a pilot biodistribution and safety study was conducted in non-human primates.

Results showed a total absence of cardiotoxicity after systemic injection with the modified vector. A study focusing on understanding the toxicity identified a mechanism explaining how the activity of calpain-3 is regulated in skeletal muscle [START_REF] Lostal | Titin splicing regulates cardiotoxicity associated with calpain 3 gene therapy for limb-girdle muscular dystrophy type 2A[END_REF]. A particular region of titin, a protein involved in the elasticity of the muscle to which calpain-3 binds, is indeed playing a role in inhibiting enzyme activity. Interestingly, this regulatory mechanism seems to be different between human and mouse hearts. With these encouraging results, a clinical trial is now in preparation at Genethon.

CONCLUSIONS AND CONTRIBUTIONS BY DOCTOR HIROYUKI SORIMACHI TO THE CALPAIN FIELD

Lastly, in the following paragraphs, we want to give the reader a notion of the extent of Hiro's legacy to the calpain field. Those who knew him remember his heartfelt laugh, his love of French food and most importantly French wine. Work in the calpain field will continue and his memory will live on through it, and in the heart of his friends and colleagues.

Dr. Hiroyuki Sorimachi was the first to demonstrate the existence of tissue-specific calpains.

In 1989, he and his colleagues screened cDNA hybridization libraries from human and rat muscle with a probe for the chicken calpain-1 and obtained a cDNA clone encoding for a novel member of the calpain family with homology of more than 50% to the ubiquitous human calpain-1 and -2. With a series of exquisite experiments for the time, they gave the first description of the structure of calpain-3 and via northern blots analyses they demonstrated that the expression of this new calpain was restricted to skeletal muscle [START_REF] Sorimachi | Molecular cloning of a novel mammalian calcium-dependent protease distinct from both m-and mu-types. Specific expression of the mRNA in skeletal muscle[END_REF]. In 1993, by screening cDNA libraries from various rat tissues, he also identified a new calpain with specific expression in the stomach (calpain-8) and showed that two isoforms of the protein were formed via alternative splicing [START_REF] Sorimachi | A novel tissue-specific calpain species expressed predominantly in the stomach comprises two alternative splicing products with and without Ca(2+)-binding domain[END_REF]. Later in the nineties, while performing the genomic cloning of CAPN8, Dr. Sorimachi and Dr. Koichi Suzuki's research group identified another novel member of the calpain family (calpain-9) and demonstrated that its mRNA expression was restricted to the digestive tract [START_REF] Lee | Molecular cloning and characterization of a novel tissue-specific calpain predominantly expressed in the digestive tract[END_REF]. Hiroyuki and Koichi also had a huge impact on our understanding of the structure of calpain. In 1998, they purified for the first time a recombinant human CAPN2 using a baculovirus system, and in the early 2000s, the crystal structure of calpain-2 was published in collaboration with Dr. Wolfram Bode [START_REF] Strobl | The crystal structure of calcium-free human m-calpain suggests an electrostatic switch mechanism for activation by calcium[END_REF]180]. Together with Dr.

Suzuki, Hiroyuki's research triggered investigations into evolutionary relationships among calpain family members and was pivotal for the characterization of this superfamily [181].

Following this pioneering work, the advent of High-Throughput Sequencing Technologies allowed for the identification of all 15 members of the calpain family listed above, half of which are predominantly tissue-specific (Table 2).

Dr. Sorimachi also contributed tremendously in unravelling the functions of several calpain family members. His most studied calpain family member was calpain-3. He was the first to describe the unusual short half-life of the muscle-specific calpain-3 and since then he went deeper into the understanding of its autolysis mechanism [101,182,183]. Dr. Sorimachi discovered calpain-3 interacts with titin and also described the mechanistic implications of this binding [START_REF] Sorimachi | Structure and physiological functions of ubiquitous and tissuespecific calpain species. Muscle-specific calpain, p94, interacts with connectin/titin[END_REF]100,101,184,185]. His laboratory also described the interaction between calpain-3 and substrates other than titin and provided evidence for a possible protein synthesis regulation by calpain-3 [186, 187] and contributed in our understanding of calpain-3 proteolytic activity and unconventional functions [START_REF] Ojima | Non-proteolytic functions of calpain-3 in sarcoplasmic reticulum in skeletal muscles[END_REF]121,123,[188][189][190]. The Sorimachi lab also discovered and characterized a gastrointestinal calpain complex formed by isoforms of calpain-8 and -9 involved in gastric mucosal defense [START_REF] Hata | Calpain 8/nCL-2 and calpain 9/nCL-4 constitute an active protease complex, G-calpain, involved in gastric mucosal defense[END_REF][START_REF] Hata | A Gastrointestinal Calpain Complex, G-calpain, Is a Heterodimer of CAPN8 and CAPN9 Calpain Isoforms, Which Play Catalytic and Regulatory Roles, Respectively[END_REF]. They described the localization of calpain-8 expression in gastric and duodenal mucosal cells, its Golgi localization and its interaction with the coatomer complex suggesting functions in membrane trafficking [191]. In addition, using a mouse model deficient for calpain-6, they were able to demonstrate how the loss of this protein promotes skeletal muscle development and regeneration [START_REF] Tonami | Calpain-6 deficiency promotes skeletal muscle development and regeneration[END_REF]. Another Sorimachi's contribution was to demonstrate that calpain has a weak substrate specificity [124,187].

