Guillaume Par

Villain

Dirigée Par

Clément Nizak

Olivier Rivoire

Agathe Urvoas

System for the generation of controlled affinity maturation trajectories of antibody fragments

Keywords: in vitro affinity maturation, in vivo mutagenesis, phage display, evolutionary trajectory, antibody fragments

Titre : Système pour la génération contrôlée de trajectoires de maturation d'affinité de fragment d'anticorps. Résumé : La maturation d'affinité est un processus évolutif rapide qui augmente l'affinité de liaison de l'anticorps avec son antigène pendant la réponse immunitaire. Pour le décrire quantitativement et comprendre comment les autres propriétés des anticorps, telles que la spécificité de liaison ou la stabilité thermique, changent au cours de ce processus, nous devons analyser de nombreuses trajectoires évolutives se produisant sous une pression de sélection connue. À ce jour, un tel jeu de données n'existe pas. En effet, les techniques actuelles utilisées pour réaliser la maturation d'affinité in vivo ou in vitro ne contrôlent respectivement pas la pression de sélection sur la liaison ou ont un faible débit. La mutagenèse in vitro couplée à la présentation sur phages prend une semaine pour réaliser un cycle de mutations et de sélection. Afin de pallier ce manque de données, l'objectif principal de ma thèse est de proposer une nouvelle technique permettant de générer des trajectoires de maturation d'affinité des anticorps efficacement et dans des conditions in vitro contrôlées. Nous avons combiné la sélection in vitro, la mutagenèse in vivo et le séquençage haut débit pour atteindre cet objectif. Notre système présente une pression de sélection contrôlée via la méthode établie de « biopanning » du « phage display ». Nous montrons que nous pouvons simplifier et accélérer la production de phages en utilisant des plasmides auxiliaires. De plus, nous réduisons considérablement le temps nécessaire à la génération de banques de mutants et limitons le temps de manipulation en utilisant le plasmide mutagène et inductible développé pour la technique d'évolution dirigée assistée par phages pour introduire des mutations aléatoires in vivo. Nous avons effectué une expérience de maturation par affinité sur plusieurs formats d'anticorps et effectué un cycle de mutation et de sélection par jour. Actuellement, le taux de mutation atteint est la principale limitation de notre technique, mais il peut être surmonté grâce à plusieurs cycles consécutif de mutagenèse. Notre technique est facilement automatisable et permet la génération de trajectoires contrôlée de maturation d'affinité d'anticorps. Mots clefs : maturation d'affinité in vitro, mutagenèse in vivo, trajectoire évolutive, fragments d'anticorps, présentation sur phage

Introduction

The main objective of this PhD thesis is to propose a new system to efficiently generate affinity maturation trajectories of antibodies in controlled in vitro conditions. This manuscript consists of 3 chapters and 4 appendices. In the first chapter, we present the motivation for this work by introducing antibodies and affinity maturation, and discussing state of the art in vitro techniques to select antibodies and in vivo techniques to efficiently evolve proteins. In the second chapter, we detail the construction of our experimental system. In the third chapter, we describe the first results we have obtained with the system and propose an analysis that allows the discrimination of sequencing errors and true mutations.

Motivation Antibodies in natural context

Structure

Antibodies are key proteins of the adaptive immune system. They are mainly expressed by B lymphocytes. An antibody's role is essentially to bind foreign antigens at one end and recruit effectors at the other end to proceed with the elimination of the foreign antigen. In short, depending on the species and conditions, organisms produce repertoires of 10 6 to 10 12 different antibodies. For each encountered foreign antigen at least one antibody in the repertoire should bind to it with sufficient affinity to trigger an efficient response. For simplification, I will refer only to the IgG isotype in this thesis. Figure 1 represents a schematic of the different parts of an antibody and a graphical representation of their general 3D structure. IgG antibodies are in a Y-shaped configuration formed by two heavy (H) and two light (L) chains linked by disulphide bonds and noncovalent interactions (1) [START_REF] Edelman | DISSOCIATION OF gamma-GLOBULIN[END_REF]. The H and L chains can furthermore be decomposed into a variable part (VH and VL around 100 amino acid long) and a constant part (CH,CL) (3) (4). The chains fold into repeated anti-parallel β-sheets with loops of variable size and shape connecting the individual strands. In the variable part, the loops are short stretches of amino acids where variation is higher than elsewhere in the molecule. These hyper-variable regions are most likely to be the sites of binding with the antigen [START_REF] Wu | An analysis of the sequences of the variable regions of Bence Jones proteins and myeloma light chains and their implications for antibody complementarity[END_REF]. They are therefore called Complementarity Determining Regions (CDR 5 to 15 amino acid long). The variable part of H and L chains consists of three CDR loops and four framework regions (FR) that act as a scaffold around the CDRs to maintain the structural stability of the variable domains of the antibody.

The Fab region is the domain responsible for binding the antigen, it regroups the VH and VL and the constant part CH and CL. There is two Fab regions in one antibody. In turn, the constant Fc (Fragment crystallisable) region triggers downstream events in the immune response.

Diversity

V(D)J Recombination

The 'V(D)J' rearrangement produces the primary repertoire of antibodies prior to the encounter with an antigen. It is the main source of diversity in antibody repertoire composition. A simplified schematic of this process is represented in Figure 2. 'V(D)J' refers to the set of 'variable' (V) genes, 'diversity' (D) genes, and 'joining' (J) genes. Each B lymphocyte expresses one variant of antibody on its membrane. The heavy and light chains are encoded in several genes: VDJ for the heavy chain, VJ for the light chain. DNA recombinases catalyse the combinational rearrangement of the V, D, and J genes to form the coding sequence of the antibody, leading to a vast diversity of the antibody repertoire. To form the heavy chain, one gene from a series of 51 'variable' (V) genes is picked and assembled with one gene from a series of 27 'diversity' (D) genes, one out of 6 'joining' (J) genes, and the 'constant' (C) gene.

To form the light chain a different pool of genes is used. One out of 40 V genes is assembled with one out of 5 J genes and the C gene.

The 'V(D)J' rearrangement results in 8262 different possibilities for the heavy chain and 200 possibilities for the light chain. Therefore, giving an overall diversity of around 10 6 possible antibodies. Furthermore, a few extra nucleotides, called N regions, can be inserted in between the gene fragments, drastically increasing the possible diversity of the antibody repertoire even further [START_REF] Nossal | The double helix and immunology[END_REF]. Saada et al. estimated the total number of possible heavy chain sequences to be 10 45 and the total number of possible light chains to be 10 12 . Thus, the total number of possible antibody sequences can be estimated to be 10 57 [START_REF] Saada | Models for antigen receptor gene rearrangement: CDR3 length[END_REF]. In a lifetime, the human immune system is able to generate at least 10 14 different antibody molecules [START_REF] Saada | Models for antigen receptor gene rearrangement: CDR3 length[END_REF]. This is just a small fraction of all the theoretically possible antibodies. The newly assembled antibody genes are expressed by B lymphocytes. B lymphocytes are considered as the 'primary repertoire'. Each B lymphocyte is linked to one antibody sequence which corresponds to a unique specificity profile to bind antigen.

Figure 2: Overview of V(D)J recombination of antibody heavy chains

One out of 27 'diversity' (D) genes recombines with one of 6 'joining' (J) genes to form the DJ exon that will recombine with one of 51 'variable' (V) genes.

Antibody will also include a constant domain (C) gene.

Maturation

The binding of an antigen to an antibody at the surface of a B lymphocyte triggers a series of events that lead to the affinity maturation of the antibody. Affinity maturation is an accelerated evolutionary process by which genes, encoding antigen-binding antibodies, undergo cycles of random mutation (somatic hypermutation) and selection to bind tighter and tighter to the target antigen. Typically, as we will detail below, antibodies accumulate ~15 to 20% mutations in their sequence and increase their affinity up to 100 to 1000-fold over the course of many (~20) cycles, occurring in vivo on a timescale of a few months [START_REF] Altan-Bonnet | Quantitative Immunology for Physicists[END_REF]. The B lymphocytes selected for antigen-binding will migrate to the germinal centre, proliferate, and undergo somatic hypermutation (SHM). The clones that express antibody variants with higher affinity towards the antigen are further selected. However, the mechanism of cell selection, regulation of cell division, and regulation of mutations remains unknown [START_REF] Gitlin | Clonal selection in the germinal centre by regulated proliferation and hypermutation[END_REF]. We will therefore introduce in the next paragraph only the mechanism of SHM.

SHM is driven by Activation-Induced cytidine Deaminases (AID) and the high transcription levels of the antibody genes [START_REF] Revy | Activation-Induced Cytidine Deaminase (AID) Deficiency Causes the Autosomal Recessive Form of the Hyper-IgM Syndrome (HIGM2)[END_REF] [START_REF] Bachl | Increased Transcription Levels Induce Higher Mutation Rates in a Hypermutating Cell Line[END_REF]. AID catalyses the deamination of cytidine (C) to uracil (U) on single-stranded DNA. Mutations accumulate in 'hotspot' positions with the sequence WRC (W=A/T, R=A/G) [START_REF] Pham | Processive AID-catalysed cytosine deamination on single-stranded DNA simulates somatic hypermutation[END_REF]. The mismatch leads to different mutational outcomes through the use of the cell's DNA replication and DNA repair mechanism (See Figure 3). The generated mutations have a bias towards transitions (A « G or T « C) over transversions (A or G « T or C) [START_REF] Neuberger | Immunity through DNA deamination[END_REF]. One mutational outcome upon DNA replication is the addition of an A opposite to the U, resulting in C → T and G → A transition mutations. Another mutational outcome is facilitated by uracil-DNA glycosylases (UNG). UNGs remove the U introduced by the AID, leaving an empty site. During DNA replication, any of the four bases can be added to the opposite of the empty site, resulting in both transition and transversion mutations. The last possible outcome that also explains the A:T mutations site is due to the behaviour of mismatch repair (MMR) mechanisms during somatic hypermutation. MutS homologs will form a complex, recognize the U:G mismatch and recruit the error prone DNA polymerase η. DNA polymerase η lacks proof-reading activity and will repair DNA with an error rate up to 10 4 -fold superior to replicative DNA polymerases [START_REF] Friedberg | Specialized DNA Polymerases, Cellular Survival, and the Genesis of Mutations[END_REF], resulting mainly in A/T mutations near the U:G lesions [START_REF] Delbos | DNA polymerase is the sole contributor of A/T modifications during immunoglobulin gene hypermutation in the mouse[END_REF]. These different mechanisms for mutational outcomes lead to an error rate of SHM of 10 -3 to 10 -2 mutations per base pair per generation. This is close to one mutation per antibody gene per generation [START_REF] Kleinstein | Estimating Hypermutation Rates from Clonal Tree Data[END_REF]. Hypermutation outcomes from a lesion in DNA induced by the AID cytidine deaminase. This can be followed by error-prone DNA repair over extended regions (∼ 20 nucleotide) of the sequence. These events may induce several simultaneous mutations at close-by positions along the sequence.

Orders of magnitude and limits of in vivo maturation

The affinity maturation is a fast evolutionary process with random mutagenesis of V-genes and selection for highest affinity antibodies. In the next paragraph, we will see what its limits and orders of magnitude are.

Affinity

Antibodies with dissociation constant (KD) values below a certain threshold do not have further selective advantage. Therefore, there is a biological limit in antibody affinity maturation by SHM. The theoretical analysis of the limits of antibody affinity for any antigen predicts that the lowest possible KD value is ≈ 10 -10 M [START_REF] Foote | Kinetic and affinity limits on antibodies produced during immune responses[END_REF]. The maximum on-rate constant (kon ≈ 10 5 -10 6 M -1 s -1 ) is restricted by the diffusion coefficients of the antibody and antigen molecules. The limiting factor for the off-rate (koff ≈ 10 -4 s -1 ) is the internalization of the membrane-bound antibody in B lymphocytes, after binding to an antigen.

Accordingly, one study of the on and off-rates of vaccinated individuals indicate that the limit could be a KD of approximately 10 -9 M. A boost vaccination induced only a small degree of sequence diversification (7.7 and 6.8 amino acid transitions) without further increase in affinity [START_REF] Poulsen | Limits for Antibody Affinity Maturation and Repertoire Diversification in Hypervaccinated Humans[END_REF]. Nevertheless, several in vitro engineered antibodies have been reported with femtomolar affinity (koff £ 10 -6 s -1 ).

Mutations

Most affinity-matured antibodies contain on average 15-20 mutations across the entire variable domains representing 7 to 10 percent of their sequence [START_REF] Tomlinson | The Imprint of Somatic Hypermutation on the Repertoire of Human Germline V Genes[END_REF]. This depends greatly on the species and the immunization protocol. In general, affinity-matured antibodies become more rigid and specific with increasing stage of maturation [START_REF] Li | Rigidity Emerges during Antibody Evolution in Three Distinct Antibody Systems: Evidence from QSFR Analysis of Fab Fragments[END_REF]. One particular case is broadly neutralizing antibodies (bnAbs) to immunodeficiency virus (HIV). bnAbs develop only in some patients 2-3 years after infection and contain a higher percentage of somatic mutations (40-110 mutations representing 20 to 50 percent of their sequence) in comparison to other antibodies isolated in the same patients [START_REF] Mikell | Characteristics of the Earliest Cross-Neutralizing Antibody Response to HIV-1[END_REF]. Structural and functional analyses reveal that somatic mutations in bnAbs do not only occur in the variable regions but are also highly abundant in FR residues (the same also holds for non-bnAbs). Those framework mutations provide increased flexibility and/or increase the surface of direct antigen contact [START_REF] Klein | Somatic Mutations of the Immunoglobulin Framework Are Generally Required for Broad and Potent HIV-1 Neutralization[END_REF].

Open questions related to antibody affinity maturation

Trade-off between affinity and specificity?

