Wen-Hui K Kuo

Howard A Chase

Exploiting the Interactions between Poly-histidine Fusion Tags and Immobilized Metal Ions

Keywords: electrochemical immuno-sensor, fusion tags, immobilized metal affinity, poly-histidine fusion tag cleavage, proteins

Immobilized metal affinity chromatography (IMAC) of proteins containing poly-histidine fusion tags is an efficient research tool for purifying recombinant proteins from crude cellular feedstocks at laboratory scale. Nevertheless, to achieve successful purification of large amounts of the target protein for critical therapeutic applications that demand the precise removal of fusion tags, it is important to also take into consideration issues such as protein quality, efficiency, cost effectiveness, and optimal affinity tag choice and design. Despite the many considerations described in this article, it is expected that enhanced selectivity, the primary consideration in the field of protein separation, will continue to see the use of IMAC in solving new purification challenges. In addition, the platform nature of this technology makes it an ideal choice in purifying proteins with unknown properties.

Finally, the unique interaction between immobilized metal ions and poly-histidine fusion tag has enabled new developments in the areas of biosensor, immunoassay, and other analytical technologies.

Introduction of immobilized metal affinity chromatography

 [START_REF] Porath | Metal Chelate Affinity Chromatography, a New Approach to Protein Fractionation[END_REF]first introduced Immobilized Metal Affinity Chromatography (IMAC) for the fractionation of serum proteins. Ever since, IMAC has become one of the most powerful tools in the field of separation science. IMAC is based on the interaction between transitional metal ions, such as Cu 2+ , Co 2+ , Ni 2+ , Zn 2+ , or Fe 3+ , immobilized on a matrix and specific amino acid side chains in proteins. Transition metal ions can be immobilized on an agarose or silica gel support derivitized by ligands such as iminodiacetic acid (IDA), N,N,N'-tris-(carboxymethyl)-ethylenediamine or nitrilotriacetic acid (NTA) groups as shown by Figure 1. The main amino acid in proteins that interacts with the immobilized transitional metal ions is histidine. This is due to the fact that the electron donor groups on the histidine imidazole ring form coordination bonds with the immobilized transition metal. The metal chelate ligands have the advantage of being more resistant to physical, chemical or microbiological degradation than other highly selective ligands such as immunoglobulin-binding proteins, which facilitates the use and storage of the immobilized metal affinity adsorbents.

Coupling immobilized metal affinity chromatography with poly-histidine tag

The fast-developing field of proteomics has promoted the use of recombinant proteins containing a poly-histidine fusion partner to expedite the isolation of the target compound [START_REF] Bolanos-Garcia | Structural analysis and classification of native proteins from E-coli commonly co-purified by immobilized metal affinity chromatography[END_REF]. These fusion tags can be attached to a wide range of different proteins. In addition, the method of tagging enables simple and accurate assay of the recombinant protein during purification. The use of poly-histidine fusion tags was first developed by [START_REF] Hochuli | Genetic Approach to Facilitate Purification of Recombinant Proteins with a Novel Metal Chelate Adsorbent[END_REF] and has indeed become the most widely used type of affinity fusion tag. Poly-histidine fusion tags range from 2 to 10 residues, with hexa-histidine tags containing 6 consecutive sequences of histidine residues, having an overall size of 0.84 kDa, being the most popular [START_REF] Terpe | Overview of tag protein fusions: from molecular and biochemical fundamentals to commercial systems[END_REF]. Peptides containing sequences of consecutive histidine residues are found to be particularly effectively bound to IMAC adsorbents. After washing of the loaded IMAC adsorbent to remove non-specifically adsorbed proteins, these peptides can be easily eluted by either lowering of the pH or by increasing concentrations of imidazole in the irrigating buffers. Following elution, the target protein can exhibit a purity higher than 95% [START_REF] Terpe | Overview of tag protein fusions: from molecular and biochemical fundamentals to commercial systems[END_REF].

Antibody fragments, labile enzymes, viruses, and membrane-associated targets present an unique set of challenges to the purification researcher that can often be solved by the use of the ubiquitous poly-histidine fusion tag that has gained popularity in purification and structural studies over the last two decades. For instance, attaching a poly-histidine fusion tag to proteins through genetic engineering has proven successful for the IMAC purification of recombinant antibody-like structures [START_REF] Bannister | Parallel, high-throughput purification of recombinant antibodies for in vivo cell assays[END_REF], complex virus-like particles [START_REF] Hu | Chimeric infectious bursal disease virus-like particles expressed in insect cells and purified by immobilized metal affinity chromatography[END_REF], protein complexes [START_REF] Kubota | Purification and characterization of photosystem I complex from Synechocystis sp PCC 6803 by expressing histidine-tagged subunits[END_REF] and viral vectors for gene therapy [START_REF] Hu | Generation of chimeric baculovirus with histidine-tags displayed on the envelope and its purification using immobilized metal affinity chromatography[END_REF].

IMAC in conjunction with poly-histidine fusion tags can be used as a highly selective primary capture step as target proteins fused with poly-histidine fusion tag can be purified without much prior knowledge of the target protein and consequently, a platform process can be developed that can be employed for a wide range of targets.

Besides aiding purification, poly-histidine fusion tags can help increase protein yield [START_REF] Sun | An improved strategy for high-level production of human vasostatin120-180[END_REF] and prevent proteolysis [START_REF] Tang | An efficient system for production of recombinant urokinase-type plasminogen activator[END_REF]. In one example, incorporation of a poly-histidine tag helped to improve rates of cell growth and protein expression levels in cells producing recombinant protein products because of the hydrophilic nature of a poly-histidine tag [START_REF] Svensson | Histidine tag fusion increases expression levels of active recombinant amelogenin in Escherichia coli[END_REF]. In fact, with minimum optimization of the poly-histidine tag sequence, most IMAC purification processes are often associated with high yield, fast development time and low costs [START_REF] Lichty | Comparison of affinity tags for protein purification[END_REF]. Likewise, a major advantage of using IMAC as a primary capture tool for protein isolation is the ability to reduce a multistep process to a single unit operation that provides highly pure, good quality end-products. The reduction in the number of processing steps not only decreases process time and costs such as the associated cleanroom residence time for therapeutic production, but also decreases the likelihood of potential protein degradation as a result of prolonged processing times, hence ensuring better product quality. Furthermore, the scalability of the technology makes it an ideal option for large-scale manufacturing.

Other advantages of IMAC include the stability of the metal chelates over a wide range of buffer conditions and temperatures, high metal ion loadings, and high protein loading capacities [START_REF] Arnold | Metal-Affinity Separations -a New Dimension in Protein Processing[END_REF]. With further developments that have increased IMAC column binding capacities and dynamics, IMAC can perform equally well, if not better, than other commonly used affinity chromatographic methods such as immunoglobulin-binding protein chromatography. Besides being cheap and robust, metal chelate ligands can complex a variety of metal ions, thus it is possible to tailor adsorption characteristics to the protein of choice. For instance, SlyD, a major host cell contaminant protein in E. coli often co-purifies with the heterologously-expressed hexa-histidine tagged proteins when using nickel-based IMAC [START_REF] Bolanos-Garcia | Structural analysis and classification of native proteins from E-coli commonly co-purified by immobilized metal affinity chromatography[END_REF]. Wuelfing et al (1994), on the other hand, found that SlyD has a markedly reduced affinity for Co 2+ than Ni 2+ and the alternative use of the former immobilized ion provides an answer to the problem of SlyD co-purification.