Finally, he and his research group made huge efforts to make calpain information accessible to all through online resources (Table 1). The readers will find in these websites many information on the history and many other aspects of the calpain field.

Figure1-

  Figure1 -Nomenclature timeline of the vertebrate calpain family. In the upper panel, boxes

Figure 2 :

 2 Figure 2: The domains of calpains. Abbreviations used for domain names: CysPc: calpain-

  IS1 residing and binding to residues within the active cleft, therefore blocking access to the catalytic cleft. In addition, autolysis leads to instability, regulating the duration of enzyme activity within the cell.Additional post translation modifications can modulate calpain activity. Phosphorylation is one of the most common protein modifications and calpains are no exception. Several findings indicate that phosphorylation plays a major role in the regulation of calpain activation[115- 117]. Tridimensional modeling revealed that phosphorylation at serine 369 in calpain-2 by protein kinase A generates constraints in domain movement, contributing to keep the protease in an inactive state[115]. In contrast, the phosphorylation of calpain-1 promotes its activity, by conformational changes yet to be characterized[116]. Interestingly, another reversible mechanism for control of proteolytic activity, acting directly through modification of the catalytic triad, has been proposed. Indeed, nitric oxide seems to reduce calpain-mediated proteolysis in cultured myotubes through nitrosylation of the cysteine of the active site[118, 119].

Finally, another indirect

  way to modulate calpain activity is to modify its interactors or substrates. For example, phosphorylation of calpastatin has been shown to decrease its inhibitory ability on conventional calpains[117]. It has recently also been found that different human cells express an alternatively spliced calpastatin (named hcast 3-25) that lacks all the exons coding for the inhibitory domains and that this form of calpastatin operates as a positive modulator of calpain-1 [120].

[

  143, 145]. The disease onset occurs in the first or second decade of life, leading to decreased ambulation and difficulties in performing daily life activities[145][146][147][148]. Recapitulation of the pathological phenotype in different calpain-3 murine models, although with reduced severity, confirmed mutations in CAPN3 are disease causing. While there is a lack of understanding of the molecular mechanism leading to the disease, analyses of these models, as well as of human biopsies, suggests an abnormal adaptation of the muscle fibers as a plausible pathological mechanism[START_REF] Duguez | Calpain 3: a key regulator of the sarcomere?[END_REF]. Dominant variants in CAPN3 were also increasingly being associated withLGMD, creating a new subtype of the disorder: LGMD-D4, with similar but more mild symptoms and a delayed onset compared to LGMD-R1 [149, 150]. Further analyses showed a loss of proteolytic activity in the mutated proteins and a dominant negative effect on wild-type calpain-3 [150]. Pathological mutations in additional calpain family members were identified some seventeen years after the initial genetic diagnosis for calpain-3 was made. In 2012, mutations in CAPN5 were reported to be responsible for autosomal dominant neovascular inflammatory vitreoretinopathy (ADNIV). This genetic disorder is characterized by a progressive retinal degeneration with presence of inflammatory T-cells and severe neovascularization, that leads to bleeding, formation of scar tissue and ultimately blindness [151]. ADNIV-causative mutations in CAPN5 were demonstrated to lead to a gain-of function representing the first example, and the only one to date, of a dominantly inherited disease caused by an hyperactive calpain [152, 153]. These gain of function mutations are located in a region encoding a flexible loop whose function is to limit substrate access to the catalytic site instead, the mutations lead to increased proteolytic activity and lower Ca 2+ -activation threshold [152, 153]. A mouse model heterozygous for one of these mutations (R243L) recapitulated the ADNIV phenotype, even if with milder symptoms, confirming the hypothesis of a dominant gain of function effect [152]. Interestingly, despite high expression of calpain-5 in other tissues, hyperactive mutation-related pathological effects were only observed in the ocular system [152]. This observation may be related to the high Ca 2+ threshold required for activation and the high Ca 2+ concentrations present in the retinal cells [154].

  An emerging strategy to cure calpain disease is the genetic correction of calpain-linked mutations in patients-derived induced pluripotent stem cells for their prospective use in an autologous cell-based therapy. Promisingly, gene editing of LGMD2A iPSC has been recently proven to be effective in restoring CAPN3 expression level and in rescuing muscle function upon transplantation of corrected iPSC-derived muscle progenitors in immunodeficient CAPN3-KO mice (iPSC)[179].

Table 2 -List of human calpain genes and their first appearance in the literature. CAPN1

 2

	and CAPN2 were purified as respective homodimers with the CAPNS1 subunit.

Table 3 :

 3

Crystal structures solved for the calpain family. (*)

[START_REF] Berman | The Protein Data Bank[END_REF]

Since the 2012 review by

  Conventional calpains require high concentrations of Ca 2+ to be activated in vitro[104, 105]. A burning question within the calpain field is, how are calpains activated within the cell where it is difficult to reach such high Ca 2+ levels? Evidences that the Ca 2+ concentration required for

	calpain-1 activation is considerably lowered by the presence of particular phospholipids [106,
	107] and that vicinity to the membrane and binding to phospholipids regulate calpain-2
	activation support the hypothesis of subcellular compartmentalization as a mechanism of
	calpain regulation [108, 109]. Calpain-5 is another calpain requiring millimolar concentration
	of Ca 2+ for its activation like calpain-2. The reason for this threshold is the presence of
	additional flexible loops leading to a particular protease core conformation associated with the
	lack of the Ca

2+

-regulated PEF domain typical of the non-classical calpains

[START_REF] Velez | Structural Insights into the Unique Activation Mechanisms of a Nonclassical Calpain and Its Disease-Causing Variants[END_REF]

. Of interest, the cells where CAPN5 is expressed have high resting Ca 2+ levels. This necessity for extreme Ca 2+ levels is likely an evolutionary adaptation providing either a specific cell defense mechanism to cells expressing high basal Ca 2+ levels such as retinal cells

[110]

or a means to avoid mislocalized and/or accidental activation when associated with spatial compartmentalization.