Specificity refers to an antibody's ability to discriminate between antigens. Many antibodies have been reported to bind several different antigens (polyreactivity), not being able to discriminate between them. How polyreactivity relates to affinity maturation is still unclear, as at least two scenarios have been reported [START_REF] Eisen | Evolving concepts of specificity in immune reactions[END_REF]. In the first one, the polyreactivity decreases significantly during affinity maturation. One proposed mechanism is that germline antibodies are conformationally flexible but become rigid during affinity maturation as their structure becomes more complementary to the antigen [START_REF] James | Antibody Multispecificity Mediated by Conformational Diversity[END_REF]. The other scenario arises from the studies of anti-HIV antibodies and shows some increase in polyreactivity during affinity maturation [START_REF] Haynes | B-cell-lineage immunogen design in vaccine development with HIV-1 as a case study[END_REF].

This indicates a reduction in specificity and would show a trade-off between affinity and specificity. The two scenarios invoke two different selective pressures during affinity maturation. Generating affinity maturation trajectories with different selection pressures can provide a clue on whether the two scenarios can be reconciled.

Evolvability of maturated and naive antibodies?

The germline antibodies are by definition the only ones transmitted to the next generation and are as such submitted to Darwinian evolution. Nevertheless, they are the starting point of a much faster evolutionary process -the affinity maturation -as we saw in the previous paragraph.

One can wonder if those germline sequences are more "evolvable" than the matured antibodies, as being able to evolve is one of their biological functions selected for by Darwinian evolution over a large timescale. Some reported a loss of mutability in one matured antibody [START_REF] Vieira | Selection and Neutral Mutations Drive Pervasive Mutability Losses in Long-Lived Anti-HIV B-Cell Lineages[END_REF] and a hint for the germline being more "evolvable" can be found in the work of our colleague [START_REF] Schulz | Parameters and determinants of responses to selection in antibody libraries[END_REF].

Those works are only based on one lineage of affinity maturation. In order to generalize these findings, there is a need to access more affinity maturation trajectories.

Flexibility and rigidity?

In direct link with the previous open questions, the mechanism of rigidification of antibodies during affinity maturation is still subject to debate. Most matured antibodies are thought to be more rigid than their germline ancestor. Nevertheless, it exists some counter examples, where the germline is more rigid than the matured antibody [START_REF] Ovchinnikov | Role of framework mutations and antibody flexibility in the evolution of broadly neutralizing antibodies[END_REF]. Once again, those are conclusions made from the limited number of evolutionary trajectories available. Most of them arise from the large studies on clinically relevant antigen, such as HIV or Herpes virus. Generating affinity maturation trajectories with a fast and easy method would allow to draw more generalizable conclusions on the evolution of flexibility or rigidity of antibodies.

To answer those questions, we need a method to generate controlled affinity maturation trajectories.

In the next chapter, we will review the state-of-the-art in vitro techniques to select antibodies and efficiently evolve proteins in the laboratory. This will be useful to establish a basis for our new method.

Current techniques

Here we present in vitro techniques for the selection of recombinant antibodies. For the past 30 years, those techniques allow to mimic and bypass the natural immune response to create antibody repertoire and select antibodies to bind antigens.

Recombinant antibody libraries

Libraries of antibodies can be obtained either from immune repertoire sequences derived from B lymphocytes of non-immunized or immunized animals, or synthetically generated by combining CDRs with one or multiple framework regions to replicate the diversity of a natural antibody repertoire. The library of antibody genes is often cloned into phagemids (described later) and subsequently transformed into Escherichia coli [START_REF] Winter | Man-made antibodies[END_REF]. We will now review the three types of antibody libraries in more detail.

Naive antibody libraries

A naive antibody library is a collection of antibodies derived from circulating B lymphocytes of non-immunized donors, such as healthy patients or animals. They are produced from multiple donors. Naive antibody libraries have a large size (up to 10 11 ) [START_REF] Clackson | Making antibody fragments using phage display libraries[END_REF]. A naive library can generate antibodies capable of targeting a wide range of antigens [START_REF] Ponsel | High Affinity, Developability and Functional Size: The Holy Grail of Combinatorial Antibody Library Generation[END_REF]. Nevertheless, the obtained antibodies have often lower affinity compared to those isolated from immune libraries [START_REF] Ponsel | High Affinity, Developability and Functional Size: The Holy Grail of Combinatorial Antibody Library Generation[END_REF].

Immune antibody libraries

Immune antibody libraries are obtained from immune donors, such as disease-infected patients or immunized animals. One unique feature of immune antibody libraries is that the sample material is obtained from activated B lymphocytes. The B lymphocytes are activated during the antigen encounter and subsequently undergo affinity maturation processes. Thus, the predisposition of antibody clones to recognize certain antigens is pronounced and allows for the isolation of high-affinity binders specific to the target antigen. In contrast to naive libraries, immune libraries are not well suited to target a large panel of antigens. The biased nature of the immune repertoire is mainly useful against the antigen of the specific disease. Immune libraries are generally generated with modest diversities [START_REF] Griffiths | Strategies for selection of antibodies by phage display[END_REF]. Therefore, new libraries are needed when targeting antigens from different diseases. There are also ethical issues related to the generation of new immune libraries: it is not moral to apply active immunization to humans and animal donors with deadly antigens (i.e toxins or immunosuppressive) to isolate their corresponding antibody.

Synthetic antibody libraries

Synthetic antibody libraries are artificially created repertoires, often by in vitro reconstruction of variable genes via CDR randomization [START_REF] Rothe | The Human Combinatorial Antibody Library HuCAL GOLD Combines Diversification of All Six CDRs According to the Natural Immune System with a Novel Display Method for Efficient Selection of High-Affinity Antibodies[END_REF]. They can be fully synthesized sequences or consist of partially natural sequences mixed with synthetic ones. The diversity of a synthetic antibody libraries can be tuned and be designed free from natural biases and redundancies. Specific frameworks can be used to ensure higher expression, improved stability, and reduced aggregation. Synthetic antibody libraries can target many different kinds of antigens [START_REF] Tiller | A fully synthetic human Fab antibody library based on fixed VH/VL framework pairings with favorable biophysical properties[END_REF].

Recombinant antibody formats

Recombinant antibodies have been dissected into minimal binding fragments. These fragments retain the target specificity of full antibodies but can easily be produced and are generally more stable [START_REF] Holliger | Engineered antibody fragments and the rise of single domains[END_REF]. We will present only 3 popular formats that will be used later in this thesis. The first format is the single-chain Fvs (scFv).). This is the largest fragment in which the VH and VL domains of the antibody are joined with a synthetic flexible glycine-serine-rich polypeptide linker. scFv fragments usually retain the antigen-binding properties of their parent antibody.

The second format, the smallest antibody fragment, is a single VH (variable heavy chain) domain (devoid of VL). These fragments are capable of binding their antigen with high affinity and specificity. However, they are thought to be poorly soluble and often prone to aggregation, but we have not noticed this inconvenience during our work for this thesis. More soluble and stable Single VH-like domains were discovered in the camelids (camels and llamas) called VHH.

Naturally, those domains are part of a full-sized antibody with an equivalent constant domain Fc and are an integral component of the immune system of those species. VHH domains display long CDR surface loops, often larger than the ones of human antibodies. VHH domains are in general soluble and stable [START_REF] Ghadessy | Directed evolution of polymerase function by compartmentalized self-replication[END_REF]. Those single domain antibody fragments are the major reagents for current analytical, diagnostic and therapetic applications [START_REF] Gonzalez-Sapienza | Single-Domain Antibodies As Versatile Affinity Reagents for Analytical and Diagnostic Applications[END_REF].

Flexible

Selection of recombinant antibodies

Phage display

Phage display is a technique that uses phage to link the phenotype of a protein of interest, here antibodies, to its genotype. Other display technologies exist, such as yeast display, surface display [START_REF] Salema | Escherichia coli surface display for the selection of nanobodies[END_REF], and ribosome display, but we will not review them in this thesis. In phage display, the protein of interest is expressed on the surface of a phage, while its gene is packed inside the phage capsid. Different phages can be used to do phage display, including lambda (39), T4, T7, Coliphage Qβ [START_REF] Skamel | In Vitro Evolution and Affinity-Maturation with Coliphage Q Display[END_REF], and filamentous phage, such as M13 and fd phage. In this thesis, we are only focusing on the M13 phage system, because it is the one that has been the most extensively used over the past decades.

In 1985, Smith first described phage display using filamentous phage and the restriction enzyme EcoRI [START_REF] Smith | Filamentous fusion phage: novel expression vectors that display cloned antigens on the virion surface[END_REF]. Later, phage display for antibodies was developed independently in Cambridge [START_REF] Mccafferty | Phage antibodies: filamentous phage displaying antibody variable domains[END_REF], Heidelberg [START_REF] Breitling | A surface expression vector for antibody screening[END_REF], and California [START_REF] Barbas | Assembly of combinatorial antibody libraries on phage surfaces: the gene III site[END_REF]. Because of the large size and complex structure of antibodies, they cannot be displayed on the phage in their entirety, but need to be fragmented.

Studies show that such smaller antibody fragments are sufficient to bind specifically to antigens (see previous paragraph). Currently, a great majority of libraries of antibodies for phage display contain Fab fragments, derivatives of human or mice VH fragments, heavy-domain of camelid or shark antibody fragments (VHH or V-NAR) [START_REF] Ghahroudi | Selection and identification of single domain antibody fragments from camel heavy-chain antibodies[END_REF], or single chain Fv fragments (scFv, synthetic fusion between VH and VL). Most of these antibodies are displayed on the pIII protein of M13 bacteriophages, but display on pVI, pVII, or pIX have also been reported [START_REF] Tornetta | Selection and maturation of antibodies by phage display through fusion to pIX[END_REF][START_REF] Hess | Orthogonal Labeling of M13 Minor Capsid Proteins with DNA to Self-Assemble End-to-End Multiphage Structures[END_REF].

Before describing the steps needed for phage display experiments, we will briefly introduce the wild-type M13 phage. The M13 phage is a filamentous phage with a length of 900 nm and a width of 6.5 nm (see Figure 4). This phage only infects E. coli strains that express the F pilus. This bacteriophage establishes a chronic infection in its host, forcing the host to continuously release new phage [START_REF] Kluge | Replication of Bacteriophage M13. Detachment of the Parental DNA from the Host Membrane and Transfer to Progeny Phages[END_REF]. The phage contains a genome of single stranded DNA (ssDNA) that consists of nine genes coding for a total of eleven proteins (pI to pXI), five coat proteins and six proteins involved in the replication. The capsid is mainly composed of the pVIII, with approximately 2700 copies that form an envelope around the genome. The top of the phage consists of pVII and pIX, while the bottom is composed of five copies of pIII and pVI. The protein protruding the furthest from the phage capsid is pIII. It is responsible for the infectivity of the phage by binding to the E. coli conjugative F pili. Additionally, if the pIII gene is expressed in E. coli, it inhibits superinfection of the host by another phage [START_REF] Boeke | Effects of bacteriophage f1 gene III protein on the host cell membrane[END_REF].

M13 phage replication cycle

The schematic of the phage replication cycle is presented in Figure 5. In the corresponding caption, the mechanism of infection, replication of the genomes in the host cell, and generation of new phage is described in detail. We will summarise here the important details for the understanding of this manuscript. The infection of the host by the phage is limited by diffusion and is mediated by pIII and the F pilus. Both are mandatory for infection. If a cell expresses pIII, it prevents its infection by another phage. The genome replication involves a step where the DNA is single stranded. Replicated single stranded DNA genomes are packed into new capsids through the recognition of a packing signal (PS). Five copies of pIII are present on the phage particles. The created phage are secreted continuously. Nevertheless the F pilus, Tol receptors and the pIII are essential for phage infection. Once the ssDNA of the phage (named (+) strand) is injected in the cell, the host machinery will synthesize the complementary strand (named (-) strand) to form the double-stranded DNA phage chromosome, termed the replicative form (RF). This process involves the RNA polymerase that will first synthesize a short RNA primer that will be used to initiate the synthesis of the (-) strand by the DNA polymerase III. To complete the double strand, DNA polymerase I will digest the RNA primer and finish the synthesis. The different phage proteins are then expressed. When pII reaches a critical concentration, it will introduce a nick in the (+) strand and bind covalently to the 5' end. The Rep helicase will displace the strand bound to pII and allow the extension of the 3' end by DNA polymerase III using the (-) strand as template for rolling circle amplification. When a round of replication is completed the old (+) strand will be cut off at the origin by a new pII. The old (+) strand is then re-circularised and can either be converted into the RF or packed into new M13 capsid depending on the stage of infection. pX is essential for the accumulation of ssDNA (+) strand nevertheless its role is still unknown despite some hint that it could be regulating pII because some labs reported the inhibition of phage replication when the number of copies of pX exceeds the number of copies of pII in the cell [START_REF] Fulford | Regulation of bacteriophage f1 DNA replication[END_REF]. In the early stage of infection all the new (+) ssDNA will be converted to RF. In later stage, the concentration of pV will be high enough for the protein to cover the whole length of the phage genome. pV binds to ssDNA and dimerizes in a 'back to back' position sequestering the DNA in a rod shape. The (+) ssDNA is entirely covered by pV excepted for a hairpin loop called the packing signal. 12 to 14 pIV monomer form a cylindrical structure that interfaces the inner with the outer membrane. The pI, pXI and pIV form the assembly complex trough their periplasmic domains. Five copies of pVII and pIX protein interact with the packing signal and attach to the pI from the assembly complex. pVII replaces pV during the translocation of the DNA through the pIV membrane pore. The translocation continues until the phage genome is completely coated with pVIII.

Three to five copies of pIII and pVI will assemble at the extremity of the filament and trigger the release of the phage from the bacterium. If either of pIII or pVI are missing several phage genomes can be loaded through the channel resulting in a longer phage particle.