Challenges in the use of IMAC and poly-histidine tag for purification

The addition of poly-histidine fusion tags can sometimes lead to undesirable protein aggregation and insoluble protein expression [START_REF] Woestenenk | His tag effect on solubility of human proteins produced in Escherichia coli: a comparison between four expression vectors[END_REF]. By manipulations of the temperature during expression and dilution refolding, this solubility problem may be able to be solved [START_REF] Bird | Manipulation of temperature to improve solubility of hydrophobic proteins and cocrystallization with matrix for analysis by MALDI-TOF mass spectrometry[END_REF]. Nevertheless, the length, composition and location of the poly-histidine tag can necessitate additional optimization depending upon the amino acid sequence of the native protein [START_REF] Chant | Attachment of a histidine tag to the minimal zinc finger protein of the Aspergillus nidulans gene regulatory protein AreA causes a conformational change at the DNA-binding site[END_REF][START_REF] Gaberc-Porekar | Histidines in affinity tags and surface clusters for immobilized metal-ion affinity chromatography of trimeric tumor necrosis factor alpha[END_REF]. For instance, the fusion tag can be attached to the N-terminus, C-terminus or both termini of virtually any protein in order to facilitate its purification by IMAC. The decision to which end the poly-histidine tag is fused depends primarily on the characteristics of the particular protein and the techniques selected to cleave the tag. Some poly-histidine tag termini become buried inside the protein core and others have a significant effect on the protein function. In such cases it is necessary to try attaching the tag to the opposite terminus. Nevertheless, it is generally difficult to predict if there is a negative influence of the tag and which terminal is likely to be least problematic for a given protein [START_REF] Goel | Relative position of the hexahistidine tag effects binding properties of a tumor-associated single-chain Fv construct[END_REF][START_REF] Lee | Expression of functional multidrug-resistance protein 1 in Saccharomyces cereviside: effects of N-and C-terminal affinity tags[END_REF][START_REF] Mason | Differential effect of a His tag at the Nand C-termini: Functional studies with recombinant human serum transferrin[END_REF].

Consequently it is best to perform functional characterization of both C-terminal and N-terminal tagged protein variants. One investigation suggests that the location of the hexa-histidine tag on the protein N-terminus or C-terminus has an influence on the binding strength of some proteins to IMAC adsorbents and a tag on both ends of the protein always increases binding strength [START_REF] Biosciences | Purification of (His)6-tagged proteins using HiTrap Chelating HP columns charged with different metal ions[END_REF]. On the other hand, [START_REF] Tsai | Effect of number of poly(His) tags on the adsorption of engineered proteins on immobilized metal affinity chromatography adsorbents[END_REF] concluded that although the capacity and binding affinity of IMAC adsorbents for fusion proteins having more than one affinity tag can, in general, be predicted depending on the size of the protein and the number of affinity tags, the structure-specific interactions between protein molecules and the surface accessibility and orientation of the affinity tags on the protein surface also need to be considered.

Similarly, it is often difficult to exclude the possibility that the affinity tag may interfere with protein activity and function [START_REF] Wu | Hexahistidine (His6)-tag dependent protein dimerization: a cautionary tale[END_REF], and especially when used in applications such as therapeutic manufacturing, the tag fused to the target protein must be subsequently removed [START_REF] Kenig | Influence of the protein oligomericity on final yield after affinity tag removal in purification of recombinant proteins[END_REF][START_REF] Rowland | Advancing TB vaccines to Phase I clinical trials in the US: regulatory/manufacturing/licensing issues[END_REF]. Some researchers suggest that the histidine tag does not need to be removed for use in structural determination [START_REF] Carson | His-tag impact on structure[END_REF]. Likewise, [START_REF] Case | Kinetic characterization of recombinant human cytosolic phosphoenolpyruvate carboxykinase with and without a His(10)-tag[END_REF] compared two variants of a recombinant human cytosolic phosphoenolpyruvate carboxykinase, one with a His10-tag and one without a poly-histidine tag; both exhibited similar kinetic properties. However, other reports suggest altered biological or physicochemical properties of the histidine tagged proteins as compared to their native counterparts [START_REF] Arnau | Cloning strategy, production and purification of proteins with exopeptidase-cleavable His-tags[END_REF][START_REF] Gaberc-Porekar | Potential for using histidine tags in purification of proteins at large scale[END_REF]. Introducing a poly-histidine tag can adversely affect the biochemical properties [START_REF] Horchani | The N-terminal His-tag and the recombination process affect the biochemical properties of Staphylococcus aureus lipase produced in Escherichia coli[END_REF], alter the binding characteristics [START_REF] Gaberc-Porekar | Histidines in affinity tags and surface clusters for immobilized metal-ion affinity chromatography of trimeric tumor necrosis factor alpha[END_REF], change protein structure conformation [START_REF] Chant | Attachment of a histidine tag to the minimal zinc finger protein of the Aspergillus nidulans gene regulatory protein AreA causes a conformational change at the DNA-binding site[END_REF], prompt protein oligomerization [START_REF] Amor-Mahjoub | The effect of the hexahistidine-tag in the oligomerization of HSC70 constructs[END_REF], and thus render the purified target unsuitable for use in X-ray crystallography or other characterization studies [START_REF] Renzi | Large-scale purification and crystallization of the endoribonuclease XendoU: troubleshooting with His-tagged proteins[END_REF]. Consequently, an important part of the design of a fusion tag is the choice of the method for removing the tag after purification. Ideally, molecular constructs contain a specific cleavage sequence included to ensure that selective cleavage can be achieved. The removal method usually then involves the use of a protease, to cleave a specific peptide linker between the tag and the target protein. However, the insertion of a cleavage site in a fusion tag can sometimes have a negative impact on the solubility of the expressed proteins [START_REF] Kurz | Incorporating a TEV cleavage site reduces the solubility of nine recombinant mouse proteins[END_REF], and this is an issue that deserves careful consideration. Some of the commonly-used cleavage enzymes include the tobacco etch virus (TEV) protease, thrombin (factor IIa, fIIa), enterokinase (EK), and factor Xa (fXa), which all belong to the protease class of endopeptidases. Some of these enzymes have been genetically modified to improve their specificity and stability such as AcTEV from Invitrogen and ProTEV from Promega.

Ideally, cleavage should be highly specific in order to avoid reducing protein activity and function. Non-specific cleavage can sometimes be resolved by using a lower enzyme to fusion-protein ratio, optimizing the temperature, pH, salt concentration, and the duration of cleavage reaction [START_REF] Jenny | A critical review of the methods for cleavage of fusion proteins with thrombin and factor Xa[END_REF]. Most commercially available cleavage enzymes need to recognize a specific amino acid sequence for reaction to occur. For instance, the most common sequence recognized by thrombin is LVPRG or LVPRGS, for enterokinase the sequence is DDDDK, and for factor Xa, it is usually IEGR [START_REF] Jenny | A critical review of the methods for cleavage of fusion proteins with thrombin and factor Xa[END_REF]. Nevertheless, cleavage by thrombin, enterokinase or factor Xa often results in unspecific internal cleavage, which is highly undesirable [START_REF] Jenny | A critical review of the methods for cleavage of fusion proteins with thrombin and factor Xa[END_REF][START_REF] Choi | Recombinant enterokinase light chain with affinity tag: Expression from Saccharomyces cerevisiae and its utilities in fusion protein technology[END_REF]Ko et al, 1994). [START_REF] Guan | Eukaryotic Proteins Expressed in Escherichia-Coli -an Improved Thrombin Cleavage and Purification Procedure of Fusion Proteins with Glutathione-S-Transferase[END_REF] found that by inserting five glycine residues between the thrombin cleavage sequence and the N-terminal tag, the unspecific internal cleavage decreased.

Similarly, TEV recognizes the sequence ENLYFQG or ENLYFQS [START_REF] Carrington | A Viral Cleavage Site Cassette -Identification of Amino-Acid Sequences Required for Tobacco Etch Virus Polyprotein Processing[END_REF]. Nevertheless, at the completion of the cleavage reaction by TEV, a G or S residue remains on the N-terminal end of the target protein [START_REF] Nallamsetty | Efficient site-specific processing of fusion proteins by tobacco vein mottling virus protease in vivo and in vitro[END_REF].

In some cases, the problem of unspecific internal cleavage stems from steric hindrance arising from the target protein structure that prevents the protease from access to the intended cleavage site [START_REF] Kim | Enhanced protease cleavage efficiency on the glucagon-fused interleukin-2 by the addition of synthetic oligopeptides[END_REF][START_REF] Shahravan | Enhancing the specificity of the enterokinase cleavage reaction to promote efficient cleavage of a fusion tag[END_REF]. Certain concentrations of denaturant were added in an attempt to improve the specificity and yield of the cleavage reactions [START_REF] Shahravan | Enhancing the specificity of the enterokinase cleavage reaction to promote efficient cleavage of a fusion tag[END_REF]. The addition of 1-4 M urea successfully improved the cleavage specificity and yield by making the protein molecules assume a more open structure. Nevertheless, to make this a general approach to improve protease specificity and yield applicable to other proteins, the tolerance of the chosen protease and target protein to denaturant concentration is important. For example, urea at 5 mM began to inhibit enterokinase activity [START_REF] Shahravan | Enhancing the specificity of the enterokinase cleavage reaction to promote efficient cleavage of a fusion tag[END_REF].