  Recessive mutations in CAPN1 were identified in 2016 as the cause of neurological

	disorders such as spastic paraplegia, spinocerebellar ataxia or as a combination of both. Spastic
	paraplegia is characterized by progressive muscle stiffness and hyperreflexia in the lower limbs
	due to corticospinal tract degeneration [155]. Spinocerebellar ataxia is characterized by
	progressive degeneration of the cerebellum leading to gait imbalance associated with limb
	incoordination [156]. Other studies correlated additional CAPN1 mutations to these
	neurodegenerative disorders, or even a combination of both [157-160]. These discoveries
	encouraged a deeper analysis of Capn1-KO mice, which were initially reported as showing no
	pathological phenotype, except for defects in platelet aggregation [161]. A more careful
	characterization of this mouse model suggested an implication of calpain-1 in a pro-survival
	pathway in cells of the cerebellum as well as in synaptic plasticity in both the hippocampus and
	cerebellum [156, 162]. Surprisingly, there is an absence of reported CAPN2 disease-causing
	mutations in humans (Table 4). Of note, this might be related to a deleterious effect of the loss
	of function of this protease during development as shown by the fact that calpain-2 knock-out
	mice are embryonic lethal, suggesting the possibility of non-detected embryonic lethal CAPN2
	mutations in humans [163, 164]. Recently, it has also been proposed that CAPN1 and CAPN2
	play opposite role in the brain: a neuroprotective action from calpain-1 and a neurodegenerative
	effect from calpain-2 [12, 165, 166]

  , thatSNPs in CAPN10 are linked to gestational diabetes mellitus[167], that mutations in CAPN12 may aggravate symptoms of congenital ichthyosis[168] and that SNPs in CAPN14 are associated with eosinophilic oesophagitis [169, 170] (Table4). It is also interestingly to note that mutations in calpastatin have been reported to lead to a peeling skin syndrome[171].

	Gene	Phenotypes associated to gene variant	Type of variant
	CAPN1	*Spastic Paraplegia[155, 159, 160] and	Causative loss-of-function mutations
		spinocerebellar ataxia [156].
	CAPN3	LGMD-R1 [143]	Causative loss-of-function mutations
		*LGMD-D4 [149]
	CAPN5	*Autosomal dominant neovascular inflammatory	Causative gain-of-function mutations
		vitreoretinopathy [152, 153, 172, 173].
	CAPN8	Gastric ulcers susceptibility [82]	Suspect that SNPs may be responsible
			for the susceptibility
	CAPN9	Gastric ulcers susceptibility [82]	Suspect that SNPs may be responsible
			for the susceptibility
	CAPN10	Type 2 diabetes [49, 167]	SNPs associated to disease
	CAPN12	*Congenital ichthyosis [168]	Mutations that aggravate the existing
			phenotype
	CAPN14	Eosinophilic oesophagitis [169, 170]	SNPs associated to disease
	CAPN15	*Microphthalmia and/or coloboma [90]	Causative loss-of-function mutations

Table 4 -List of genetic calpain variations associated with human pathologies. New

 4

associations since last review from Sorimachi and colleagues in 2016 (*)

[START_REF] Ono | Calpain research for drug discovery: challenges and potential[END_REF]

.

ACKNOWLEDGMENTS

We would like to thank Evelyne Gicquel and Celia Thevenard for their help during the preparation of the manuscript.

We would like to acknowledge the support of the Foundation Coalition to Cure Calpain 3 and the Association Lumyosa.


Stefan Nemanja
email: nemanja.stefan.perovic@ucd.ie

Member, IEEE Le-Nam Perović

Senior Member, IEEE Marco Di Tran
email: nam.tran@ucd.ieandmark.flanagan@ieee.org

Fellow, IEEE Mark F Renzo
email: marco.di-renzo@universite-paris-saclay.fr

Senior Member, IEEE Flanagan

N S Perović

L N Tran

M F Flanagan

M Di Renzo

Optimization of RIS-aided MIMO Systems via the Cutoff Rate

Keywords: Channel cutoff rate (CR), mutual information (MI), multiple-input multiple-output (MIMO), optimization, reconfigurable

The main difficulty concerning optimizing the mutual information (MI) in reconfigurable intelligent surface (RIS)aided communication systems with discrete signaling is the inability to formulate this optimization problem in an analytically tractable manner. Therefore, we propose to use the cutoff rate (CR) as a more tractable metric for optimizing the MI and introduce two optimization methods to maximize the CR, assuming perfect knowledge of the channel state information (CSI). The first method is based on the projected gradient method (PGM), while the second method is derived from the principles of successive convex approximation (SCA). Simulation results show that the proposed optimization methods significantly enhance the CR and the corresponding MI.