Phage and phagemid

Phage vector

In the earliest reports about phage display, antibody genes in the form of scFv were cloned directly into the M13 genome as a fusion protein with the pIII. This phage vector carries all the genes needed for infection, replication, and assembly, as well as the sequence of the antibody.

To amplify phage libraries, it is sufficient to infect E. coli with the modified phage. This makes phage vector easy to use in selection, particularly in large scale selection, where libraries are selected against several targets in parallel [START_REF] Hallborn | Automated screening procedure for high-throughput generation of antibody fragments[END_REF]. The phage vector is said to be "multivalent", because they display 3 to 5 copies of the antibody at the tip of the phage capsid. When the antibodies are multimerized, the apparent affinity is higher than the one of monomeric antibodies. This is called the "avidity effect" and can bias the selection process towards low affinity binders that multimerize well.

Avidity: describes the strength of the combination of multiple antigen-antibody interactions.

An increase in avidity is obtained with multimerization of antibodies, which in turn leads to an interaction with increased stability as compared to monovalent antibodies. One independent model can be " #$$ = ( " ) ( where n is the number of monomers. The effective affinity of the multimers is higher than the one of the monomer, resulting in tighter interaction with the antigen.

Phagemid and Helper phage

Most antibody phage display systems nowadays utilize a two-component system: a phagemid vector and a helper phage. The phagemid is a plasmid that contains three basic elements (53):

• an antibiotic marker and an origin of replication for the selection and replication of the plasmid in E. coli,

• the gene encoding the antibody-pIII fusion,

• the phage origin of replication and packing sequence.

The phagemid cannot produce phage on its own. It requires the genetic information from the helper phage. The widely used helper phage M13KO7 is derived from M13 phage with an impaired origin of phage replication that is 100-fold less efficient for packing of the phage genome. It contains all the necessary genes for the replication and formation of an infectious phage capsid. Additionally, it includes an antibiotic cassette that facilitates selection. Upon superinfection by the helper phage, E. coli (containing the phagemid) will produce all the phage proteins. The phagemid packing sequence will outcompete the one from the helper phage, meaning that the majority of the capsid will carry the phagemid DNA. The phage will integrate the wild-type pIII from the helper phage together with the antibody-pIII fusion from the phagemid. Nevertheless, the wild type pIII is preferably incorporated rather than the antibody fusion. This results in a population of phage with 90 to 99 % that do not display an antibody. It is possible to engineer the wild-type pIII to be cleaved by proteases and thus, render the nondisplaying phage un-infectious. The vast majority of the phage that carry an antibody will present a single copy. This is called monovalent display. Usually the antibodies selected in a monovalent format possess higher affinity than the ones selected from multivalent format [START_REF] O'connell | Phage versus phagemid libraries for generation of human monoclonal antibodies[END_REF].

To restore the same multivalent display as the phage vector system, a new helper phage, called "Hyperphage", that does not encode for pIII, has been developed. In phage display, all the pIII copies of the phage must come from the phagemid, thus being fused to the antibody.

The phagemid vector has a higher transformation efficiency than the phage vector, enabling the construction of large libraries. The drawback of this method is that it needs one more step of infection with the helper phage to produce phage. This step requires the monitoring of bacterial density, because the infection through the F pilus is only possible at specific culture density (exponential growth). The population of phage for selection are heterogeneous in their capsid (display or not of the antibody in the case of monovalent display), but also in their genotype (the packaged genome can either be the phagemid or the helper phage). Even though the packing sequence of the helper phage is impaired, the number of packed helper phage genomes can sometimes reach or exceed the number of packed phagemid in the phage population.

Helper plasmid

To eliminate the need of helper phage for the production of phagemid-containing phage, the team of Bradbury developed 3 plasmids based on the M13 phage [START_REF] Chasteen | Eliminating helper phage from phage display[END_REF]. The phage origin of replication and the packing sequence is replaced by a plasmidic origin of replication and an antibiotic marker. Therefore, the helper plasmid cannot be packed into a phage and is selectable with the appropriate antibiotic. E. coli are co-transformed with one of those helper plasmids and a phagemid. They are used to produce phagemid-containing phage with different properties.

The first plasmid contains all the genes of the phage and allows the production of phagemidcontaining phage with monovalent display, similarly to the one produced with the help of M13KO7. The drawback of this plasmid is that the cells containing this plasmid have a reduced infectability due to the constant expression of pIII. The second plasmid uses a truncated version of pIII. It results in the production of phage that are mainly non-infective due to the truncated pIII, except for the ones that display the antibody-pIII fusion from the phagemid. This is also a monovalent display. The third plasmid has a deletion of the gene 3. The phage are produced in totality with the pIII from the phagemid. This is a multivalent display [START_REF] Chasteen | Eliminating helper phage from phage display[END_REF]. The drawback of this plasmid is that the titre of produced phage is very low compared to the production with the classical helper phage or the other types of helper plasmid.

Helper plasmid: Full pIII Truncated pIII ∆pIII

Output phage

Monovalent +bald Monovalent + uninfective Multivalent

Infectability of host cell

Low Normal Normal

Titre of phage

High Low Low

Biopanning days to be performed.

The second part of phage display is the selection (56), also called biopanning (depicted in Figure 7). The E. coli library, containing the phagemids with diverse antibodies, is grown and infected with the helper phage. After one hour, the culture is centrifuged and washed to remove the excess of helper phage. Afterwards, it is grown overnight under appropriate antibiotic conditions to produce phage that display one antibody on their surface, with its corresponding gene packed in the capsid. The phage are harvested, purified, and incubated in the presence of the target of interest (antigen), often attached to a solid surface. During this capturing step, the conditions (concentration, incubation time, temperature, pH, presence of competing agents) are perfectly controlled. In order to remove the unbound and non-specific phage, the solid surface is extensively washed. This should result in bound phage with good affinity. Nevertheless, there is always some background of non-specifically bound phage. Finally, the bound phage are eluted, often through acidic or basic pH or with proteases. The resulting phage are used to infect fresh E. coli to serve as a new library. Due to the large diversity and the background noise, three to five rounds of biopanning are usually performed in order to obtain antibodies that bind the target with high affinity.

This can be justified by a small model:

• the maximum number of input phage is 10 12

• non-specific antibodies pass a round of selection with a probability of 10 -6 (reported by Sébastien Boyer in his thesis ( 57))

• specific antibodies pass a round of selection with a probability of 10 -4 (10 -1 to 10 -4 reported in ( 57))

The above scheme describes the change in composition of a library composed of one specific Phage display biopanning of recombinant antibody libraries against target antigens is a major approach in the pharmaceutical industry to produce therapeutic antibodies, which represent a market of >$100 billion/year [START_REF] Ecker | The therapeutic monoclonal antibody market[END_REF].

Maturation

For many applications in research and therapy, antibodies with affinity in the lower nanomolar or sub-nanomolar range are needed [START_REF] Saeed | Antibody Engineering for Pursuing a Healthier Future[END_REF]. For this reason, methods mimicking natural antibody affinity maturation have been developed [START_REF] Thie | Affinity Maturation by Phage Display[END_REF]. They are based on the introduction of mutations into an already specific antibody gene. Traditionally, they employ random mutagenesis as a means to introduce variation in antibody sequences. In the following, we will describe the current strategies to introduce mutation either in vitro or in vivo.

In vitro mutagenesis

After a first step of biopanning, a new secondary library is generated by amplifying the recovered clones with error-prone PCR [START_REF] Renaut | Affinity Maturation of Antibodies: Optimized Methods to Generate High-Quality ScFv Libraries and Isolate IgG Candidates by High-Throughput Screening[END_REF]. The secondary library is then processed through several additional rounds of affinity selection under more stringent conditions [START_REF] Huang | Selection of recombinant anti-SH3 domain antibodies by high-throughput phage display[END_REF]. In many cases, to improve the antibody affinity from nanomolar to the picomolar range, several iterations of mutation/selection cycles are required to optimize multiple CDRs and Framework regions. Each mutagenic cycle takes about a week to process, introducing on average one mutation per gene.

Typically, five rounds are performed (see Figure 8). To reach similar levels of maturation as the natural system, one would need to perform 15 cycles, taking around 3 months. If one would like to reach the same level of maturation as broadly neutralizing antibodies, at least 40 cycles should be done, taking about a year to generate one affinity maturation trajectory.

Affinity maturation through phage display and in vitro mutagenesis is the standard approach in the pharmaceutical industry to optimize the affinity of recombinant antibodies for downstream therapeutic applications.

In vivo mutagenesis

Natural mutation rate

E. coli error-prone replication

DNA polymerases have high nucleotide selectivity, they can discriminate between correct and incorrect nucleotides during the insertion step of the polymerization. Their proofreading ability makes DNA replication a very accurate process (see Figure 9).

Mutator strain

Using E. coli mutD5 (71) that is deficient for mismatch repair mechanism mutD (the proofreading exonuclease domain of DNA Pol III) two teams mimicked affinity maturation. In the first one, the mutation rate reported seems extremely high compared to more recent literature [START_REF] Irving | Affinity maturation of recombinant antibodies using E. coli mutator cells[END_REF]. In their protocol they do 3-4 passages in the mutator strain before the biopanning process. After biopanning they recover the phage with the non-mutator strain TG1. After 6 rounds of biopanning the reported mutants have only 1 to 2-point mutations [START_REF] Irving | Affinity maturation of recombinant antibodies using E. coli mutator cells[END_REF], meaning either the selection is very powerful (which is unlikely because after the 6 round 8 out of 10 clones are false positive in Phage ELISA) or the mutation rate is much lower than reported.

In the second one, they did 12 rounds of mutagenesis with increasing selection pressure and 3 extra rounds of selection with very high selection pressure without mutations. Isolating mutant with 4-point mutations with a 100-fold increase KD from 320nM to 3.2nM. Here the mutation rate reported is 4.7×10 -4 substitutions per base pair for 28 replications of phage corresponding to 1.7 × 10 -5 substitutions per base pair per generations [START_REF] Low | Mimicking somatic hypermutation: affinity maturation of antibodies displayed on bacteriophage using a bacterial mutator strain[END_REF] To my knowledge this is the first reconstruction of affinity maturation trajectories. It seems like the last 3 rounds of selection are crucial to isolate the best mutants and that the best mutations are very underrepresented in the mutating populations.

Even if those approaches where quite successful, I do not have knowledge of any following studies using mutator strains to introduce mutation during the selection. This can be due to several disadvantage of using a mutator strain: the mutator strains are quite unstable and grow very slowly, the production of mutated phage is quite low -both teams had to propagate the phage in a non-mutator strain to be able to produce a titre of phage suitable for the selection.

The mutation rate is quite low and doing in vitro mutagenesis followed by 3 rounds of selection seems more efficient to evolve antibodies in the laboratory.

Orthogonal plasmid replication

To remediate to the instability of mutator strain, the idea of this technique is to introduce mutation more specifically in a plasmid using the characteristic of its replication mechanism.

Basically, the ColE1 origin requires two DNA polymerases to replicate: the DNA polymerase I (Pol I) that will initiate the replication by extending an RNA primer and the DNA polymerase III (Pol III) that will finish the replication. The idea is to use a low fidelity variant of the pol I (that contains 3 mutations reducing its fidelity) to introduce mutation in the region close to the origin of replication. In the first paper the authors evolve the TEM-1 β lactamase to hydrolyse the third generation antibiotic aztreonam. ( 74) They did a first round of mutation without selection followed by 2 rounds of low selection pressure and one of high selection pressure.

After this they isolate 46 clones that all contained mutations at the same position corresponding to the most frequently observed mutation in clinical resistant isolate. They did a first estimate of the mutation frequency assuming a uniform distribution of mutations along the gene to 8.1 × 10 -4 bp -1 . In a later paper they characterized thoroughly the mutation rate and the distances to the origin of replication. They report a mutation rate 5.5 × 10 -4 substitution per base per nucleotide happening in a 250 bp window in the leading strand. They also identify the reason for the mutations happening further in the gene. Indeed, the lagging strand is replicated differently from the leading strand. The synthesis always goes from 3' to 5' but the lagging stand is in the reverse direction for the pol III to be able to synthesize the new strand, a RNA primer is first created by the primase. Pol III synthesize small stretch of DNA until the previous primer. DNA pol I will then remove the RNA primer and finish the synthesis of the DNA. ( 75)

The authors reported that the effect is only high during the first 16h. Most probably the mismatch repair system of the cell is triggered by the excess of error-prone DNA pol I that will also make mistakes during the replication of the lagging strand of the genome of bacteria triggering over-expression of mismatch repair mechanisms and thus reducing the effective mutation rate of the plasmid. Few limits of this method are the fact that it is limited to ColE1related plasmid, restrained to temperature-sensitive pol I strains, and the mutation rate is highly dependent on the distance from the origin of replication.

CAS9

The CAS9 protein is an exonuclease that introduces double-strand breaks in DNA guided by an RNA molecule. This protein is part of the CRISPR system (immune system of prokaryotes)

and has been extensively studied and engineered. The key property of this protein that we will be interested in for the next two paragraphs is its ability to bind to DNA at a specific sequence that is specified by a guide RNA.

Deaminase

Recently, several teams developed fusion proteins with the deactivated CAS9 (dCAS9), that is not cleaving the DNA, with a deaminase. [START_REF] Hess | Directed evolution using dCas9-targeted somatic hypermutation in mammalian cells[END_REF] The first one is the AID deaminase that is involved in the somatic hypermutation of antibodies. Then there are the engineered ones by David Liu. These systems are mainly used and developed for gene therapy due to their high efficiency and their very narrow window of action. Nevertheless, they are quite efficient and now we can have access to all the possible substitutions. This development not focused on directed evolution still presents a great potential to be used in the context of in vivo mutagenesis once the window of action of the mutation increases.