Enabling the wider use of IMAC and poly-histidine tags for purification

As mentioned earlier, major issues that arise when using histidine tag approaches to protein purification include the precision and efficiency required to remove the fusion tag at the end of the purification process. Some additional limitations concerning the removal of fusion tags are non-specific cleavages yielding truncated forms of the target protein, addition of extra amino acids, partial removal of the tag, and the presence of contaminating proteases and cleaved peptides contaminating the target protein. Indeed, cleavage by endopeptidases has not been observed to be very specific and was suspected to generate complex mixtures of variants [START_REF] Gaberc-Porekar | Potential for using histidine tags in purification of proteins at large scale[END_REF]. Furthermore, the use of endopeptidases is costly and, as scales increase, the cost could become prohibitive. In order to address these problems, the selective enzymatic removal of tags using exopeptidases has been studied. Such strategies mainly utilize dipeptide aminopeptidase I (DAPase) to successively cleave the N-terminal fusion tag until a dipeptide stop point in the sequence is encountered [START_REF] Arnau | Cloning strategy, production and purification of proteins with exopeptidase-cleavable His-tags[END_REF]. The precise removal of poly-histidine tags to yield the target protein without any additional residues has been widely proven and further variants of this strategy have been engineered to work with a diversity of sequences [START_REF] Block | Production and comprehensive quality control of recombinant human Interleukin-1 beta: A case study for a process development strategy[END_REF][START_REF] Kenig | Influence of the protein oligomericity on final yield after affinity tag removal in purification of recombinant proteins[END_REF]. The use of exopeptideases provides a convenient solution to the occurrence of non-specific cleavage within the target protein encountered with endopeptidases, avoiding unnecessary destruction of the protein product. In addition, the enzyme-to-substrate ratio required for the use of exopeptidases in comparison to conventional endopeptidase systems is much lower, making the overall process more cost-effective [START_REF] Arnau | Cloning strategy, production and purification of proteins with exopeptidase-cleavable His-tags[END_REF].

However, any protease added to achieve the processing of poly-histidine tags must be removed at the end of the cleavage reactions due to the immunogenic nature of these enzymes and other problems arising from their presence contaminating the target protein. In most cases, this involves an additional chromatography step as shown by the downstream process in Figure 2. The first IMAC column produces purified proteins containing fusion tags and following batch enzymatic digestion, the mixture containing native de-tagged protein and contaminants arising from tagged processing are then separated using a second IMAC column. The latter process is feasible as the proteases used are often genetically-modified such that they themselves contain a hexa-histidine fusion tag, thus allowing the clearance of the enzymes by the second subtractive IMAC column step. Alternatively, it is also possible to immobilize non-tagged proteases onto solid supports allowing their separation from the desired product by simple filtration or centrifugation. Similarly, non-tagged thrombin, fXa, and EK, which are often produced directly from natural sources, can be removed by affinity chromatography using adsorbents such as benzamidine-agarose.

Cleavage efficiency should also be high as this directly affects the extent of recovery of the native target protein [START_REF] Kenig | Influence of the protein oligomericity on final yield after affinity tag removal in purification of recombinant proteins[END_REF]. The extent of recovery of multimeric proteins can be significantly lower than that of monomeric proteins as the cleavage efficiency achieved with the former proteins has an increased influence on the overall recovery. This is because only one residual partially cleaved subunit is needed to result in adsorption of an entire multimeric protein on the second subtractive IMAC column during the contaminant clearance step shown in Figure 2 (Schafer et al, 2002).

An apparent inefficiency observed in most downstream processes is the large number of sequential unit operations involved, resulting in poor overall process economics.

It is highly desirable to combine the multiple steps involved in the processing of tagged proteins, especially the steps associated with tag removal, and hence intensify multiple unit operations to give a more streamlined process. As an illustration, the potential of adopting on-column detagging strategies is attractive as these strategies are often advantageous due to their ease of integration with purification operations, which leads to improved process economics and yield by reducing the number of separate downstream unit operations. After the capture step from the crude feedstock, the on-column cleavage of the tag from the tagged protein was performed by dispensing the solution of cleavage enzyme into the chromatography column and allowing the cleavage reaction to proceed. In most cases, on-column cleavage strategies have been designed for integration with endopeptidase systems [START_REF] Bhikhabhai | Production of milligram quantities of affinity tagged-proteins using automated multistep chromatographic purification[END_REF][START_REF] Dian | Strategies for the purification and on-column cleavage of glutathione-S-transferase fusion target proteins[END_REF][START_REF] Hefti | A novel purification method for histidine-tagged proteins containing a thrombin cleavage site[END_REF]. Nevertheless, on-column cleavage with tagged exopeptidases was also found to be feasible, although some inherent process complexities do exist [START_REF] Kuo | Process Intensification for the Removal of Poly-Histidine Fusion Tags from Recombinant Proteins by an Exopeptidase[END_REF].

The poly-histidine-tagged exopeptidase cannot normally gain access to the N-terminal end of the fusion tag on the target protein as both entities are tightly bound to the surface of the IMAC adsorbent. However, as shown in Figure 3, the introduction of imidazole into the buffer in which the cleavage reaction takes place can induce a dynamic binding situation for the poly-histidine tagged species. The consequent on-off binding characteristics of the poly-histidine tagged target protein then enable the exopeptidase to diffuse between the surface of the IMAC adsorbent and the fusion tag on the target protein, thus allowing the exopeptidase to bind to the N-terminal end of the fusion tag and thus allow the cleavage reaction to proceed. After incubation in the column, the cleaved, untagged protein can then be recovered simply by washing the column, followed by elution of the cleaved dipeptides and any remaining un-cleaved proteins.

This procedure avoids the need for additional steps to ensure separation of tag fragments and detagging enzymes. A further advantage is the ability to recover the protein in a suitable buffer that facilitates the subsequent downstream step. This can reduce the processing time as well as maintain product quality for proteins that are less stable to extensive pH titration and concentration/diafiltration conditions.

Likewise, the ability to recover the product in buffers containing no imidazole has also proved to be beneficial as the imidazole normally added to elute poly-histidine tagged proteins from IMAC can affect the subsequent analyses by NMR, interfere in studies involving crystallography as well as increase the likelihood of protein aggregation [START_REF] Hefti | A novel purification method for histidine-tagged proteins containing a thrombin cleavage site[END_REF].

Finally, while some general patents regarding the use of poly-histidine fusion tags have expired, there may still be licensing issues associated with the commercial use of fusion tag sequences containing at least two adjacent histidine residues such as the use of the 6xHis tag developed by F. Hoffmann-La Roche Ltd. Another potential difficulty is that when used for the production of pharmaceuticals, Ni 2+ commonly used in IMAC would be considered a hazard if immobilized nickel ions were leached from the IMAC adsorbent during processing as nickel may produce an allergic reaction. Nevertheless, IMAC using ligands that demonstrate very low levels of leaching of chelated ions such as the tetradentate ligand NTA can be a technology compatible with the requirements of a medicinal manufacture [START_REF] Block | Production and comprehensive quality control of recombinant human Interleukin-1 beta: A case study for a process development strategy[END_REF].

Other applications of immobilized metal affinity or poly-histidine tag

Besides exploiting the interaction between metal ions and poly-histidine tags for protein purification, several other applications employ the specific interactions of immobilized metal ions and poly-histidine tags. For example, poly-histidine fusion tags with high affinity to metal ions are also useful tools to immobilize and stabilize peptides or proteins on surfaces [START_REF] Noji | Direct observation of the rotation of F-1-ATPase[END_REF]. Proteins fused with poly-histidine tags can become specifically orientated when bound to surfaces derivatized with immobilized nickel with high binding energy and good stability and Martin, 2005). In the fabrication of biosensors, appropriate orientation of the immobilized immunoreagents is a key step. Vallina-Carcia et al (2007) have demonstrated the possibilities of electrochemical immunosensing through the orientation of poly-histidine tagged, anti-pneumolysin antibody fragments by nickel-histidine complexation or histidine-anti-histidine-antibody interactions. In both cases, gold films were chosen as the electrode material as they were considered to be ideal solid surface for protein immobilization and electrochemical transduction. Such electrochemical methods of detection have the advantages of sensitivity, miniaturization potential, and simplicity.

