Michael Busby

Lukas Kurt

Josef Stadler

Paul Ko Ferrigno

Jason J Davis
email: jason.davis@chem.ox.ac.uk

Optimisation of a Multivalent Strep-tag for Protein Detection

Keywords: Strep tag, Multivalency, Microarray, Surface Plasmon Resonance, FRET

The Strep tag is a peptide sequence that is able to mimic biotin's ability to bind to streptavidin. Sequences of Strep tags from 0-5 have been appended to the N-terminus of a model protein, the Stefin A Quadruple Mutant (SQM) peptide aptamer scaffold, and the recombinant fusion proteins expressed. The affinities of the proteins for streptavidin have been assessed as a function of the number of tags inserted using a variety of labelled and label free bioanalytical and surface based methods (western blots, microarray assays and surface plasmon resonance spectroscopy). The binding affinity increases with the number of tags across all assays, reaching nanomolar levels with 5 inserts, an observation assigned to a progressive increase in the probability of a binding interaction occurring. In addition a novel interfacial FRET based assay has been developed for generic Strep tag interactions, which utilises a conventional microarray scanner and bypasses the requirement for expensive lifetime imaging equipment. By

and haemagglutinin (HA) [START_REF] Kolodziej | Epitope tagging and protein surveillance[END_REF][START_REF] Tai | A broad-host-range vector system for cloning and translational lacZ fusion analysis[END_REF], fusion protein tags such as gluthatione S-transferase (GST) and maltose binding protein [START_REF] Duplay | Sequence of male gene and of its product, the maltosebinding protein of Escherichia coli K12[END_REF][START_REF] Smith | Single-step purification of polypeptides expressed in Escherichia coli as fusions with glutathione S-transferase[END_REF], and short affinity tags such as the 6xhistidine tag and poly-arginine tag [START_REF] Porath | Metal chelate affinity chromatography, a new approach to protein fractionation[END_REF][START_REF] Terpe | Overview of tag protein fusions: from molecular and biochemical fundamentals to commercial systems[END_REF]. The interaction between streptavidin and biotin possesses one of the lowest dissociation constants in biology and is of particular interest because it occurs even when the biotin is covalently attached to proteins [START_REF] Diamandis | The biotin-(strept)avidin system: principles and applications in biotechnology[END_REF]. Each streptavidin monomer binds one biotin molecule with femtomolar affinity, one of the strongest non-covalent interactions observed in nature [START_REF] Green | Avidin and streptavidin[END_REF]. Over the past 15 years several groups have developed peptide binders to the tetrameric streptavidin protein [START_REF] Lamla | Searching sequence space for high-affinity binding peptides using ribosome display[END_REF][START_REF] Schmidt | Molecular interaction between the strep -tag affinity peptide and its cognate target, streptavidin[END_REF][START_REF] Schmidt | The random peptide library-assisted engineering of a Cterminal affinity peptide, useful for the detection and purification of a functional Ig Fv fragment[END_REF][START_REF] Weber | Crystal structure and ligand-binding studies of a screened peptide complexed with streptavidin[END_REF][START_REF] Wilson | The use of mRNA display to select high-affinity protein-binding peptides[END_REF] and a variety of protein engineering approaches have been applied to tailor affinity for specific applications [START_REF] Klumb | Energetic roles of hydrogen bonds at the ureido oxygen binding pocket in the streptavidin biotin complex[END_REF][START_REF] Chilkoti | Site-directed mutagenesis studies of the highaffinity streptavidin-biotin complex: contributions of tryptophan residues 79, 108, and 120[END_REF][START_REF] Freitag | A structural snapshot of an intermediate on the streptavidin-biotin dissociation pathway[END_REF]. This provides a simple and clean system for protein purification as a streptavidin binding peptide can be selectively eluted from a streptavidin affinity column using appropriate concentrations of biotin. A further advantage of streptavidin as an affinity reagent is its relative low production cost. This in turn has led to the commercialisation of a wide variety of streptavidin-derived materials such as enzymes, plates, microarray slides, beads and more. Schmidt and Skerra were among the first to develop a streptavidin binding peptide when they identified the Strep-tag in a peptide library screen [START_REF] Schmidt | The random peptide library-assisted engineering of a Cterminal affinity peptide, useful for the detection and purification of a functional Ig Fv fragment[END_REF]. The Strep-tag, although useful for the one-step purification of recombinant proteins [START_REF] Schmidt | One-step affinity purification of bacterially produced proteins by means of the ''Strep tag'' and immobilized recombinant core streptavidin[END_REF], could only be used in the C-terminus of the protein, as a free carboxy-terminus is required for binding. This deficit motivated the development of the StrepII-tag, which is similar in length and can be included anywhere within the protein sequence [START_REF] Schmidt | Molecular interaction between the strep -tag affinity peptide and its cognate target, streptavidin[END_REF]. Both these tags contain the conserved HPQ amino acid motif, which is found in many streptavidin binding peptides and is chiefly involved in mediating the interaction with streptavidin. Although the shortness of the Strep-tags makes them very attractive as biochemical tools, both bind streptavidin with relatively low affinity (37µM and 72µM, respectively [START_REF] Schmidt | Molecular interaction between the strep -tag affinity peptide and its cognate target, streptavidin[END_REF]). This poses a problem in cases where higher affinity is required e.g. in recombinant protein microarrays, or in the detection of

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Strep-tagged proteins in Western Blots. Consequently, several groups have set out to find higher affinity alternatives to the Strep-tags [START_REF] Lamla | Searching sequence space for high-affinity binding peptides using ribosome display[END_REF][START_REF] Wilson | The use of mRNA display to select high-affinity protein-binding peptides[END_REF]. Using a mRNA display library, Wilson and coworkers, for example, have identified a 38 amino acid long peptide (streptavidin-binding peptide, SBP), which binds with nanomolar affinity [START_REF] Wilson | The use of mRNA display to select high-affinity protein-binding peptides[END_REF].

Juntilla and co-workers have explored the possibility of Strep-tag multivalency by using two tandem StrepII-tags separated by a 12 amino acid linker, a configuration named a StrepIII-tag, in the purification of protein complexes with streptavidin coated beads [START_REF] Junttila | Single-step Streptag® purification for the isolation and identification of protein complexes from mammalian cells[END_REF].

An improved detection of the relevant protein complex when using the StrepIII-tag compared to the StrepII-tag was observed on a Western blot assay. However, it remains to be shown whether the increased effectiveness of multiple combinations of StrepIItags results in multivalency effects, where the measured binding affinity is determined by a combination of stable interactions, or simply by a probability effect, where low affinity interactions occur at high frequency because the binding sites are present in excess. Multivalency [START_REF] Mathai | Polyvalent Interactions in Biological Systems: Implications for Design and Use of Multivalent Ligands and Inhibitors[END_REF] effects are commonly observed in nature, where they have evolved to strengthen a diverse array of biochemical interactions. These include polyvalent antibodies binding to an array of antigens on a pathogenic microorganism [START_REF] Dimmock | Multiple mechanisms of neutralization of animal viruses[END_REF][START_REF] Oda | Reevaluation of stoichiometry and affinity/avidity in interactions between anti-hapten antibodies and mono-or multi-valent antigens[END_REF], or indeed the attachment of a virus particle coated in binding proteins (e.g.

haemagglutinin) to multiple receptors on a host cell [START_REF] Lees | Polyacrylamides bearing pendant alpha-sialoside groups strongly inhibit agglutination of erythrocytes by influenza A virus: multivalency and steric stabilization of particulate biological systems[END_REF]. In the laboratory, recent exploitations of polyvalency include the development of a self-assembling peptide polymer, used to improve the attachment rate of nanoparticles to human cells [START_REF] Simnick | Morphing low-affinity ligands into high-avidity nanoparticles by thermally triggered self-assembly of a genetically encoded polymer[END_REF], or the development of a dual 6xHistidine tag (two 6xHis tags in tandem) in order to improve binding of recombinant proteins to Ni 2+ coated microarray slides [START_REF] Steinhauer | Improved affinity coupling for antibody microarrays: Engineering of double-(His)6tagged single framework recombinant antibody fragments[END_REF].

In the present study we have taken further the development of the strepIII-tag [START_REF] Junttila | Single-step Streptag® purification for the isolation and identification of protein complexes from mammalian cells[END_REF] in setting out to investigate the role of binding affinity effects in the performance of the StrepII-tag. We have combined up to 5 StrepII-tags (each of sequence WSHPGFEK) separated by three copies of a regular spacer peptide(GGGS) (Fig. 1A). The tags herein have been appended to the N-terminus of a model protein using the Stefin A Quadruple

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Mutant (SQM) peptide aptamer scaffold protein [START_REF] Hoffmann | Structure-function studies of an engineered scaffold protein derived from stefin A. I: Development of the SQM variant[END_REF]. Peptide aptamers are engineered proteins designed to possess the binding specificities and affinities of antibodies, without the requirement for a host animal or complex post-translational modifications for their production. In our system, the SQM protein is expressed in E coli with an amino-terminal tail designed to contain a variety of functional domains that enable subsequent purification or modification of the expressed protein. One of these domains introduces a unique cysteine residue [START_REF] Davis | Peptide aptamers in label-free protein detection: 1. Characterization of the immobilized scaffold[END_REF] that allows the dimerisation of the SQM proteins under oxidising conditions. We use a subscript " assessed by western blot, surface plasmon resonance and fluorescent microarrays. In addition to obvious application to chromatographic purification, the basic format of a surface microarray based FRET assay [START_REF] Nagl | Microarray analysis of protein-protein interactions based on FRET using subnanosecond-resolved fluorescence lifetime imaging[END_REF] is also described for the peptide aptamer scaffold. The advantages of a FRET assay over a single (target labelled) fluorophore equivalent are the inherent reductions in artefactual noise (such as protein autofluorescence) and the confirmation that the two binding partners are in an intimate sub 10 nm binding sphere [START_REF] Nagl | Fluorescence analysis in microarray technology[END_REF][START_REF] Oh | Inhibition assay of biomolecules based on fluorescence resonance energy transfer (FRET) between quantum dots and gold nanoparticles[END_REF]. Such a platform is also inherently "label free", as has shown to be useful for DNA microarrays [START_REF] Lubin | Folding-based electrochemical biosensors: The case for responsive nucleic acid architectures[END_REF][START_REF] Wang | Molecular engineering of DNA: molecular beacons[END_REF].

Materials and Methods

Plasmid and DNA manipulation for bacterial expression

The streptavidin binding tags were cloned into a modified pET30a+ vector (Novagen, USA; residue [START_REF] Davis | Peptide aptamers in label-free protein detection: 1. Characterization of the immobilized scaffold[END_REF]), amino-terminal to the SQM peptide aptamer scaffold (pET30a+-SQM, [START_REF] Hoffmann | Structure-function studies of an engineered scaffold protein derived from stefin A. I: Development of the SQM variant[END_REF]). The Strep-tag (WSHPGFEK) was generated by annealing the primers This reaction will yield a mix of plasmids, some containing the desired reverting mutation to encode the amino acids GGGS, others having some GTGS and some GGGS.