EvolvR

In this paper [START_REF] Halperin | CRISPR-guided DNA polymerases enable diversification of all nucleotides in a tunable window[END_REF] the authors propose a method to introduce mutations on a targeted region of DNA using a fusion protein between a DNA nicking enzyme (nCAS9) and an error-prone polymerase (PolI3M). Using this fusion protein, they achieve a mutation rate of 10 -5 bp -1 to 10 -6 bp -1 in a 56 bp window around the nick. To achieve this, the authors have extensively optimized the parts of this chimeric protein. nCas9 is a nicking endonuclease guided by RNA:

it cuts the leading strand DNA at a site corresponding to the guide RNA. In this setting the authors introduce mutations to decrease its affinity for DNA. For the error prone polymerase, they use a mutated DNA polymerase I (also known as Klenow DNA polymerase) which is naturally involved in the repair of DNA. This polymerase processes a nicked strand of DNA to synthesize a new strand by displacing the older strand. Nevertheless, the Klenow polymerase has a low processivity, meaning that it stops the synthesis of DNA and falls off quite rapidly (20 nt per binding event). To remediate this issue, they also fuse the Polymerase I with the thioredoxin binding domain of the bacteriophage T7 that is known to confer higher processivity (up to 300 nt per binding event) to the polymerase I. [START_REF] Halperin | CRISPR-guided DNA polymerases enable diversification of all nucleotides in a tunable window[END_REF] This EvolvR system offers a quite new combination of features compared to current directed evolution systems. It can introduce all four bases although it still has as strong bias for introducing purines (adenine and thymine) in more than 80% of the mutations. Processivity is still an issue for introducing mutations in an entire gene but one can imagine an improvement of this system using different kinds of DNA polymerases.

MutaT7

This paper [START_REF] Moore | A Processive Protein Chimera Introduces Mutations across Defined DNA Regions In Vivo[END_REF] directly solves the problem of short windows of action of dCAS9-AID fusion.

The idea of this paper is to fuse the AID deaminase with the T7 RNA polymerase. The high specificity of the T7RNAP for its promotor allows this system to direct the mutator effect of the AID to a specific gene under the T7 promotor. In this setting the processivity of the T7RNAP is also a problem but this time it is because it is too processive and continues to induce mutations after the end of the gene. The authors proposed to add an array of terminators to stop most of the T7RNAP. The mutation rate reported 5.7 × 10 -5 substitution per base per generation but is limited to C → T mutations. The authors proposed to add a T7 promotor in the reverse direction to access G → A mutations [START_REF] Moore | A Processive Protein Chimera Introduces Mutations across Defined DNA Regions In Vivo[END_REF]. A better approach could be to make RNA polymerase fusion with the engineered deaminase describe previously.

PACE

Recently, a general approach for the directed evolution of proteins in vivo, using the M13 phage, was developed: Phage-Assisted Continuous Evolution (PACE) (represented in Figure 10). This technique is a phage selection system that requires a link between the desired phenotype and the phage replication. The gene of interest replaces the gene III of the M13 genome of the "selection phage". The gene III on the other hand, is inserted into an accessory plasmid and transformed into E. coli host cells. The key for PACE experiments is to link the desired phenotype to the phage replication. Therefore, pIII production must depend on the activity of the evolving gene carried by the "selection phage" [START_REF] Esvelt | A system for the continuous directed evolution of biomolecules[END_REF]. A population of "selection phage" encoding the gene of interest replicates by infecting E. coli host cells containing the accessory plasmid. These host cells continuously flow through the "lagoon", a fixed-volume vessel. Only the phage particles that assemble with pIII are infectious and propagate fast enough to stay in the "lagoon". Genetic variation is obtained by a "mutagenic plasmid" (MP) that increases the mutation rate of any replicating DNA in the host cells. An arabinose-inducible pBAD promoter allows a tight control of the expression of the mutator genes. The precise details of the MP are described in the next paragraph. The in vivo mutagenesis is only induced when the host cells enter the lagoon [START_REF] Badran | Development of potent in vivo mutagenesis plasmids with broad mutational spectra[END_REF]. Mutations accumulate within the packaged phage genome, containing the target gene. They also accumulate in the E. coli host strain. However, the bacteria are continuously discarded, and new, uninfected, and unmutated bacteria are continuously provided. PACE allows a production of 24 to 30 generations of phage per day. The evolutionary pressure can be modulated by the efficiency with which gene III is expressed. Some negative selection pressure can be applied by linking the undesired activity to the inhibition of phage infectivity using a non-functional pIII variant [START_REF] Carlson | Negative selection and stringency modulation in phage-assisted continuous evolution[END_REF].

However, there is only a small window of pIII expression that can sustain a replicating population of phage, limiting the modulation of the selection pressure. In the "lagoon", an evolving gene is encapsulated in a "selection phage". The phage will infect the fresh host cell arriving at a constant inflow. The cell will replicate and mutate the evolving gene. If the gene is functional, the host cell will produce infective phage that can sustain in the "lagoon". A non-functional gene result in non-infective phage that will be diluted out.

In principle, PACE can evolve any gene that can be linked to pIII production in E. coli. [START_REF] Bryson | Continuous directed evolution of aminoacyl-tRNA synthetases[END_REF]. PACE also enables the directed evolution of protein-protein interactions: the authors evolved variants of a toxin against a cell receptor from an insect to overcome insect toxin resistance [START_REF] Badran | Continuous evolution of Bacillus thuringiensis toxins overcomes insect resistance[END_REF] ; the group of Dickinson evolved a split T7 RNA polymerases for biosensor applications [START_REF] Pu | Evolution of a split RNA polymerase as a versatile biosensor platform[END_REF]. Recently, PACE has been used to increase the solubility of disulphide-free scFv and cysteine deaminase [START_REF] Wang | Continuous directed evolution of proteins with improved soluble expression[END_REF].

PACE is currently limited to cytosolic proteins (which is not the case of antibodies in general)

with general limitations of bacterial expression such as protein solubility, post translational modifications and disulphide bond formation. The selection pressure is only qualitative and finding the regime of maintaining a replicating phage population seems to be the key to a successful directed evolution experiment. Moreover, several of this group's studies abandon the lagoon part and prefer classical non continuous serial dilutions [START_REF] Suzuki | Crystal structures reveal an elusive functional domain of pyrrolysyl-tRNA synthetase[END_REF][START_REF] Hu | Evolved Cas9 variants with broad PAM compatibility and high DNA specificity[END_REF][START_REF] Roth | Phage-Assisted Evolution of Bacillus methanolicus Methanol Dehydrogenase 2[END_REF]. The mutations are not only targeted at the gene of interest and might affect any other genetic elements of the system, increasing the probability of selecting false positive variants using this approach.

Mutagenesis plasmid

With the objective of creating an in vivo mutagenesis method with a tuneable mutation rate and a broad mutagenesis spectrum, Badran created a series of inducible plasmids with different mutational potency. For the most potent one, MP6, they report an average of 6.2 × 10 -6

substitutions per bp per generation on the E. coli genome and expect a mutation rate of 7.2 × 10 -3 substitutions per bp per generation for the M13 bacteriophage [START_REF] Badran | Development of potent in vivo mutagenesis plasmids with broad mutational spectra[END_REF].

MP6 can achieve this high mutation rate by impairing the following mechanisms (see Figure 11):

• the proofreading of DNA PolIII by expressing dnaQ926, a dominant negative-variant of DNA PolIII proofreading domain.

• the mismatch repair mechanism by disrupting the methylation system. MP6 introduces the DNA adenine methyltransferase (dam) and seqA that bind to hemi-methylated DNA (one strand methylated, while the other strand is unmethylated).

• the base selection by expressing emrR. This inhibits the multidrug resistance pumps, retaining the intermediates of mutagenic nucleobases, toxic to the cell.

• the base excision by expressing the Uracil-DNA glycosylase inhibitor and PmCDA1.

The latter is the cytidine deaminase from sea lamprey, an analogue to the AID cytidine deaminase involved in antibody maturation.

Objectives

The goal of this thesis is to develop a fast and easy system to generate affinity maturation trajectories of antibody fragments in the laboratory. To achieve this goal, we set the following objectives for our system:

1 Have a well-controlled selection pressure 2 Be rapid (within a few weeks)

3 Introduce one mutation per gene per generation 4 Be parallelizable with some automations

In order to meet these objectives, we use the biopanning technique from phage display coupled with the in vivo mutagenesis from PACE. All the steps from our affinity maturation protocol can be conducted with commercially available robots, allowing easy automation and parallelization.

To check our understanding of the selection, we derive a basic model of the phage interaction with its target of the form of a simple, reversible, bi-molecular binding reaction. This model can be found in the appendix. The main results are quite intuitive and show the following: (I)

We should reduce the time of incubation of phage with its target during affinity maturation to decrease the binding constant kon (II) We should prolong the washing step to increase the dissociation constant koff.

In vivo mutagenesis

The critical part for the in vivo affinity maturation of antibodies is to introduce in vivo mutations. This will drastically reduce the time for the generation of mutant libraries and limit the hands-on time. To this aim, we use the system developed for PACE in the Liu Lab. It works with a phage and thus, is easily tuneable to be used with phage display selection. For our system, we use the E. coli host cells S1030 and the mutagenic plasmid MP6 (Addgene plasmid # 69669) from the PACE system of Badran & Liu (see p38-41 for more detail). We received the S1030 strain as a generous gift from the Liu lab; it is now also available on Addgene under the plasmid # 105063. The important features of this strain for us are the F' plasmid, the tight repression of lac operators (by lacI q ), and its optimization for the usage of arabinose-inducible promotors (pBAD). The F' plasmid, selectable with tetracycline, contains the gene for the F pilus (important for the infectibility of the cells by phage). Another interesting feature of the S1030 strain is its optimization for the avoidance of biofilms in a continuous culture setting which can be useful for automation in future work.

The mutagenic operon on the mutagenic plasmid are under the control of the arabinose inducible promotor pBAD. The MP6 plasmid features the highest mutagenic potency reported to date, resulting in a mutation rate of 0.7 mutations per kb per generation. This is a good starting point for an evolution experiment with antibody fragments, given their gene size range of 350 bp to 750 bp.

We transform S1030 with the MP6 plasmid giving rise to the strain 'SM' (resistant to tetracycline and chloramphenicol). To check for in vivo mutations with this strain, we infected arabinose-induced 'SM' cells with the kanamycin resistant helper phage M13KO7 (see p. [START_REF] Schulz | Parameters and determinants of responses to selection in antibody libraries[END_REF][START_REF] Ovchinnikov | Role of framework mutations and antibody flexibility in the evolution of broadly neutralizing antibodies[END_REF]. The M13KO7 is similar to the natural M13 phage, but with a genome behaving like a plasmid due to an additional origin of replication. We extracted the genomes of the newly produced helper phage and sequenced a 1kb region to count the number of mutants.

# mutants / seq. colonies SM infected non-induced induced 3 hours 0/6 1/6 16 hours 1/6 2/6 Due to the very low number of samples see Table 1; the mutations rate can only be roughly estimated to 0.1 (std=0.3) for the non-induced and 0.3 (std=0.4) for the induced substitutions per kb. This is a low mutation rate but quite encouraging to move forward with this system.

The mutation rate could be improved by screening different inducer concentrations, induction time before infection or optimal OD600 of induction. Nevertheless, this results are consistent with more rigorous analysis of NGS sequencing data (See p 68).

Model antibodies

Because of the large size and complex structure of antibodies, they cannot be displayed on the phage in their entirety but need to be fragmented. As detailed in the paragraph p. 21-22 studies

shows, that such smaller antibody fragments are sufficient to bind specifically to antigen. As a model antibody we chose a VHH anti-eGFP and one VHH anti-mCherry (gift from Sandrine Moutel in the Perez lab) [START_REF] Moutel | NaLi-H1: A universal synthetic library of humanized nanobodies providing highly functional antibodies and intrabodies[END_REF]. This choice is motivated by their targets. They are easily produced and are fluorescent, so they are easy to detect. The development of the technique has mostly been done using these antibody fragments as proof of principle.

Other quite easily accessible targets are DNA targets. They can be ordered directly as biotinylated oligos and are therefore easy to use with streptavidin coated surfaces. Previously, antibody fragments binding DNA hairpins have been selected in the lab. ( 57) We picked 2 VH fragments (Top1 and Top3) that have been selected for their binding to a single-stranded DNA loop [START_REF] Schulz | Parameters and determinants of responses to selection in antibody libraries[END_REF] and a scFv (C1) that has been selected to bind double-stranded DNA fragment [START_REF] Modi | Recombinant antibody mediated delivery of organelle-specific DNA pH sensors along endocytic pathways[END_REF].

Antibody fragment

Construction of the phagemids

The phagemids that are currently used in most of the phage display experiments are made to easily express soluble antibody fragments. For that, they possess an amber stop codon (TAG) placed between the antibody fragment and the pIII gene. They require a supE suppressor strain to express the full-length antibody-pIII fusion protein. SupE strains possess a special transfer tRNA that can incorporate a glutamine at the position of an amber stop codon, thus enabling read trough. Our E. coli host strain S1030, from the PACE system, is not a suppressor strain and we had to modify the phagemid, changing the amber stop codon to a glutamine codon. As a starting point, we used the vector pHEN2-3myc developed in the Perez lab. We changed the TAG stop codon to a glutamine codon CAG. We named this new phagemid pHEN3. The plasmid map is represented in Figure 12. We sub-cloned all the antibody fragments from their original vector to pHEN3 between the restriction sites NcoI and NotI. This drastically reduces the time needed for a round of phage display.

Phage production in mutator strain

We use the phage particles to infect 'SM' strain to produce 'SMP' strain. 'SMP' strain contain the F' plasmid, the mutagenic plasmid and the phagemid. They are resistant to tetracycline, chloramphenicol and ampicillin.