I. INTRODUCTION

The recently developed reconfigurable intelligent surfaces (RISs) have the potential to shape and control the radio wave propagation in wireless networks, which makes them a promising candidate for future beyond-5G communication systems. RISs consist of a large number of small, low-cost, and nearly-passive elements each of which can reflect the incident signal through an adjustable phase shift, thereby modifying the wavefront of the scattered wave [START_REF] Renzo | Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead[END_REF]. Changing the wavefront of the reflected signals enables us to shape how the radio waves propagate through the channel, and thus improve key system performance metrics such as the achievable rate. The main body of research work in this area concentrates on the achievable rate optimization for single-user [START_REF] Wu | Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming[END_REF] and multi-user [START_REF] Nadeem | Asymptotic max-min SINR analysis of reconfigurable intelligent surface assisted MISO systems[END_REF] multiple-input single-output (MISO) communication systems. Another significant body of research work has focused on the achievable rate optimization for multiple-input multiple-output (MIMO) systems equipped with RISs in single-user [START_REF] Perović | Channel capacity optimization using reconfigurable intelligent surfaces in indoor mmWave environments[END_REF], [START_REF]Achievable rate optimization for MIMO systems with reconfigurable intelligent surfaces[END_REF] and multi-user [START_REF] Pan | Multicell MIMO communications relying on intelligent reflecting surfaces[END_REF], [START_REF]Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF] communications. In the aforementioned papers, the transmitted symbols are distributed according to a circularly-symmetric complex Gaussian distribution, which is a capacity achieving distribution. However, in practice the transmitted symbols are usually chosen from a discrete signal constellation and thus the present solutions cannot be used to establish realistic bounds on the achievable data rate in practical RIS-aided communication systems.

The main difficulty concerning the practical achievable rate, i.e., the analysis and optimization of the mutual information (MI) in RIS-aided communication systems with discrete signaling, is the inherent difficulty of formulating this optimization problem in an analytically tractable manner. Hence, MI optimization and analysis were considered in only a few publications. In [START_REF] Karasik | Adaptive coding and channel shaping through reconfigurable intelligent surfaces: An information-theoretic analysis[END_REF], the authors considered the mutual information optimization for an RIS-aided system, where the transmit information is encoded into in-phase and quadrature (IQ) symbols transmitted from a single transmit antenna and also into the (discrete) RIS phase shifts. However, the proposed optimization approach in [START_REF] Karasik | Adaptive coding and channel shaping through reconfigurable intelligent surfaces: An information-theoretic analysis[END_REF] is not directly implementable to MIMO systems that transmit multiple IQ symbols in parallel. The mutual information analysis for a MISO communication system, where the transmit information is encoded into the IQ symbol and a subset of active RIS elements, is presented in [START_REF] Lin | Reconfigurable intelligent surfaces with reflection pattern modulation: Beamforming design and performance analysis[END_REF]. The receiver device is equipped with a single receive antenna in [START_REF] Lin | Reconfigurable intelligent surfaces with reflection pattern modulation: Beamforming design and performance analysis[END_REF], so the proposed optimization method is not applicable to multi-stream MIMO systems.

Motivated by this, we propose to use the channel cutoff rate (CR) as a surrogate metric for optimizing the MI of RISaided multi-stream MIMO communication systems, assuming perfect knowledge of the channel state information (CSI). The idea comes from the fact that the channel CR serves as a practical upper limit on the information rate for reliable communications [START_REF] Massey | Coding and modulation in digital communications[END_REF]. In mathematical terms, the relation between the channel CR R 0 and the codeword error probability P e can be formulated as P e ≤ e -n(R0-R) , where n is the codeword length and R is the information rate in bits per channel use (bpcu). In other words, for very long coded sequences (i.e., n → ∞), P e can be made arbitrarily small as long as R < R 0 . Since the channel capacity is the theoretical upper limit of the information rate for reliable communications, the CR is usually employed as a practical lower bound on the channel capacity. More specifically, it can be shown that a lower bound on the MI is determined by the CR as [START_REF] Perović | Optimization of the cut-off rate of generalized spatial modulation with transmit precoding[END_REF]Eq. (36)]

MI ≥ N r (1 -log 2 e) + R h 0
where R h 0 corresponds to the CR evaluated at half the noise power. In practical terms, the use of the channel CR facilitates the otherwise intractable optimization of the channel capacity (i.e., MI) and provides us with a tool to optimize modulation techniques for communication systems.

Against this background, the contributions of this paper are listed as follows:

1) Instead of optimizing the MI directly, which results in an intractable problem, we propose to use the CR as a surrogate metric for the MI optimization. We show that the CR can lead to a deterministic optimization problem for which efficient numerical algorithms can be derived. 2) To maximize the CR for the considered system, we formulate a joint optimization problem of the precoding matrix and the RIS elements' phase shifts. Since the problem of interest is nonconvex and our numerical experiments indicates that it contains many local optima (see Section V), we propose two local optimization methods for solving this problem. The first method is a projected gradient method (PGM) which admits closedform expressions in each iteration. The second method is derived from the principles of successive convex approximation (SCA). To achieve this, we prove that the nonconvex unit-modulus constraints on the RIS elements can be convexified without loss of optimality. Hence, one of the proposed methods can avoid being trapped in an unsuitable local optimum, and thus will provide a near-optimal solution for a given channel realization. 3) We present simulation results which show that the proposed optimization methods, by maximizing the CR, can substantially increase the MI. Moreover, the CR optimization produces superior MI results compared to the achievable rate optimization for Gaussian signaling.