Consequently several colonies have to be screened until the desired plasmid encoding all GGGS is found. This was achieved with Strep3-and Strep4-pET30a+-SQM, but not with Strep5-pET30a+-SQM, which consequently remained with the amino acid sequence GTGS at the beginning of 3 of its spacers. A schematic diagram of the constructs used in this study is shown (Fig 1A).

Gel electrophoresis, gel staining and Western Blotting

StrepX-SQM 1 was separated on a 15% polyacrylamide SDS gel followed by either transfer onto nitrocellulose membrane (Pierce, UK) for western blotting, or incubation in Coomassie blue stain (40% ethanol, 10% isopropanol, 10% acetic acid, 0.25% w/v Coomassie brilliant blue) for 1h at room temperature, followed by incubation in destain A secondary anti-mouse antibody (Jackson Immunolabs) was used at 1:12,000 in TBST. Western blots were quantified using the Quantity One Software (Biorad) and data processed using Excel for Macintosh (Microsoft).

Surface Plasmon Resonance (SPR)

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Surface coverages in ng mm -2 were calculated from the relationship 1mDeg =122ng mm - 2 [START_REF] Su | Comparison of surface plasmon resonance spectroscopy and quartz crystal microbalance techniques for studying DNA assembly and hybridization[END_REF]. Kinetic parameters were calculated with Kinetic Evaluation Software (Ecochemie).

Microarray and FRET Assays

All proteins were labelled using standard protocols supplied by the dye manufacturer, Atto-tec Gmbh. Concentrations were checked using a Nanodrop spectrophotometer For these assays, array features were printed in repeats of three and at two concentrations (5 µM and 10 µM).

Results

Strep-tag/streptavidin interaction in a Western

Surface Plasmon resonance measurements

A more quantitative analysis of relative binding affinities of the StrepX-SQM 2 systems in their native state (oxidised, dimeric, with no denaturation) was sought through surface plasmon resonance analysis. The series of StrepX-SQM 2 samples were flowed at 1 µM over a pre-formed streptavidin-Biotin-SAM (Fig. 3A). The maximum equilibrium coverage taken from sensorgrams was plotted after subtraction of any non-specific binding contributions measured using Strep0-SQM 2 and the relative affinities were displayed as pM cm -2 . Consistent with the western blot data, binding is observed to increase with additional Strep-tags from 1-5, though the rate of increase is observed to progressively fall. The percentage increase in the number of moles of streptavidin immobilised between Strep1-SQM 2 and Strep2-SQM 2 was 124% whereas between Strep4-SQM 2 and Strep5-SQM 2 the increase was much smaller at 13%. This observation is consistent with the onset of steric saturation as the surface density of Strep tags

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11 increases (Fig. 3B) [START_REF] Bonanno | Steric Crowding Effects on Target Detection in an Affinity Biosensor[END_REF]. Analyses repeated under reducing conditions were equivalent (data not shown).

These quantitative trends were observed to be fully consistent on inverting the interfacial configuration, that is on exposing immobilised StrepX-SQM 2 on a COOH-OEG terminated SAM to solution phase streptavidin (0.05 to 2 µM). Equilibrium dissociation constants, K D , obtained for Strep1-SQM 2 and Strep5-SQM 2 (Fig. 3C) were 1.49 ± 0.01µM and 260 ± 0.18nM respectively (Fig. 3D). The associated R max values for the interaction were 186.61 ± 0.70 mDeg for the Strep1-SQM 2 surface and 877.07 ± 0.17 mDeg for that presented by Strep5-SQM 2 . When comparing the R max for the initial StrepX-SQM 2 surface and the R max for the saturated binding data it becomes apparent that, although capture ratios scale with the number of Strep-tags, they remain below 1:1 for both conditions. Thus, the capture ratio for Strep1-SQM 2 = 0.18:1 while that for Strep5-SQM 2 = 0.43:1. This is consistent with both expected steric limitations and observed electrophoretic mobility shift assays carried out in solution where only 1:1 complexation was observed (data not shown). These analyses, of course, do not shed light on how many tags per SQM molecule are involved in the binding to any one streptavidin but prior work, at typical surface presentations of biotin, have confirmed that streptavidin binds predominantly through two of its sites [START_REF] Jung | Binding and dissociation kinetics of wild-type and mutant streptavidins on mixed biotin-containing alkylthiolate monolayers[END_REF][START_REF] Pérez-Luna | Molecular recognition between genetically engineered streptavidin and surface-bound biotin[END_REF][START_REF] Haeussling | Biotin-functionalized selfassembled monolayers on gold: surface plasmon optical studies of specific recognition reactions[END_REF], a configuration demonstrably useful in building up surface structures through accessing the initially unused streptavidin sites pointing away from the surface.

Affinity measurements in a protein microarray format

In Incubation with fluorescently labelled streptavidin and subsequent fluorescence intensity analysis confirmed a broadly linear increase in binding affinity with tag number (Fig. 4B), tailing off slightly at higher tag numbers. Interestingly, in this format, the strep0-SQM 2 gave a zero background response, indicating that the microarray surface was entirely specific.

Streptavidin FRET Assay

A microarray streptavidin FRET assay, was developed using the microarray surface described previously. In this configuration, the capture and target biomolecules are labelled with complementary (FRET pairing) dye molecules. In the proximity afforded by binding, donor excitation-dependant fluorescence is detected at the acceptor emission wavelength. In a comparative analysis of binding interactions between streptavidin and Strep0-SQM 2 and Strep5-SQM 2 previously discussed, one would expect that "no FRET"

and "strong FRET" behaviours will be observed respectively (if nonbonded dye-dye interactions are negligible, Fig. 5A). The dye pair choice was optimised so as to both maximise spectral overlap at convenient frequencies and generate a Förster radius large enough to accommodate the dimensions of the proteins used herein (Atto550/Atto647N, Ro = 6.5 nm, Atto-tec.com). In all experiments, dye-to-protein labelling ratios were maintained at 1:1, so as to remove potential contributions from labelling variance to any observed variations in emission.

The first microarray lines (Fig. 5B) show Strep0-SQM 2 -Atto550 spots used as donor only leakage controls, to assess whether the donor contributes fluorescence at the measured emission wavelength of the acceptor. As noted earlier, streptavidin-Atto647N binding is not expected here. The absence of acceptor dye is confirmed by the strong donor signal present, confirming no FRET and no crosstalk. The second line of the array confirms the presence of streptavidin bound acceptor (streptavidin-Atto647N) bound to Strep5-SQM 2 -Atto550, as excitation of the SQM bound donor results in emission from Atto647.

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The bottom microarray line is a Strep5-SQM 2 /streptavidin-Atto647N pairing. Here, only the Atto647N dye is present which acts as the acceptor leakage control; again no signal is seen in either the donor or the FRET channel. These results confirm that the dyes are acting orthogonally to each other and the assay is not detecting crosstalk. It also confirms that the FRET signal is due to binding interactions between the two labelled proteins (see discussion for further details).

Discussion

The first experiments on the multivalent effects of multiple Strep-tag units enhancing streptavidin binding affinity came from Hengsakul and Cass, who showed that, when coupling the original Strep-tag [START_REF] Schmidt | Molecular interaction between the strep -tag affinity peptide and its cognate target, streptavidin[END_REF][START_REF] Schmidt | The random peptide library-assisted engineering of a Cterminal affinity peptide, useful for the detection and purification of a functional Ig Fv fragment[END_REF] to the C-terminus of dimerising alkaline phosphatase, a dissociation constant of 0.5 µM was measured [START_REF] Hengsakul | Alkaline Phosphatase-Strep tag fusion protein binding to streptavidin: resonant mirror studies[END_REF]. Though K d values obtained were lower than reported with a single "original" Strep-tag (72 µm) [START_REF] Schmidt | Molecular interaction between the strep -tag affinity peptide and its cognate target, streptavidin[END_REF], an observation assigned to the presence of two tags in the dimeric phosphatase, direct comparisons cannot be made as experimental conditions in the two studies were distinct. Though the StrepIII-tag, comprised of two StrepII-tags, is reported to engender improved streptavidin binding affinity, reported results are wholly qualitative [START_REF] Junttila | Single-step Streptag® purification for the isolation and identification of protein complexes from mammalian cells[END_REF]. We have demonstrated herein that combining 2 StrepII-tags separated by a 12 amino acid linker and fused to an empty peptide aptamer scaffold protein does improve streptavidin binding, by approximately five fold, compared to that of the single tag.

Furthermore, the addition of a third, fourth and fifth StrepII-tag leads to further sequential binding increases, though the advantage of this progressively diminishes.

These observations have been reproducibly confirmed under carefully controlled conditions across three assay formats. They have also been confirmed in either "inversion" of the surface binding assay (the SQM fusion protein on the surface capturing solution phase streptavidin or vice versa).