SMP infected In media non-induced induced

Amp only 10 5 10 6 Cm Amp 10 6 -10 7 10 4 -10 7

Tet Cm Amp 10 9 10 9

Table 2: Number of phage particle produced helper phage in 16h

We induced the cells at OD 0.4 with 25mM Arabinose and infected them with M13Ko7. We recovered newly formed helper phage by infecting S1030 with serial dilution of the supernatant. We plated S1030 on Ampicillin and counted the colonies. The CFU per ml of supernatant are reported for each condition.

Mutations

We sequenced 12 colonies for each condition. None of them was mutated. The mutation rate is too low to be detectable with such a low number of samples, meaning that we cannot use this system for the in vivo affinity maturation of antibodies because the mutation rate is too low.

This low mutation rate is probably due to the fact that the phagemid is in double stranded state most of the time because of it plasmidic origin of replication and therefore do not allow the deaminase to process it for mutagenesis. In the earlier case the phage genome replicated mainly through the phage replication mode with this essential step of ssDNA where the cytidine deaminase can proceed to the deamination. We did not prove this hypothesis, but we decided to tweak the system to have the phagemid replicate, as soon as it enter the cells, preferably using the phage replication mode with the ssDNA step. This would increase the mutation rate due to the mechanisms of action of the deaminase from MP6. For the vector carrying the antibody fragment to replicate and mutate as soon as it enters the cells, we had two options: using a phage vector or use a helper plasmid. If we had used a phage vector, the mutations would have accumulated also in the genome of the phage and might have introduced biases in the selections. Therefore, we chose to use a helper plasmid. The helper plasmid needs to express all genes of the phage except for pIII to allow the cell to be infected. This will lead to multivalent display on the phage particle. The high valency will change the apparent affinity of the displayed antibody due to avidity effect. This does not prevent us to do in vivo affinity maturation of antibodies, nevertheless we need to be aware that we could select for a better avidity effect of the antibody fragments. Before moving to helper plasmid, we first produced multivalent display phagemid with the hyperphage to see if our phagemid is compatible with this new display valency. 'SMP' cells infected by hyperphage produced 10 9 phage particles per ml containing the phagemid per ml after 16h and only 10 6 particles containing the hyperphage genome. As we can produced multivalent phage particles with our phagemid pHEN3 we can move on to a helper plasmid system.

Phage production

We transformed this plasmid into S1030 to produce the SH15 strain. We transformed MP6 into SH15 to make SH15M. We infected with the phage containing phagemid and plated SH15M to make SH15MP. In the absence of tetracycline this strain produced overnight 10 4 per ml phage containing phagemid (counting colonies resistant to ampicillin) and none containing the helper plasmid (no colony resistant to kanamycin). We produced them with tetracycline and there the phage count raised drastically to 10 7 -10 8 . (See Table 3) This was the first hint that there are probably interferences between the F plasmid and the Helper plasmid origin of replication.

VHH -αeGFP VHH -αmCherry VHH -αeGFP VHH -αmCherry

Table 3: Number of phage particles produced in 16h

The CFU per ml of supernatant are reported for each condition. In red the phage containing phagemid (resistant to ampicillin) and in green phage containing helper phage or plasmid (resistant to kanamycin). The results reported in the last two line are difficult to reproduces.

Davis Rich Medium

We checked whether using the culture medium reported in the PACE paper (Davis Rich)

explains the difference in mutation rate. This medium has been optimized to grow S1030 in a continuous culture setting. We sequenced clones produced in this medium, induced or not with Arabinose and IPTG none of the 8 sequenced clone per condition where baring mutations. This is probably due to the repression of our system by the 25 mM Glucose of the medium. We, therefore, chose to continue using 2xYT as our medium because it is much easier to make and for now, we do not need continuous cultures. Noticeable result is that with the helper plasmid producing at 37C gives higher titre than at 30C.

Harvard media + Conditions particle count CFU/ml

Mutants / Sequenced SP + M13Ko7 30 We induced and/or infected cell at OD 0.4 with the reported conditions. We recovered newly formed phage particle by infecting S1030 at OD 0.4 with serial dilution of the supernatant. We plated S1030 on Ampicillin counted and picked colonies. We miniprepped and Sanger sequenced. The number of mutants over the sequenced clones are reported for each condition.

Display level

We checked the display level by doing a western blot (Figure 15) with the phage particles using anti-pIII antibody to detect the level of expression of pIII vs VHH -pIII fusion. The input phage containing phagemid produced with M13KO7 are, as expected, mainly composed by the wild type non fused pIII. With the glucose condition, the titre of phage is too low to see the pIII on the membrane (Glucose is a repressor for the lac promotor under which the pIII fusion is). For the Arabinose condition we see a band at the wild type pIII and one with the fused antibody. This is unexpected. In theory only the fused pIII should be visible. The wild type pIII must come from a contamination with the helper phage M13KO7 carried over from the production of the input phage containing phagemid. To avoid this problem, we can use phagemid produced with the hyperphage or phagemid produced directly with the helper plasmid. The Arabinose IPTG condition only present fused pIII. This is expected.

Origin incompatibility

With the SH15M system phage production or infection was not very reproducible Table 3. The growth on tetracycline + Kanamycin is very slow and only growing on kanamycin yield in the loss of infectibility. It is very likely that the p15a origin of replication is incompatible with the one of the F plasmids. We therefore replace the p15a origin of replication of the helper plasmid with the pSC101 origin of replication. We transformed S1030 with this new helper plasmid to build SH101 strain. We transform SH101 with the different antibody phagemid (SH101P) or with MP6 to make the SHM strain. The number of phagemid produced by SH101P is reproducible and does depend on the antibody displayed. We noticed that production phage particle in 2xYT+Amp+Km from a fresh colony of SH101P gave 10 10 -10 11 CFU/ml with the phagemid inside with only 10 5 colonies growing on Km (probably with the helper phagemid but not sequenced). Using directly glycerol stock of SH101P yield to lower particle production 10 8 -10 9 CFU/ml with 10 We recovered newly formed helper phage by infecting S1030 with serial dilution of the supernatant. We plated S1030 on Ampicillin and counted colonies.

Binding

The last thing we needed to check is that the multivalent display of our antibody fragment still allows to bind to their target. For that we perform phage ELISA with the different phagemid against all the targets (Figure 17). We can see that the C1 scFv bind to its target but has also a strong background. The Top1 binds to its target probably better than the Top3. The VHH anti-mCherry gives only a weak signal for mCherry. The VHH anti GFP give a stronger signal for eGFP and mCherry. It has also a strong background. The multivalent phagemids are binding specifically to their respective target nevertheless some present a strong background.

Scheme of the system

Figure 18 represent a schematic of our system. SH101 are renamed SH for simplicity. A full protocol can be found in the appendices. Bound phage are eluted and used to infect fresh SHM E. coli, completing a full cycle.

Conclusion

We have built a system to enable affinity maturation of antibodies fragments displayed on a phagemid. The phagemid can be produced very easily using the Helper plasmid carriyng strain SH101. Those phage particles harbour antibody fragment on all five pIII copies. The multivalency does not prevent the antibody fragment from binding its target, even in the case ð Efficient and tuneable mutation rate.

o We did not achieve this objective as we are not able to reproduce the mutation rate reported by Badran, nevertheless several generations of mutations can be performed to increase the mutation rate.

4 Be parallelizable with some automations.

ð All steps can be done with robots o This is the case for all our steps through the use of turbidostats or robots

Results

Affinity maturation experiments

In order to validate our system, we performed 8 affinity maturation experiments with 2 conditions called "Selection" and "Drift" (described later) and 4 different antibody fragments as initial starting points.

The first two antibodies are Top1 and Top3 that are the best and the third best binder to a ssDNA loop, respectively [START_REF] Schulz | Parameters and determinants of responses to selection in antibody libraries[END_REF]. These were previously selected from a library of VHs with 4 positions of the CDR3 randomized. Top1 and Top3 differ in 2 nucleotide points mutations, of which one is a silent mutation. The second mutation results in the amino acid K at position 107 in Top1

and the amino acid T in Top3.

The third antibody fragment that we used as an evolutionary starting point in our affinity maturation experiment is scFv C1. It has previously been shown that scFv C1 binds a specific dsDNA fragment [START_REF] Modi | Recombinant antibody mediated delivery of organelle-specific DNA pH sensors along endocytic pathways[END_REF].

In order to represent the broad diversity of antibodies format, the fourth antibody fragment in our experiment is the VHH anti-eGFP. This specific antibody fragment has been selected with phage-display to bind the protein eGFP [START_REF] Moutel | NaLi-H1: A universal synthetic library of humanized nanobodies providing highly functional antibodies and intrabodies[END_REF].

For all four antibody fragments, we produced an initial clonal phage population using the SH101P strain. These clonal phage populations are used to infect SHM cells to induce in vivo mutagenesis and amplification, entering into the selection or the "Drift" cycle. We performed 10 rounds of "Selection" and 10 rounds of "Drift" for each antibody fragment (see Figure 19).

The phage titre after amplification and after selection are reported in supplementary Table 7.

In the selection cycle, phage displaying antibody fragments are bound to immobilized target on magnetic beads. The phage that display non-specific antibody fragments are washed away through 10 subsequent washes. Afterwards, the phage with specific antibody fragments bound to the target are eluted and counted. These selected phage are used to infect fresh SHM cells.

Those cells will produce new mutated phagemids. These mutated and amplified phage enter into the next cycle of selection.

The "Drift" cycle represents a simple control of the experiment of affinity maturation without explicit selection for binding. Instead of the binding to a target, we perform a 1000-fold dilution of the phage, followed by in vivo mutagenesis and amplification. This control might certainly comprise some natural selection on the level of the amplification and reproduction of the phage.

Nevertheless, we named it "Drift", as no external selection pressure for binding is applied. where no external selection for binding is applied.

Note on "Drift":

We called the control experiment "Drift" because we do not apply external selection pressure for binding. This is not the same as drift in the sense of neutral genetic drift where no selection is applied to one gene. In the "Drift" some natural selection for amplification and reproduction of the phage might occur without our control.

After performing the 10 rounds of "Selection" and "Drift" for the four initial antibody fragments, we noticed that the population of VHH anti-eGFP went extinct after the 1 st round of selection. Subsequent rounds were therefore not reported. The "Drift" cycles for VHH anti-eGFP yielded similar phage titer as the experiments with the other antibody fragments.

Sequencing several clones from round 5 and 10 with Sanger sequencing however, showed that the VHH gene was lost.

For further characterization of the processes during our affinity maturation experiments, we sequenced rounds 5 and 10 of the "Selection" and the "Drift" for the VH Top1 and Top3, and the scFv C1 with Illumina MiSeq v3 2x300 bp. The experiments with scFv C1 is sequenced in two parts: C1 VH and C1 VL. The number of paired-end reads before and after preprocessing are reported in Table 8.

lowest probability to have a sequencing error at a given position. In Figure 22, we can see that the quality of the sequencing is highly dependent on the sequencer cycle regardless of the sequence of the antibody fragment. For some positions we therefore obtain a lower amount of sequences with sufficient quality than for other positions. To take this into account, we analyse the frequency of substitutions per position: the number of non-wild type sequences over the number of high-quality sequences per position. By doing so, we are losing the link between two positions. This means that now we can only look at single-point substitutions for our analysis and we will therefore not be able to conclude on higher order mutants.

To attenuate the effect of cycle specific quality drop for future sequencing experiments, we suggest shifting the sequences by a couple of bases using different lengths of primers during the preparation for sequencing. A longer primer will shift the starting position of the sequence of interest to a later sequencing cycle. Using a pool of primers with different lengths will result in the same position being read at several different cycles of the sequencing run. This will maximize the quality for each position by retaining only higher QC data, providing a data set less biased to the sequencing run cycle. We aggregated all the data from the "Selection" experiments to fit our model of error rate distributions. It is generally acknowledged that most mutations are deleterious and under high selection pressure, most of them will be discarded [START_REF] Keightley | Toward a realistic model of mutations affecting fitness[END_REF]. Thus, we hypothesize that for the "Selection" part of our experiment most of the substitutions are sequencing errors.

Nevertheless, the data will also contain a small amount of real mutations that were selected for.

As described above, these selected mutations should have a substitution rate higher than the error rates. In order to take this into account, we filtered out the outliers from the gamma distribution using an Adjusted Boxplot for Skewed Distributions [START_REF] Hubert | An adjusted boxplot for skewed distributions[END_REF]. We kept all the positions within the 95% adjusted confidence interval. This method has been specifically developed to accurately identify outliers in skewed distributions like the gamma distribution. Nevertheless, for this filtering to be valid, according to the method we need to verify that no more than 5% of the data is found to be outliers [START_REF] Hubert | An adjusted boxplot for skewed distributions[END_REF]. This is true for all substitution types for the aggregated "Selection" experiments. We then fitted a gamma distribution for each type of substitution.

To check that our workaround is valid, we compared the obtained distributions of error rates to a data set without mutations, previously sequenced by our colleague (see next paragraph).

Comparison with previous sequencing results

In our experiments, we limited the sequencing to the rounds 5 and 10 of the "Drift" part and the "Selection" part. We do not have a non-mutating population in our data set.

As we want to be confident in the distributions of sequencing error rates predicted with all our "Selection" experiments, we need to compare our findings with previously sequenced nonmutating populations. Our colleague Steven Schulz produced such a data set [START_REF] Schulz | Parameters and determinants of responses to selection in antibody libraries[END_REF]. He sequenced VH fragments, including the Top1 and Top3 of our experiment, without the addition of random mutations. The main difference between his data set and ours is that the samples have been sequenced with 2 different sequencing kits. Steven's sequences come from a Miseq V2 2x250pb kit, whereas ours come from a Miseq V3 2x300pb. Both kits sequence DNA composed of the natural bases (A, T, C, and G) and the physical properties of the sequencing steps are the same. We therefore expect the error rates of Steven's data to show the same bias as observed for our data (i.e. T to G has a narrow distribution and C to A a broader one). The gamma distributions of the error rates, separated by type of substitution and obtained as described above, are depicted in Figure 23.