II. SYSTEM MODEL

A. System Model

An aerial view of the considered system is shown in Fig. 1. It contains a transmitter equipped with N t antennas and a receiver equipped with N r antennas, where we assume N t ≥ N r . The separations between the adjacent antennas in the transmit and the receive antenna array are s t and s r , respectively. These antenna arrays are placed on parallel vertical walls, which are at a distance D from each other 1 . To mitigate the blockage (i.e., attenuation) of the direct link, an RIS, whose midpoint is at a distance d ris from the plane containing the transmit antenna array, is installed. The RIS is placed on a vertical wall that is perpendicular to both the transmit and the receive antenna array. We assume that the RIS, the transmit antenna array and the receive antenna array are approximately at the same height. The RIS consists of N ris reflecting elements, which are placed in a rectangular formation such that the separation between the centers of adjacent RIS elements in both dimensions is s ris = λ 2 , where λ denotes the wavelength. The distance between the midpoint of the transmit (respectively, receive) antenna array and the plane containing the RIS is l t (respectively, l r ).

The signal vector at the receive antenna array is given by

y = HPx i + n, (1)
1 This system geometry described in the paper is adopted for ease of exposition, and to provide the reader with a concrete use case. However, the optimization approach proposed in this letter is applicable to any system geometry. where H ∈ C Nr×Nt is the channel matrix, P ∈ C Nt×Nr is the transmit precoding matrix and x i ∈ C Nr ×1 is the transmit symbol vector. The elements of x i are chosen from a discrete symbol alphabet of size M with unit average symbol energy.

The number of different transmit symbol vectors is N s = M Nr . We assume that the precoding matrix P preserves the average power of the transmit signal (i.e., Tr(PP H ) = N r ).

The noise vector n ∈ C Nr×1 consists of independent and identically distributed (i.i.d.) elements that are distributed according to CN (0, σ 2 ), where σ 2 denotes the noise variance. Since an RIS is present in this system, the channel matrix can be expressed as

H = β -1 DIR H D + β -1 INDIR H 2 F(θ)H 1
, where H D ∈ C Nr×Nt represents the direct link between the transmitter and the receiver, H 1 ∈ C Nris×Nt represents the link between the transmitter and the RIS, and H 2 ∈ C Nr×Nris represents the link between the RIS and the receiver. The distancedependent path loss for the direct link is β -1 DIR and the free space path loss (FSPL) for the indirect link is β -1 INDIR . Signal reflection from the RIS is modeled by

F(θ) = diag(θ) ∈ C Nris×Nris , where θ = [θ 1 , θ 2 , . . . , θ Nris ] T ∈ C Nris×1 .
Since we assume that the reflection is without any power loss, we may write θ l = e jφ l (i.e., |θ l | = 1) for l = 1, 2, . . . , N ris , where φ l is the phase shift induced by the l-th RIS element. In this letter, we assume perfect knowledge of the CSI.

III. PROBLEM FORMULATION

Since the considered system is a discrete-input continuous-output memoryless channel (DCMC), the MI is given by [12, Eq. (7.2.1)]

MI = Ns i=1 y p(y, x i ) log 2 p(y|x i ) Ns j=1 p(y, x j ) dy, (2)
where the conditional probability density function is

p(y|x i ) = 1/ πσ 2 Nr exp - y -HPx i 2 σ 2 . (3)
Assuming that all the transmit symbol vectors are equally probable p(x i ) = 1/N s (i = 1, . . . , N s ), we obtain

MI = log 2 N s - 1 N s Ns i=1 E n log 2 Ns j=1 exp(ψ i,j ) , (4)
where

ψ i,j = (-HP(x i -x j ) + n 2 + n 2 )/σ 2 .
We remark that the expression for the MI in (4) leads to an intractable stochastic optimization problem, if one wishes to maximize the MI directly. The fact that the MI expression in ( 4) is neither convex nor concave with respect to P and θ also adds to the difficulty. Also, the feasible set for θ l , which satisfies |θ l | = 1, is nonconvex. To overcome these issues, we propose the use of the CR as an auxiliary metric whose optimization is well-aligned with that of the MI.

A. Derivation of the CR

The CR of the considered system for equiprobable symbol vectors x i is given by [START_REF] Massey | Coding and modulation in digital communications[END_REF]

R 0 = -log 2 1 N 2 s y Ns i,j=1
p(y|x i )p(y|x j )dy . (5)

By inserting (3) into (5) and after some algebraic manipulations, the CR of the considered system can be written as

R 0 = -log 2 1 N 2 s Ns i,j=1 exp - Φ i,j (θ, P) 4σ 2 , ( 6
)
where Φ i,j (θ, P) = HPe i,j 2 and e i,j = x ix j .

B. Optimization Problem

Upon close inspection of ( 6), it is easy to see that the CR optimization problem can be formulated as

minimize θ,P f (θ, P) = Ns i,j=1 exp - Φ i,j (θ, P) 4σ 2 (7a) subject to Tr(PP H ) = N r (7b) θ l = 1, l = 1, 2, . . . , N ris .
(7c) Since the objective of ( 7) is nonconvex, finding a globally optimal solution is very difficult. Hence, in the next section, we present two local optimization algorithms to solve [START_REF]Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF].

IV. PROPOSED OPTIMIZATION METHODS

A. Projected Gradient Method (PGM)

The first proposed method is based on the PGM [START_REF] Boyd | Convex optimization[END_REF], which consists of the following iterations:

θ n+1 = P Θ (θ n -µ 1 ∇ θ f (θ n , P n )), (8a)
P n+1 = P P (P n -µ 2 ∇ P f (θ n+1 , P n )), (8b)
where ∇ θ f (θ, P) and ∇ P f (θ, P) are the gradients of f (θ, P) with respect to θ * and P * , respectively 2 , and µ 1 and µ 2 are the corresponding step sizes. Also, P Θ (•) and P P (•) denote the projection onto Θ and P, respectively, which are detailed in the sequel. Note that we use the complex-valued gradients [START_REF] Hjörungnes | Complex-Valued Matrix Derivatives With Applications in Signal Processing and Communications[END_REF], the explicit forms of which are provided in the next lemma.

Lemma 1. The gradients of f (θ, P) with respect to θ * and P * are given by

∇ θ f (θ, P)=- 1 4σ 2 Ns i,j=1 e - Φ i,j (θ,P) 4σ 2 vec d (H H 2 HPi,j P H i,j HH 1 ) (9a) ∇Pf (θ, P)=- 1 4σ 2 H H HP Ns i,j=1 e - Φ i,j (θ,P) 4σ 2 ei,je H i,j , (9b)
where P i,j = Pe i,j , H1 = β -1 INDIR H 1 and vec d (A) denotes the vector comprised of the diagonal elements of A.

Proof: See Appendix A. 1) Projection Operations: Next we show that the projection operations in (8a) and (8b) can be calculated in closed form. Note that the constraint θ l = 1 states that θ l lies on the unit circle in the complex plane. Thus, for a given point θ ∈ C Nris×1 , P Θ (θ) is given by

θl = θ l |θ l | θ l = 0 e jφ , φ ∈ [0, 2π] θ l = 0 , l = 1, . . . , N ris . ( 10
)
2 Here x * denotes the complex conjugate of x.

In particular, θl can be any point on the unit circle if θ l = 0, and thus P Θ (θ) is not unique. Similarly, the constraint Tr(PP H ) = N r implies that the projection of the precoding matrix P P (P) is given by

P = P N r /Tr(PP H ). ( 11
)
2) Backtracking Line Search: Appropriate choices of the step sizes in (8a) and (8b) are instrumental for ensuring the convergence of the PGM. Ideally, each step size should be inversely proportional to the Lipschitz constant of the corresponding gradient. However, the optimal Lipschitz constants are difficult to find for our considered problem. Therefore, we utilize the Armijo-Goldstein backtracking line search to determine the step sizes µ 1 and µ 2 at each iteration.

Let L 0 > 0, δ > 0 be a small constant, and ρ ∈ (0, 1). The step size µ 1 in (8a) is found as L o ρ kn , where k n is the smallest nonnegative integer such that

f (θ n+1 , P n ) ≤ f (θ n , P n ) -δ||θ n+1 -θ n || 2 . ( 12
)
The step size µ 2 is found similarly. The proposed line search procedure ensures that the objective sequence strictly decreases after each iteration. Thus, the PGM is guaranteed to converge to a stationary point of ( 7), which is, however, not necessarily a globally optimal solution.

B. Successive Convex Approximation (SCA)

Since ( 7) is a nonconvex problem, the proposed PGM can become trapped in an unsuitable local optimum. Thus, it is desirable to check the obtained solution with another local optimization method that is developed by applying a different optimization paradigm. Our expectation is that the two proposed local optimization methods can complement each other and one of them can escape unsuitable local optima. In particular, the second proposed method is derived from the SCA, which has been shown to be a powerful tool for a range of nonconvex optimization problems.

First, we relax the equality constraints in [START_REF]Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF], leading to the following program:

minimize f (θ, P) = Ns i,j=1 exp - Φ i,j (θ, P) 4σ 2 (13a) subject to Tr(PP H ) ≤ N r (13b) |θ l | ≤ 1, l = 1, 2, . . . , N ris .
(13c) It is easy to see that Φ i,j (θ, αP) > Φ i,j (θ, P) for any α > 1. Thus, the constraint in (13b) must hold with equality at the optimum. Otherwise, we can always scale up P and achieve a strictly smaller objective. Similarly, it can be shown that (13c) holds with equality at the optimum (see Appendix B). These two observations imply that the set of optimal solutions of ( 13) is the same as that of [START_REF]Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF]. Also, the aforementioned relaxation ensures the convexity of the feasible sets for θ and P, while the objective function f (θ, P) remains nonconvex.

Let us suppose that we have already obtained a feasible point (P n , θ n ). To find P n+1 based on the SCA method, we need to find a convex upper bound on f (θ n , P). Since Φ i,j (θ n , P) is convex with respect to P, we have

Φ i,j (θ n , P) ≥ Φ i,j (θ n , P n ) + ∇ P Φ i,j (θ n , P n ), P -P n △ = Φi,j (θ n , P n ; P) (14)
where X, Y = 2ℜ(Tr(X H Y)). In fact, Φi,j (θ n , P n ; P) is an affine approximation of Φ i,j (θ n , P) around P n . Next, P n+1 is a solution to the following convex problem:

min P Ns i,j=1
exp -Φi,j (θn, Pn; P)

4σ 2 Tr(PP H ) ≤ Nr . (15)
To find θ n+1 , we again need to derive a convex upper bound on f (θ, P n+1 ). It is easy to see that Φ i,j (θ, P n+1 ) is convex with respect to θ and thus we have

Φ i,j (θ, P n+1 ) ≥ Φ i,j (θ n , P n+1 )+ ∇ θ Φ i,j (θ n , P n+1 ), θ -θ n △ = Φi,j (θ n , P n+1 ; θ). (16)
Next, θ n+1 is a solution of the following convex problem:

minimize θ Ns i,j=1
exp -Φi,j (θ n , P n+1 ; θ)

4σ 2 (17a)
subject to |θ l | ≤ 1, l = 1, 2, . . . , N ris . (17b) Both convex optimization problems in ( 15) and ( 17) can be solved by off-the-shelf convex solvers. By minimizing an upper bound in each iteration, the SCA-based method generates a decreasing objective sequence. It can shown that the SCA-based method converges to a stationary point of [START_REF] Boyd | Convex optimization[END_REF].

C. Complexity Comparison

We next provide a brief analysis of the complexity of the proposed methods based on the required number of complex multiplications. The per-iteration complexity of the PGM method is O(N 2 s N ris (N t + N r )). To solve subproblems [START_REF] Björnson | Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces[END_REF] and (17), we first need to compute [START_REF] Hjörungnes | Complex-Valued Matrix Derivatives With Applications in Signal Processing and Communications[END_REF] and [START_REF] Danufane | On the path-loss of reconfigurable intelligent surfaces: An approach based on Green's theorem applied to vector fields[END_REF], and this requires approximately the same per-iteration complexity as the PGM. By treating [START_REF] Björnson | Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces[END_REF] and (17) as generic convex problems, the worst-case complexity for solving [START_REF] Björnson | Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces[END_REF] and ( 17) is O(N 4 t N 4 r ) and O(N 4 ris ), respectively. The SCA-based method generally requires higher computational complexity to return a solution than the PGM method. The reason is that convex solvers commonly use an interior point method to solve problems [START_REF] Björnson | Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces[END_REF] and (17) which involve exponential cones. The memory requirement and the computational complexity increase very quickly with the problem size. Consequently, the SCA-based method is not suitable for large-scale scenarios, i.e., when N s or N ris is large. For such cases, the PGM is the only viable option. For problems of moderate size, the SCA-based method is a good choice since it is a descent method without a line search. On the other hand, the PGM method requires a line search to ensure a good convergence rate.

V. SIMULATION RESULTS

In this section, we evaluate the CR and the MI of the proposed optimization algorithms with the aid of Monte Carlo simulations. More precisely, we utilize θ and P obtained by optimizing the CR to calculate the MI according to the expression in [START_REF] Perović | Channel capacity optimization using reconfigurable intelligent surfaces in indoor mmWave environments[END_REF], where the expectation is evaluated by generating random noise vectors. Also, we compare the MI optimization results in the case of discrete and Gaussian signaling. In the following simulations, all of the channel matrices are modeled according to the Rician fading channel model with Rician factor K, as specified in [START_REF]Achievable rate optimization for MIMO systems with reconfigurable intelligent surfaces[END_REF]. Also, we assume no spatial correlation3 exists among the elements of matrices H 1 and H 2 . The distance-dependent path loss for the direct link is β DIR = (4π/λ) 2 d αDIR 0 , where d 0 is the distance between the transmit array midpoint and the receive array midpoint, and the path loss exponent of the direct link is denoted by α DIR . The far-field FSPL for the indirect link β -1

INDIR can be computed according to [START_REF] Danufane | On the path-loss of reconfigurable intelligent surfaces: An approach based on Green's theorem applied to vector fields[END_REF] as

β -1 INDIR = λ 4 (l t /d 1 + l r /d 2 ) 2 /(256π 2 d 2 1 d 2 2 )
, where d 1 is the distance between the transmit antenna array midpoint and the RIS center, and d 2 is the distance between the RIS center and the receive antenna array midpoint.

In the following simulation setup, the parameters are f = 2 GHz (i.e., λ = 15 cm), s t = s r = s ris = λ/2 = 7.5 cm, D = 500 m, l t = l r = 20 m, d ris = 30 m, N t = 8, N r = 2, α DIR = 3, K = 1, M = 4, and σ 2 = -110 dB. The RIS consists of N ris = 225 elements placed in a 15 × 15 square formation. The line search procedure for the PGM utilizes the parameters L 0 = 10 3 , δ = 10 -3 and ρ = 1/2. For the SCAbased method, we use the CVX tool with MOSEK as the internal software package to solve [START_REF] Björnson | Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces[END_REF] and (17). The initial values of θ and P are randomly chosen for both methods. All results are averaged over 30 independent channel realizations.

In Fig. 2, we show the CR and the MI results of the proposed PGM and SCA for scenarios in which the direct link is present and in which the direct link is blocked. As benchmark schemes, we consider the presented system without the RIS, whose precoding matrix is optimized by using the proposed methods. The two proposed optimization methods achieve approximately the same results and need only a few iterations to reach the optimum. In general, the MI shows the same behavioral trend as the CR, but the MI is always larger than the CR. For both metrics, the proposed algorithms achieve a similar gain through the parameter optimization, which is around 2 bpcu and 3 bpcu if the direct link is present and blocked, respectively. These results justify our initial claim that the CR presents a suitable metric to optimize the MI. In addition, removing the RIS from the considered system causes

Fig. 1 .