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The physical basis of the increase in target pair binding can be explained in terms of multivalence [START_REF] Mathai | Polyvalent Interactions in Biological Systems: Implications for Design and Use of Multivalent Ligands and Inhibitors[END_REF][START_REF] Oda | Reevaluation of stoichiometry and affinity/avidity in interactions between anti-hapten antibodies and mono-or multi-valent antigens[END_REF]. The increase in tag number results, specifically, in both a higher local concentration of binding points, and a greater receptive surface area at which the large 52.8 kDa streptavidin molecules can be accommodated. The added advantage of streptavidin having four binding pockets increases the potential network of interactions that may occur, though it is likely that only 1-2 binding pockets per streptavidin are bound to any one SQM receptor at any single point in time [START_REF] Jung | Binding and dissociation kinetics of wild-type and mutant streptavidins on mixed biotin-containing alkylthiolate monolayers[END_REF][START_REF] Pérez-Luna | Molecular recognition between genetically engineered streptavidin and surface-bound biotin[END_REF][START_REF] Haeussling | Biotin-functionalized selfassembled monolayers on gold: surface plasmon optical studies of specific recognition reactions[END_REF]. An approximately linear increase in affinity is achieved on adding more tags through the Strep1-Strep5 [START_REF] Clegg | Fluorescence resonance energy transfer[END_REF][START_REF] Heim | Engineering green fluorescent protein for improved brightness, longer wavelengths and fluorescence resonance energy transfer[END_REF][START_REF] Lakowicz | Principles of Fluorescence Spectroscopy[END_REF][START_REF] Kwan | A fluorescence resonance energy transfer-based binding assay for characterizing kinase inhibitors: Important role for Cterminal biotin tagging of the kinase[END_REF][START_REF] Lebakken | Development and applications of a broad-coverage, TR-FRETbased kinase binding assay platform[END_REF] few have translated to solid surfaces [START_REF] Nagl | Microarray analysis of protein-protein interactions based on FRET using subnanosecond-resolved fluorescence lifetime imaging[END_REF][START_REF] Kim | Energy transfer-based multiplexed assay of proteases by using gold nanoparticle and quantum dot conjugates on a surface[END_REF][START_REF] Kim | Dual-mode fluorophore-doped nickel nitrilotriacetic acid-modified silica nanoparticles combine histidine-tagged protein purification with site-specific fluorophore labeling[END_REF] due to the complexity of experiment design. In addition, the majority of interfacial FRET based assays have relied on the use of life time imaging instrumentation [START_REF] Nagl | Microarray analysis of protein-protein interactions based on FRET using subnanosecond-resolved fluorescence lifetime imaging[END_REF][START_REF] Kim | Dual-mode fluorophore-doped nickel nitrilotriacetic acid-modified silica nanoparticles combine histidine-tagged protein purification with site-specific fluorophore labeling[END_REF], which although convenient and sensitive is also highly specialised. Examples which rely on photoluminescence intensity have utilised quantum dot/ gold nanoparticle pairs, which due to their size may interfere with complex protein-protein interactions [START_REF] Kim | Energy transfer-based multiplexed assay of proteases by using gold nanoparticle and quantum dot conjugates on a surface[END_REF]. The work described herein presents a self consistent methodology for configuring surface based assays between two protein pairs that can be performed on a conventional microarray scanner, with small molecule modification of the capture and target proteins. The data also confirms that the two binding partners are in an intimate sub 10 nm binding sphere. These factors contribute favourably to the potential applications of the assay towards the capture of more complex recombinantly expressed biomolecules with appended Strep-tag tails or towards the development of more sophisticated and sensitive assays based with potential label free applications [START_REF] Lubin | Folding-based electrochemical biosensors: The case for responsive nucleic acid architectures[END_REF][START_REF] Wang | Molecular engineering of DNA: molecular beacons[END_REF].

Conclusion

A progressive improvement in binding affinity between the StrepII-tag and streptavidin can be engineered in association with a StrepX-SQM 2 scaffold. On increasing the number of Strep-tags to 5, nanomolar K d `s are afforded and can be replicated across several interfacial assay formats. A surface based microarray based on FRET between the capture molecule and the target has also been demonstrated, highlighting this new highly sensitive, selective assay platform. Future work should explore the use of different length spacers between the tags, which might further improve affinity.

Applications of the FRET assay can be progressed with the introduction of FRET assays that detect more complex and physiologically relevant targets. In addition even higher

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sensitivity and limits of detection are obtainable based on lifetime dependant analysis [START_REF] Nagl | Microarray analysis of protein-protein interactions based on FRET using subnanosecond-resolved fluorescence lifetime imaging[END_REF].

3 The

 3 tagged StrepX-SQM 2 and streptavidin targets, the conjugate is primed for label free FRET based displacement assays. use of protein affinity tags to facilitate the purification and detection of recombinant proteins has become indispensable to the field of protein chemistry and beyond. The most common classes of tags include antibody epitope tags, such as Myc

F 7 3, 4 , 5 F

 745 into KpnI cut pET30a+-SQM vector. Strep-GGGS(x3)-Strep (strep2) was generated with flanking KpnI sites by PCR-amplifying the Strep containing pET30a+ vector using the following primers: was purified, digested with KpnI and ligated into the KpnI cut pET30a+-SQM vector. A mutagenesis PCR reaction was carried out in order to remove the upstream KpnI restriction site flanking the Strep-tag insert using the primers F: 5'-CCAGATCTGAGTACCTGG-3' R: 5'-CCAGGTACTCAGATCTGG-3' In order to make 3, 4, and 5 Strep tags, we first annealed the primers: F: 5'-TCTGGGTACCGGTTCTGGTGGCGGTTCTGGTGGCGGTTCTTGGAGC-3' R: 5'-CGTCGGTACCCTTTTCGAACTGCGGGTGGCTCCAAGAACCGCC-3', followed by a 'fill-in' reaction using Phusion polymerase (NEB, UK). This yields the element GTGSGGGSGGGS-Strep (i.e. spacer plus tag), flanked by KpnI restriction sites. The amino acid sequence GTG instead of GGG at the beginning of the spacer is necessary as the encoding DNA sequence contains the upstream KpnI restriction site. After the PCR reaction the DNA was digested with KpnI, followed by a ligation reaction with KpnI cut Strep-pET30a+-SQM containing a 10-fold excess of the insert. This yields a Strep-pET30a+-SQM vector with varying additional copies of the Strep-tag (i.e. a total of A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT and more Strep-tags). Finally, to reverse the GTGS part of the spacer (generated by the KpnI restriction site) back to the desired GGGS sequence a PCR mutagenesis reaction was performed on the new constructs using the following oligonucleotides:

(

  the staining solution without Coomassie blue) overnight at 4 o C. For western blotting the membrane was blocked in 3% fat-free powdered milk in TBST (Tris buffered saline pH 8.5 with 0.05% Tween-20) at 4 o C overnight and then briefly rinsed in TBST. The membrane was then incubated with 1 µg/ml streptavidin-HRP (Horseradish peroxidase; Invitrogen, USA) for 1 hour at room temperature and washed 3 times for 10 minutes each in TBST. A chemi-luminescent HRP substrate (Perbio, UK) was then applied, excess rinsed off in TBS and the membrane air-dried and exposed to photographic film. For the loading controls the membrane was re-probed using an anti-S-tag antibody (1:5000; A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8 Novagen, USA), which recognises a unique S-tag encoded by the pET30 vector upstream of the Strep tags.

(

  Thermo Scientific). Microarray assays were performed by spotting (BioOdessey, Biorad Corp) the StrepX-SQM 2 samples on amine coated glass slides (Genetix) using 100 µm capillary pins. Samples were printed at concentrations of 10 µM in print buffer which comprised of PBST (phosphate buffered saline + 3% Tween 20) and 10% glycerol. Array features were printed in repeats of four and the entire array was repeated three times across the slide. Spotted volumes were allowed to incubate on the slide for 60 min prior to a blocking step in 1% BSA in PBST for 60 min. Atto 647N-NHS ester labelled streptavidin (1:1 labelling ratio) was then incubated at a concentration of 7 nM in 1% BSA in PBST for 40 min using 50 µl volume Lifterslips (Thermo Scientific). Slides were washed in PBST twice for 10 min then twice in deionised water for 2 min. The slides were dried under a stream of nitrogen and scanned at a resolution of 5 µm with 543 nm lasers under the Cy3 detection protocol or 633 nm lasers under the Cy5 detection protocol (ScanArray Express, Perkin Elmer). Spot intensities were analyzed with the integrated ScanArray software. FRET assays were performed under identical conditions to microarray binding assays, vide supra, though now both capture and target molecule are labelled, with donor (Atto 550) and acceptor (Atto 647N) fluorophores, respectively.

  Blot format A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 10 Initially the relative changes in binding affinity between one and multiple Strep-tags for streptavidin were assessed in a western blot format. Heat denaturation and SDS stabilisation of the StrepX-SQM proteins in the presence of DTT produce linear, monomeric forms of the proteins containing 0-5 Strep-tags that are separated electrophoretically and transferred onto nitrocellulose. Therefore in this assay, the interactions occur between reduced monomeric StrepX-SQM immobilised on a membrane and soluble HRP-conjugated streptavidin, although some re-folding can occur during the transfer step. HRP metabolises a chemi-luminescent substrate and relative binding affinities are measured via signal intensity on photographic film (Fig. 2A). The use of 2 Strep-tags increases binding to streptavidin by approximately 5-fold compared to a single Strep tag according to this method. A further significant increase in affinity is observed when 3 tags are used (12-fold compared to a single Strep-tag and 2.5 fold compared to Strep2). Further small improvements are observed with Strep4 and Strep5 (Fig. 2B, p>0.05; Student's t-test used).

  order to further study the binding behaviour of the multiple Strep-tags and pave the way for a surface based FRET sensor, the binding efficiencies of the Strep-tag systems were investigated in a microarray format. StrepX-SQM 2 samples were spotted from 10 µM solutions onto amine modified glass slides in repeated arrays to compensate for potential slide inhomogeneities. An excessive concentration was used to ensure that all A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 12 the active surface binding sites were saturated with the StrepX-SQM 2 protein.

15 Finally

 15 series and associated K d values progressively fall into the nanomolar regime. Based on the linear behaviour, the effect is deemed to be probabilistic resulting from greater capture of individual streptavidin molecules. The discrepancy observed herein between the Strep1-SQM 2 and original calculated StrepII-tag K d values may be reflective of interactions involving the scaffold protein. Mammen et al[START_REF] Mammen | Polyvalent interactions in biological systems: implications for design and use of multivalent ligands and inhibitors[END_REF], have presented the relationship between poly and monovalent binding constants (K poly and K mono respectively) as K poly = (K mono )α N , where α quantifies the "degree of cooperativity" and N is the potential number of individual interactions. Applied here to the data presented in section 3.2 (Strep1-SQM 2 vs Strep5-SQM 2 ), α is resolved as being very close to 1.0, consistent with these observations being probabilistic rather than cooperative. All experimental formats utilised indicate a progressive reduction in the advantage conferred by adding more tags across the StrepX-SQM 2 series. This observation is entirely consistent with a progressive steric saturation of binding sites presented by the StrepX-SQM 2 protein (the steric footprint of streptavidin covering large portions of the active surface), preventing any further streptavidin molecules from binding and is thus related to similar observations made on increasing biotin surface density at planar monolayer surfaces[START_REF] Bonanno | Steric Crowding Effects on Target Detection in an Affinity Biosensor[END_REF]. Such behaviour has been quantitatively defined herein by SPR, where the maximum molar ratio of streptavidin that may bind to the Strep5-SQM 2 is observed to be 0.43:1. For the Strep1-SQM 2 surface the maximum streptavidin binding is 0.18:1.A C C E P T ED M A N U S C R I P T ACCEPTED MANUSCRIPT , the strong tagged StrepX-SQM 2 streptavidin association can be utilised in establishing robust interfacial FRET interactions. Though numerous examples of the applications of FRET based bioassays exist

Fig. 1 . 6 Fig. 2 . 22 C.Fig. 3 . 24 Fig. 4 .Fig. 5 .

 1622232445 Fig. 1. A. Multiple cloning site region of the pET30a+-SQM construct used in this study. All streptavidin binding tags were inserted into the KpnI restriction site. Approximate location of the cysteine residue (cys) is indicated. B. Purified samples of StrepX-SQM separated on SDS-PAGE with and without DTT followed by staining with Coomassie blue. Dimer formation mediated by the cysteine residue is readily observed as a molecular weight shift (increase) in the oxidised state (-DTT). Numbers 1-5 signify number of Streptags (X) in the monomer. C. Schematic representation of dimer formation by StrepX-SQM.

Acknowledgements

This work was funded through BBSRC Grant BB/F011296/1 to JJD and PKF. LKJS and PKF gratefully acknowledge the support of AstraZeneca.


Yvonne Schmitz

Olaf Wolkenhauer
email: wolkenhauer-olaf.wolkenhauer@uni-rostock.de

Katja Rateitschak
email: rateitschak-katja.rateitschak@uni-rostock.de

Nucleo-cytoplasmic shuttling of APC can maximize -catenin/TCF concentration

Keywords: mathematical model, intracellular transport, steady state analysis, Wnt pathway, retention

β-catenin is the key player of the canonical Wnt pathway. Its activity is mainly regulated via protein degradation.

In the nucleus, its interaction with TCF initiates target gene expression. Although the functional relevance is unclear, it has been shown that β-catenin antagonists are also capable of nucleo-cytoplasmic shuttling. The focus of our systems biology analysis lies on the β-catenin subcellular distribution regulated by the antagonist and scaffolding protein APC. We address the following questions: Can the concentration of the transcription factor complex [β-catenin/TCF], which is considered as the output of the pathway, be maximized by APC nucleo-cytoplasmic shuttling and how is retention of β-catenin by APC influencing this output?

We established a mathematical model based on experimental findings to examine the influence of nucleocytoplasmic shuttling of APC and retention of β-catenin by APC on the output of the pathway. The model is based on ordinary differential equations and includes protein shuttling between the two compartments nucleus and cytoplasm as well as protein complex formation in each compartment. We discuss how the steady state concentration of [β-catenin/TCF] is influenced by APC shuttling and retention. The analysis of the model shows that the breakdown of β-catenin cytoplasmic retention induced by APC shuttling can enhance nuclear accumulation of β-catenin and hence maximize the output of the pathway.

Using mathematical modelling, we demonstrate that in certain parameter ranges, the steady state concentration of [β-catenin/TCF] benefits from APC shuttling. The inhibitory effect of APC is alleviated due to shuttling of APC. Surprisingly, our study therefore indicates that the nucleo-cytoplasmic shuttling of APC can have a beneficial effect on the output of the pathway in steady state, although APC is in general a β-catenin antagonizing protein.

Furthermore, we show that saturated protein translocation can under certain conditions be modelled by pure diffusion. A difference in the shuttling rate constants of sufficient orders of magnitude leads to an accumulation in either compartment, which corresponds to saturation in translocation.

Introduction 1.The Wnt Signalling Pathway

The canonical Wnt signalling pathway has been studied extensively, since it has been experimentally proven to have a major impact on cell proliferation, migration and fate decisions as apoptosis and differentiation [START_REF] Cadigan | Wnt signaling: a common theme in animal development[END_REF][START_REF] Huang | Wnt/β-catenin signaling: new (and old) players and new insights[END_REF][START_REF] Cadigan | Wnt signaling from development to disease: Insights from Model Systems[END_REF][START_REF] Inestrosa | Emerging roles of Wnts in the adult nervous system[END_REF]. Dysfunctions within the pathway can lead to serious consequences, such as osteoporosis, cancer, and neurodegenerative diseases (reviewed in [START_REF] Moon | Wnt and β-catenin signalling: diseases and therapies[END_REF][START_REF] Logan | The Wnt signaling pathway in development and disease[END_REF][START_REF] Reya | Wnt signalling in stem cells and cancer[END_REF][START_REF] Klaus | Wnt signalling and its impact on development and cancer[END_REF][START_REF] Nusse | Wnt signaling and stem cell control[END_REF]).

The protein β-catenin is the central signalling molecule of the canonical Wnt signalling pathway [START_REF] Kikuchi | Regulation of β-catenin signaling in the Wnt pathway[END_REF]. Its complex formation in the nucleus with transcription factors from the LEF/TCF family activates target genes [START_REF] Behrens | Functional interaction of β-catenin with the transcription factor LEF-1[END_REF][START_REF] Mosimann | β-catenin hits chromatin: regulation of Wnt target gene activation[END_REF]. The principal regulatory mechanism that controls the nuclear accumulation of β-catenin is the activity of the so-called destruction complex, consisting of Axin, Adenomatous polyposis coli (APC) [START_REF] Kishida | Axin, a negative regulator of the Wnt signaling pathway, directly interacts with Adenomatous Polyposis Coli and regulates the stabilization of β-catenin[END_REF] and the kinases glycogen synthase kinase 3 (GSK3) [START_REF] Yost | The axis-inducing activity, stability, and subcellular distribution of β-catenin is regulated in Xenopus embryos by glycogen synthase kinase 3[END_REF][START_REF] Hart | Downregulation of β catenin by human Axin and its association with the APC tumor suppressor, β-catenin and GSKβ[END_REF] as well as casein kinase 1 (CK1) [START_REF] Liu | Control of β-catenin phosphorylation/degradation by a dual-kinase mechanism[END_REF]. As these proteins counteract β-catenin, we refer to them as β-catenin antagonists. In the absence of a Wnt signal (the 'off' state; see Figure 1, left), the destruction complex phosphorylates β-catenin. Thereby, β-catenin is targeted for ubiquitylation and subsequent degradation by the proteasome [START_REF] Aberle | β catenin is a target for the ubiquitinproteasome pathway[END_REF]. Thus, in the 'off' state βcatenin levels are kept low. Upon binding of Wnt ligands to their receptors Frizzled [START_REF] Bhanot | A new member of the Frizzled family from Drosophila functions as a Wingless receptor[END_REF] and LRP5/6 [START_REF] Mao | Lowdensity lipoprotein receptor-related protein-5 binds to Axin and regulates the canonical Wnt signaling pathway[END_REF], the pathway is activated (Figure 1, right). The presence of an extracellular Wnt stimulus ('on' state) inhibits the activity of the destruction complex. Thereby it induces cytoplasmic β-catenin accumulation and β-catenin's entry into the nucleus, where it acts as a transcriptional co-factor [START_REF] Tolwinski | A nuclear function for Armadillo/β-catenin[END_REF]. However, not only β-catenin is able to translocate into and out of the nucleus. A surprisingly high number of cytoplasmic Wnt regulators are shuttling proteins that distribute both, in the cytoplasm and the nucleus, where they may, directly or indirectly, interact with β-catenin and modulate its transcriptional activity [START_REF] Willert | Wnt signaling: is the party in the nucleus?[END_REF]. This holds especially for the β-catenin antagonists APC [START_REF] Neufeld | Adenomatous polyposis coli protein contains two nuclear export signals and shuttles between the nucleus and cytoplasm[END_REF][START_REF] Rosin-Arbesfeld | The APC tumour supressor has a nuclear export function[END_REF], Axin [START_REF] Cong | Nuclear-cytoplasmic shuttling of Axin regulates subcellular localization of β-catenin[END_REF][START_REF] Wiechens | Nucleo-cytoplasmic Shuttling of Axin, a negative regulator of the Wnt-β-catenin pathway[END_REF] and the kinases GSK3 [START_REF] Franca-Koh | The regulation of Glycogen Synthase Kinase-3β export by Frat/GBP[END_REF][START_REF] Bijur | Glycogen synthase kinase -3β is highly activated in nuclei and mitochondria[END_REF][START_REF] Caspi | Nuclear GSK-3β inhibits the canonical Wnt signalling pathway in a β-catenin phosphorylation-independent manner[END_REF] and CK1 [START_REF] Willert | Wnt signaling: is the party in the nucleus?[END_REF].

Next we give a short introduction to nucleo-cytoplasmic shuttling mechanisms and we summarize experimentally derived hypotheses on β-catenin and APC shuttling.

Nucleo-cytoplasmic shuttling

The contents of the eucaryotic nucleus are separated from the cytoplasm by the nuclear envelope. All molecules enter or exit the nucleus through nuclear pore complexes (NPCs) [START_REF] Zilman | Efficiency, Selectivity, and Robustness of Nucleocytoplasmic Transport[END_REF]. Small molecules [START_REF] Sierra | The APC tumor suppressor counteracts β-catenin activation and H3K4 methylation at Wnt target genes[END_REF] kDa [START_REF] Macara | Transport into and out of the Nucleus[END_REF][START_REF] Kopito | Reversibility in nucleocytoplasmic transport[END_REF]) can diffuse freely across the NPC, while large macromolecules in general are transported if they exhibit a particular transport signal sequence, such as the nuclear localization signal (NLS) or nuclear export signal (NES) [START_REF] Zilman | Efficiency, Selectivity, and Robustness of Nucleocytoplasmic Transport[END_REF]. These sequences are required for nucleo-cytoplasmic transport mediated by importin/exportin receptors, such as importin-β and CRM1, respectively [START_REF] Xu | Nucleocytoplasmic shuttling of signal transducers[END_REF]. The process of actual translocation across the NPC generally occurs either by diffusion or facilitated transport [START_REF] Xu | Nucleocytoplasmic shuttling of signal transducers[END_REF].

β-catenin and APC shuttle using different mechanisms

Unlike NLS-mediated import and despite β-catenin's ∼ 90 kDa, it enters the nucleus independently of the importin receptor. Instead, β-catenin can interact directly with the NPC and can translocate on its own [START_REF] Fagotto | Nuclear localization signal-independent and importin/karyopherin-independent nuclear import of β-catenin[END_REF][START_REF] Yokoya | β-catenin can be transported into the nucleus in a RAN-unassisted manner[END_REF]. This is possible because, like importin-β, β-catenin contains several HEAT-like repeats, which are both, necessary and sufficient for nuclear import [START_REF] Xu | Nucleocytoplasmic shuttling of signal transducers[END_REF]. Its nuclear export is "rather nebulous and somewhat controversial" [START_REF] Städeli | Transcription under the control of nuclear Arm/β-catenin[END_REF]. On the one hand, there is experimental evidence that β-catenin is exported out of the nucleus on its own, using interactions with the nucleoporins to pass through the NPC independently of the CRM1 exportin pathway [START_REF] Wiechens | CRM1-and Ran-independent nuclear export of β-catenin[END_REF][START_REF] Eleftheriou | Nuclear export of human β-catenin can occur independent of CRM1 and the Adenomatous Polyposis Coli Tumor suppressor[END_REF]. On the other hand, it has been proposed to bind to APC or Axin to exit the nucleus [22-25, 38, 39]. APC is a large ∼ 310 kDa protein. It exhibits at least two NES interacting with the CRM1 nuclear export factor and has been proven to shuttle between cytoplasm and nucleus [START_REF] Neufeld | Adenomatous polyposis coli protein contains two nuclear export signals and shuttles between the nucleus and cytoplasm[END_REF][START_REF] Rosin-Arbesfeld | The APC tumour supressor has a nuclear export function[END_REF],

but the functional relevance of this is still controversial [START_REF] Henderson | The ins and outs of APC and β-catenin nuclear transport[END_REF][START_REF] Bienz | The subcellular destinations of APC proteins[END_REF][START_REF] Brocardo | APC shuttling to the membrane, nucleus and beyond[END_REF]. It was proposed that the nuclear export of APC controls the level and hence the transcriptional activity of nuclear β-catenin [START_REF] Henderson | Nuclear-cytoplasmic shuttling of APC regulates β-catenin subcellular localization and turnover[END_REF][START_REF] Rosin-Arbesfeld | Nuclear export of the APC tumour suppressor controls β-catenin function in transcription[END_REF]. Most recent findings, however, indicate that APC (and also Axin and Axin2) enriches β-catenin in the cytoplasm, but does not accelerate the nucleo-cytoplasmic shuttling of β-catenin, i.e. increases the rate of β-catenin nuclear import or export [START_REF] Krieghoff | Nucleo-cytoplasmic distribution of β-catenin is regulated by retention[END_REF]. From their results, Krieghoff et al. [START_REF] Krieghoff | Nucleo-cytoplasmic distribution of β-catenin is regulated by retention[END_REF] concluded that β-catenin antagonists, such as APC, mainly regulate β-catenin subcellular localization by retaining it in the compartment in which they are localized, rather than by active transport into or out of the nucleus.

Scope of the paper

To sum up, several approaches have been followed in order to understand the functional role of nucleocytoplasmic shuttling of Wnt pathway components, in particular APC as a β-catenin antagonist. Different experiments resulted in different hypotheses. The functional relevance of antagonist shuttling is yet to be determined.

Contrary to the so far mentioned experimentally focussed studies, we choose a theoretical approach. In the literature one can find several mathematical models describing the canonical Wnt signalling pathway.

The models are based on ordinary differential equations and focus on different aspects and influences of the pathway (e.g. [START_REF] Lee | The roles of APC and Axin derived from experimental and theoretical analysis from the Wnt Pathway[END_REF][START_REF] Cho | Wnt pathway mutations selected by optimal β-catenin signaling for tumorigenesis[END_REF][START_REF] Wawra | Extended analyses of the Wnt/β-catenin pathway: Robustness and oscillatory behaviour[END_REF][START_REF] Mirams | A multiple timescale analysis of a mathematical model of the Wnt/β-catenin signalling pathway[END_REF][START_REF] Van Leeuwen | Elucidating the interactions between the adhesive and transcriptional functions of β-catenin in normal and cancerous cells[END_REF]). However, none of them accounts for nucleo-cytoplasmic shuttling of Wnt pathway components.

In our work we investigate the impact of nucleo-cytoplasmic APC shuttling on the β-catenin/TCF concentration using mathematical modelling. Thus, our model mimics an activated Wnt pathway in steady state, as β-catenin levels are kept very low in the Wnt-off state (see Figure 1). We neglect β-catenin degradation and membrane associated events, as we focus on the influence of APC shuttling on the β-catenin/TCF 2 Results and Discussion

Nucleo-cytoplasmic shuttling of β-catenin and APC

We established a simple biochemical reaction network describing nucleo-cytoplasmic shuttling of β-catenin (β-cat), the main protagonist of the canonical Wnt signalling pathway, and its antagonists APC. The biochemical reaction network is depicted in Figure 2. Both key players are able to shuttle between the nucleus and the cytoplasm (ν 1 and ν 2 , respectively) [START_REF] Neufeld | Adenomatous polyposis coli protein contains two nuclear export signals and shuttles between the nucleus and cytoplasm[END_REF][START_REF] Wiechens | CRM1-and Ran-independent nuclear export of β-catenin[END_REF][START_REF] Rosin-Arbesfeld | Nuclear export of the APC tumour suppressor controls β-catenin function in transcription[END_REF]. Furthermore, they can associate to and dissociate from a common complex [β-cat/APC] within the cytoplasm (ν 3 ) as well as within the nucleus (ν 4 ) [START_REF] Krieghoff | Nucleo-cytoplasmic distribution of β-catenin is regulated by retention[END_REF].

These complexes, however, are not able to cross the nuclear envelope [START_REF] Wiechens | CRM1-and Ran-independent nuclear export of β-catenin[END_REF][START_REF] Eleftheriou | Nuclear export of human β-catenin can occur independent of CRM1 and the Adenomatous Polyposis Coli Tumor suppressor[END_REF][START_REF] Krieghoff | Nucleo-cytoplasmic distribution of β-catenin is regulated by retention[END_REF].

Additionally, nuclear β-catenin can interact with TCF (ν 5 ) to activate transcription [START_REF] Cong | Nuclear-cytoplasmic shuttling of Axin regulates subcellular localization of β-catenin[END_REF]. We hence consider the complex [β-cat/TCF] as the output of both, the Wnt pathway [START_REF] Lee | The roles of APC and Axin derived from experimental and theoretical analysis from the Wnt Pathway[END_REF] and our shuttling model.

Next, we translated the reaction network into a system of ordinary differential equations (ODEs) describing temporal changes of protein concentrations as functions of interactions and transport processes. The simplifying assumptions are presented in the Methods section. The network depicted in Figure 2 includes neither a time dependent input nor output. Thus, we can study it in steady state. The calculation of the steady state concentration for the network components is presented in the Methods section.

The protein β-catenin not only exhibits the intrinsic ability to enter the nucleus, but also to move bidirectionally across the nuclear envelope, proposing a free, nondirectional nuclear translocation model for β-catenin [START_REF] Xu | Nucleocytoplasmic shuttling of signal transducers[END_REF][START_REF] Wiechens | CRM1-and Ran-independent nuclear export of β-catenin[END_REF]. We will therefore consider pure diffusion for nuclear translocation of β-catenin.

The nucleo-cytoplasmic shuttling of APC, however, depends on indirect interaction with the NPCs and occurs in both directions by facilitated transport [START_REF] Neufeld | Adenomatous polyposis coli protein contains two nuclear export signals and shuttles between the nucleus and cytoplasm[END_REF][START_REF] Rosin-Arbesfeld | The APC tumour supressor has a nuclear export function[END_REF]. In our approach, we investigate the influence of nucleo-cytoplasmic shuttling of APC on the output of the Wnt signalling by studying different shuttling mechanisms:

In a first step, we neglect the nucleo-cytoplasmic shuttling of APC, which means that APC is only located in the cytoplasm. This system serves as a reference system in order to examine the influence of APC shuttling.

We then study nucleo-cytoplasmic shuttling of APC and its effect on the output of the pathway. For the sake of simplicity, we first assume diffusion of APC across the nuclear envelope as proposed for nucleo-cytoplasmic shuttling of β-catenin. We start by setting all dissociation constants and total concentrations according to measurements and estimations conducted by the Kirschner group in the Wnt off-state in Xenopus oocytes extracts to [START_REF] Lee | The roles of APC and Axin derived from experimental and theoretical analysis from the Wnt Pathway[END_REF], see Table 1. These numbers serve as the default parametrization and hence form the starting point for the investigation of the effect of nucleo-cytoplasmic antagonist shuttling. Next, we analyze the influence of cytoplasmic and nuclear retention of β-catenin by APC, by varying the respective dissociation constants. Then, we choose a more complicated, but more realistic approach for a APC's shuttling mechanism and assume facilitated nuclear export, which is based on a Michaelis Menten type rate law.

Absence of APC shuttling

Absence of APC shuttling is described by setting the antagonist shuttling constants to zero (see Eqs.( 16)).

The steady state analysis of the ODE model is performed in the Methods section and leads to the dependency of the steady state concentration [β-cat/TCF] * (see Table 2) on the ratio of rate constants for β-catenin nucleo-cytoplasmic shuttling (K 1 = k-1 k1 ) depicted in Figure 3a. The parameter K 1 is defined as the ratio of the shuttling rate constants of β-catenin, which is equal to the ratio of free protein concentrations in the cytoplasm and nucleus calculated in steady state (Eqs. [START_REF] Aberle | β catenin is a target for the ubiquitinproteasome pathway[END_REF]). The smaller the value of K 1 , the higher the concentration of free β-cateninin the nucleus. If more β-catenin is located in the nucleus, the output of the pathway is maximized in both presented cases. This meets our expectations, as only nuclear β-catenin is able to interact with TCF and therewith initiate target gene expression. However, a certain plateau of [β-cat/TCF] * ∼ 19 nM cannot be exceeded. This is due to the fact that the binding of β-catenin to TCF is limited by the value of its dissociation constant K 5 . This constant regulates the level of the plateau height:

decreasing K 5 increases the height of the plateau.

In Figure 3a, the blue curve is obtained using the default parametrization (Table 1). The green curve in comparison is obtained if K 3 is reduced to 30 nM. This lower value corresponds to a higher affinity of cytoplasmic β-catenin to bind APC and therefore describes a higher cytoplasmic retention of β-catenin by APC. In this parametrization, β-catenin's binding to TCF and to APC occur equally likely. We see that the variation in values for K 3 does not influence the height of the minimal or maximal plateau but shifts the curve to smaller values of K 1 . For K 1 = 1, the steady state concentration [β-cat/TCF] * is decreased by half, because of the increased competition of the antagonist with the transcriptional factor for binding to β-catenin. Our results therefore show that cytoplasmic retention of β-catenin by its antagonist APC influences the output of the model without antagonist shuttling.

Diffusive nucleo-cytoplasmic shuttling of APC

In the following, we investigate the effects of nucleo-cytoplasmic antagonist shuttling on the steady state concentration [β-cat/TCF] * . First, we assume diffusive shuttling for β-catenin and APC. All translocation processes across the nuclear envelope are modelled as simple as possible, meaning that transport is based on diffusion. Thus, nucleo-cytoplasmic shuttling is described by linear rate equations, see Eqs. [START_REF] Mao | Lowdensity lipoprotein receptor-related protein-5 binds to Axin and regulates the canonical Wnt signaling pathway[END_REF].

The standard (default) parametrization

We start by setting all dissociation constants and total protein concentrations to their default values (see Table 1). As a result of the steady state analysis (see Methods), we obtain that [β-cat/TCF] * depends on the biochemical binding and shuttling parameters. The steady state concentration [β-cat/TCF] * as a function of the nucleo-cytoplasmic shuttling rate constant ratios (K 1 and K 2 ) is presented in Figure 3b.

In this parametrization, the shuttling ratio of β-catenin (K 1 ) has a much higher impact on the output than the shuttling ratio of APC (K 2 ). The shape of the curve is qualitatively very similar to the result obtained with the reference model presented in Figure 3a. For this model, it is also more favorable for the output of the pathway, if more β-catenin is located in the nucleus. The plateau has the same height as the plateau of the reference model. APC shuttling parameter values between K 2 = 0.01 and K 2 = 100 exhibit an almost identical output, even if the parameter values of K 2 are further increased or decreased (data not shown). Thus, for the default parametrization, the steady state concentration [β-cat/TCF] * is only slightly influenced by APC shuttling. This is due to the large dissociation constants describing the binding of APC to β-catenin, i.e. K 3 = K 4 = 1200 nM (see Table 1), which exceeds the dissociation constant of TCF to β-catenin by a factor of 40. Hence, β-catenin has a much higher affinity to bind to TCF than to APC and the concentration of the complex [β-cat/APC] * can neither reach high numbers in the cytoplasm nor in the nucleus. Subsequently, β-catenin can neither be retained in the nucleus nor cytoplasm by its antagonist APC. Thus, the influence of antagonist shuttling is negligible.

In the next section, we study the question, whether lower dissociation constants can account for a greater impact of APC shuttling on the steady state concentration [β-cat/TCF] * in comparison to the values of the reference model.

The impact of the dissociation constants

For the default parametrization, the affinity of β-catenin to bind to APC is low: K 3,4 = 1200 nM (experimentally based estimation by the Kirschner group [START_REF] Lee | The roles of APC and Axin derived from experimental and theoretical analysis from the Wnt Pathway[END_REF]). However, it has been demonstrated that phosphorylation enhances the ability of APC to interact with β-catenin in vitro [START_REF] Rubinfeld | Binding of GSK3β to the APCβ-Catenin Complex and regulation of complex assembly[END_REF]. Moreover, Sierra et al. [START_REF] Sierra | The APC tumor suppressor counteracts β-catenin activation and H3K4 methylation at Wnt target genes[END_REF] concluded from their experiments that β-catenin cannot bind unphosphorylated APC efficiently and that CK1 phosphorylation of APC might induce high-affinity binding to β-catenin and even trigger its dissociation from LEF/TCF. Seo and Jho proposed in [START_REF] Seo | Axin-independent phosphorylation of APC controls β-catenin signaling via cytoplasmic retention of β-catenin[END_REF] that accumulation of cytoplasmic β-catenin induces phosphorylation of APC and that phosphorylated APC retains β-catenin. In accordance with these findings, we investigate the influence of small values for dissociation constants (K 3,4 = 30 nM) between β-catenin and APC. Lower dissociation constants correspond to a higher binding affinity and hence higher possible compartmental retention. We choose these specific values to account for a "fair" competition between TCF and APC for β-catenin binding. It is still selected arbitrarily. However, our results do not change qualitatively if other values (10 to 120 nM) were chosen. Since APC phosphorylation can occur in both compartments, nucleus and cytoplasm, we change the two different parameter values independently: we first decrease K 3 to consider an enhanced cytoplasmic retention and second decrease K 4 to account for an enhanced nuclear retention of β-catenin by APC. An additional justification for different dissociation constants in nucleus and cytoplasm could also be different biochemical compositions of the two compartments. In a last step we show the combined effect of nuclear and cytoplasmic retention of β-catenin by APC.

For K 3 = 30 nM, the binding affinity of β-catenin to APC in the cytoplasm is equal to the binding affinity of β-catenin to TCF in the nucleus (K 5 = 30 nM). The steady state analysis presented in the Methods section leads to the dependency of [β-cat/TCF] * on the nucleo-cytoplasmic shuttling rate constant ratios presented in Figure 3c. The dependency on the shuttling ratio of β-catenin (K 1 ) is plotted for different parameter values of K 2 , which captures the shuttling ratio of APC. The shape of all curves is similar to the ones obtained with the default parametrization. In contrast to the default case presented in Figure 3b however, the curves for different K 2 are visibly shifted by different extent along the x-axis. This leads to a qualitatively different dependency of the output of the model on the shuttling of APC. The curves are shifted such that they intersect in one specific point 3c). Thus, nucleo-cytoplasmic shuttling of APC can increase the output of Wnt signalling in comparison to the reference model. We can explain our results as follows (see Figure 4): If APC is highly concentrated in the nucleus (K 2 is small, green curves), and at the same time the dissociation constant of nuclear APC to β-catenin (K 4 = 1200 nM) is large, only a small amount of β-catenin can bind to APC (see Figure 4f). Thus, most of the APC proteins remain free and unbound in the nucleus (Figure 4e). This in turn means, that the high binding affinity of cytoplasmic APC and β-catenin cannot be fully utilized due to a low cytoplasmic APC concentration (Figure 4b), hence the concentration of [β-cat/APC] in the cytoplasm is small (Figure 4c). As a result, more β-catenin remains unbound to APC (Figure 4a and4d)

K 1 ≈ 3 • 10 -2 .
and can hence bind to TCF (Figure 3c). Cytoplasmic retention of β-catenin is down-regulated due to APC nucleo-cytoplasmic shuttling. Therefore, in this parametrization Wnt signalling benefits from antagonist shuttling.

In order to investigate the impact of nuclear retention, we now increase the affinity of β-catenin to APC in the nucleus, such that it is equal to the affinity of β-catenin to bind to TCF in the nucleus (K 4 = K 5 = 30 nM). The results of the steady state analysis based on these parameter values are shown in Figure 3d. The steady state concentration of [β-cat/TCF] * is plotted with respect to changes in the parameter values K 1 and K 2 . The black curve where K 2 = 100 also equals the reference case (Figure 3a). The curves show that the output of our diffusive shuttling model can only be reduced with these parameter values, in contrast to the results presented in Figure 3c. This is due to an enhanced competition of APC and TCF for β-catenin binding in the nucleus.

Next, we choose all three dissociation constants to have the same value (K 3 = K 4 = K 5 = 30 nM). The steady state concentration [β-cat/TCF] * is shown in Figure 3e. We see a combination of the above discussed effects. For K 1 < 1 the effect of nuclear retention is dominant, as more β-catenin is located in the nucleus:

Higher nuclear concentration of APC hence corresponds to higher competition of nuclear APC and TCF for β-catenin binding and therefore reduces the output. For K 1 > 1, the effect of cytoplasmic retention is prevalent; Cytoplasmic retention is down-regulated due to nuclear import of APC. Interestingly, the [βcat/TCF] * concentration benefits from a decreased competition of cytoplasmic APC and TCF for β-catenin binding, although the binding affinity of β-catenin to APC in the nucleus is equally high. This means that [β-cat/TCF] signalling can even benefit from APC shuttling if β-catenin is retained by APC also in the nucleus. The intersection point in Figure 3e corresponds to the situation in which nuclear and cytoplasmic retention are equally powerful.

To sum up, diffusive nucleo-cytoplasmic shuttling of APC has a significant influence on the steady state concentration of [β-cat/TCF] * . Our results demonstrate that the shuttling of the antagonist can yield an enhanced output of the system. We have shown that this enhancement can be induced by a high binding affinity of APC and β-catenin in the cytoplasm, which can be caused by phosphorylation of APC [START_REF] Rubinfeld | Binding of GSK3β to the APCβ-Catenin Complex and regulation of complex assembly[END_REF].

Phosphorylation of APC can be induced by a high cytoplasmic β-catenin concentration [START_REF] Seo | Axin-independent phosphorylation of APC controls β-catenin signaling via cytoplasmic retention of β-catenin[END_REF]. The latter is supported by a high APC shuttling ratio leading to an accumulation of free APC in the nucleus. However, this effect is also visible if the nuclear [β-cat/APC] concentration is high due to enhanced binding affinity between the two proteins in the nucleus. Therefore the question arises, if [β-cat/TCF] * can further be maximized by saturated nuclear export and low binding affinity of β-catenin and APC in the nucleus, which is studied in the following section.

Facilitated export of APC

Facilitated export of APC is modelled by a Michaelis-Menten type function (see Eqs.( 24)). This is reasonable, since APC exhibits a NES [START_REF] Neufeld | Adenomatous polyposis coli protein contains two nuclear export signals and shuttles between the nucleus and cytoplasm[END_REF][START_REF] Rosin-Arbesfeld | The APC tumour supressor has a nuclear export function[END_REF], and its export therefore depends on the CRM1 exportin pathway. It is the formation of the transportin-protein complex, that restricts export rates and can lead to saturation [START_REF] Timney | Simple kinetic relationships and nonspecific competition govern nuclear import rates in vivo[END_REF].

This holds also true for the nuclear import of APC. However, we have seen in the previous section, that especially nuclear accumulation of APC yields a maximization of the output of the system. Therefore, and for the sake of simplicity, we restrict our analysis to the saturation of nuclear export. The nuclear import of APC remains diffusive. The steady state analysis of the ODE model is presented in the Methods section.

For our analysis, we set K 3 = K 5 = 30 nM, K 4 = 1200 nM and K 1 = 1, as chosen in the previous sections.

For facilitated export, two extreme cases can be considered: First, for k m >> X 4 , the nuclear export of APC changes linearly with the concentration of nuclear APC (X 4 ), which corresponds to diffusive shuttling.

In this case, we obtain

K lin = v max k m k 2 = K 2 (1)
in steady state. Second, for k m << X 4 , the nuclear export is independent of X 4 ; it is therefore saturated.

The nucleo-cytoplasmic shuttling of APC is in this case determined by

K sat = v max k 2 . ( 2
)
Note that K sat is independent of k m . In Figure 3f, the dependency of [β-cat/TCF] * on the parameters describing the shuttling of APC is shown.

In the linear extreme case k m >> X 4 , our results depending on K lin are equal to the results obtained assuming diffusive APC export (left border of the scatter-plot), which meets our expectations (see Eq.( 1)).

For decreasing k m , more solutions are obtained on the high plateau, but the height of this plateau does not exceed the results we obtain with the diffusive model. For small half-saturation constants (k m < 1 nM), the concentration of [β-cat/TCF] * is completely determined by K sat (not shown), as expected from Eq.( 2). In this case, export is saturated, hence more free APC is located in the nucleus and the steady state solution we obtain is located on the high plateau, which is in agreement with our results for the diffusive model. However, the maximum value of [β-cat/TCF] * ≈ 14 nM cannot be exceeded due to saturated export.

We therefore conclude that the concentration of [β-cat/TCF] * cannot be further maximized in response to facilitated and hence saturated export of APC.

We have also considered facilitated import. This results in a more complicated system which is numerically more challenging, but did not gain new insights.

Our results thus show, that it is possible to simulate the effects of saturation by using pure diffusion, if the shuttling rate constant ratio is chosen appropriately. A difference in the shuttling rate constants of significant orders of magnitude leads to an accumulation of the protein in the respective compartment, which corresponds to saturation in translocation.

Conclusions

Using mathematical modelling, we investigated the impact of nucleo-cytoplasmic shuttling of APC on the We propose the following mechanism to explain our results: In the Wnt-on state, where β-catenin levels are high, β-catenin can cause phosphorylation of APC and phosphorylated APC in the cytoplasm may retain β-catenin [START_REF] Seo | Axin-independent phosphorylation of APC controls β-catenin signaling via cytoplasmic retention of β-catenin[END_REF]. From their results, Seo and Jho [START_REF] Seo | Axin-independent phosphorylation of APC controls β-catenin signaling via cytoplasmic retention of β-catenin[END_REF] concluded that this is to regulate excessive [β-cat/TCF] signalling. Our results, however, allow for an alternative interpretation as we additionally considered nucleocytoplasmic shuttling of APC: a high APC shuttling ratio can lead to a breakdown of β-catenin retention in the cytoplasm, therefore enhance nuclear accumulation of β-catenin and hence [β-cat/TCF] signalling.

The breakdown of cytoplasmic retention by APC can even lead to an enhanced [β-cat/TCF] concentration, if phosphorylated APC retains β-catenin also in the nucleus. This can be explained by a sufficiently high concentration of nuclear β-catenin available to bind TCF. The inhibitory effect of APC is alleviated due to shuttling of APC. In other words, our study indicates that the nucleo-cytoplasmic shuttling of APC has a beneficial effect on the steady state output of pathway, although APC is an antagonizing protein. Nucleocytoplasmic shuttling of antagonists may also play a relevant role in other pathways and the advantage it brings may even be a general property in signalling.

Next, we investigated facilitated nuclear export of APC, which is based on a Michaelis Menten type rate law.

We showed that maximization of the output, which we have obtained by considering diffusive shuttling of β-catenin and APC, cannot be further increased by facilitated antagonist export. Our results demonstrate that it is possible to simulate the effects of saturated translocation by pure diffusion. A difference in the shuttling rate constants of significant orders of magnitude leads to an accumulation of the protein in the respective compartment. Based on this result, further modelling approaches of nucleo-cytoplasmic shuttling can be simplified.

Methods

Mathematical modelling of β-catenin and APC shuttling

The biochemical reaction network is presented in Figure 2. We have translated this network into a system of ordinary differential equations (ODEs). Each protein and protein complex concentration is represented by a variable in the mathematical model (see Table 2). Taking the five interactions into account, the dynamics of the biological system, i.e. the time-dependent changes of the concentrations of the proteins and protein complexes are determined by the following eight coupled ODEs:

Ẋ1 = -ν 1 -ν 3 Ẋ2 = -ν 2 -ν 3 Ẋ3 = +ν 1 -ν 4 -ν 5 Ẋ4 = +ν 2 -ν 4 (3) Ẋ5 = +ν 3 Ẋ6 = +ν 4 Ẋ7 = -ν 5 Ẋ8 = +ν 5
where the reaction rates v i on the right-hand side are functions of the protein or complex concentrations, each one of them describing biochemical reactions or transport processes:

ν 1 = F 1 (X 1 , X 3 ) ν 2 = F 2 (X 2 , X 4 ) ν 3 = k 3 X 1 X 2 -k -3 X 5 (4)
ν 4 = k 4 X 3 X 4 -k -4 X 6 ν 5 = k 5 X 3 X 7 -k -5 X 8 .
Binding and dissociation processes (ν 3 , ν 4 and ν 5 ) are described with mass-action kinetics [START_REF] Heinrich | The regulation of cellular systems[END_REF]. The functions F 1 (X 1 , X 3 ) and F 2 (X 2 , X 4 ) describe nucleo-cytoplasmic transport processes of β-catenin and APC, respectively. To investigate the influence of the nucleo-cytoplasmic shuttling on the output of the system, we consider different translocation mechanisms which are presented in the following.

Rapid equilibrium approximations and conservation equations

Independent of the translocation mechanism, one can apply the following approximations and conservation equations in order to simplify the model: We assume that the binding and dissociation processes considered in the network (i.e. β-catenin to APC and β-catenin to TCF, see Figure 2) will approach the quasi-equilibrium rapidly, leading to:

K 3 = X 1 X 2 X 5 = k -3 k 3 (5) K 4 = X 3 X 4 X 6 = k -4 k 4 (6) K 5 = X 3 X 7 X 8 = k -5 k 5 . ( 7
)
The biochemical reaction network (see Figure 2) implies the existence of three conservation equations.

This means, that total amounts of molecules are conserved. In our schema these conserved quantities correspond to the total number of molecules of β-catenin, APC, and TCF. If more than one compartment is considered like in our case, conservation sums can only be applied to molecule numbers, not to concentrations. To calculate the corresponding protein concentrations, the compartmental volumes must be taken into account. For the sake of simplicity, we assume that the volumes and capacities of nucleus and cytoplasm are of the same size. This assumption leads to the following expressions exhibiting the double total concentrations B, A and T , respectively:

total β-catenin: 2B = X 1 + X 3 + X 5 + X 6 + X 8 (8)
total APC: 2A

= X 2 + X 4 + X 5 + X 6 (9)
total TCF: 2T

= X 7 + X 8 . ( 10
)
The set of differential equations can be simplified by considering the conservation equations and the rapid equilibrium approximations to the binding reactions. This results in a subdivision of dependent variables.

The dependent variables are algebraic functions of other variables, which are determined as solutions of the remaining equations.

Elimination of X 7 and X 8 :

These dependent variables can be expressed as a function of the variable X 3 using the conservation condition for TCF (Eq.( 10)) and the equilibrium condition for β-catenin to TCF binding (Eq.( 7)). The combination of both yields

X 7 = 2K 5 T K 5 + X 3 and X 8 = 2T X 3 K 5 + X 3 . ( 11
)
Elimination of X 5 and X 6 :

Both dependent variables are eliminated by the equilibrium conditions for the binding of β-catenin to APC in the cytoplasm and nucleus, respectively (Eqs.( 5) and ( 6)).

X 5 = X 1 X 2 K 3 and X 6 = X 3 X 4 K 4 . ( 12
)

Differential equations for the remaining variables

Substituting the dependent variables of Eqs.( 11) and ( 12), into the ODE system (3), the dynamics of the remaining variables are determined by the nucleo-cytoplasmic reaction rates:

Ẋ1 = -F 1 (X 1 , X 3 ) = -Ẋ3 Ẋ2 = -F 2 (X 2 , X 4 ) = -Ẋ4 . ( 13
)
The connection between the variables is given by the conservation relations of β-catenin and APC, described by Eqs.( 8) and ( 9), where the dependent variables of Eqs.( 11) and ( 12) have been substituted: total β-catenin:

2B = X 1 + X 3 + X 1 X 2 K 3 + X 3 X 4 K 4 + 2T X 3 K 5 + X 3 (14)
total APC:

2A = X 2 + X 4 + X 1 X 2 K 3 + X 3 X 4 K 4 (15)
Equations ( 13) -( 15) depend on four variables (X 1 , ..., X 4 ) and can thus be solved in steady state for different nucleo-cytoplasmic transport mechanisms. The other four dependent variables can afterwards be calculated with Eqs.( 11) and ( 12).

Steady states of the ODE model

Since we investigate a closed system without a stimulus, transcription or degradation, all transport processes are in steady state. To determine the steady states, i.e. the stationary solutions of the model, we calculate the states in which the time derivatives vanish: Ẋi = 0. We annotate the steady state concentration by an asterisk. By considering the conservation equations and substituting X * 1 and X * 2 (Eqs.( 13)), we obtain two algebraic equations, that depend on X * 3 and X * 4 . Independent of the transport mechanism we choose, substitution always leads to an algebraic equation in which X * 3 (i.e. nuclear β-catenin) is the only remaining variable. It has the form of a polynomial with respect to X * 3 , and depends on dissociation and shuttling constants, and total protein concentrations. The solution of the polynomial can be determined in general terms using computer algebra based on symbolic maths (e.g. MAPLE or MATLAB). However, the particular expressions are far too complex to tell us much about the systems behavior. Therefore we determine the stationary solution numerically using the "roots" (for cubic polynomials) or "fzero" (for polynomials of higher order) function in MATLAB (The root finding algorithm used in the "fzero" function depends on the initial value of search. Therefore one has to make sure to find every biological sensible basin of attraction.).

We investigated parameter space by generating randomized parameter values for those parameters describing nucleo-cytoplasmic shuttling. In the following we consider and analyze different transport mechanisms.

No antagonist shuttling

This system serves as a control in order to investigate the influence of antagonist shuttling on X * 8 = [βcat/TCF]. In this case, the antagonist APC resides in the cytoplasm and is not able to cross the nuclear envelope.

F 1 (X 1 , X 3 ) = k 1 X 1 -k -1 X 3 F 2 (X 2 , X 4 ) = 0 ( 16
)
X 4 = X 6 = 0
Hence, the system depicted in Figure 2 simplifies significantly. The system reduces from eight to six variables.

In steady state we find:

X * 1 = k -1 k 1 X * 3 = K 1 X * 3 X * 2 = 2K 3 A K 3 + X * 1 ( 17
)
X * 5 = 2AX * 1 K 3 + X * 1 .
Substituting these solutions into Eq.( 14), we obtain the steady state solutions as the roots of a cubic polynomial with respect to X * 3 :

X * 3 3 (K 1 + 1)K 1 +X * 2 3 2K 1 (A -B + T ) + (K 1 + 1)(K 3 + K 1 K 5 ) (18)
+X * 3 2K 1 K 5 (A -B) + 2K 3 (T -B) + K 3 K 5 (K 1 + 1) -2BK 3 K 5 = 0.
Therefore, the steady state concentration of X * 8 is also only, but explicitly, dependent on the dissociation and shuttling constants as well as the conserved total protein concentrations.

A cubic polynomial has one to three solutions. We find that there exist three solutions, which raises the question of their ranges and stability. If two biological meaningful and stable steady states coexist, the system exhibits bistability. Hence hysteresis-like behavior can be expected when crossing bifurcation points.

However, in every parametrization two of the three steady states solutions exhibit at least one negative protein concentration. We investigated parameter space by generating parameter values for the shuttling parameters, i.e. K 1 ∈ [10 -4 , 10 4 ]. In a biological sensible range, the system remains monostable. This is therefore a numerically obtained finding, which originates from the strict condition that all variable and parameter values must be positive in order to be biologically meaningful.

Free diffusive shuttling of APC

The simplest assumption for nucleo-cytoplasmic shutting is diffusion, which is described by a linear rate equation:

F 1 (X 1 , X 3 ) = k 1 X 1 -k -1 X 3 F 2 (X 2 , X 4 ) = k 2 X 2 -k -2 X 4 (19)
The index of the shuttling constants (k

1 , k -1 , k 2 , k -2
) is positive for transport into the nucleus and negative for the transport back to the cytoplasm. In steady state we obtain the following relations:

X * 1 = k -1 k 1 X * 3 = K 1 X * 3 X * 2 = k -2 k 2 X * 4 = K 2 X * 4 (20)
By considering the conservation equations ( 14) and ( 15) and substituting X * 1 and X * 2 as presented in Eqs. [START_REF] Tolwinski | A nuclear function for Armadillo/β-catenin[END_REF], we obtain two algebraic equations, which depend on X * 3 and X * 4 . Solving Eq.( 15) for X * 4 and substituting into Eq.( 14) leads to the following equation in which X * 3 (i.e. nuclear β-catenin) is the only remaining variable. It also has the form of a cubic polynomial with respect to X * 3 :

X * 3 3 (k 1 + 1)C 1 +X * 2 3 2C 1 (A -B + T ) + C 2 + (K 1 + 1)K 5 C 1 (21) +X * 3 2K 5 C 1 (A -B) + 2(T -B)(K 2 + 1) + C 2 K 5 -2K 5 B(K 2 + 1) = 0.
where

C 1 = K 1 K 2 K 3 + 1 K 4 ( 22
)
and

C 2 = (K 1 + 1)(K 2 + 1). (23)

Figures Figure 1 -The Wnt signalling pathway

Left: The Wnt-off state. Cytoplasmic degradation keeps β-catenin levels low. Right: The Wnt-on state.

The destruction complex is inhibited. β-catenin and its antagonists, for example APC can shuttle between cytoplasm and nucleus. In the nucleus, β-catenin forms a transcription factor complex with TCF.

Figure 2 -Biochemical reaction network

Biochemical reaction network describing β-catenin and APC nucleo-cytoplasmic shuttling and protein complex formation of β-catenin and APC as well as β-catenin and TCF. The reactions are numbered 1 to 5. In Table 2 the protein names are translated into state variables (X 1 to X 8 ) as used in the Methods section. The steady state concentration [β-cat/TCF] * is depicted in dependence on the ratio of shuttling rate constants of β-catenin (K 1 ) and APC (K 2 ). a) The reference model without APC shuttling. The blue curve is obtained using the default parametrization (Table 1); for the green curve we chose K 3 = K 5 = 30 nM to account for increased cytoplasmic retention. b-e) Diffusive APC shuttling: b) The default parametrization:

The black curve is equal to the blue curve in Figure 3a. For this parametrization, the steady state analysis indicates that the shuttling rate constant ratio of β-catenin has a much higher impact on [β-cat/TCF] * than the ratio of shuttling rate constants of APC. c) K 3 = 30 nM. The black curve is equal to the green curve in The steady state concentrations are plotted in dependence on the ratios of the shuttling rate constants of β-catenin (K 1 ) and its antagonist APC (K 2 ), respectively.

Tables Table 1 -The default parametrization following Lee et al., 2003 [44]

Numeric values and description of the parameters of the model in its default state. The K i -parameters are dissociation constants K i = k -i /k i , the others (A, B, T ) illustrate total protein concentrations. These parameters are crucial for simplifying the system using rapid equilibrium approximations and conservation equations.

Parameter

  output of the Wnt signalling pathway, to answer the following questions: Does nucleo-cytoplasmic shuttling of APC influence the [β-cat/TCF] concentration, which we consider as the output of the pathway? Can the concentration of [β-cat/TCF] benefit from APC shuttling? How is the output influenced by cytoplasmic and nuclear retention of β-catenin by APC?To answer these questions, we established a simple biochemical reaction network based on experimentally obtained results, which describe the interactions and nucleo-cytoplasmic shuttling of β-catenin, the main protagonist of the canonical Wnt signalling pathway, and its antagonist APC. The network was translated into a mathematical model based on ordinary differential equations (ODEs). We assumed diffusive transport for nuclear translocation of β-catenin. To examine the influence of nucleo-cytoplasmic shuttling of APC on the output of Wnt signalling, we studied different shuttling mechanisms.The analysis of the ODE model leads to the following results: The steady state concentration of [β-cat/TCF] can be enhanced in response to nucleo-cytoplasmic shuttling of APC. If the nuclear import rate constant of APC exceeds its export rate constant, the accumulation of APC in the nucleus can cause a breakdown of βcatenin retention by APC in the cytoplasm. As a consequence, free β-catenin accumulates in the cytoplasm and thus in the nucleus, which leads to a maximization of the transcription factor complex [β-cat/TCF] in contrast to a reference model without APC shuttling. This effect is also visible if the nuclear concentration of [β-cat/APC] is high due to enhanced binding affinity between the two proteins in the nucleus. Thus our findings show that Wnt signalling can benefit from nucleo-cytoplasmic shuttling of APC.

Figure 3 -

 3 Figure 3 -Influence of nucleo-cytoplasmic shuttling of APC on [β-cat/TCF] *

Figure 3a :

 3a Figure 3a: The steady state concentration of [β-cat/TCF] * crucially depends not only on β-catenin, but also on APC shuttling. d) K 4 = 30 nM: The black curve is equal to the green curve in Figure 3a. The steady state concentration [β-cat/TCF] * only decreases in comparison to the reference curve. e) K 3 = K 4 = 30 nM: For K 1 < 1 the effect of nuclear retention is dominant, for K 1 > 1 cytoplasmic retention is mainly determining the steady state concentration of [β-cat/TCF] * . f) Facilitated export of APC, K 3 = 30 nM and K 1 = 1: The steady state concentration [β-cat/TCF] * is presented in dependence on the ratios of the shuttling rate constant of APC (K lin = v max /k 2 k m ). Shuttling parameters are generated randomly. Every dot corresponds to the solution for a specific parametrization.

Figure 4 -

 4 Figure 4 -Influence of diffusive shuttling on the protein and complex concentration of β-catenin and APC in the cytoplasm (top row) and nucleus (bottom row) for K 3 = 30 nM.

Figure 1 Figure 2 K

 12 Figure 1

Figure 4

 4 Figure 3

  concentration. The experimental findings obtained by Krieghoff et al.[START_REF] Krieghoff | Nucleo-cytoplasmic distribution of β-catenin is regulated by retention[END_REF] provide the structural basis of our model: β-catenin and APC shuttle independent of each other; the complexes are however unable to cross the nuclear envelope. Our key questions of interest are: Does the nucleo-cytoplasmic shuttling of

APC influence the [β-catenin/TCF] concentration, which we consider as the output of Wnt signalling? Can the [β-catenin/TCF] concentration be maximized by APC shuttling? How is the output of the pathway influenced by cytoplasmic and nuclear retention of β-catenin?

  For K 1 values smaller than K 1 , the steady state concentration of [β-cat/TCF] * is higher, if the shuttling constant of APC is larger, but it cannot exceed the concentration obtained with the reference model. For small values of K 1 , most of the β-catenin proteins are located in the nucleus. In this case, a higher concentration of [β-cat/TCF] * is obtained, if APC remains in the cytoplasm, because cytoplasmic APC and nuclear TCF cannot compete directly for β-catenin binding.

However, for K 1 > K 1 we see that the concentration of [β-cat/TCF] * is larger than the reference concentration of [β-cat/TCF] * (Figure

  name Default value Description

	K 3	1200 nM	binding of β-catenin to APC (cytoplasm)
	K 4	1200 nM	binding of β-catenin to APC (nucleus)
	K 5	30 nM	binding of β-catenin to TCF (nucleus)
	A	100 nM	Total concentration of APC
	B	35 nM	Total concentration of β-catenin
	T	15 nM	Total concentration of TCF

Table 2 -Definition of the eight state variables of the system

 2 Squared brackets denote protein complexes. The indices n and c denote nuclear and cytoplasmic protein concentrations, respectively. In the text, variables in steady state are annotated by an asterisk, i.e.

	State variable	Original variable	Interpretation
	as used in Methods as used in Results and Figure 2
	X 1	β-cat c	free cytoplasmic β-catenin
	X 2	APC c	free cytoplasmic APC
	X 3	β-cat n	free nuclear β-catenin
	X 4	APC n	free nuclear APC
	X 5	[β-cat/APC] c	cytoplasmic retention complex [β-catenin/APC]
	X 6	[β-cat/APC] n	nuclear retention complex [β-catenin/APC]
	X 7	TCF	free nuclear transcription molecules
	X 8	[β-cat/TCF]	transcription complex [β-catenin/TCF]

* .

Acknowledgements

We thank Felix Winter for help and fruitful discussion regarding numerical difficulties and Simone Frey for her careful reading of the manuscript. We furthermore acknowledge helpful input from Mukhtar Ullah and Benjamin Bader. Y.S. is supported by the DFG Graduiertenkolleg 1387 -dIEM oSiRiS (Die Integrative Entwicklung von MOdellierungs-und SImulationsmethoden für Regenerative Systeme). K.R. is supported by the Bundesministerium für Bildung und Forschung through the FORSYS partner programme (grant number 0315255). Y.S and O.W. acknowledge support from the Helmholtz Society as part of the MDC systems biology network (MSBN: Systems Biology of cardiovascular and neurodegenerative disease processes).

We investigated parameter space by generating parameter values for the shuttling parameters, i.e. K 1,2 ∈ [10 -4 , 10 4 ]. As discussed in the previous case without APC shuttling, we again obtain only one positive, and therefore biological meaningful solution.

Facilitated nuclear export of APC

In contrast to the previous cases, the process of the nuclear export of APC across the NPC is now considered as facilitated transport [START_REF] Xu | Nucleocytoplasmic shuttling of signal transducers[END_REF]. It is reasonable to assume that the concentration of APC is in excess of the NPC.

Therefore, the Michaelis-Menten approach based on the quasi-steady state assumption can be applied [START_REF] Millat | Approximations and their Consequences in Dynamic Modeling of Signal Transduction Pathways[END_REF].

In case of facilitated export of APC, the search of the steady states leads to the following dependencies:

Substituting X * 1 and X * 2 in Eqs.( 14) and [START_REF] Hart | Downregulation of β catenin by human Axin and its association with the APC tumor suppressor, β-catenin and GSKβ[END_REF], we obtain two algebraic equations, which depend on X * 3 and X * 4 . Considering Eq.( 15) with its substitutes, we see the following equation in which X * 3 (i.e. nuclear β-catenin) is still depending on X * 4 :

where

and

Substituting these equations into the conservation equation ( 14), we end up with a polynomial of 6th order with respect to X * 4 , which we omit to write down. However, we may obtain up to six different roots and therefore steady state solutions. We investigated parameter space by generating randomized parameter values for the shuttling parameters, i.e.