Overall, the sequencing bias is similar. When comparing the average of the error rates for each type of substitution between the two data sets, we see a clear correlation (linear regression, R 2 =0.80) (see Figure 24). This confirms that our choice to ue the aggregated "Selection" data is valid for the majority of the types of substations. It is worth noticing though, that the sequencing with the V3 kit has an average sequencing error larger than the sequencing with the V2 kit. This is likely due to the increase in loading density of the Illumina chips rendering more difficult to discriminate clusters.

Limits to our estimation of sequencing errors in the case of C to T and G to A substitutions

At each round of mutagenesis, we imposed to our sequences a systematic mutation rate that is globally of the same order of magnitude as the sequencing error rate. This systematic mutation rates also have some bias and in some type of substitutions might be visible and above the error rate. That would mean that our fitted sequencing error rates might probably be an overestimate and might include a second distribution of systematic mutations rate introduced at each round by the mutagenesis step. The only substitutions types that might present that characteristic in our data set are that C to T and G to A. A major mechanism of mutation in our system is the cytidine deaminase that change the C to T (or G to A on the complement strand) could explain this behaviour. This behaviour is occurring only in my dataset (with mutagenesis induced)and not in the dataset from Steven (sequencing error only) as shown in Figure 25.

As an exception to the general behaviours, we will now focus on two particular types of substitutions. With the linear regression, we can see that the C to T and G to A data points are quite far off the fitted line. We showed earlier that the mutation rate, imposed on our system at each round during the mutagenesis, has the same order of magnitude that the sequencing error.

The main mechanism of mutation in our system is the cytidine deaminase that switches the C to T (or G to A on the complement strand). This specific mechanism could have a rate slightly higher than the sequencing error rate explaining the displacement of the C to T and G to A data points. Therefore, we also applied the linear regression to the two different sequencing data sets (with Miseq V2 2x250pb and with Miseq V3 2x300pb), omitting the C to T and G to A data points. The resulting fit has an even higher coefficient of determination (R 2 =0.93), indicating a strong correlation between the sequencing error rates of the two data sets.

In the next paragraph, we will try to justify that we might be overestimating the error rates in the cases of the C to T and G to A substitutions and propose a first solution.

We manually "adjusted" the error rate distribution (red curve), so it corresponds better to the one expected from the linear regression (red crosses in Figure 24). The previously fitted distributions of error rate are probably composed of 2 distributions: one emerging from the true sequencing error and the second from the systematic mutation accumulated at this round. We can sketch another gamma distribution to represent the systematic mutation rates. The sum of those distributions (blue curve) seems to better capture the shape of the overall substitution distribution. Nevertheless, those are manually drawn curve so we cannot conclude on their significance. We will therefore use the previously fitted distributions for the rest of the analysis.

We should, nonetheless, acknowledge that the previously fitted distributions are wider with a higher mean than the true sequencing error only distribution and therefore we might slightly overestimate the distribution of the sequencing error rates. To prove this statement, in future work, one should include in the same sequencing run a control with non-mutated wild type sequences and use them for fitting the sequencing error rates of the experiment.

To conclude, our approach provides good fit for the sequencing error rates, coherent with previously sequenced samples where only the sequencing error where present. the selections experiments sequenced with Miseq V3 kit 2x300, the green curve is the gamma distribution obtain with the same fitting protocol as all the other substitutions fits, the red one is a manually adjusted fit for sequencing error to bring the mean closer to the expecting value according to the linear regression in Figure 24, the back gamma distribution is a sketch of gamma distribution for the mutations added by the mutagenesis plasmid at each round and the blue curve is the sum of the adjusted sequencing error distribution and the mutations distribution, it looks like a better fit than the original green gamma distribution.

The lower part is the data from non-mutating VH sequenced previously with For some fits, especially in the "Drift" the predictions of sequencing error rates seem to be off by a translation factor. This can be explained by the fact that "real" mutations are present in the drift with a basal rate on which is added the error rate, displacing the apparent distribution to higher frequency, nevertheless this shifted distribution are strong indications of the presence of real mutations in the drifts. This observation might reveal an overestimation the scaling factor leading to an underestimation of the probability to be a sequencing error rate. lines are the threshold above which we consider the substitution rate to be a real mutation rate and not a sequencing error rate at a probability less than 5%. For some fits, especially in the drift the predictions of sequencing error rates seem to be off by a translation factor. This can be explained by the fact that "real" mutations are present in the drift with a basal rate on which is added the error rate, displacing the apparent distribution to higher frequency, nevertheless this shifted distribution are strong indications of the presence of real mutations in the drifts.

Figure 29: Type of codon mutations

Upper plots: In blue are the expected outcomes types of amino acid mutations based on the sequence if the mutations are uniformed distributed; we generated all possible nucleotide mutations from the initial sequences and reported the corresponding type of mutation at the amino acid level. In grey is the expected distribution if the mutation rate is biased like in Figure 28; we generated all possible nucleotide mutation from the initial sequences with a weigh factor corresponding to bias in Figure 28 and reported the corresponding type of mutation at the amino acid level. In red are the observed mutations. Lower plot global Ka/Ks ratio (97) for all samples separated by "Drift" and "Selection" experiments. The Ka/Ks ratio is used commonly to estimate the presence of selection. Ka is ratio between the observed sense amino acid different mutations (plain red) and the one expected under the uniformly represented mutations model (blue). Ks is ratio between the observed synonymous mutations (red) and the predicted ones (blue).

Interpretation of the specific mutations

If we now look at the amino acid mutants as a function of the sequence position the results are quite difficult to interpret. Sequence logo of selected mutations can be found in Figure 30 for the "Drift" and Figure 31 for the selection.

As we have seen previously that the selection pressure in our experiment is mainly purifying, the presence of those specific mutants can be explained by two mechanism. The first one is "Genetic drift" or the Sewall Wright effect: the change in the frequency of a mutant during our experiment could be due to random sampling [START_REF] Masel | Genetic drift[END_REF]. Nevertheless, as our population of phage is large during our experiment, the mutation rate is low, and the number of generations is small this effect should be negligible. The second one is positive selection and we will present argument in favour of this mechanism in the following paragraph. Nevertheless, to be able to conclude on the exact nature of the mechanism that makes specifics mutant emerge one should compare the mutants with the initial genes experimentally.

The first counter intuitive argument for positive selection is that in all the case there is a significant presence of stop codons. Those sequences can be quite advantageous at the current round as they hijack the system. The link between phenotype and genotype is not strictly conserved in our system. Indeed, one parent phagemid can produce functional antibody-pIII protein and get mutated during replication. This replicated mutated phagemid can get encapsulated in a phage displaying the parental antibody-pIII and pass the round of selection.

Presenting a stop codon will provide a transient large fitness advantage because the phagemid can get replicated and do not proportionally express the fusion protein that is known to be toxic for the cell membrane at high level. Those stop codon containing sequences will be purged during the next rounds as they will not be able to produce functional antibody-pIII fusion protein.

The second interesting result is that we found the Top1 sequence in the Top3 "selection" at round 5. This is the only sequence that emerged from sequencing errors. It is kwon that the Top1 is better than the Top3 as it has a higher selectivity (probability to pass one round of selection) [START_REF] Schulz | Parameters and determinants of responses to selection in antibody libraries[END_REF]. Nevertheless, Top1 has only a slightly better selectivity than the Top3 (2-fold) and it is not possible to differentiate it presence in the round 10 from sequencing error.

Nonetheless, in the round 10 of "selection" for Top3 we identify 7 mutations that should result in an antibody with a higher affinity. To show that we will need to perform further experiments.

For the "Drift" it seems that the mutants are present among all the antibody sequence. The position that are most amplified are in most of the cases in a CDR region suggesting that even in the absence of selection for binding those regions play a role in the solubility and/or production of the antibodies. It would be interesting to test those sequences to characterise the increased of amplification.

"Drifts" For the selection, one can notice that there is no functional selected sequence above the threshold in the case of the Top1. In the case of the Top3 we can see that at the round 5 the only sequence above the background noise is a K that is exactly the sequence of the Top1. In the round 10 this sequence is in the sequencing error noise, but other sequences arose with higher frequency.

Notably the H at the last position of the CDR3. This position has not been screened by previous experiments, but the emerging histidine is coherent with the previous observation of selecting histidine or positively charged amino acid in the CDR3 that could interact with the negatively charge DNA loop used as a target for the selections. Each letter corresponds to its amino acid, stop codon are represented with "*"

Top1

Conclusion and perspective

We provided a simple method to differentiate real mutations from sequencing errors. With this method, we can successfully analyse the sequencing result of our affinity maturation trajectory.

We showed that our experiment is governed by purifying selection. Nevertheless, we identified antibody fragments with potentially higher affinity for their antigens or properties that lead to better amplification. Further studies are needed to prove this increase in binding affinity and/or amplification. To this end, we suggest building a mini-library consisting of all the identified mutants. Perform an amplification in non-mutating strain and sequence before and after amplification to compute a relative enrichment of the sequences. This will characterise quantitatively how sequences amplify. To show the increase of binding affinity we would proceed the same way an perform with the mini-library one round of selection by biopanning.

We will sequence before and after the binding of the target. The relative enrichment will give us a hint of the relative increase of affinity of each sequences.

Our analysis as only been performed on single-point mutants, we could analyse this data by looking at couple of position to access the double point mutants. Preliminary analysis show that there are no significant double mutants in the "Selection" whereas in the "Drift" double mutants have very low frequency compared to the single-point mutants.

Conclusion

We develop a new system for affinity maturation of antibody fragments using the biopanning technique from phage display coupled with the in vivo mutagenesis from the plasmid developed by Badran for PACE. The biopanning provides our system with a well-controlled selection pressure. We provide a simplistic model of the selection pressure applied as a function of incubation time and washing time, as well as antibody biochemical parameters (see appendix).

Our system introduces mutations in vivo with a rate of ~0.15 mutations per kb. This is still far from the objective of one mutation per gene per generation. Nevertheless, one can proceed to several rounds of drifts before the selection and achieve the desired mutation rate. In our experiment we performed 10 cycles of selection and 10 cycles of amplification/mutation in 10 days. Nevertheless, cycles of amplification/mutation can be drastically reduced from overnight production to 2-3 hour productions. Our system is easily parallelisable with some automations:

the selection part uses magnetic beads, which can be handled with biopanning robots; the amplification/mutation phase requires little hands-on time and can be automatized using liquid handling robots and/or a turbidostat. Overall, our system meets most of our objectives and can easily be further optimized.

The concentration of the complex as a function of time is:

[φ] 2 = *$$ + *( ([φ] 0 + [] 0 ) 2 *( - 2 *( ⋅ tanh = 2 > + *$$ + *( ([φ] 0 + [] 0 ) -2 *( [φ] 0 1 + tanh = 2 > ⋅ *$$ + *( ([φ] 0 + [] 0 ) -2 *( [φ] 0 With = ? = *$$ + *( ([φ] 0 + [] 0 )> 3 -4( *( ) 3 [φ] 0 [] 0 , and [φ] 0 , [] 0 , [φ] 0 the initial
concentrations of phage, targets and complex, respectively.

In the case of large excess of target, we can now define the fitness of a phage as a function of time f(t), the proportion of phage bound at time t.

() = [φ] 2 [φ] 0
The initial conditions are set and limited by the experimental constrains:

• we will do the experiment in 1mL of liquid;

• there is a maximal amount of target one can bind to beads. In the following, we used a value of 100 pmol that corresponds to 0.5 mg of beads fully saturated with target.

• The phage production will not exceed 10 10 phages During the phage display experiment, two steps are crucial for the control of the biopanning:

First, we bind the phages having different affinities to the target molecules. The effect of this step is mainly selection on kon. The figures show intuitively that the more we wait, the wider range of kon we select, and we lose power of discrimination for the very fast kon. The second step is washing to remove the non-specific phages. For the model we consider that at t = 0 all the phages are bound to the target and compute the proportion of it still bound to the target after being in solution that does not contain free phages. The complete selection pressure is the product of the 2 independent steps (see Figure 32).

Number of rounds required

Now we can look at the proportion of one antibody with a probability k-times better than the background after r rounds. Here are reported the parameters found for the fitting of sequencing error rate as a gamma distribution for each type of substitutions for our data set.

Substitution type Shape parameter (a ) Rate parameter (q)

A to C Here are reported the parameters found for the fitting of sequencing error rate as a gamma distribution for each type of substitutions for Steven's data set.

Supplementary Figures

Errors rate prediction and mutations thresholds

Reproductivity of drifts Top1 vs Top3

Figure 1 :

 1 Figure 1: 3D Structure and schematic of an antibody

Figure 3 :

 3 Figure 3: Mutational outcomes after the action of Activation-Induced cytidine Deaminase

Figure 4 :

 4 Figure 4: Schematic of M13 phage particle

Figure 5 :

 5 Figure 5: Schematic of the phage replication cycle

Figure 6 :

 6 Figure 6: Schematic of the different phage particles in this thesis

Figure 7 :

 7 Figure 7: Schematic of a phage display biopanning experiment

  antibody and one non-specific antibody in large 10 5 -fold excess over successive selection/amplification cycles. Green are the number of specific antibodies; black the number of non-specific antibodies. During this process no evolution is happening (no mutation). The best binder already present in the library is 'fished out', mimicking the first step of the immune response. The number of specific antibodies first surpasses the number of non-specific antibodies after the third selection.

Figure 8 :

 8 Figure 8: Phage display mutagenesis cycle

  E. coli can differentiate the newly synthesized strand by using DNA methylation. The DAM enzyme (deoxyadenosine methylase) methylates adenine of GATC sites on the DNA. Immediately after the replication, the DNA is hemimethylated: only the older strand is methylated. If a DNA mismatch occurs the MutS protein recognizes the site and recruits MutL proteins that will activate MutH, an endonuclease that cleaves the unmethylated strand at the GATC sequences. The UvrD helicase and one of several exonucleases remove the nicked strand, containing the erroneous base. The DNA is then resynthesized by the DNA polymerase III. Due to the efficiency of the three main mechanisms to secure high fidelity: base selection, exonucleolytic proofreading, and post replicative mismatch correction, E. coli has a mutation rate of 10 -10 substitution per base per generation.