It is also possible to assemble quantum dot bio-conjugates by conjugating proteins to quantum dots via a poly-histidine tag [START_REF] Goldman | Self-assembled luminescent CdSe-ZnS quantum dot bioconjugates prepared using engineered poly-histidine terminated proteins[END_REF]. Quantum dots are luminescent semiconductor nano-crystals that have been actively studied due to their potential in overcoming the drawbacks of traditional organic fluorophores. The conjugation of quantum dots to poly-histidine tagged proteins is based on the coordination of histidine residues to zinc metal ions on the surface of cadmium selenide/zinc sulphide nano-crystals. Again, such an approach to conjugation can be generalized to a wide range of proteins through the poly-histidine fusion tag technology. Consequently, adoption of this procedure opens up the prospect of forming quantum dot bio-conjugates for applications such as imaging, immunoassay, and other diagnostics.

The ability of metal ions to chelate with terminal poly-histidine tags in recombinant proteins can be extended to the fabrication of affinity membranes [START_REF] Ke | Preparation of the immobilized metal affinity membrane with high amount of metal ions and protein adsorption efficiencies[END_REF].

Compared to packed-bed columns typically used in protein purification, affinity membranes provide benefits such as no intra-particle diffusion, short axial-diffusion paths, low pressure drops, and no possibility of bed compaction. Alternatively, for analytical applications, [START_REF] Ahmed | Controlling the orientation of immobilized proteins on an affinity membrane through chelation of a histidine tag to a chitosan-Ni 2+ surface[END_REF] coated chitosan, which has nickel-binding properties, onto the surface of polycarbonate track-etched membranes. Recombinant proteins fused with poly-histidine tags were then immobilized to the membrane surface by binding to Ni 2+ ions adsorbed to the chitosan. The ability of the metal-immobilized proteins to capture their specific antibody was then compared with those immobilized directly onto the membranes using glutaraldehyde reaction chemistry. The histidine-metal immobilization method provided enhanced control of protein orientation and maintained good bioactivity which led to higher capacity, affinity, and sensitivity. Such affinity membranes with high selectivities can increase the sensitivity of multi-membrane blotting methodologies, such as layered expression scanning (LES) for high-throughput analysis of proteomic profiles, as well as conventional blotting when only trace amounts of the target protein is present [START_REF] Ahmed | Controlling the orientation of immobilized proteins on an affinity membrane through chelation of a histidine tag to a chitosan-Ni 2+ surface[END_REF].

A poly-histidine tag can also be used as an epitope tag for antibody binding. An epitope tag is a short stretch of amino acids that is a part of a molecule, usually a protein, that can be specifically bound by an antibody. Use of a poly-histidine epitope can facilitate the visualization of the target molecule via a gel, Western blot or an immunofluorescence label. Monoclonal and polyclonal antibodies have been raised against poly-histidine fusion tags by immunizing mice or rabbits with poly-histidine tagged fusion proteins [START_REF] Noedl | Simple histidine-rich protein 2 double-site sandwich enzyme-linked immunosorbent assay for use in malaria drug sensitivity testing[END_REF]. Consequently, by adding an epitope tag onto any target protein, a universal method for following the given protein can be established by using a generic assay for detecting the known molecular handle that has been added. Indeed, [START_REF] Huang | Enzyme-linked immunosorbent assays for insulin-like growth factor-I using six-histidine tag fused proteins[END_REF] studied the binding kinetics between hexa-histidine-tagged insulin-like growth factor-I (IGF-I) and anti-hexa-histidine antibody using surface plasmon resonance (SPR). They concluded that by using an enzyme-labelled anti-hexa-histidine antibody as a tracer, a

competitive sandwich ELISA has the potential to be used as a generic immunoassay for the detection of a wide range of polypeptides or proteins tagged with a hexa-histidine sequence.

Finally, IMAC ligands have been used in protein chips for the enrichment of proteins analyzed by surface-enhanced laser desorption/ionization time-of-flight mass spectrometry (SELDI-TOF-MS) [START_REF] Kong | Using proteomic approaches to identify new biomarkers for detection and monitoring of ovarian cancer[END_REF]. Also as an enrichment method for mass spectrometry, [START_REF] Jalili | Enhancement of ionization efficiency and selective enrichment of phosphorylated peptides from complex protein mixtures using a reversible poly-histidine tag[END_REF] derivatized phosphorylated peptides with a poly-histidine tag and later enriched the tagged peptides on a nano-scale IMAC column. The subsequent cleavage of the enriched tagged peptides with Factor Xa to release the target peptides imparted even higher selectivity to the process. The eluted peptides were then mapped using mass spectrometry. This approach to the enrichment of phosphopeptides at low concentrations produced better results than those achieved by other, more common enrichment methods. Similarly, Kaur-Atwal et al ( 2007) described the potential of interfacing capillary IMAC with mass-spectrometric detection. The direct combination of these techniques allowed the integration of on-line isolation of histidine-containing peptides and mass spectrometric detection in a high-throughput manner. IMAC beads with diameters of 20 m were packed into a 250 m diameter fused silica column and the resulting capillary IMAC column was used successfully in isolating the histidine-containing peptides in order to simplify complex peptide mass spectrometric analysis.

Conclusions

The technology of exploiting the interactions between poly-histidine fusion tags and immobilized metal ions has proved to be a successful and generic procedure for the isolation of antibody fragments, membrane proteins, and fragile enzymes that are otherwise difficult to purify. Incorporating a fusion tag on the target protein is beneficial as the protein purification method can be chosen without much prior knowledge of the target protein. Such flexibility is also a preferred quality of any downstream purification technology as multiproduct processing facilities are most commonly sought-after today. Furthermore, advances in bio-separation demand more cost-effective methods that have the potential of meeting the most stringent product requirements. For instance, lowering the cost of protein therapeutics has become more important as healthcare funders are increasingly reluctant to pay for expensive biotechnological pharmaceuticals [START_REF] Aggarwal | What's fueling the biotech engine?[END_REF]. In addition, in order to design and manufacture protein therapeutics for sale as bio-generics, new technologies are needed to control production costs and thus achieve lowered pricing.

Reducing the costs of manufacture can also result in these therapeutics becoming accessible to third-world nations.

Despite the many advantages of using IMAC and poly-histidine tag in fulfilling these promises, there remain several obstacles that need to be overcome prior to the wider adoption of this technology. The challenge of removing the fusion tag post-purification is an important issue as well as the potential validation steps to show that tags are removed correctly and remnants are not present in the final product.

This review has discussed various approaches to overcome these shortcomings.

Finally, the ability to prepare a wide range of recombinant proteins fused with poly-histidine tags and to capture these proteins in a highly specific manner are key advantages of the technology, making it generic and highly valuable to any analysis based on the binding of biomarker molecules.

FiguresFigure 1 :

 1 Figures

Figure 2 :Figure 3 :

 23 Figure 2: Typical two-step IMAC purification process involving the use of poly-histidine tagged recombinant proteases


Alberto D'onofrio
email: alberto.donofrio@ifom-ieo-campus.it

Piero Manfredi
email: manfredi@ec.unipi.it

Piero Poletti
email: poletti@fbk.eu

The impact of vaccine side effects on the natural history of immunization programmes: an imitationgame approach

Keywords: Epidemiological models, vaccination choices, imitation dynamics, vaccine side effects

When the incidence and prevalence of most common vaccine preventable childhood infectious diseases are constantly low, as is the case in many industrialized countries, the incidence of vaccine-associated side effects might become a key determinant in vaccine demand. We study an SIR transmission model with dynamic vaccine demand based on an imitation mechanism where the perceived risk of vaccination is modelled as a function of the incidence of vaccine side effects. The model shows some important differences compared to previous game dynamic models of vaccination, and allows noteworthy inferences as regards both the past and future lifetime of vaccination programmes. In particular it is suggested that a huge disproportion between the perceived risk of disease and vaccination is necessary in order to achieve high coverages. This disproportion is further increased in highly industrialised countries. Such considerations represent serious challenges for future vaccination programmes.