 1 Fig. 1. Aerial view of the considered communication system.

  The direct link is blocked.

Fig. 2 .

 2 Fig. 2. The CR (R 0 ) and the MI of the proposed PGM and SCA methods.

M = 4 ,

 4 CR-based opt. M = 16, CR-based opt. M = 4, Gauss-based opt. M = 16, Gauss-based opt. Achievable rate (Gauss) (a) The direct link is present.

Spatial correlation models for RIS channel matrices have recently become available in the literature (e.g., in[START_REF] Björnson | Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces[END_REF]).

The work of N. S. Perović and M. F. Flanagan was funded by the Irish Research Council under grant number IRCLA/2017/209. The work of L. N. Tran was supported in part by a Grant from Science Foundation Ireland under Grant number 17/CDA/4786. The work of M. Di Renzo was supported in part by the European Commission through the H2020 ARIADNE project under grant agreement number 871464 and through the H2020 RISE-6G project under grant agreement number 101017011.

In Fig. 3, we compare the MI when discrete and Gaussian signaling are considered. Specifically, we plot the MI where the precoding matrix and the RIS coefficients are obtained by implementing the CR optimization (referred to as CR-based optimization), and the MI where the precoding matrix and the RIS coefficients are obtained based on the achievable rate optimization for Gaussian signaling (referred to as Gaussian signaling based optimization) [START_REF]Achievable rate optimization for MIMO systems with reconfigurable intelligent surfaces[END_REF]. Moreover, we present the achievable rate for Gaussian signaling [START_REF]Achievable rate optimization for MIMO systems with reconfigurable intelligent surfaces[END_REF]. As can be seen, the CR-based optimization yields a higher MI than the Gaussian signaling based optimization, particularly for a small size of symbol alphabet. As expected, the achievable rate for Gaussian signaling is always higher than the MI. However, for M = 16 the gap between the MI obtained by using the discrete optimization and the achievable rate is less than 0.2 bpcu.

VI. CONCLUSION

In this letter, we have investigated the use of the CR as a simple and meaningful metric for optimizing the MI in RISaided MIMO communication systems. Since the maximization of the CR is a nonconvex optimization problem, we proposed two local optimization methods. The first method is based on the PGM which uses closed-form expressions in each iteration. The second method is derived from the principles of SCA. Simulation results show that both optimization methods produce very similar performance results and that the CR is indeed a suitable substitute metric for optimizing the MI in RIS-aided MIMO communication systems. This work can be extended to the case where the transmitted data is simultaneously encoded into the IQ symbols and the RIS phase shifts, as well as to the case where statistical CSI is used for the CR optimization.

exp -Φi,j (θ, P) 4σ 2 ∇ θ Φi,j (θ, P). (18) Also, the complex differential of Φ i,j with respect to F(θ) = diag(θ) and F * (θ) can be expressed as

With the aid of a few algebraic steps, we obtain dΦ i,j (θ, P) = vec

Substituting ( 22) into (18), we obtain (9a).

In a similar way, the gradient of f (θ, P) with respect to P * is equal to

exp -Φi,j (θ, P) 4σ 2 ∇PΦi,j (θ, P). (

The complex differential of Φ i,j with respect to P and P * is

= Tr{(H H HPe i,j e H i,j ) T dP * + e i,j e H i,j P H H H HdP} (24) where (a) is obtained from [START_REF] Hjörungnes | Complex-Valued Matrix Derivatives With Applications in Signal Processing and Communications[END_REF]Table 4.3]. Similarly, we have

Substituting ( 25) into (23), we obtain (9b).

APPENDIX B CONNECTION BETWEEN (7c) AND (13c)

Let θ * an the optimal solution to [START_REF] Boyd | Convex optimization[END_REF] and suppose that there is some l such that |θ * l | < 1. Note that we can write Φ i,j (θ

where a i,j ≥ 0, b i,j and c i,j are the resulting constants obtained by rearranging Φ i,j (θ * , P) with respect to θ * l . Now consider the point θl = -θ * l which is also feasible to (13c). Also, we define θ = [ θ1 , θ2 , . . . , θNris ] T ∈ C Nris×1 such that θk = θ * k for k = 1, . . . , N ris and k = l, and θl = -θ * l . It is easy to see that Φ i,j ( θ, P) = a i,j |θ * l | 2 -2ℜ b i,j θ * l + c i,j . Thus we can conclude that 2ℜ b i,j θ * l ≥ 0, otherwise Φ i,j ( θ, P) > Φ i,j (θ * , P), and thus, a strictly smaller objective can be obtained. Next, since 2ℜ b i,j θ * l ≥ 0 we immediately have that Φ i,j (αθ * l , P) = α 2 a i,j |θ * l | 2 + α2ℜ b i,j θ * l + c i,j ≥ a i,j |θ * l | 2 + 2ℜ b i,j θ * l + c i,j = Φ i,j (θ * l , P) for any α > 1. That means, if |θ * l | < 1 then we can always scale θ * l up to achieve a better objective.