Figure 9 :

 9 Figure 9: Mismatch repair From the Nobel prize in chemistry 2015: popular science background, The Royal Swedish academy of sciences.

Figure 10 :

 10 Figure 10: Phage Assisted Continuous Evolution cycle

Figure 11 :

 11 Figure 11: The mutagenic plasmid MP6

Figure 12 :

 12 Figure 12: Plasmid map of pHEN3

Figure 13 :

 13 Figure 13: Kinetic of phage particles production in SP

4

 4 | 0(<10 2 ) 10 4 | 0(< 10 2 ) 10 8 -10 9 | 10 5 10 8 -10 9 | 0(< 10 2 )

Figure 15 :Version 3 :

 153 Figure 15: Western Blot revealed with anti-pIII antibody

Figure 16 :

 16 Figure 16: Western Blot revealed with anti-pIII antibody

Figure 17 :

 17 Figure 17: Phage ELISA revealed with anti-pVIII antibody

Figure 18 :

 18 Figure 18: Schematic of our system

ðo

  of a scFv fragment (largest fragment tested). The inducible mutator version SHM introduces mutation in a replicative phagemid at 1/10000 substitution per base per generation. 1 Have well-controlled selection pressure. Selection in vitro. è Link phenotype and genotype. o We are using phage display in vitro biopanning: § the selection conditions are well controlled, and the genotypes and phenotype are linked through the phage 2 Be rapid. We showed that we can accelerate drastically the time it takes to generate phage containing phagemid taking advantage of the phage production kinetic or through the use of helper plasmid removing the time-consuming superinfection step by helper phage. ð In vivo mutagenesis o We are using in vivo mutagenesis trough the MP6 plasmid from PACE 3 Introduce one mutation per gene per generation.

Figure 19 :

 19 Figure 19: Schematic of the affinity maturation experiment

Figure 22 :

 22 Figure 22: Fraction with highest sequencing quality

Figure 23 :

 23 Figure 23: Example of typical fits on substitutions

Figure 24 : 71 (R 2

 24712 Figure 24: Means of fitted gamma distributions per type of substitutions

Figure 25 .

 25 Figure 25.

Figure 25 :

 25 Figure 25: Outliers fits C to T and G to A substitutions

Miseq

  Proposed sketch of mutation rates distribution Sum sequencing error and mutation rate distribution

Figure 26 :

 26 Figure 26: Top3 round 10 for the "Selection" experiments

Figure 27 :

 27 Figure 27: Top3 round 10 for the "Drift" experiments

Figure 30 :Figure 31 :

 3031 Figure 30: Sequence logo of "Drift" for the Top1 and Top3 experimentThe frequency reported in percent are the one above the highest threshold found for each type of substitutions to not bias this figure with amino acid that are accessible with types of substitution for which we have a better resolution. To facilitate comparation between samples the scale is cut at 1% nevertheless some positions present higher frequency. (Top1 round 10 at position 39 up to 1.3%;Top3 round 5 at position 39 up to 1.6% and position 101 up to 1.2%; Top3 round 10 at position 39 up to 4.7% and position 101 up to 7.1%.). It seems like there is a strong similarity between the round 5 and 10 but also between the Top1 and Top3 experiments. Each letter corresponds to its amino acid, stop codon are represented with "*"

Figure 32 :

 32 Figure 32: Fitness (proportion of phage bound to the target) after the different step of biopanning

  Now if we look at the frequency of better in the population ( 4#22#) + 45678)*9(: ) we optain: initial frequency. If we take f0= 10 -7 corresponding to a k-times better mutant present 1000 in a pool of 10 10 phages we can draw Figure33.

Figure 35 : 1 -2) Day 2 - 1 )

 35121 Figure 35: Expected distribution of mutants

Figure 36 :

 36 Figure 36: Top1 round 5 for the "Selection" experiments

Figure 37 :

 37 Figure 37 : Top1 round 10 for the "Selection" experiments

Figure 38 :

 38 Figure 38 : Top1 round 5 for the "Drift" experiments

Figure 39 :

 39 Figure 39: Top1 round 10 for the "Drift" experiments

Figure 40 :

 40 Figure 40: Top3 round 5 for the "Selection" experiments

Figure 41 :

 41 Figure 41: Top3 round 5 for the "Drift" experiments

Figure 42 :

 42 Figure 42: C1 scFv VH part round 5 for the "Selection" experiments

Figure 43 :

 43 Figure 43: C1 scFv VH part round 10 for the "Selection" experiments

Figure 44 :

 44 Figure 44: C1 scFv VH part round 5 for the "Drift" experiments

Figure 45 :

 45 Figure 45: C1 scFv VH part round 10 for the "Drift" experiments

Figure 46 :

 46 Figure 46: C1 scFv VL part round 5 for the "Selection" experiments

Figure 47 :

 47 Figure 47: C1 scFv VL part round 10 for the "Selection" experiments

Figure 48 :

 48 Figure 48: C1 scFv VL part round 5 for the "Drift" experiments

Figure 49 :

 49 Figure 49: C1 scFv VL part round 10 for the "Drift" experiments

Figure 52 :

 52 Figure 52: Presence of mutants after 10 rounds of drift for the top1 and top3 Each letter represents an amino acid, the red dotted line is the linear regression y=x gives a R 2 of 0.62 indicating that there is a large reproducibility for each amino acid frequency in each experiment. Each histogram represents the density of point present in an interval. There is 5% of unique sequence only present in Top1 while there is 20% only in Top3. The dashed dotted line represent the highest threshold above which we can differentiate all types of substitutions from sequencing error, therefore the reported amino acid below might have a bias towards the types of amino acid that are accessible with substitutions for which with have a better power of differentiation of mutation from sequencing errors.

Table 1 : Number of mutants in newly produced helper phage

 1 We induced the "SM" cells at OD 0.4 with 25mM Arabinose and infected them with 1:10 M13Ko7. We recovered newly formed helper phage by infecting S1030 with the supernatant. We plated S1030 on kanamycin and picked 6 colonies per condition to mini prep for Sanger sequencing. The number of point mutants are reported for each condition.

Table 5 : Number of mutants produced in 16h

 5

Table 6 : Phagemid titre produced in 16h with SH101P

 6

	Antibody Fresh colony Glycerol Stock
	VHH -αeGFP	10 11	10 8
	VHH -αmCherry	10 10	10 8
	VH Top1	10 10	10 9
	VH Top3	10 11	10 8
	scFv C1	10 10	10 8

3

Kanamycin resistant CFU.

Table 7 :

 7 Counts of phage particlesDn are the drifts round n. Bn are the phages bounds to the beads round n. Pn are the produced phage after amplification of the beads round n bound phage and the input for the binding assay round n+1. For the drift phage titer is manly stable at 1010-11 . For the selection scFv and the VH, the phage titer has a

	tendency to increase with the number of rounds. The population of the VHH went
	extinct after 1 round of selection. We sequence 6 VHH clones from the drift round
	5 and 10, all presented a deletion in the antibody gene fragment. No further
	analysis has been performed on this antibody fragments.

Table 8 :

 8 Number of readsNumber of paired-end reads of the Illumina MiSeq v3 2x300 bp sequencing for the experiments with the initial antibody fragments VH Top1, Top3, and scFv C1.Rounds 5 and 10 of the selection and the "Drift" were sequenced.

	109

Table 9 : Parameters for the gamma distributions our data set

 9

Table 10 : Parameters for the gamma distributions Steven's data set

 10

		2.2823368459083215	3.375082917018203e-06
	A to G	3.1477084681788896	1.81401279284444e-05
	A to T	2.763018756813718	8.763112939059982e-06
	C to A	4.198333530361725	3.223765973450445e-05
	C to G	2.4959907816046045	8.316943664034441e-06
	C to T	1.9020880683932544	2.7682058171376536e-05
	G to A	3.4786362682590526	1.470751379549682e-05
	G to C	3.009828734391864	6.9030353062536395e-06
	G to T	4.030207002263053	3.5730956564261095e-05
	T to A	2.0792231716958485	1.064536781539258e-05
	T to C	3.284118570035688	1.8694647417592777e-05
	T to G	1.6479819891123317	5.931257471922536e-06

Remerciements

I would like to thank my wife Judith, for all her support every day, the scientific discussion that

Summary

In vitro mutagenesis provides a perfect control of the generated library. This control comes with the costs of the reagents and the time to prepare new libraries, as all the techniques require the highly time-consuming step of cloning. In vitro mutagenesis is therefore not compatible with the efficient generation of affinity maturation trajectories in the laboratory.

In vivo mutagenesis on the other hand, is inexpensive and minimizes hands-on time, because the organism will do the mutagenesis for you. This comes at the expanse of the control of the generated libraries, as most of previously established techniques are biased and present poor mutation rates. Nevertheless, the recently described mutagenetic plasmid from PACE seems to have a mutation rate that is good enough for affinity maturation experiments and could potentially be of use to accelerate in vitro affinity maturation by providing in vivo mutagenesis. ð Efficient and tuneable mutation rate. [START_REF] Edelman | The covalent structure of an entire gammaG immunoglobulin molecule[END_REF] Be parallelizable with some automations.

Construction of the system

ð All steps can be done with robots

In vitro Selection

The main objective and principal motivation for this project is to generate affinity maturation trajectories with well-controlled selection pressure. To satisfy this objective, our system must have an in vitro selection, where we can control all the parameters of the selection. We will use streptavidin-coated magnetic beads to attach a controlled concentration of target molecules, either through the use of biotinylated target or a tag. We will incubate the target-coated beads with a controlled amount of antibody molecules for a controlled amount of time. Additionally, the buffer and biophysical parameters of the system, such as temperature, pH, ionic concentration, etc., will be well controlled. To remove the unspecific antibodies, we will proceed to wash the beads several times with the buffer. These steps are fully automatable through the use of robots.

To be able to perform directed evolution on the antibodies, we need to link the phenotypes (here the binding properties) in our controlled setup with the genotypes (the sequence of our antibodies). For that, we will use phage display. This allows us to easily link the antibody molecule to its gene through the phage. More detail on phage display can be found in the motivations.

Version 1: Helper phage

Phage production in S1030

In order to check if we are still able to produce phage with the amber codon removed, we cloned the phagemid VHH anti-mCherry and VHH anti-eGFP into S1030 to form 'SP' cells.

We produced phage by superinfecting 'SP' with helper phage M13KO7. Those phage contain either the phagemid or the genome of the helper phage. In order to count those phage, we infected S1030 with the supernatant of the "SP" production. We selected the phage containing phagemid by plating on ampicillin plates, the helper phage by plating on kanamycin plates. In 16h, we produce 10 10 phage containing phagemid per ml of culture. The same amount of phage (10 10 ) carry the helper phage genome.

Additionally, we did a preliminary kinetic analysis of phage production to see if we can shorten the production time of the phage particles. We reached 10 8 phage containing phagemid in 2 to 3 hours. This means that we can drastically reduce the time that is taken to do a round of phage display reducing the production time of the phage containing phagemid to 2 to 3 hour instead of overnight production.

Version 2: Helper plasmid

We built the first helper plasmid by deletion of the origin of replication and the pIII gene in the M13KO7 helper phage genome. Transforming this helper plasmid into "SP" produced a strain that produces overnight 10 4 colonies with the phagemid (resistant to ampicillin) and 10 5 with the helper plasmid (resistant to kanamycin). This plasmid probably still contains a quite efficient packing signal. As we want the helper plasmid to not be packed into the phage but stay in the cells, we asked Brodel to send us their version of the ∆pIII ∆PS helper plasmid (deletion of pIII and of the packing signal). They kindly provided it. This is an unpublished plasmid made during the work for this paper [START_REF] Brödel | Intracellular directed evolution of proteins from combinatorial libraries based on conditional phage replication[END_REF]. The helper plasmids contain a Kanamycin resistance cassette and all the protein coding genes to of the M13 phage except for the pIII. They cannot be packed into phage particle. They differ from their origin of replication.

Mutations

To check if the helper plasmid restored the mutations in the produced phagemid we infected SH15M with phagemid produced with M13KO7. We produced phage particles overnight under different induction conditions: Glucose 25mM + Arabinose 25mM to mimic the condition of PACE, Arabinose 25mM, Arabinose 25mM + IPTG 300μM to also induce the antibody-pIII production.

Phage helper used, Conditions

Count CFU/ml Mutants / Sequenced We induced SH15M at OD 0.4 with the reported conditions and infected them with phagemid produced with M13Ko7 or Hyperphage. We recovered newly formed helper phage by infecting S1030 with serial dilution of the supernatant.

We plated S1030 on Ampicillin and picked colonies the DNA of which was Sanger sequenced. The number of mutants over the sequenced clones are reported for each condition.

We collected the supernatant, infected fresh S1030 and plated them on Ampicillin. We picked 8 colonies for each condition. The number of mutants is depicted in Table 4. We also sequenced 12 clones from the Arabinose condition where the phagemid used to infect SH15M was multivalent (produced with the Hyperphage). The number of mutants are on the order of 1 every 10kb which is one order of magnitude lower than the one reported in the Badran paper [START_REF] Badran | Development of potent in vivo mutagenesis plasmids with broad mutational spectra[END_REF].

Display level

We checked the display level by doing a western blot with the phage particles using anti-pIII

antibody. An annotated picture of the blot is represented in Figure 16. We can see that the system behaves has expected. The helper phage present only the wild type pIII. The phage containing phagemid produced by the M13KO7 presents mainly the wild type pIII. The phage containing phagemid produced by Hyperphage is multivalent. The phage containing phagemid produced by the Helper plasmid is multivalent. The phage containing phagemid that went through one round of amplification/mutation in the mutator strain is multivalent. The number of phage produced with helper plasmid is low and the band for the antibody-pIII are very thin.

Analysis

The goal of this analysis is to characterize the mutations in our affinity maturation experiments, and to see if there is selection towards specific mutants.

One major challenge is that the deep sequencing technologies are error prone and we do not a

priori know what mutants we generate with our technique. The next paragraphs will explain how we estimate the mutation rate and discriminate real mutations from sequencing errors.

Pre-processing

From the Illumina MiSeq v3 2x300 bp sequencing, we obtain paired-end reads. Before analysing the mutations and sequencing errors, we have to pre-process that large amount of data. The data contain the read with their nucleotide sequence and a position-wise QC score.

The first question to ask is how much you can trust that a sequence is the correct one. For that we can use the quality score (QC) given by the sequencer. It is a prediction of the probability of an error (Perror) in base calling. The highest value of QC is 38. This corresponds to an error probability of approximately 0.16 substitutions per kb.

The first read is at most 300 bp and start at the 3' end of the antibody fragment gene. It has in general a good quality score for the first 250 bases. The second read is at most 300 bp long and start 5' beginning of the antibody fragment gene. It is overall of poorer quality than the first read but still has a good quality for the first 150 bases. In the case of the C1 scFv, the VH and the VL domains have been treated as independent fragments as the full length exceeds the length we can sequence.

We truncate each sequence as soon as two consecutive bases have a QC ≤ 20 (1% chance of being an error). As our fragment have a length of around 400 bp, we then align the 2 paired reads and compute the overlap. We disregard sequences without an overlap. For the other sequences, we check the alignment of the 2 reads within the overlap and keep those that are in agreement (see Figure 20).

This method of pre-processing keeps approximately 50% of the raw sequencing data and is thus sufficient for further analysis. Nevertheless, there are several improvements that could be made to the pre-processing in order to keep more data. Currently, only paired-end reads with full nucleotide agreement of the overlap are considered. Allowing a few mistakes and only disregarding these erroneous positions, would increase the number of pre-processed sequences.

The number of these sequences could be further increased by even considering non-overlapping paired-end reads. Although they do not cover the whole sequence, they can still contain valuable information for parts of the sequence. Our first sequence analysis here, however, requires the two paired-end reads to comprise the whole antibody fragment sequence. As we will ask what the typical number of substitutions per sequences is.

Characteristic mutation rate

We compare each pre-processed sequence with the sequence of its respective gene of initial antibody fragment (wild-type) and report the observed number of nucleotide substitutions per sequence. We can retrieve a substitution rate ( * ) by fitting a Poisson distribution to the observed substitutions per sequence. This substitution rate is composed of x times the real mutation rate ( 1 ) added every round in the in vivo mutagenesis step, and the sequencing error rate ( # ).

# + 1 = * Two assumptions underlie this equation: (1) There is no selection and mutations are equally likely to occur. [START_REF] Edelman | DISSOCIATION OF gamma-GLOBULIN[END_REF] The number of diverse sequences after mutagenesis is sufficiently large to prevent the fixation or loss of random mutations.

We hypothesize that these two assumptions could be met by the "Drift" experiments. We thus computed the average number of nucleotide substitutions per sequence for all the sequenced samples at round 5 and 10. We then fitted a Poisson distribution to retrieve * for round 5 and 10 (see Figure 21). The mutation rate we can compute is then 0.13 mutations per kb. This rate is 50-fold less than the expected mutation rate predicted in [START_REF] Badran | Development of potent in vivo mutagenesis plasmids with broad mutational spectra[END_REF]. This correspond to one mutation every 20 genes. Nevertheless this mutation rate is consistent with the one observed by sanger sequencing during the construction of the system (see p). The error rate we can compute is 0.18 mutations per kb. Which is close to the error probability estimated with the highest value of QC above. The mutation rate is the limiting factor of our technique. It is on the same order of magnitude as the sequencing error for the highest QC of reads. For this reason, we have to do an analysis to discriminate true mutation from sequencing errors.

Differentiation of true mutations

Position by position analysis

To identify true mutations, we analysed the sequences position by position. For each sequence, we kept only the nucleotides with the highest QC of 38. This will provide us the data with the

Fitting error rates

As we do the analysis position by position, we are looking at the frequency (or rate) of substitutions per position and not at the number of substitutions per sequence. We have to discriminate substitution rates that correspond to a typical sequencing error from a rate corresponding to a real mutation. The hypothesis is that a sequencing error can happen anywhere on the sequence, thus each position should have a similar error rate (depending on the original nucleotide). On the other hand, if at a position a mutation is selected, the apparent substitution rate will be different from the positional error rate (it should be higher). In order to estimate the error rate, we analysed the distribution of the substitution rates. For each sample, the distribution seems to have various contributions from each type of substitutions. The individual distributions for each type of substitution are different: some are very narrow, like for T to G substitutions, or broader, like for the C to A substitutions. We therefore asked the question of what the typical error rate for each type of substitution is. This question is fundamentally different from the one we asked in the previous paragraph (see Characteristic mutation rate). We now look at substitution rate per position, not at the number of substitutions per sequence. Nevertheless, we can assume the same underlying stochastic process. Previously, we saw that the number of expected substitutions per sequence follows a Poisson distribution, which is defined on discrete numbers. In the case of substitution rate per position, however, our data are continuous. Empirically, an appropriate distribution to use in this case is the gamma distribution. The gamma distribution involves a 'shape' parameter a, which determines how the data varies, with a ≤ 1 representing large variation and a > 1 little variation. If a is greater than 1, the distribution is bell-shaped, otherwise it follows an L-shape. The gamma distribution has been used extensively to model mutation rates in phylogeny data sets [START_REF] Yang | Among-site rate variation and its impact on phylogenetic analyses[END_REF] and provides a good fit with our data (see Figure 23).

All our samples were sequenced in one sequencing run. The sequencing errors depend mainly on the chemistry of the base pairing and the fluorescent readout of each base call. There is no apparent reason to believe that the rates of sequencing errors should be different for each sample. We therefore assume, that the error rates are the same for all our samples. Ideally, samples consisting of one sequence without mutations would be the perfect data set to identify the distribution of error rates. Unfortunately, we did not sequence any sample where no mutations were added to the system. We therefore had to find a workaround compensating this lack of data.

Application to our sequencing data

We assume that the distribution of error rates is the same for all the samples. We have fitted gamma distributions to the error rates of aggregated samples (from the "Selection" part) that contain sequencing errors only, after we removed the mutations through an outlier analysis. Our The main challenge here is to rescale the fitted gamma distribution to explain the part of the data resulting from sequencing errors. We assume that the rates that emerge from sequencing error are the lower ones. To find a good scaling parameter, we focus on the data that is below an arbitrary threshold. The reason to fit the distribution partially is that true mutations should present higher rates than sequencing errors and therefore would bias strongly the scaling in the tail of distribution. We define this threshold as the rate of the 90 th percentile of the fitted gamma distribution. (90% of the variables of the gamma distribution are bellow this threshold.) We found the scaling parameter by fitting the part of the gamma below the threshold to the part of the data below the threshold. The obtained, scaled gamma distributions are our predictions for the sequencing error rates. The scaling factor represent the proportion of the data that are due to sequencing errors.

Using the prediction of sequencing error rate, we now wonder what are the lowest rates that represent a true mutation within an acceptable margin of error. Therefore, we computed the probability of the observed substitution rate to be a sequencing error rate based on the scaled distribution. For that we took the ratio between the integral of the distribution of predicted sequencing error rates and the integral of the distribution of the observed substitution. This ratio gives us the probability to identify wrongly a rate as coming from a real mutation (false positive rate). From this probability, we choose as threshold the rate for which we make mistakes less than 5% of the time. We can now discriminate true mutation from sequencing errors. Below the threshold there is a too high risk that the observed substitution rate is coming from a sequencing error. Above the threshold we are confident at more than 95% that this rate is coming from a true mutation. Examples of this process can be found in Figure 26 and Figure 27. All the fits and threshold are reported in the appendices.

The "Selection" and the "Drift" present different profiles: the "Selection" is dominated by sequencing error whereas the "Drift" presents more true mutations.

Characteristic mutation bias

As we have seen in earlier paragraphs the sequencing errors are biased with the bias dependent on the type of substitution. After the filtering, only true mutations are left, we can see that the bias is different from the one of the sequencing errors. This indicates that they come from two distinct process and comforts that our analysis to separate sequencing errors to mutations is working.

Evidence of selection

We are able to identify true mutations from sequencing error and characterized the mutation rate we can wonder if there is evidence for selection in this dataset.

To see evidence of selection in our dataset we looked at the type of amino acid substitution present in the true mutations.

Figure 29 shows that the mutation pattern is different form just random mutations, the mutation bias of the technique does not explain the differences in the type of mutation. In all the cases silent mutations are overrepresented. This is an indication of selection pressure applied on the sequences. In the "Selections", the silent mutations are even more represented.

The Ka/Ks ratio ( 97) is used commonly use to estimate the presence of selection. Ka is ratio between the observed sense amino acid different mutations and the ones expected under the condition than all the mutations are equally represented (It does not account for nonsense mutation i.e. stop codons). Ks is ratio between the observed synonymous mutations and the ones expected under the condition than all the mutations are equally represented.

Here we found Ka/Ks ratio less than 1. It implies purifying or stabilizing selection [START_REF] Yang | Statistical methods for detecting molecular adaptation[END_REF].

However, one should be careful with this type of ratio as combination of positive and purifying selection at different positions within the gene may cancel each other out. Our observed Ka/Ks value can mask the presence of positive selection or lower the magnitude of the observed purifying selection.

The selection and "Drift" value of Ka/Ks ratios are significantly different that one (performing 1 sample t test give a p-value of respectively 4.5 10 -3 and 4. 3 10 -5 ). This is a strong indication that purifying selection is happening even in the case of the "Drift".

We do not observe a significant difference between the distribution of the Ka/Ks ratio of the selection and "Drift" (two samples Kolmogorov-Smirnov test give a p-value of 0.28)

Through the analysis of bias in types substitutions at the amino acid level, we have seen that there is evidence of a purifying selection occurring in both selection and "Drift" experiments. This is not due to the bias of nucleotide mutation reported in the previous paragraph.

Perspectives

The main limitation of our system is currently the mutation rate. We propose two ways to overcome this limitation:

The first one is to increase the amount of cycles by automatizing the process. Our system can be automatized because the hands-on time is drastically reduced compared to classic in vitro mutagenesis techniques. Proceeding to several rounds on drift in between the selections will produce the desired mutation rate. The time to produce new phage can be reduced from overnight to 2-3 hours as shown in Figure 13. One can probably cycle the drifts with cycles of dilution/amplification 8 to 12 times per day or use a turbidostat for the mutagenesis and harvest it daily. Nowadays, it exist low cost turbidostats (99) (100). The selection phase is also fully automatable thanks to the use of magnetic beads.

The second way is biochemical: one can try to remove the coli origin of replication of the phagemid to force ssDNA phage replication. The drawback of this will be that it will be much more difficult to titre the phage to follow their evolution over time. It would also be difficult to isolate clonal antibodies. Nevertheless, it should not prevent the generation of affinity maturation trajectories through the use of deep sequencing, as the phagemid DNA can be amplified directly via PCR. One can try to couple the MP6 with the MutaT7 system. The deaminase can be replaced by newly evolved, more efficient versions.

The multivalent display can also be tweaked with the use of the mutagenic plasmid DP6 also constructed by Badran for PACE [START_REF] Badran | Development of potent in vivo mutagenesis plasmids with broad mutational spectra[END_REF]. It contains the same mutagenic machinery as MP6, with the addition of the pIII gene under a pTet promotor (inducible with atc).

Appendices Modelling of the selection

Selection pressure

To check our understanding of the selection pressure, we decided to explore a very simple model of binding. The production of phage is much slower than the binding to the target, so we hypothesize that there is no change in concentration of phage over time. We also hypothesize that the medium is well mixed and that, even if the target is physically linked to magnetic beads, they can defuse freely. With these constrains in mind, one can write a basic model of the phage interaction with its target of the form of a simple, reversible, bimolecular binding reaction: A phage φ binds to its target T to form the complex φT.

φ+ T ⇌φT

The rate of production of the complex φT depends on the probability of a phage to encounter a target and the association rate constant kon. The complex can dissociate at a rate koff, the probability that the complex will fall apart in a unit of time. One can write the following differential equation modeling the change of the complex over time:

Where [φ] and [T] are the concentration of unbound phages and unbound target, respectively.

These are available for the formation of the complex [φT] at a given point in time. Similarly, the concentration of unbound phages [φ] and unbound target [T] follow these equations:

Due to the conservation law, this set of equation can be reduced to one. The kinetics then only depends on the initial conditions. Steven Schulz solved these equations analytically. Figure 33 and Figure 34 show that after 10 rounds we should be able to see antibodies only twice as good with a frequency of 10 -4 if they were present in 1000 copies in the initial population of 10 10 phages.

Mutants per round without selection

Assuming that there is no selection, the frequency of mutants should follow a Poisson distribution with parameter l = mutation rate × number of rounds. Figure 35 shows this distribution for the 10 rounds for a mutation rate of 1 mutation every 10 genes. To design the primers so that they all have the same annealing temperature of 64˚C we used the online tools NEB calculator (from v1.9.6 to v1.11.0)

PCR protocol

Supplementary Tables

Sequences logos

"Drift" The frequency reported in percent are the one above the highest threshold found for each type of substitutions to not bias this figure with amino acid that are accessible with types of substitution for which we have a better resolution. To facilitate comparation between samples the scale is cut at 1% nevertheless some positions present higher frequency.

"Selection" For the selection, in the round 5 no functional sequences are present for the VH, at the round 10 some mutations are present in majority in the CDR regions. For the VL mutations do not show a preference for CDR regions.