The impact of vaccine side effects on the natural history of immunization programmes: an imitation-game approach

Introduction

Although forms of exemption to vaccination have always existed [START_REF] Salmon | Compulsory vaccination and conscientious or philosophical exemptions: past, present and future[END_REF], the "natural history" of vaccination programmes, as part of the historical pathway through which humankind progressively rids itself of infectious diseases, has always been pervaded by a high degree of optimism [START_REF] Chen | Epidemiologic Methods in Immunization Programs[END_REF]. However, recently this optimistic view has increasingly been challenged: for example, opposition to the whole-cell pertussis vaccine [START_REF] Gangarosa | Impact of anti-vaccine movements on pertussis control: the untold story[END_REF], the thimerosal case [START_REF] Luman | Impact of thimerosal-related changes in hepatitis B vaccine, birth-dose recommendations[END_REF], and the MMR scare [START_REF] Roberts | Reasons for non-uptake of measles, mumps and rubella catch up immunisation in a measles epidemic and side effects of the vaccine[END_REF] can be considered evidence that in industrialized countries, the success story of vaccination is feeding back on itself. This is the consequence of two different processes. On the one hand, the high degree of herd immunity achieved by decades of successful immunization programmes has reduced the incidence of many infections to negligible levels. On the other hand the large, and increasing, number of vaccines routinely administered every year yields steady flows of vaccine-associated side effects [START_REF]Immunization safety[END_REF], [START_REF]Guidelines for managers of immunization programmes on reporting and investigating adverse events following immunization[END_REF]. In the US approximately 30,000 reports of Vaccine Adverse Events are notified annually, with 10-15% classified as serious [START_REF]Vaccine safety[END_REF].

In such circumstances the perception of the public will likely rank the perceived risk of suffering a vaccine side effect (VSE) as much higher than the corresponding risk of infection [START_REF] Bauch | Vaccination and the theory of games[END_REF][START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF][START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF][START_REF] Onofrio | Fatal SIR diseases and rational exemption to vaccination[END_REF]. A common example is poliomyelitis in industrialized countries. For example in Italy during 1980-2000 the number of vaccine-induced polio cases was three times higher than wild polio cases [START_REF]Istituto Superiore di Sanità, Surveillance of adverse events[END_REF]. Under voluntary vaccination high degrees of herd immunity might therefore incentivise vaccination free riding [START_REF] Brito | Externalities and compulsory vaccinations[END_REF], [START_REF] Reluga | Evolving public perceptions and stability in vaccine uptake[END_REF], i.e. the parents' decision not to vaccinate children after comparing the perceived risk of disease and the perceived risk of vaccine side effects [START_REF] Bauch | Vaccination and the theory of games[END_REF][START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF][START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF]. Vaccination free riding [START_REF] Brito | Externalities and compulsory vaccinations[END_REF][START_REF] Fine | Individual versus public priorities in the determination of optimal vaccination policies[END_REF][START_REF] Geoffard | Disease eradication: private versus public vaccination[END_REF][START_REF] Bauch | Vaccination and the theory of games[END_REF][START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF][START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF][START_REF] Onofrio | Fatal SIR diseases and rational exemption to vaccination[END_REF][START_REF] Onofrio | Vaccine demand driven by vaccine side effects: dynamic implications for SIR diseases[END_REF][START_REF] Reluga | Evolving public perceptions and stability in vaccine uptake[END_REF][START_REF] Reluga | An SIS epidemiology game with two subpopulations[END_REF][START_REF] Galvani | Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum[END_REF] makes eradication impossible (unless special contact structures are considered [START_REF] Perisic | Social Contact Networks and Disease Eradicability under Voluntary Vaccination[END_REF]) and triggers stable oscillations in the infection prevalence. Previous studies on the impact of vaccination free riding on endemic infections have focused on scenarios where the vaccine demand is driven by the time changes in the perceived risk of disease, measured through the current (or past) infection prevalence. Based on the above-cited literature on VSEs we instead suggest that in industrialized countries, where the incidence of common vaccine preventable infections is very low, the available information on vaccine side effects might become the main driving force of vaccine demand. The only paper devoted to this issue is [START_REF] Onofrio | Vaccine demand driven by vaccine side effects: dynamic implications for SIR diseases[END_REF], where, however, vaccine demand was phenomenologically modelled as a decreasing function of the perceived risk of suffering a vaccine-associated side effect. The perceived risk of VSEs was in its turn evaluated by the public from available information on current and past trends of side effects attributed to the vaccine. In this paper we shift the focus onto the impact of VSEs on the dynamics of immunization programmes for endemic infections, using an SIR transmission model with voluntary vaccination choice. Vaccination choices are described by an evolutionary game as in [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF], where the vaccine uptake p(t) is determined by an imitation process between agents (the parents of the children to be vaccinated) who are divided into "vaccinators" and "non-vaccinators". However, in [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] the perceived payoff of the "non-vaccinators" is proportional to the prevalence of the disease, whereas the payoff of "vaccinators" is constant, i.e. independent of VSEs. The novelty of the present work is that the payoff of "vaccinators" is proportional to the incidence of vaccine-associated side effects. In turn, this incidence is assumed to be proportional to the actual vaccine uptake. We believe that this model represents a more appropriate description of the future evolution of immunization programmes in voluntary vaccination regimes. Indeed, the fact that available information on vaccine side effects might become the main driving force of vaccine demand is strongly supported by the empirical evidence, e.g. the case of England and Wales, where due to the MMR scare [START_REF] Roberts | Reasons for non-uptake of measles, mumps and rubella catch up immunisation in a measles epidemic and side effects of the vaccine[END_REF], first dose measles uptake fell for several years from 94% to about 75%. Compared to [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] we also expand our basic model to include (a) nonlinear perceived costs of infection; (b) the possibility that the perceived costs of infection and vaccination are evaluated by the public using past values of state variables, for example due to information delay [START_REF] Onofrio | Fatal SIR diseases and rational exemption to vaccination[END_REF] or to the perception of long-term vaccine side effects [START_REF] Onofrio | Vaccine demand driven by vaccine side effects: dynamic implications for SIR diseases[END_REF].

Materials and Methods

Dynamic vaccine demand and vaccine side effects

We consider a family of models of the spread of a non-fatal SIR infection controlled by voluntary vaccination with a "perfect" vaccine administered in a single dose at birth and giving life-long immunity:

S = μ(1 -p) -μS -βSI (1)
I = βSI -(μ + ν)I (2) p = k 1 ΔEp(1 -p) ( 3
)
where: S, I are the susceptible and infective fractions, and p the vaccinated proportion among newborn children; β > 0, μ > 0, ν > 0 denote the transmission, the mortality and the recovery rates. We assume that, in absence of vaccination, the disease is endemic, i.e. R 0 = β/(μ + ν) > 1.

The dynamics of p obeys a learning by imitation [START_REF] Hofbauer | Evolutionary Games and Population Dynamics[END_REF] process where k 1 is the "imitation" coefficient and switching between the decisions to vaccinate or not to vaccinate, is determined by the payoff gain ΔE(t). The latter is given by the difference between the perceived payoff of vaccinators -ρ V (t), where ρ V (t) is the perceived risk of suffering a VSE, and the perceived payoff of non-vaccinators: -ρ I (t), where ρ I (t) is the perceived risk of suffering serious illness due to infection. We note that, irrespective of the specific forms of ΔE(t), the family of models (1-2-3) has the following three equilibria: (i) an unstable disease-free equilibrium with no vaccinators A = (1, 0, 0); (ii) a pure-vaccinator disease-free equilibrium B = (0, 0, 1); (iii) the pre-vaccination endemic equilibrium

C = S SIR = R -1 0 , I SIR = μ(1 -R -1 0 )/(μ + ν), 0 .
The stability of B and C and the existence of further equilibria depend on the specific types of the payoff gain. Modelling of the dynamics of p by an imitation game was introduced by Bauch in the seminal paper [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF], where a specific model of the family (1-2-3) was proposed, where: i) the perceived payoff of vaccinators is constant:-ρ V = -r V ; and ii) the perceived payoff of non-vaccinators is proportional to the infective prevalence I(t): -ρ I (t) = -r I mI(t), where mI(t) is an estimate of the current risk of infection, and r I is the risk of serious disease as a consequence of infection. Hence:

ΔE(t) = r I mI(t) -r V = r V (ϑI(t) -1) ,
where ϑ = mr I /r V is proportional to the relative cost of the non-vaccinator strategy. In Bauch's model the B equilibrium is unstable and there is a fourth equilibrium: the post-vaccination equilibrium D = R -1 0 , ϑ -1 , p where:

p = (1 + ν/μ) I SIR -ϑ -1 . At ϑ = ϑ 0 = I -1
SIR there is a transcritical bifurcation between C and D. In turn, the stability/instability of D depends on the product k 1 (ϑϑ 0 ). The assumption in [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] that the perceived risk of VSEs ρ V (t) is constant can be justified if the public correctly evaluates this risk as the ratio between total VSEs per unit time, given by α (μN ) p(t) (α ∈ (0, 1)) where N is the total population size (so that μN is the yearly number of births), and α the per-capita probability of incurring a VSE during a single vaccine administration, and the total number of vaccinations administered for that specific disease μN p(t). We instead suppose that the public evaluates the risk of VSEs by using the available information on the total number of vaccineassociated side effects, or, which is equivalent, on the ratio between the total number of vaccine associated side effects and the total number of newborn children per unit of time. We therefore set:

ρ V (t) = αp(t) ( 4 )
The implication of (4) is that periods of large vaccine uptake negatively feed back, through an increase in the incidence of VSEs, into the proportion of parents favourable to vaccination. We model the perceived risk of infection as an increasing function h 1 (•) ≥ 0 of the information index M , introduced in [START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF][START_REF] Onofrio | Fatal SIR diseases and rational exemption to vaccination[END_REF], that summarizes the publicly available information on the infection:

ρ I (t) = h 1 (M (t)). (5)
The index M , used by agents to evaluate the risk of infection, may model not only current, but also past information. The case h 1 (0) > 0 accounts for the scenario where the disease is locally eliminated but disease re-emergence by external reintroduction is feared. In the simplest case, the information index M (t) is related to the current prevalence of the infection as follows:

M (t) = h 2 (I(t)) , ( 6
)
where h 2 (•) > 0 and h 2 (•) > 0. Using (4)-( 5)-( 6) the dynamics of p can be written as follows:

p = k (h(I) -p) p (1 -p) ( 7
)
where

h(I) = α -1 h 1 (h 2 (I)), k = αk 1 . First, note that from: p ≥ k (h(0) -p) p (1 -p) it follows that h(0) ≥ 1 ⇒ p(t) → 1
, so that we shall not consider this trivial case. On the contrary, if h(0) < 1 then p(t) ≥ h(0) holds asymptotically, i.e. the collective coverage will at least reach the minimal level h(0). Symmetrically, inequality p ≤ kh Max p (1p) means that p(t) is bounded by a logistic dynamics with kinetics constant kh Max . Finally, if imitation dynamics is faster than the infection time scale:

k >> (μ + ν) -1 , then p(t) is at quasi-equilibrium: p(t) ≈ min (h (M (t)) , 1) , ( 8
)
which is the phenomenological relationship first proposed in [START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF].

The importance of time delays

The assumption that ΔE depends only on current values of I and p is an approximation. For example agents might perceive that vaccines are responsible for VSEs arising with long time delays [START_REF] Onofrio | Vaccine demand driven by vaccine side effects: dynamic implications for SIR diseases[END_REF], as might be the case with auto-immune diseases [START_REF] Molina | Infection, vaccines and other environmental triggers of autoimmunity[END_REF]. Moreover, delays of different nature, concerning the information on the spread of the disease, may involve both p(t) and M (t) [START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF]. Thus in the case in which the perceived risk of vaccination is evaluated on past VSEs:

ρ V (t) = t -∞ Q H (T v )αp(t -T v )dT v (9)
where Q H is a delay kernel. An important kernel is the exponentially fading memory:

Q H (x) = b exp (-bx
) , b > 0, which allows reduction to ODE since it holds that:

ρ V = b (αp(t) -ρ V ) . ( 10
)
Scenarios such as long-term VSEs may require different kernels, more concentrated on past periods, such as the Erlang functions: A n x n-1 exp(-bx), also allowing reduction to ODEs.

Results

Endemic equilibria and their stability

In this section we shall investigate the model ( 1)-( 2)- [START_REF] Onofrio | Fatal SIR diseases and rational exemption to vaccination[END_REF] where no delays are present. Considering a generic h(I), the system (1)-( 2)-( 7) has three equilibrium points A (unstable), B (unstable) and C, which, unlike [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF], is always unstable since the linearized equation for p reads η = kh (I SIR ) η. Moreover, two further equilibria are induced by the specific ΔE we introduced:

• A disease-free equilibrium with positive vaccine uptake E df e = (1h(0), 0, h(0)).

• A new behaviour-related endemic equilibrium:

E beh = R -1 0 , I e , h (I e ) ,
where I e is the unique solution of the equation:

h(I) = 1 -R -1 0 - μ + ν μ I. ( 11
)
Remark 3. [START_REF] Bauch | Vaccination and the theory of games[END_REF] The equilibrium E beh is equal to the endemic equilibrium of the model studied in [START_REF] Onofrio | Vaccine demand driven by vaccine side effects: dynamic implications for SIR diseases[END_REF], where, however, p is not a state variable.

Since all equilibria are independent of k, which is an epidemiologically relevant parameter (we recall that it is proportional to the imitation coefficient k 1 determining the time-scales of the imitation phenomenon), this makes it natural to choose k as a bifurcation parameter. Our main results (demonstrated in the Appendix ) are as follows: holds then there are two positive values k 1 and k 2 > k 1 such that:

A) If
1. If 0 < k < k 1 or k > k 2 then E beh is LAS; 2. At k = k 1 and at k = k 2 there are Hopf bifurcations; 3. If k ∈ (k 1 , k 2 ) then E beh is unstable. C) In the case k ∈ (k 1 , k 2 )
the orbits x(t) = (S(t), I(t), p(t)) are oscillatory in the sense of Yabucovich [START_REF] Efimov | Yakubovich's oscillatority of circadian oscillations models[END_REF]. This intuitively means that for sufficiently large t all state variables are permanently oscillating, with regular or irregular oscillations. Formally, for j = 1, 2, 3 it holds that:

minlim t→+∞ x j (t) < maxlim t→+∞ x j (t). D) If h 0 > p c then E df e is globally asymptotically stable (GAS); if h 0 < p c then E df e is unstable.

Remark 3.2 Note that, on dividing both the r.h.s. and the l.h.s. of (13) by βμI e , after trivial calculations we obtain that to have Hopf bifurcation at E beh the derivative of h at I e has to be at least of order O( ν/μ).

Remark 3.3 Property D shows that, unlike [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF], elimination is possible in our model, although only if in absence of the disease the vaccinator payoff is so large as to yield a vaccine uptake greater than the elimination threshold.

To better understand the phenomenon of epidemic oscillations driven by vaccination choice we compare the above results with the recent literature on this subject. Unlike [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] where large values of k (ϑϑ 0 ) induce sustained oscillations around the endemic state, in our model oscillations are possible in an intermediate window of values of the imitation coefficient k. This means that both slow and fast imitation are stabilizing forces. Moreover, we note that the stability condition [START_REF] Gangarosa | Impact of anti-vaccine movements on pertussis control: the untold story[END_REF] has the same formal structure as that of the delayed model in [START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF], where dependence of the current vaccine uptake on past prevalence of the infection was required to obtain oscillations. Indeed, imitation is a (nonlinear) adjustment process introducing a delay, whose characteristic time scales are determined by the speed with which the vaccine uptake reacts to changes in the payoff gain. Heuristically, close to the equilibrium eq. ( 7) becomes:

p ≈ kh (I e ) (1 -h (I e )) (h (I) -p) = Ψ (h (I) -p) ( 14
)
which may be read as an exponentially fading memory mechanism with average delay 1/Ψ. This suggests a route to estimate the elusive imitation coefficient k, by preliminarily estimating the average delay 1/Ψ. Finally, we note that the Yabucovitch oscillatority is a global result, unlike the Hopf bifurcation theorem, which is local. Although the nature (periodic, quasi-periodic or chaotic) of Yabucovitch oscillations cannot be determined a priori, this makes the predictions of our model very general. Finally, it is easy to show that the inclusion of time delays does not affect the location and stability of equilibria A, B, C and E df e , whereas the stability of E beh may be affected, as seen in the numerical simulations below (see also the Appendix ).

Analysis of selected subcases

We report numerical illustrations of noteworthy sub-cases of epidemiological interest. The basic reproduction number R 0 is set to 10; the recovery rate is set to either ν = 0.1 days -1 (pertussis) or ν = (1/7) days -1 (measles), which correspond to an average duration of the infectious period of respectively 10 days (pertussis) and one week (measles). Finally the life expectancy L = 1/μ is fixed either at L = 50 years (used in [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] to more closely reproduce UK pertussis data) or at L = 75 years, which is more representative of mortality in modern industrialized countries.

The basic unlagged model: the case of linear h(I)

Letting h(I) = ϑI, the infection prevalence at E beh is:

I e = I e (ϑ) = 1 -1 R0 1 + ν μ + ϑ = I SIR 1 + ν μ 1 + ν μ + ϑ
and the vaccine uptake is:

p e (ϑ) = ϑI e (ϑ) = p c ϑ 1 + ν μ + ϑ (15)
As regards the stability of E beh , it is of interest to assess the relative role of the two main behavioral parameters introduced by our model: ϑ and k. By applying [START_REF] Geoffard | Disease eradication: private versus public vaccination[END_REF] it is easy to show that a ϑ * exists such that for ϑ > ϑ * then ( 13) is fulfilled and the two branches k 1 (ϑ) and k 2 (ϑ) of the bifurcation curve in the space (k, ϑ) exist and are analytically computable. Below ϑ * the equilibrium E beh is (at least) locally stable.

For example in the case ν = 0.1 days -1 , μ = 1/50 years -1 it is (ϑ * , k * ) ≈ (280, 0.042), from which the two branches k 1 (ϑ) and k 2 (ϑ) of the Hopf curve depart. For ϑ = 1000 (implying I e = 3. 18 × 10 -4 and p e = 0.32) we obtain k -1 2 = 4.75 days and k -1 1 = 312 days a range difficult to interpret. However, the heuristic average imitation delay 1/Ψ correspondingly ranges between three weeks and about four years, close to the values found in [START_REF] Onofrio | Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases[END_REF]. The bifurcation curve in the (k, ϑ) and (k -1 , ϑ) parametric spaces are shown in Fig. 1.

The basic unlagged model: the case of linear-affine h (I)

The case h(I) = ϑ 0 + ϑI is of interest since it assumes that even in scenarios of zero prevalence agents perceive a positive risk ϑ 0 of infection re-emergence. We get:

I e (ϑ) = 1 -R -1 0 -ϑ 0 1 + ν μ + ϑ , p e (ϑ) = ϑ 0 + ϑ 1 -R -1 0 -ϑ 0 1 + ν μ + ϑ
Thus the perception of such a risk has a positive effect on equilibrium uptake. The dependence of the bifurcation curve on ϑ 0 may be non-monotone, as in the right-hand panel of Fig. 1.

Substantive implications of vaccine side effects for vaccination programmes 4.1 The epidemiological transition and vaccination payoff

We use equilibrium results from the simple case of linear h(I) to explore the impact of human progress on the natural history of vaccination programmes. Note that in the parametric set we are considering the dimensionless quantity ν/μ ranges from about 1826 (pertussis in a population with low lifeexpectancy) up to about 3913 (measles in a population with high life-expectancy). Thus, achieving a large equilibrium uptake requires very large ϑ, i.e. of the order of ν/μ. Since ϑ = r I m/α, it follows that to achieve large equilibrium uptakes for measles or pertussis, the perceived cost of serious disease has to be at least three order of magnitude higher than the perceived cost of VSE. Though this seems surprising, we feel it is consistent with what was still observed at the beginning of the 20th century, when the risk of serious sequelae following measles or scarlet fever was extremely large (100-250 deaths per 100000 cases of disease), and the absence of vaccines was keeping the risk of infection very high. In such circumstances even a large probability of suffering a VSE from a vaccine could have been tolerated by the community. This also suggests that industrialized countries could face serious difficulties in maintaining high vaccine uptake in the future. Indeed, completion of the epidemiological transition [START_REF] Solomon | The epidemiological transition revisited: new compositional models for causes of death by age and sex[END_REF], i.e. the historic change in the cause composition of mortality in industrialized countries from infectious and nutritional diseases to chronicdegenerative ones, led to a dramatic fall in the levels of serious morbidity and mortality from all infectious diseases. Together with the current higher-than-ever degrees of herd immunity allowed by decades of sustained vaccination, this has reduced the perceived risks of serious disease from most infections to negligible levels. Finally, as regards the role played by life expectancy on the steady state E beh , our results suggest that the achievement of a given equilibrium uptake in industrialized countries (i.e. with a very high life expectancy) requires a much larger value of ϑ compared to developing countries. This is consistent with the fact that in regimes with low mortality (also as a consequence of vaccinations), agents demand a vaccine only if the relative risks of suffering a VSE are very low. We stress that our predictions are equilibrium ones based on a simple deterministic model with homogeneous mixing. Nonetheless more realistic models would not substantially affect them.

Simulations

In our simulations we set h(I) = ϑI and, to better emphasize our main messages, we allow immigration of infectives, according to two different hypotheses: a) a small constant transfer Imm from the susceptible to the infective state, representing a steady flow of infections as a consequence of international travelling; b) a few new infections appear at some stage once and for all, in order to mimic the possibility of infection resurgence. Indeed, in the oscillatory regime the infection prevalence I(t) may reach extremely low values, which calls into question the appropriateness of a deterministic model. This drawback is avoided by assumption a, also used in [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF]. Imm is set to one infective individual per week in a population of 5 × 10 6 individuals.

The basic unlagged model

We first assess the impact of VSEs on the transient infection dynamics triggered by a new vaccination introduced at the pre-vaccination endemic state, and under assumption a. The vaccine is introduced at time t = 0, with initial vaccine uptake set to 0.95 > p c . Vaccine side effects occur from the beginning of the programme. We set ϑ = 15000 (implying p e ≈ 0.71, k = 0.002 days -1 and 1/Ψ ≈ 7 years). Note that k is small since it is the product between the natural imitation rate and the low probability α of suffering a VSE. As predicted by (13), the system converges, in epidemiologically reasonable time scales, to a stable limit cycle. With reference to Fig. 2, the vaccine uptake (right-hand panel) starts declining soon after the programme starts, due to the onset of VSEs, and falls below the critical threshold p c in less than four years. Thus the effective reproduction number R E (t) = R 0 * S(t) (lefthand panel) initially declines but then increases and exceeds the unit threshold at t ≈ 8 years, yielding a new epidemic outbreak at t ≈ 10 years. During this rather long 'honey-moon' period the circulation of the infection is essentially sustained by immigration. Note that p(t) > p c for a rather long period of time (about 36 per cent of total time). During such periods routine vaccination surveys would reveal a satisfactorily high coverage. Therefore they could not explain the endemic persistence of infection. The role played by k is illustrated in Fig. 3, which considers the values: k = 0.0005 days -1 , k = 0.002 days -1 (as before), k = 0.0035 days -1 , and which shows that: i) the average uptake is not significantly affected by k and remains close to p e ; ii) both the amplitude of oscillations of p(t) and the fraction of total time where p(t) > p c increase in k; iii) the duration of the period between the start of the programme and a new epidemic outbreak is decreasing in k. For example, the 'low' k value (k = 0.0005 days -1 ) yields oscillations that are of small amplitude and that for a small portion of their period are such that p(t) > p c . Moreover, there is an interval of 17 years about before a new epidemics, which is only a few years long for k = 0.0035 days -1 . This is the consequence of the slow spread of information occurring for low k, which slows down the reactivity of the vaccine uptake to changes in the payoff gain. Note that if no external infections are introduced, then for small k the infection prevalence is close to zero. This suggests that oscillations might produce stochastic elimination of infection. We now study the introduction of a new vaccine for which no vaccine side effects are initially known. In this case it seems reasonable to assume that an intermediate period might exist during which no perception of VSEs arises. We consider α = 0 for t < t 1 where t 1 = 10 years, and that the starting point is the pre-vaccination steady state. Of course, the vaccine uptake start increasing since the payoff vaccination is initially positive (α = 0). If VSEs raise after a very long time, or no VSEs are reported, then stochastic elimination is possible. By contrast, if evidence of VSEs emerges, the scenario reduces straightaway to the cases treated above. If the imitation process is very slow, VSEs appear before reaching a sufficient coverage and a sub-optimal vaccination coverage is achieved (left-hand panel of Fig. 4). Instead for a larger value of k, the infective fraction is very close to 0 such that stochastic elimination may occur (central panel of Fig. 4). Finally, for an intermediate range of k there is onset of oscillations, but at the end of their transitory the minimum I(t) is very close to 0 and elimination can again occur (right-hand panel of Fig. 4). Finally, we simulated the case where eradication was achieved thanks to a period of compulsory vaccination, after which vaccination becomes voluntary. In this case VSEs induce individuals to switch to the 'non-vaccinator' strategy, thereby producing a decrease in vaccination coverage. This in turn increases the probability of infection re-emergence from imported cases. If a few new external infections are introduced (assumption b) before R E (t) = 1, then the occurrence or not of stochastic transient elimination depends again on k, with patterns similar to the previous case (see Fig. 5.a,5.b). Moreover, if individuals take into account a non zero risk of infection re-emergence from importation (the linear affine case h(I) = ϑ 0 + ϑI), the population is more 'protected' and the impact of the external infections can be reduced also for large values of k (see Fig. 5.c).

Impact of information-related delays

An exponentially distributed lag in the p term yields a model given by ( 1)-( 2) complemented by:

p = k (h(I) -H)) p (1 -p) (16)
and by [START_REF] Fine | Individual versus public priorities in the determination of optimal vaccination policies[END_REF]. Fig. 6 reports the stability regions in the (k, b) plane for ϑ = 260, ϑ ≈ 279.5 (which is close to the critical value ϑ * in absence of delays) and ϑ = 500. Note that for ϑ = 500 > ϑ * for all b there are two values k l (b; ϑ) and k r (b; ϑ) such that for k l (b; ϑ) < k < k r (b; ϑ) the endemic state E beh is unstable and it is also an easy matter to show that Yabucovitch oscillations can arise. Similar plots are obtained in all cases where ϑ > ϑ * . This can be easily explained since for very large b the lagged model reduces to the unlagged one that, for ϑ > ϑ * , has an instability interval. Interestingly, the right panel of 6 for ϑ = 260 < ϑ * shows instability regions that are uniquely due to the information delay, because in absence of the delay (i.e. for large b) there is no instability, as shown in the previous sub-sections. This confirms that the addition of time delays in the p term (mimicking delayed onset of VSEs or delayed information acquisition) cumulates with the imitation delay in triggering instabilities. Fig. 7 reports the time course of vaccine uptake p(t) for ϑ = 260 for two distinct cases. In the left-hand panel the behaviour in absence of delay is reported, showing convergence to the endemic state. The right-hand panel reports the behaviour for a delay of 250 days (b = 0.004 days -1 ) in the occurrence of side effects, showing instead convergence to a limit cycle.

Let us now consider an exponentially fading memory in the perceived risk of infection: let M stand for past prevalence, and a the corresponding delaying rate. We set h 1 (M ) = ϑM , and h 2 (I) = I. In Fig. 8 we plotted the stability regions in the (k, a) plane for ϑ = 260, ϑ ≈ 279.5 (which is close to the critical value ϑ * ) and ϑ = 500. Finally, in the case where exponentially fading memories appear both in the perceived risk of vaccination (with rate b) and the perceived risk of infection (with rate a), numerical simulations showed a wide pattern of periodic behaviour, including very long periods (an example with alternating low and high peaks is reported in the left-hand panel of Fig. 9), and also aperiodicity (as in the right-hand panel of Fig. 9)

Concluding remarks

We investigated an SIR transmission model with voluntary vaccination. Unlike [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF] we consider a dynamic perceived risk of vaccination proportional to the trends of VSEs. Mathematical analysis of the model confirms already known results, such as the onset of behaviour-triggered steady oscillations about the post-vaccination endemic state. In addition it shows some noteworthy differences compared to findings elsewhere. For example global elimination may, or may not, be possible depending on the actual magnitude of the baseline perceived risk of disease relative to the risk of vaccine side effects. As regards the pre-vaccination equilibrium, differently from [START_REF] Bauch | Imitation dynamics predict vaccinating behaviour[END_REF], it is always unstable, so that any vaccination programme will initially be successful, at least partially. Moreover oscillations occur, provided the relative risk of disease is large enough, in a bounded window of the imitation parameter. In addition, we also considered the impact of time delays on the vaccination payoff gain, which is a realistic feature. Finally, the model allows noteworthy inferences on the future lifetime of vaccination voluntary programmes. It suggests that in order to achieve high equilibrium uptakes for e.g. measles, the perceived cost of the disease must be at least three orders of magnitude higher than the perceived cost of vaccine-associated side effects, and moreover that this requirement is further increased under situations with large life expectancies. Both these facts seem to be consistent with real world observations, and suggest that maximal vaccination effort by international institutions should target increasing vaccine uptake in less developed countries where the risk of serious sequelae is still high, and life expectancy is still low. Simulations finally provide further interesting insights on the effect of behavioural parameters on vaccine uptake.

where Ψ is as in eq. ( 17), and the coefficients are third order polynomials in the delay parameter b. Thus, for each b there are at most two bifurcation values for k, whereas for each k there may be at most three bifurcation values for b.

  βI e μh (I e ) < (μ + βI e ) (μ + βI e ) + 2 βI e (μ + ν) (12) then E beh is locally asymptotically stable (LAS) irrespective of the imitation speed k; B) On the contrary, if βI e μh (I e ) > (μ + βI e ) (μ + βI e ) + 2 βI e (μ + ν) (13)

Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: Bifurcation curve for linear and linear affine h(I), in the case R 0 = 10, ν = 1/10 days -1 , μ = 1/50 year -1 . (left-hand panel) Bifurcation curve for linear h(I) = ϑI in the parameter space (k, ϑ). (Central panel) Bifurcation curve for linear h(I) = ϑI in the parameter space (k -1 , ϑ) emphasizing patterns for small imitation-related delays. (Right-hand panel) Bifurcation curve for linear affine h(I) = ϑ 0 + ϑI plotted for three values of ϑ 0 . As ϑ 0 increases the minimal threshold of ϑ is nonmonotone.

Figure 5 :Figure 6 :

 56 Figure 5: Left and central panels: dynamics of I(t) (black line), R 0 S(t) (dotted black line), p(t) (dashed black line) for ϑ = 5000, ϑ 0 = 0 and p(0) = 0.95, S(0) = 1p(0), I(0) = 10 -5 . Left-hand panel k = 0.001 days -1 . Central panel: k = 0.01 days -1 . The black line is the threshold p c . Righthand panel: as the central panel but for ϑ 0 = 0.7, p(0) = 0.80 and S(0) = 0.07. Other parameters: R 0 = 10, ν = 1/10 days -1 , μ = 1/50 years -1 , ϑ = 5000.

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Effects of an exponential fading memory in the perceived risk of vaccination with ϑ = 260. Left-hand panel: the case of no delay. Right-hand panel: the delayed case with b = 0.004 days -1 , corresponding to an average delay of 250 days. Other parameters: R 0 = 10, ν = 1/10 days -1 , μ = 1/50 years -1 .

Acknowledgements We are greatly indebted to an anonymous referee for her/his suggestions that helped to improve this work significantly.

Appendix

The biomathematical properties listed in section 3 may be demonstrated as follows: Property A Let us define, to simplify the notation, the parameter:

Linearizing at E beh yields the following characteristic polynomial with positive coefficients:

for which the Routh-Hurwitz condition yields this inequality in the variable Ψ:

Since the coefficients of power 0 and 2 of eq. ( 19) are positive, and taking also into account the sign of the coefficient of power 1, it is easy to show that if [START_REF] Gangarosa | Impact of anti-vaccine movements on pertussis control: the untold story[END_REF] holds then E beh is LAS. Property B If (13) holds then there are two positive values 0

2 then E beh is unstable and k 1 and k 2 are Hopf points.

After some algebra it is easy to verify that at the Hopf points there are Hopf bifurcations since the nonzero speed condition Re (dλ/dk)| λ=±iω Hopf = 0 is fulfilled. Property C As regards the Yabucovitch oscillations, note that: i) the bounded set

is positively invariant and attractive; ii) E beh is unstable, and E df e , which is in the boundary of A, has as the stable manifold the line (w, 0, 1) w ∈ [0, 1] to which E beh does not belong, excluding heterocline orbits. Thus we may apply the Yabucovitch theorem [START_REF] Efimov | Yakubovich's oscillatority of circadian oscillations models[END_REF]. Property D Since asymptotically p(t) ≥ h(0), it follows that for large times:

i.e. asymptotically S(t) ≤ (1h(0)). This in turn implies: