D Muraro

A Larrieu

M Lucas

J Chopard

H Byrne

C Godin

J King

A multi-scale model of the interplay between cell signalling and hormone transport in specifying the root meristem of Arabidopsis thaliana

Keywords: gene networks, plant hormones, tissue differentiation

The growth of the root of Arabidopsis thaliana is sustained by the meristem, a region of cell proliferation and differentiation which is located in the root apex and generates cells which move shootwards, expanding rapidly to cause root growth. The balance between cell division and differentiation is maintained via a signalling network, primarily coordinated by the hormones auxin, cytokinin

and gibberellin. Since these hormones interact at different levels of spatial organisation, we develop a multi-scale computational model which enables us to study the interplay between these signalling networks and cell-cell communication during the specification of the root meristem. We investigate the responses of our model to hormonal perturbations, validating the results of our simulations against experimental data. Our simulations suggest that one or more additional components are needed to explain the observed expression patterns of a regulator of cytokinin signalling, ARR1, in roots not producing gibberellin.

By searching for novel network components, we identify two mutant lines that affect significantly both root length and meristem size, one of which also differ-

Introduction

Arabidopsis thaliana is a small annual weed that has been widely used in plant research as a model organism. Several features make this particular plant amenable to genetic and molecular studies. These include its small size (about 20 cm), relatively short generation time (8-10 weeks), high fecundity and small, compact genome [START_REF] Page | The art and design of genetic screens: Arabidopsis thaliana[END_REF]. The growth of the Arabidopsis root is sustained by the root apical meristem. This is a region of dividing cells near the tip of the root. Within this are four undifferentiated cells known as the quiescent centre (QC), which has properties analogous to a stem cell niche in animal systems (i.e. slowly dividing, pluripotency). The QC is surrounded by differentiated cells that divide several times and form the rest of the meristem. After a number of divisions, cells have moved shootwards and enter the transition zone where they stop dividing and start elongating in the longitudinal direction. In order to maintain root stability, the specification of these tissues must be maintained during growth. The plant hormone families auxin, cytokinin and gibberellin play key roles in controlling this developmental programme [START_REF] Moubayidin | Cytokinin-auxin crosstalk[END_REF].

Upon seed germination, and at early stages of root development, increased cell proliferation causes the expansion of the root meristem until approximately 5 days post germination (5 dpg) [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]. This process is influenced by interactions between several different signalling pathways and is coordinated by the plant hormone families auxin, cytokinin and gibberellin [START_REF] Moubayidin | Cytokinin-auxin crosstalk[END_REF], [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]. Auxin (IAA) plays a central role in root development, regulating the positioning and formation of the meristem, and stimulating mitotic activity [START_REF] Blilou | The PIN auxin efflux facilitator controls growth and patterning in Arabidopsis roots[END_REF], [START_REF] Ioio | Cytokinins determine Arabidopsis rootmeristem size by controlling cell differentiation[END_REF].

Auxin is transported from the shoot to the root apex through cell membranes by the combined action of influx and efflux transporters, named AUX/LAX and PIN proteins respectively. It then accumulates in the QC and columella cells whence it is redistributed shootwards (via the root cap and the epidermal cells), where it promotes cell proliferation [START_REF] Kramer | Auxin transport: a field in flux[END_REF], [START_REF] Perrot-Rechenmann | Cellular responses to auxin: division versus expansion[END_REF].

Cytokinin (CK) is antagonistic to auxin and regulates the activity of the meristem by negatively modulating auxin transport [START_REF] Ruzicka | Cytokinin regulates root meristem activity via modulation of the polar auxin transport[END_REF], [START_REF] Marhavý | Cytokinin modulates endocytic trafficking of PIN1 auxin efflux carrier to control plant organogenesis[END_REF]. Exogenous application of cytokinin, or over-expression of the IPT5 (Isopentenyltransferase 5) gene, which is responsible for cytokinin biosynthesis, reduces mitotic activity by promoting cell differentiation [START_REF] Ioio | A genetic framework for the control of cell division and differentiation in the root meristem[END_REF], [START_REF] Chavarria-Krauser | A cellular growth model for root tips[END_REF].

Gibberellin (GA) is involved in several developmental processes in plants, including stem and leaf growth, seed germination and flowering time [START_REF] Sun | Molecular mechanism of gibberellin signaling in plants[END_REF]. In roots, it is active primarily during the early stages of development, when it sustains auxin transport and promotes cell proliferation [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF].

While cross-talk among these three hormone families is still poorly understood, a network scheme that integrates their signalling pathways has been proposed [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]. In this scheme, gibberellin levels act as a switch: high gibberellin levels antagonise cell differentiation during early growth (3 dpg), whereas at about 5 dpg, a decrease in gibberellin levels in the adult root reduces cell proliferation in the meristem.

Several mathematical models have been proposed to investigate the formation of plant hormone gradients during root development in a multi-cellular framework [START_REF] Grieneisen | Auxin transport is sufficient to generate a maximum and gradient guiding root growth[END_REF], [START_REF] Band | Root gravitropism is regulated by a transient lateral auxin gradient controlled by a tipping-point mechanism[END_REF], [START_REF] Muraro | The role of auxin and cytokinin signalling in specifying the root architecture of Arabidopsis thaliana[END_REF], [START_REF] Band | Systems analysis of auxin transport in the Arabidopsis root apex[END_REF], [START_REF] Cruz-Ramírez | A bistable circuit involving SCARECROW-RETINOBLASTOMA integrates cues to inform asymmetric stem cell division[END_REF] and their hormonal cross-talk at the singlecell level [START_REF] Liu | Modelling and experimental analysis of hormonal crosstalk in Arabidopsis[END_REF], [START_REF] Muraro | The influence of cytokinin-auxin cross-regulation on cell-fate determination in Arabidopsis thaliana root development[END_REF]. However, the influence of the integrated cross-talk pathway between auxin, cytokinin and gibberellin on meristem development has not been modelled in root geometries derived from a segmented root image. The situation is complicated by hormone fluxes between cells that influence, and are influenced by, the dynamics of the signalling network in each cell. As described above, auxin flux is the result of a multi-scale process which includes cross-talk between cytokinin and gibberellin. Moreover, a heterogeneous spatial distribution of auxin concentration has different effect on the biosynthesis of the other hormones leading, as a result, to different responses in their signalling networks. Inclusion of these interactions in a multi-scale model permits analysis of the ways in which auxin flux may vary when other root hormones are perturbed and investigation of the time evolution of the network response in different root tissues. Finally, the results of our simulations can be qualitatively validated using confocal or reporter-based images. In this paper, we develop a multi-scale model of the cross-talk between auxin, cytokinin and gibberellin and use it to study their roles in the development of the meristem of Arabidopsis roots. We validate our simulations against confocal and reporter-based images of untreated roots. Additional model simulations of hormonal treatments reveal that one or more additional regulators are needed to inhibit the expression of the protein ARR1 in the meristem. By selecting, from a gene expression map, genes whose expression profiles are more highly expressed in this root region, we identify experimentally two mutants that affect meristem size. RT-qPCR data generated for these mutants show that one of them also affects the expression of SHY2, a key component of the signalling network, extending the network of known interactions.

Biological and modelling background

Control of root meristem size and root growth between 3 dpg and 5 dpg is influenced by the cross-talk between cytokinin and auxin [START_REF] Moubayidin | Cytokinin-auxin crosstalk[END_REF], [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]; this occurs through signalling pathways converging on the SHY2 (SHORT HYPOCOTYL 2) gene, a member of the Aux/IAA gene family, which is part of the auxin response network [START_REF] Ioio | Cytokinins determine Arabidopsis rootmeristem size by controlling cell differentiation[END_REF] (see Figure 1). Moubayidin and co-workers showed that cell differentiation is promoted by high levels of the SHY2 protein [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]. SHY2 regulates meristem size by altering the balance between auxin and cytokinin signalling, the former acting as an inhibitor and the latter as a promoter of SHY2 levels. It achieves this effect by repressing some members of the PIN protein family, which are auxin efflux transporters, and (in so doing) slows down PIN-mediated auxin transport. SHY2 expression is promoted by cytokinin signalling as follows. Cytokinin binds reversibly to the AHK receptors, which auto-phosphorylate, forming the complex Ck:AHKph. Subsequently, phosphorylated AHKs phosphorylate AHPs inducing their translocation to the nucleus where they phosphorylate in turn a family of transcription factors, the B-type ARRs (mainly ARR1 and ARR12 in the root meristem). We model this process by assuming that the complex Ck:AHKph transfers its phosphoryl group to ARR1 and ARR12 [START_REF] Moubayidin | Cytokinin-auxin crosstalk[END_REF]. Both ARR1 and ARR12 promote SHY2 expression. ARR12 expression is not regulated by gibberellin, whereas ARR1 is down-regulated by gibberellin, which promotes degradation of a member of the DELLA family, namely RGA. The biosynthesis of auxin and gibberellin is also influenced by other hormones, as follows. Cytokinin promotes auxin biosynthesis [START_REF] Jones | Cytokinin regulation of auxin synthesis in Arabidopsis involves a homeostatic feedback loop regulated via auxin and cytokinin signal transduction[END_REF], auxin up-regulates the expression of genes involved in gibberellin metabolism [START_REF] Frigerio | Transcriptional regulation of gibberellin metabolism genes by auxin signaling in Arabidopsis[END_REF] while suppressing cytokinin biosynthesis [START_REF] Nordström | Auxin regulation of cytokinin biosynthesis in Arabidopsis thaliana: a factor of potential importance for auxin-cytokinin-regulated development[END_REF]. At early stages of growth (3 dpg), high levels of gibberellin selectively repress ARR1 and, as a result, cell division dominates cell differentiation. At 5 dpg, the meristem reaches maturity and a decrease in the rate of gibberellin biosynthesis relieves the repression of ARR1, leading to a decrease in the rate of cell proliferation [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF].

We have developed a multi-scale model, based on differential equations, to investigate the interactions among the signalling pathways associated with auxin, cytokinin and gibberellin in a longitudinal root section. The model embeds the signalling pathways described above in every cell of a multicellular geometry derived from a segmented root image. As we will show in the Results section, our model contains an unknown component which acts as a repressor of ARR1. In what follows we will denote this subcellular component X. A schematic diagram of this network and of the representative root tissue are presented in Figure 1.

The species and the reactions incorporated in our model include those proposed by Moubayidin et al. [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF], accounting for phosphorylation and for regulation of hormonal biosynthesis. The reactions that constitute our signalling network are listed in the Supplementary Material. From this set of reactions we derive a system of ordinary differential equations governing the dynamics of the interaction network. The computational tools used to simulate our model are described in the Materials and Methods section, whereas a detailed description of the model reactions and equations, together with the results of a parameter sensitivity analysis, is reported in the Supplementary Material.

Based on previous single-cell models [START_REF] Muraro | The role of auxin and cytokinin signalling in specifying the root architecture of Arabidopsis thaliana[END_REF], [START_REF] Muraro | The influence of cytokinin-auxin cross-regulation on cell-fate determination in Arabidopsis thaliana root development[END_REF], [START_REF] Rice | Reconstructing biological networks using conditional correlation analysis[END_REF] we model regulation of mRNA levels in a particular cell using Michaelis-Menten kinetics. Auxin transport is modelled using a modified version of the Flux based polarization model developed by Stoma et al. [START_REF] Stoma | Fluxbased transport enhancement as a plausible unifying mechanism for auxin transport in meristem development[END_REF] (see equations ( 1) and (2) below), where we specify, as part of the root template, the sides of the cell on which PIN proteins are present. The boundary conditions applied to the cells at the most shootwards end of the spatial domain are that auxin concentration is constant for the vasculature cells, and that there is zero efflux (in a shootwards direction) from the other cells. For simplicity, we follow [START_REF] Grieneisen | Auxin transport is sufficient to generate a maximum and gradient guiding root growth[END_REF] in ignoring the contribution of auxin influx transporters.

In Arabidopsis, the PIN family consists of eight members [START_REF] Zazímalová | Auxin transporters -Why so many? Cold Spring Harb[END_REF]. In our model, we group this family into a single, representative protein whose spatial distribution is consistent with PIN expression data from confocal images and the multi-cellular model proposed in [START_REF] Grieneisen | Auxin transport is sufficient to generate a maximum and gradient guiding root growth[END_REF]. Since the network controlling the size of the meristem is known to act on the PIN proteins located in the vascular tissues (PIN1, 3, 7), we assume that only the PINs in this tissue are transcriptionally repressed by SHY2, and we assign constant PIN transcription in the external layers. In particular, PIN1, 3, 7 are expressed in the vascular tissue and PIN3 is also expressed in the columella; we also account for SHY2 in the columella as it appears to be weakly expressed in this region both at 3 dpg and at 5 dpg in untreated roots and in roots grown 4 hr on 5 mM transzeatin (Zt) (see Fig- [START_REF] Scacchi | Spatio-temporal sequence of cross-regulatory events in root meristem growth[END_REF]). Within each cell, PIN proteins are distributed on those cell walls where the proteins are known to be present (see Supplementary Figure 1).

The concentrations of PIN proteins at these specified membrane domains vary depending on the flux of auxin through the cell walls. Between two neighbouring cells, the flux due to passive auxin diffusion follows a discrete form of Fick's First Law, whereas active auxin transport depends on the efficiency of PIN transporters [START_REF] Mitchison | The polar transport of auxin and vein pattern in plants[END_REF]. More precisely, denoting by [Aux] i the concentration of auxin in cell i, V i the cell size, N i the set of cells neighbouring cell i and S i,n the cell wall between cells i and n, we assume that the auxin concentration in cell i is governed by the equation

d[Aux] i dt = - 1 V i n∈Ni S i,n (J P i→n + J T i→n ) + p Aux F (i) Aux -d Aux [Aux] i , (1)
where p Aux , d Aux denote the production and degradation rate of auxin,

J P i→n = P Aux ([Aux] i -[Aux] n )
is the flux of auxin due to passive transport and P Aux represents the permeability of the cell wall;

J T i→n = T Aux ([Aux] i [PIN p ] i,n - [Aux] n [PIN p ] n,i
) is the flux of auxin due to active transport by PIN proteins of concentration [PIN p ] i,n on S i,n and T Aux is the rate of active flux;

F (i)
Aux is a functional form controlling auxin biosynthesis given by

F (i) Aux = b Aux + ([Ck] i /θ Ck ) 1 + ([Ck] i /θ Ck ) (2)
where b Aux is a parameter regulating basal auxin production in all tissues and θ Ck is the cytokinin binding threshold. Cytokinin and gibberellin fluxes are governed by equations similar to those for auxin, except that active transport is neglected and zero flux boundary conditions are imposed on the cells on the boundary of the computational domain.

Auxin controls cytokinin bio-synthesis [START_REF] Nordström | Auxin regulation of cytokinin biosynthesis in Arabidopsis thaliana: a factor of potential importance for auxin-cytokinin-regulated development[END_REF], interacting with IPT5 and IPT7 [START_REF] Miyawaki | Expression of cytokinin biosynthetic isopentenyltransferase genes in Arabidopsis: tissue specificity and regulation by auxin, cytokinin, and nitrate[END_REF], and most auxin-resistant mutants also show changes in their cytokinin sensitivity [START_REF] Leyser | Mutations in the AXR3 gene of Arabidopsis result in altered auxin response including ectopic expression from the SAUR-AC1 promoter[END_REF], [START_REF] Rashotte | Expression profiling of cytokinin action in Arabidopsis[END_REF]. The regions of biosynthesis of the cytokinin biosynthesis gene IPT5 (ATP/ADP-isopentenyltransferase 5) are localised in the vasculature at the elongation zone and in the root apex [START_REF] Moubayidin | Cytokinin-auxin crosstalk[END_REF], whereas AtIPT7::GFP fluorescence was observed in the vascular stele of the elongation area [START_REF] Takei | AtIPT3 is a key determinant of nitrate-dependent cytokinin biosynthesis in Arabidopsis[END_REF]. Transcriptomic and reporter analyses suggest that gibberellin biosynthesis occurs predominantly in the meristem [START_REF] Band | Growth-induced hormone dilution can explain the dynamics of plant root cell elongation[END_REF], [START_REF] Fu | Gibberellin-mediated proteasome-dependent degradation of the barley DELLA protein SLN1 repressor[END_REF]. We model this by assuming tissue specific self-activation of gibberellin biosynthesis (see Supplementary Material).

As in most models of signalling networks based on differential equations, the predicted qualitative outcome of the model is dependent on the choice of the parameter values (e.g. production and degradation rates, protein-DNA binding thresholds), many of which are unknown [START_REF] Liu | Modelling and experimental analysis of hormonal crosstalk in Arabidopsis[END_REF], [START_REF] Rausenberger | Photoconversion and nuclear trafficking cycles determine phytochrome A's response profile to far-red light[END_REF]. Following [START_REF] Liu | Modelling and experimental analysis of hormonal crosstalk in Arabidopsis[END_REF], [START_REF] Muraro | Integration of hormonal signaling networks and mobile microRNAs is required for vascular patterning in Arabidopsis roots[END_REF], we have based our estimates of the ratio between the rates of auxin transport and permeability on those used in previous models [START_REF] Grieneisen | Auxin transport is sufficient to generate a maximum and gradient guiding root growth[END_REF], [START_REF] Muraro | Integration of hormonal signaling networks and mobile microRNAs is required for vascular patterning in Arabidopsis roots[END_REF], [START_REF] Laskowski | Root system architecture from coupling cell shape to auxin transport[END_REF], whereas other values have been chosen manually to ensure agreement with patterns experimentally observed. In order to determine the degree to which parameter variation affects the outcome of our model of hormonal cross-talk we have performed a global parameter sensitivity analysis at a single-cell level in all root regions in which the parameters have different, tissue specific values. In this way, we identified those parameters to which the model is most sensitive, these being mainly associated with gibberellin, RGA and auxin regulation. Guided by the results of this single-cell analysis, we used the multi-cellular model to perform a multi-cellular local sensitivity analysis with respect to these parameters, together with those associated with transport and permeability. Since SHY2 is regulated negatively by auxin and positively by the cytokinin pathway, we also used the multi-cellular model to study sensitivity to parameters associated with such regulation. The parameters identified by the single-cell sensitivity analysis mainly affect gibberellin, RGA, ARR1, SHY2 and Ck:AHK; the parameters associated with transport and permeability mainly influence auxin, cytokinin, phosphorylated ARR12 and gibberellin; the parameters associated with SHY2 regulation mainly influence SHY2, auxin, cytokinin. We also investigated whether tissue-specific permeability rates (P Aux , P CK , P GA , see Supplementary Material) may affect the formation of the simulated protein patterns. To this end, we extended our sensitivity analysis by randomly varying the values of P Aux , P CK , P GA in different cell types up to two-fold in comparison to default values. These changes did not significantly affect the simulation results. The patterns obtained when varying these parameters, considering as default the parameter values used to simulate an untreated root (Supplementary Movie 5), are reported in the Supplementary File 'Permeability Sensitivity.pdf'.

A detailed description of this analysis is reported in the Supplementary Material. The simulated SHY2 pattern is more highly expressed in the vascular tissues above the meristem and in the columella than elsewhere, in agreement with SHY2::GUS reporter images (see Figures 2A,2C, Supplementary Movie 1 and Moubayidin et al. [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]). The inhibitory role of gibberellin means that RGA reaches higher concentration levels in the elongation zone; consequently, ARR1 mimics the higher expression of its activator RGA in the elongation zone (see Supplementary Movie 1). Our model enables us to predict the spatial distributions and dynamics of auxin, cytokinin and gibberellin, providing a visualisation of hormones whose concentrations cannot be easily measured experimentally but which affect other measurable network components. In particular, auxin is actively transported through the vasculature from the basal tissues to the root cap where it is re-directed towards the stele through the external root layers (see Supplementary Movie 1). These auxin dynamics are consistent with those generated from other multi-cellular models of auxin transport, such as [START_REF] Grieneisen | Auxin transport is sufficient to generate a maximum and gradient guiding root growth[END_REF], that do not include a subcellular interaction network. Guided by our simulations of the untreated root, we now apply hormonal perturbations to our computational model and compare the results with confocal and staining images obtained from experiments performed under similar conditions.

Results

Model predictions for roots grown on transzeatin

The number of cells in the root meristem shows an average decrease under treatment for 12 hours with 5µM of a member of the cytokinin family (transzeatin, Zt) [START_REF] Ioio | A genetic framework for the control of cell division and differentiation in the root meristem[END_REF]. We now investigate how an increase in the rate of biosynthesis of cytokinin may affect the patterns observed in untreated roots (see Figure 2 and Supplementary Movie 2a). Figure 2B reveals that SHY2 expression at 5 dpg is clearly visible in the vasculature of the elongation zone and is more strongly induced by the higher cytokinin levels. This induction reduces the size of the region in which PINs are expressed in cytokinin-treated roots, prompting greater cell differentiation (see Figures 2G-L) [START_REF] Ioio | A genetic framework for the control of cell division and differentiation in the root meristem[END_REF]. In our computational simulations, over-production of cytokinin in its biosynthesis regions leads to an increase in the phosphorylation of ARR1 and ARR12. The higher levels of active ARRs (ARR1ph, ARR12ph) promote SHY2 expression in the region of cytokinin biosynthesis (see Figure 2D and Supplementary Movie 2a). SHY2

represses PIN genes in the vasculature and causes down-regulation of PIN expression in the elongation zone (see Figure 2N and Supplementary Movie 2a).

We also use our model to predict how the flux of auxin may be affected by cytokinin over-expression and the subsequent degradation of PIN proteins in the vascular tissue. In our simulations, the greater the abundance of the PINs, the higher the flux of auxin; consequently, the degradation of PIN proteins, caused by cytokinin over-expression, hinders the transport of auxin, which tends to accumulate where PIN levels are low, namely in the elongation zone (see Figure 2F). Simulating the same treatment by varying where the basal boundary of the meristem is defined did not show qualitative differences in the expression pattern (see Supplementary Movie 2b). SHY2::GUS expression in root meristems at 5 dpg of untreated roots (A) and of roots grown on 5 µM transzeatin (Zt), (B). C-D) SHY2 protein expression in our multi-scale model in untreated roots (C) and in roots with higher cytokinin biosynthesis rate (D); cytokinin activates SHY2 transcription via the phosphorylation of ARR1 and ARR12, enhancing SHY2 protein levels in the vasculature at the elongation zone. E-F) Simulated auxin flow in untreated roots (E) and in roots with higher cytokinin biosynthesis rate (F). Auxin flows from the apical vasculature to the root cap, part of auxin then being redirected towards the shoot via the epidermal cells. When cytokinin is over-expressed, the flux is hindered by the degradation of PIN proteins and auxin tends to accumulate into regions of low PIN activity, namely the elongation zone. G-L) PIN1,3,7:GFP expression in root meristems of untreated roots at 5 dpg (G,I,K) and in cytokinin treated roots (5 µM Zt), (H,J,L). M-N) PIN protein expression in our multi-scale model in untreated roots (M) and in roots with higher cytokinin biosynthesis rate (N); the higher activity of SHY2 down-regulates PIN expression in the vascular tissue. A detailed description of the equations and parameters used in our simulations is reported in the Supplementary Material. In Figures A-B) and G-L) blue and white arrowheads indicate the quiescent centre and the beginning of the cortex elongation zone, respectively; these figures are reproduced with permission from [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF], [START_REF] Ioio | A genetic framework for the control of cell division and differentiation in the root meristem[END_REF]. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article).

The joint action of auxin and gibberellin reproduces the observed distribution of RGA

Since RGA levels are low where the concentration of gibberellin is high, the RGA:GFP reporter can be used to estimate the levels and spatial distribution of gibberellin. When Moubayidin et al. [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF] analysed confocal images to deter-245 mine the average pixel intensity of the RGA:GFP reporter at 3 and 5 dpg they found RGA:GFP fluorescence to be significantly higher at 5 dpg than at 3 dpg, but not uniform in all root tissues, being concentrated in the elongation zone and the cortex (see Supplementary Figure S2I [3]). RGA expression depends on gibberellin biosynthesis which, in turn, is promoted by auxin [START_REF] Frigerio | Transcriptional regulation of gibberellin metabolism genes by auxin signaling in Arabidopsis[END_REF]. In our model, most of the auxin flux is through the vascular tissue, so auxin accumulates in the columella and in the QC. Subsequently, some auxin flows back via the epidermis. We assume that the combination of the self-and auxin-mediated activation of gibberellin determines the spatial distribution of RGA. At 3 dpg gibberellin is present at higher concentrations [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]. For this reason, we suppose that the network response of the untreated root at 3 dpg is analogous to the network response of a root over-producing gibberellins and we compare simulations having gibberellin biosynthesis higher than that in an untreated root with confocal and staining images of roots at 3 dpg. The simulation results are shown in Figures 3C,3D and in Supplementary Movie 3a. At 3 dpg, the concentration of RGA is uniformly low because of the over-production of gibberellin (see Figures 3A,3C and Supplementary Movie 3a), whereas at 5 dpg the heterogeneous distribution of gibberellin in the root tissues causes the concentration of RGA to peak in the cortex (see Figures 3B,3D and Supplementary Movie 1).

Simulating gibberellin over-expression by varying where the basal boundary of the meristem is defined did not lead to qualitative differences in the expression pattern (see Supplementary Movies 3a, 3b).

Roots grown on gibberellin and PAC

ARR1:GUS expression in untreated roots is localised in the elongation zone (see Figure 3E). Its expression is not detectable when the root is grown on gibberellin (see Figure 3F), whereas it is enhanced when gibberellin levels are reduced using a pharmacological approach (PAC treatment; see Figure 3G).

While our simulations under gibberellin treatment (Supplementary Movies 3a, 3b) yield lower levels of ARR1 expression than those of untreated roots (Supplementary Movie 1), and these lower levels could not be detectable with the ARR1:GUS reporter (as shown in Figure 3F), regulation via RGA alone cannot explain the spatial distribution of ARR1 in PAC treated roots: the depletion of gibberellin biosynthesis would cause both RGA and ARR1 to be distributed approximately uniformly throughout the root (see Supplementary Movie 4), at higher levels than in an untreated root (see Supplementary Movie 1). This result is incompatible with the staining image of ARR1 for roots grown on PAC, which shows that its expression is lower in the meristem than in the elongation zone. In order to reproduce the observed expression pattern, we assume the existence of an extra regulator (or regulators) whose (combined) effect is to repress ARR1 in the meristem, via either direct control of its transcription or down-regulation of RGA in the meristem. When this regulator (denoted by the letter X in Figure 1 and equations ( 1)-( 4) in the Supplementary Material) is included, our model reproduces the experimentally observed patterns of ARR1 under all treatments without influencing substantially the patterns of the other network components (see Figures 3H,3I, 3J and Supplementary Movies 5-8).

Similar simulation results are obtained if we assume direct repression of ARR1 by X or indirect repression via RGA (see Supplementary Movie 9). Independently of the inclusion of X, our model also predicts lower levels of ARR1 in the vascular tissue in the simulation of the untreated root at 5 dpg (see Figure 3H). These lower levels of ARR1 in the vasculature are due to RGAmediated activation of ARR1 and may depend either on another regulatory mechanism not included in the model or on technical limitations of staining images that show ARR1 expression in external tissues.

Novel components regulate root and meristem size

Based on our simulation results, we selected 77 genes from a gene expression map [START_REF] Birnbaum | A Gene Expression Map of the Arabidopsis Root[END_REF] whose expression profiles were most anti-correlated with ARR1 and, as a consequence, were more highly expressed in the meristem. We then identified at least one homozygous T-DNA insertion line within the coding sequence of 34 of these 77 genes [START_REF] Scholl | Seed and molecular resources for Arabidopsis[END_REF]. Comparison of the root lengths of the mutants with those of the WT led to the identification of 12 genes with a significantly different phenotype. Lines whose phenotype was confirmed in two subsequent generations were further selected.

This last selection led to the identification of three lines, which have mutations in the following three genes: an ELMO/CED-12 family protein (AT3G60260, that we term aag1-1 for ARR1 anticorrelated gene 1), the CID3 ctc-interacting domain 3 protein (AT1G54170, aag2-1 ) and a transcription factor (AT1G68920, aag3-1 ). aag1-1 and aag3-1 mutants developed longer roots, whereas aag2-1 developed shorter ones (see Figure 4). Consistently, meristem length was shorter for aag2-1 and longer for aag3-1 (results were not significant for aag1-1). In order to test whether the expression of several regulators of meristem length was affected in the mutants, their expression in roots tips was analysed using RT-qPCR. In aag1-1 mutants several network components were down-regulated (ARR1, ARR10 and SHY2). In aag2-1 mutants only SHY2 was affected while no network components showed differential expression in aag3-1 mutants compared to the WT. These results suggest that compensatory mechanisms not included in the pathways analysed are also responsible for the absence of effects of other mutants on ARR1 expression. Since our selection of mutants was not able to identify a gene playing the role of regulator X, we have analysed two further model setups. We investigated whether PAC treatment could specifically increase RGA levels in the elongation zone; however, simulation results showed that ARR1 is still expressed in the meristem, although at lower levels than in the elongation zone (see Supplementary Movie 10). We also instead assumed that ARR1 transcription is restricted to the elongation zone because RGA activation requires an X transcription factor which is expressed only in this area; under this hypothesis our simulation shows that ARR1 is nearly absent in the meristem (see Supplementary Movie 11). Taken together, these results suggest lines of further research. Single asterisks indicate statistical significance with p-value < 0.1, whereas the double asterisk indicates a p-value < 0.05.

Discussion

The organisation of the root meristem in Arabidopsis is established in the first five days following germination. During this time period, hormonal crosstalk regulates the balance between cell proliferation, differentiation and elongation. The regulation of these patterning events in the root apex involves complex interactions between three classes of signalling molecules, namely auxin, cytokinin and gibberellin [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]. Such interactions involve cross-talk at the subcellular and tissue scales, the former being mediated via a signalling network and the latter via auxin transport in different tissues which then influences levels of other network components.

When studying such systems, mathematical modelling can be used to perform in silico experiments designed to determine whether the current state of knowledge of the system is sufficient to explain the experimental data. It also permits visualisation of how the dynamics of hormonal signalling may change in response to specific perturbations, such visualisation not being easy to perform in vivo.

In this paper, we proposed a multi-scale computational model to investigate the specification of the root apical meristem in Arabidopsis thaliana. Although mathematical models have previously been used to analyse auxin transport in multi-cellular frameworks [START_REF] Grieneisen | Auxin transport is sufficient to generate a maximum and gradient guiding root growth[END_REF], [START_REF] Stoma | Fluxbased transport enhancement as a plausible unifying mechanism for auxin transport in meristem development[END_REF], to our knowledge the cross-talk between auxin, cytokinin and gibberellin has not previously been modelled in a multicellular geometry. Our model accounts for two different levels of regulation:

active and passive (diffusive) transport of hormones between cells and molecular interactions at the subcellular scale. Our model can reproduce the gene expression patterns observed experimentally in untreated roots and in those grown on transzeatin. The model simulations suggest that the signalling network proposed in [START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF] may not be sufficient to explain the spatial distributions of all of its components in roots grown on PAC. In particular, a regulator of cytokinin signalling (ARR1) has a specific expression pattern that we could not reproduce when performing model simulations. To account for this difference, we introduced an additional inhibitor of ARR1 in our model and showed that it yields expression patterns similar to those observed in vivo.

In order to provide biological support for this model prediction we looked for candidate genes that could fulfil this function. Using published microarray datasets, we identified those genes whose expression profiles are most strongly anti-correlated with ARR1 in the root apical meristem. Of these mutants 3 lines (aag1-1, aag2-1, aag3-1 ) had root sizes that were either longer (aag1-1 and aag3-1 ) or shorter (aag2-1 ) than their WT counterparts. RT-qPCR data generated for these lines showed that aag1-1 and aag2-1 mutants express lower levels of SHY2. The aag1-1 mutant also has a reduced expression of ARR1 and ARR10, suggesting an indirect regulation of SHY2 expression through ARR1. None of the mutated genes acts as a repressor of ARR1 in the manner required for the unknown component X, suggesting either that this component is not part of the selected genes or that its role might be played by the combined action of distinct genes. Moreover, the effects of such genes on the phenotype suggests that other signalling pathways may affect the definition of the meristematic region. In addition, the influence of two of these mutations (aag1-1 and aag2-1 ) on the expression levels of other components that control meristem size (SHY2 and ARR1) suggests that these genes may act either directly, or indirectly via ARR1, as activators of SHY2, thereby extending the network of known interactions.

We conclude that computational modelling of the type performed here can serve as a valuable tool for predicting how specific perturbations affect the spatio-temporal dynamics of the Arabidopsis root, yielding results that may not easily be visualised in the laboratory, and for determining when additional components are needed to interpret experimental data. Since the hormone concentration in each cell depends both on regulation of active transport by PIN proteins and single-cellular hormonal cross-talk, analysis of such dynamics requires a multi-scale approach. By selecting a biological hypothesis and interpreting our simulation results, we identified candidate genes that may play an important role in determining root and meristem size. Our findings also highlight the potential benefits of interdisciplinary research. It provides a coherent theoretical framework for testing whether the current understanding of how a system functions is compatible with observed behaviours and for identifying where additional experimental investigation is needed. Our analysis may stimulate further investigation, from the experimental side, to analyse in more detail the spatio-temporal influence of the identified mutants on the other network components and, from the theoretical side, to extend the modelling framework to include further components of cell signalling and cell-to-cell communication, and into more realistic three-dimensional geometries.

Materials and Methods

Computational tools

The tissue representation is extracted from the OpenAlea framework [START_REF] Pradal | Ope-nAlea: A visual programming and component-based software platform for plant modeling[END_REF].

OpenAlea is an open source software project for Plant Architecture modelling.

Libraries and tools are primarily based on the high level, object-oriented script language, Python. Cell wall geometry and PIN distribution are manually reproduced from the experimental data of a root tissue using the open source software Inkscape [START_REF] Hiitola | Inkscape 0.48 Essentials for Web Designers book[END_REF]. A section of the root geometry is represented in Supplementary Figure 1. A detailed description of the reactions included in our model, the equations and associated parameters is presented in the Supplementary Material, together with the parameter sensitivity analysis.

Mutant lines

Homozygous SALK lines for the ARR1 anti correlated genes were ordered from the Nottingham Arabidopsis Stock Centre (NASC). The mutant lines with the original stock numbers are reported in the Supplementary Workbook.

Seed sterilisation, seedling growth and root length measurements

Seeds were surface sterilized for 5 minutes in 50% bleach, 0.1% triton X-100, and then washed three times with sterile ddH2O. Seeds were stratified at

ures 1 a

 1 ,b,g,h in Moubayidin et al.[START_REF] Moubayidin | The rate of cell differentiation controls the Arabidopsis root meristem growth phase[END_REF]). Low levels of SHY2::GUS were also detected in the columella by Scacchi et al. (Figure2A

Figure 1 :

 1 Figure 1: Mathematical model. Schematic diagram of our computational model. A signalling network comprising auxin, cytokinin and gibberellin (A) is embedded within each cell of a multi-scale model of a root tissue (B). Green arrows indicate activations, red arrows inhibitions, reactions involved in phosphorylation and the formation of the complex Ck:AHK are represented with black arrows. Different colours in root cells represent different tissues; root cap tissues are represented in red, the vascular tissue is represented in green, whereas outer root layers are represented in blue. A detailed tissue schematic is contained in the Supplementary Material. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article).

190 3 . 1 .

 31 Model validation As a first step, we compared protein expression patterns obtained by simulating our model with confocal and GUS-stained images of transgenic reporter lines of untreated roots. As shown in Figure 2 and in Supplementary Movie 1, our model can reproduce the patterns observed experimentally in the untreated 195 root for SHY2, ARR1, RGA and for PIN proteins expressed in the vasculature (PIN1::PIN1:GFP reporter).

Figure 2 (

 2 Figure 2 (previous page): Untreated roots and roots grown on transzeatin. A-B)

Figure S2 .Figure 3 :

 S23 Figure S2. RGA Activity Increases at Late Stages of Root Meristem Development and Controls Its Growth (Related to Figure 2) (A) ARR12:GUS expression in root meristems at 3 dpg. (B) ARR12:GUS expression in root meristems at 3 dpg grown 16 hr on 10 μM gibberellin (GA). (C) ARR12:GUS expression at 5 dpg. (D) ARR12:GUS expression in root meristems at 5 dpg grown 16 hr on 10 μM GA. (E) qRT-PCR of ARR1 in 3 dpg and 5 dpg wild-type roots. (F) qRT-PCR of ARR12 in 3dpg and 5 dpg wild-type roots.RGA: GFP (5dpg)

Figure 4 :

 4 Figure 4: Novel components regulating root and meristem size. A) Root length of mutants relative to untreated roots. aag1-1 and aag3-1 mutants developed longer roots, whereas aag2-1 developed shorter roots compared with untreated roots; however, only difference in root length in aag2-1, aag3-1 is statistically significant (p-values < 10 -4 ). B) Meristem size of mutants relative to untreated roots. aag2-1 mutants developed shorter meristems (p-value < 0.05), whereas aag3-1 developed longer meristems compared with untreated roots (p-value ≈ 0.07). C) Relative expression levels of several genes involved in meristem size in aag1-1, aag2-1 and aag3-1 mutants compared with untreated roots. ARR1, ARR10 and SHY2 are expressed to lower levels in aag1-1 mutants (p-values < 0.03), whereas only SHY2 expression is affected in aag2-1 (p-value < 0.05). None of these genes are significantly affected in aag3-1.

4 0 for 2

 2 days to synchronise germination. Root length was determined using 20 images of WT and mutant seedlings taken 7 days post germination with a Canon PowerShot G10 camera and analysed using Fiji (Fiji Is Just ImageJ) (http://fiji.sc/Fiji) (ImageJ version 1.41b) and Microsoft Excel 2007 (Microsoft Corporation, Redmond, USA).5.4. RNA extraction and RT-qPCRRNA was extracted from plant tissues using Trizol Reagent (Invitrogen) and cleaned up using the RNeasy kit (Qiagen). Poly(dT) cDNA was prepared from 0.5µg total RNA with Superscript II reverse transcriptase (Invitrogen) and analyzed on a LightCycler 480 apparatus (Roche Diagnostics) with the Quantace SYBRGREEN mix (Quantace) according to the manufacturers' instructions.Targets were quantified with specific primer pairs designed with the Universal Probe Library Assay Design Center (Roche Applied Science). All individual reactions were done in quadruplicate and data were statistically analyzed with Microsoft Excel 2007 (Microsoft Corporation, Redmond, USA). Expression levels were normalized to At1G04850. The primer sequences that were used for the genotyping and RT-qPCR are reported in the Supplementary Workbook.

Acknowledgments

We gratefully acknowledge the Biotechnology and Biological Research Council and the Engineering and Sciences Research Council for financial support as part of the CISB programme award to CPIB. J.R. King gratefully acknowledges the funding of the Royal Society and Wolfson Foundation. The authors thank S. Úbeda-Tomás, L. Band, M. Bennett and J. Fozard for their support and feedback in the development this work.


Michael Wessely
email: michael.wessely@inria.fr

Theophanis Tsandilas
email: theophanis.tsandilas@inria.fr

Wendy E Mackay
email: wendy.mackay@inria.fr

Stretchis: Fabricating Highly Stretchable User Interfaces

Keywords: Stretchable interfaces, personal fabrication, sensing technologies, custom-shaped displays, wearables. ACM Classification H.5.2. Information Interfaces and Presentation

Figure 1. Stretchis are highly stretchable user interfaces that include touch and proximity sensors and electroluminescent displays (a). Stretchis are transparent (b); can be stretched to fit to the geometry of different physical objects (c); and can act as on-skin user interfaces (d).

INTRODUCTION

Early user interfaces separated input hardware (keyboard, mouse, stylus) from output displays. Later research moved beyond dedicated input devices, detecting touch on everyday objects and the human body [START_REF] Sato | Touché: Enhancing touch interaction on humans, screens, liquids, and everyday objects[END_REF]. Today's smartphones and tablets combine input and output, allowing users to interact directly via multi-touch displays. Unfortunately, these technologies are rigid, expensive and complex to manufacture.

What if we could create inexpensive, lightweight, interactive surfaces that can be embedded in or attached to nearly any physical object? We are particularly interested in ultrathin, stretchable user interfaces that can embed rich interaction onto a wide variety of objects. To this end, we need ultra-flexible and stretchable substrate materials that can adapt to complex object geometries, doubly curved and shape-changing surfaces, and fabrics. We also need highly deformable sensors and displays that remain functional even when the underlying substrates are under strain.

Recent research on printed electronics has made considerable progress in this direction, but has not yet reached a complete solution. For example, iSkin [START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF] sensors allow limited stretching, up to 30%, but do not provide visual output. Other research [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] demonstrates how to print flexible electroluminescent displays, but the fabrication approach does not support stretchable substrates. This paper offers a set of key innovations toward our goal of creating inexpensive, ultra-thin, stretchable user interfaces. We introduce Stretchis: stretchable, fully customizable interfaces for physical objects and fabrics. As in previous research [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF][START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF], we use silicon-based organic polymers (PDMS) as the base material for embedding sensors. However, we introduce a different fabrication method that supports both sensor input (touch and proximity) and visual output. In particular, we present a novel multilayer screenprinting method that extends the PrintScreen [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] approach.

Our method deals with the hydrophobicity of PDMS [START_REF] Ginn | Polymer surface modification using microwave-oven-generated plasma[END_REF][START_REF] Haubert | Pdms bonding by means of a portable, low-cost corona system[END_REF], which allows printing of transparent water-based conductive inks on stretchable silicon-based substrates. Makers can create complex multilayer Stretchi components digitally by using common design tools such as GradSoft EAGLE and Adobe Illustrator. They can also customize the visual aesthetics by adding inks of arbitrary colors and shapes, without interfering with the interactive components.

Stretchis can be very thin (≈ 200 µm) and can be produced in large formats and at low cost. They are rollable, foldable, and stretchable. We show that Stretchi sensors and displays remain functional even when their length is doubled, stretching over 100%. We demonstrate how these features can add interactivity to diverse surfaces, including fabrics, shape-changing surfaces, and human skin.

In summary, our key contributions are:

1. We show how to embed sensor input and visual output in transparent and highly stretchable silicon-based substrates by using a novel, inexpensive fabrication method that even non-expert makers can use. Our method deals with the hydrophobicity of PDMS, allowing to print water-based conductive inks on silicon-based substrates.

2. We introduce a multi-layer fabrication approach for printing displays, capacitive sensors, and aesthetics such that sensors and displays are independent and do not interfere with each other. We demonstrate the usefulness of our approach by fabricating stretchable prototypes that cover ubiquitous, mobile, and on-skin interaction.

3. We evaluate the behavior and reliability of our stretchable sensors and displays under variable strain configurations and show that they can be stretched by more than 100%.

RELATED WORK

Our work combines results from materials science on stretchable electronics and research on methods for prototyping and fabricating flexible sensors and displays.

Stretchable Electronics

Research on fabricating elastic electronics includes electrically conductive fluids [START_REF] Kim | A multiaxial stretchable interconnect using liquid-alloy-filled elastomeric microchannels[END_REF] and gels that can be embedded in stretchable substrates [START_REF] Kaltenbrunner | Arrays of ultracompliant electrochemical dry gel cells for stretchable electronics[END_REF]. Other elastic circuit technologies are based on rigid conductors such as nanotubes [START_REF] Cao | Ultrathin films of single-walled carbon nanotubes for electronics and sensors: A review of fundamental and applied aspects[END_REF], graphene [START_REF] Kim | Large-scale pattern growth of graphene films for stretchable transparent electrodes[END_REF], and wavy circuit wiring [START_REF] Lacour | Stretchable gold conductors on elastomeric substrates[END_REF]. Other researchers [START_REF] Vosgueritchian | Highly conductive and transparent pedot:pss films with a fluorosurfactant for stretchable and flexible transparent electrodes[END_REF][START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF] produce transparent circuits by depositing poly (3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) on PDMS. Although most of this research focuses on new materials and methods for creating stretchable conductors, Lu et al. [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF] presented rapid techniques for fabricating a variety of laser-patterned conductive material on thin films of PDMS, including silver or carbon-filled PDMS (cPDMS), EGaln, and transparent PEDOT:PSS ink. Elastic electronics support a variety of applications, such as deformable input devices [START_REF] Bächer | Computational design of customized deformable input devices[END_REF] and stretchable keyboards [START_REF] Xu | Stretch not flex: programmable rubber keyboard[END_REF].

A few researchers have investigated stretchable OLEDs [START_REF] Sekitani | Stretchable active-matrix organic light-emitting diode display using printable elastic conductors[END_REF], electrochromic [START_REF] Yan | Stretchable and wearable electrochromic devices[END_REF], and electroluminescent [START_REF] Wang | Highly stretchable and self-deformable alternating current electroluminescent devices[END_REF] displays. Others [START_REF] Wang | Extremely stretchable electroluminescent devices with ionic conductors[END_REF][START_REF] Yang | Electroluminescence of giant stretchability[END_REF] have created transparent and highly stretchable (up to 1500%) electroluminescent devices based on ionic hydrogel conductors which exhibit impressive strain resistance. However, these are less conductive than electronic conductors, and it remains unclear how to adapt these fabrication methods for non-expert makers.

We are inspired by stretchable electronic conductor research by Lipomi et al. [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF], Lu et al. [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF], and Wang et al. [START_REF] Wang | Highly stretchable and self-deformable alternating current electroluminescent devices[END_REF]. However, we use a more flexible fabrication method for printing transparent PEDOT:PSS sensors on stretchable PDMS substrates in combination with electroluminescent displays. Our method does not require specialized treatments or laser engravers to dispose and pattern the conductive material, making it better adapted for quick prototyping by designers, non-expert practitioners, and Human-Computer Interaction (HCI) researchers.

Fabricating Flexible Sensors and Displays

HCI researchers are increasingly interested in flexible sensors and displays. For example, Savage et al. [START_REF] Savage | Fabricating custom capacitive touch sensors to prototype interactive objects[END_REF] created a toolkit to support the design, fabrication, and programming of flexible copper-based capacitive sensors. Kawara et al. [START_REF] Kawahara | Instant inkjet circuits: Lab-based inkjet printing to support rapid prototyping of ubicomp devices[END_REF] introduced a rapid-prototyping approach for printing conductive ink on paper with off-the-shelf inkjet printers. Gong et al. [START_REF] Gong | Printsense: A versatile sensing technique to support multimodal flexible surface interaction[END_REF] extended this approach to allow for a larger range of sensing capabilities, including proximity, pressure, and folding. More recently, Dementyev et al. [START_REF] Dementyev | Sensortape: Modular and programmable 3d-aware dense sensor network on a tape[END_REF] demonstrated a flexible and cuttable sensing tape that carries a dense array of sensors and other electronic components.

The above approaches support use of flexible, e.g., foldable and bendable, yet non-stretchable substrates. To this end, Sugiura et al. [START_REF] Sugiura | A thin stretchable interface for tangential force measurement[END_REF] produced a thin elastic interface made of stocking fabric that can sense deformations caused by pressing or stretching the fabric.

Inspired by Lu et al. [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF], Weigel et al. [START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF] introduced a fabrication method for integrating customizable carbon-based press and touch sensors into stretchable PDMS substrates. They can laser-engrave custom patterns of conductive material into PDMS and showed that their sensors are still usable even if they are stretched by 30%. The key limitation is the reliance on back carbon-based conductive ink, which is incompatible with printable displays and limits the designers' freedom to customize the interface.

Previous HCI work on flexible displays has focused primarily on integrated devices [START_REF] Lahey | Paperphone: Understanding the use of bend gestures in mobile devices with flexible electronic paper displays[END_REF] rather than on solutions for fabricating custom-made displays. The PrintScreen framework [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] was the first to make fabrication of interactive electroluminescent displays accessible to non-expert makers. Their approach is applicable to a variety of flexible substrates, including paper, leather, and ceramics. Our fabrication method extends this work to include new applications with highly stretchable transparent substrates, allowing greater flexibility and freedom when designing and instrumenting interactive sensors.

STRETCHIS: REQUIREMENTS AND CHALLENGES

The key requirements and challenges for fabricating Stretchis include:

Form Factor. Stretchis must resist strain and remain interactive when adapted to a wide variety of objects. For example, makers may want to add interaction to curved surfaces, such as a door handle; deformable objects, such as a lamp's flexible neck; elastic joints, such as a mobile phone cover; as well as their own clothing or bodies.

Substrate Properties. The base material of a Stretchi must be able to host sensors and displays but also expose its physical properties and affordances, such as its thickness, flexibility, and stretching behavior. PDMS is inexpensive, malleable and especially suitable for fabricating flexible and transparent substrates for printed electronics [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF][START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF]. Given these properties, we decided to use PDMS as the base material for Stretchis.

Input and Output Capabilities. Stretchis must accommodate both sensors and displays in the same stretchable substrate. Olberding et al. [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] use layers of transparent conductive ink to illuminate a layer of phosphor and to sense touch. However, their fabrication method does not support stretchable silicon-based substrates, because PDMS is hydrophobic and thus incompatible with water-based conductive inks. Another limitation is the use of the same layer of conductive ink to support both illumination and touch sensing, which significantly constrains the design of input and output widgets on interactive surfaces.

Fabrication Process. Stretchis should support rough prototyping and encourage customization, especially for users without access to cutting-edge equipment: (i) HCI practitioners who must quickly prototype interactive devices and ubiquitous interactive displays; (ii) designers who work on artistic installations, create wearables for interactive performances, or bring interaction to sharable objects and public spaces; and (iii) hobbyists who want to add interactive parts to their fabricated objects.

MULTI-LAYER FABRICATION APPROACH

The Stretchi platform includes four functional layers that serve as the building blocks for each customizable interface:

1. Base Substrate Layer -serves as the base material, consisting of pure PDMS, that provides stability and determines the physical properties of the interface, including thickness, color, and stretching behavior, as well as support for embedding Stretchi interfaces into porous materials, such as fabrics.

2. Sensing Layer -consists of a layer of conductive ink that adds sensing capabilities, patterned and shaped according to the specific input requirements. Stretchis currently support touch and proximity sensing.

3. Display Layer -provides electroluminescent visual output. Each display layer is composed of two layers of conductive ink, two layers of binding ink, and a layer of a phosphor/PDMS mix which emits light.

Aesthetics Layer -consists of colored inks in arbitrary 2D

shapes to add visual effects to the interface.

The multi-layer approach increases flexibility, since layers are independent and can be customized and printed during different phases of the fabrication process.

Adding Stretchable Conductors to PDMS

The stretchable base material places special demands on the ink, which must not only be conductive, but stretch without failing. We could apply other methods for fabricating stretchable sensors based on liquid metal (EGaIn) conductors [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF] and carbon-filled PDMS (cPDMS) [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF][START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF] to the Stretchi's sensing layer. Unfortunately, these inks are not transparent, which limits the display and aesthetic possibilities; and the methods require expensive specialized equipment, such as a laser engraver, to print the conductive material.

First, the conductors are not transparent -this constrains the way someone designs and adds displays and aesthetics to the interface. Second, to pattern conductive material, these methods rely on specialized equipment such as laser engraversyet, our goal was to avoid imposing the use of such equipment.

Other studies of stretchable materials [START_REF] Vosgueritchian | Highly conductive and transparent pedot:pss films with a fluorosurfactant for stretchable and flexible transparent electrodes[END_REF][START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF] indicate that transparent conductive inks based on PEDOT:PSS (GWENT D2070209P6) can be added to stretchable PDMS surfaces and are resilient to strain. In particular, Lipomi et al. [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF] report that PEDOT:PSS films retain conductivity even after a strain of over 100%, where 188% is the break point. In addition, for strain levels up to 30%, these films are reversibly stretchable, where resistance increases by a factor of 5 after 1000 stretches. However, applying PEDOT:PSS on PDMS requires specialized treatments difficult to integrate into a lightweight fabrication process. Applying water-based inks such as PE-DOT:PSS on PDMS substrates is especially challenging because the surface of PDMS is hydrophobic. Printed shapes lose their homogeneity, and the liquid ink forms drops on the surface of the PDMS (similar to the lotus effect).

Simple methods for treating the surface to increase water acceptance include corona [START_REF] Haubert | Pdms bonding by means of a portable, low-cost corona system[END_REF] and plasma treatments [START_REF] Ginn | Polymer surface modification using microwave-oven-generated plasma[END_REF]. Unfortunately, these methods require highly specialized equipment and laborious procedures. Moreover, the effect is temporary: depending upon the method, the surface becomes hy- drophobic again in a few minutes to one hour. This complicates the printing process and hinders usability of the fabricated sensors, especially if the PEDOT:PSS ink peels off. The plasma treatment permits a maximal stretching of 10% before a critical breakdown occurs [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF].

After experimenting with various treatments, we developed a simple, inexpensive and permanent method for applying water-based inks onto hydrophobic surfaces such as PDMS.

Since direct application of the conductive ink is not possible, we use a stretchable binding layer as an interface between the PDMS and the ink. AQUAPLAST-DIY Transparent Binder is an inexpensive (< $15/kg), off-the-shelf transparent ink commonly used for fabric printing. Not only does it have excellent printing properties on PDMS, it remains fully stretchable. This transparent binder contains a lower percentage of water than the PEDOT:PSS ink, which reduces the repellent effect. Also, because it is highly viscous, the ink does not form drops on the PDMS surface, making it suitable as as a base for applying conductive ink.

Figure 3 compares two different prints: (Left) water-based ink printed on pure PDMS, and (Right) the same ink printed on PDMS through a binding layer. Note that the print pattern of the transparent conductive material on pure PDMS is disconnected and has many holes. The binding layer allows the conductor to be printed homogeneously.

Embedding Stretchable Displays

Since we can now add transparent conductive inks on PDMS, we can also extend Olberding et al.'s method [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] and print electroluminescent displays on stretchable PDMS substrates. The general principles of electroluminescence are simple [START_REF] Kitai | Thin Film Electroluminescence[END_REF]: Phosphor particles are sandwiched between two electrodes. Applying an AC voltage to the electrodes establishes an electric field between the two plates, which activates the phosphor particles and causes them to emit light. The method offers great freedom for the design and customization of ubiquitous interactive displays [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF].

We adapt Wang et al.'s [START_REF] Wang | Highly stretchable and self-deformable alternating current electroluminescent devices[END_REF] fabrication method to produce a stretchable phosphor layer. First, we mix phosphor particles (KPT D310B) into fluid transparent two-component silicon in a ratio of 2 to 1, and screen print the mix. After curing, the layer is stretchable and connects directly to the underlying layers. Each stretchable display is composed of five or six sub-layers (Figure 4). The phosphor/PDMS mix is printed be- tween two layers of conductive material made of stretchable transparent ink. As explained earlier, we must also print a binding layer for each conductive layer. We then print an isolation (PDMS) layer on top of the phosphor layer to protect the two conductive layers from electrical breakdowns.

Sensing

We adapt Olberding et al.'s [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] time-multiplexing approach to avoid interference between electroluminescent displays and attached sensing layers. The display is deactivated during the sensing cycle by turning off its energy supply and setting both its electrodes to high impedance. In the current implementation, internal capacitors take 1.7 ms to discharge. The Stretchi is then free of electric noise, allowing us to choose from a variety of sensing methods.

For touch and proximity sensing, we use Arduino's Capacitive Sensing Library [START_REF] Badger | [END_REF]. One sensing cycle takes less than 2 ms, resulting in 3.7 ms total turn-off time for the display.

To avoid flicker, we turn on the display for 10 ms. This corresponds to a frame rate of 73 Hz, which is not visible to the human eye. Deactivation of the display leads to a 27% reduction in brightness. We compensate for this either by increasing the display's AC voltage or the AC signal's frequency.

Although we currently demonstrate only touch and proximity sensing, our fabrication approach can accommodate other sensing techniques for printed electronics. For example, we could apply one-layer [START_REF] Gong | Printsense: A versatile sensing technique to support multimodal flexible surface interaction[END_REF] or multi-layer sensing methods [START_REF] Woo | A thin all-elastomeric capacitive pressure sensor array based on micro-contact printed elastic conductors[END_REF] to sense pressure, folding or stretching. A major strength of our approach is that Stretchi sensors are designed and printed independently of each other and of the display. Unlike Olberding et al.'s [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF] method, the display does not constrain the number, geometry or layout of the sensors.

Fabrication Process

The Stretchi fabrication process consists of five basic steps:

Design the Stretchi -The designer uses graphic design software such as Adobe Illustrator or Gimp or circuit design software, such as EAGLE, to draw the components of the interface (Figure 5a). Each virtual layer corresponds to physical Stretchi structures: the PDMS substrate, the sensors, the displays, and the aesthetics or graphical features (see Figure 2). Binding and isolation layers are normally larger than the transparent conductors to ensure that the conductors are well covered and thus protected. Produce the Stretchable Base Material -Ready-to-print sheets of PDMS of varying colors, elasticity, and thickness (20 -400 µm) are commercially available, e.g., at Wacker GmbH, at a cost of less than $10/sheet. However, we can also fabricate custom PDMS substrates with inexpensive materials and tools. We can fully control the form and composition of the PDMS with fluid two-component silicon, which remains liquid for an hour after mixing. The PDMS surface can be shaped by spin coating, molding, or using a thin film applicator, to achieve the desired form and thickness. Color pigments can also be mixed into the silicon to create custom colors. The silicon cures at room temperature in 24 hours or, if placed on a hot plate at 90 • C, cures in about 20 minutes.

Print Sensors, Displays, and Aesthetics -We use screen printing [START_REF] Cossu | Silk Screen Basics: A Complete How-to Handbook[END_REF] to add display and sensing layers to the base layer of the PDMS substrate. Screen printing is a wellknown, easy-to-use, and low-cost method for printing functional inks on a variety of substrates. We use a standard offthe-shelf manual screen-printing table that costs under $200 (Figure 5b). We recommend using high resolution nets (100T -140T) to ensure high-quality printed shapes.

The designer prints each successive display and sensing layer, with sufficient time between each step for the printed material to cure (Figure 5c) and cool down. A good practice is to move the squeegee fast and with low pressure while printing to protect previously printed layers. Off-the-shelf screenprinting inks, e.g., TEXPRINT-AQ, have properties similar to the binding layer but include opaque or semi-transparent color pigments. The designer can build layers of color to create various visual effects and custom graphic designs.

Shape the Stretchi -The designer can manually shape the Stretchi with scissors or laser cut it for more precise control [START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF]. To produce 3D shapes such as cylinders or structures that combine multiple Stretchis together, the designer can glue them together with fluid silicon or with a special silicon glue.

Add Rigid Electronics, Power, and Control -Stretchis may include additional electronic components, such as integrated circuits, LEDs, and various rigid sensors (Figure 5d, 5e). They can also be wired with non-transparent conductive materials, such as cPDMS [START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF]. Carbon Powder (e.g., Cabot Black Pearls 800) is mixed with fluid PDMS (8kΩ/cm 2 ) in a ratio of 8%. Before curing, the cPDMS is applied as a connector between a Stretchi and the electronic component. The designer has various options for powering and controlling the Stretchi's sensors and displays. Our prototypes use an Arduino Uno microcontroller.

Displays require high voltage (≈ 150 V) but low-current AC, so we use a compact inverter to transform 12VDC to high voltage AC. The designer can also create mobile Stretchi applications by using more compact power and control units. For example, a very small inverter IC (e.g., SIPEX SP4405) can generate AC in combination with a dedicated mobile controller unit, such as an Arduino nano or Intel Curie controller, and be powered with very small button battery cells.

APPLICATION EXAMPLES

How can designers use Stretchis to embed interaction onto objects or into the physical environment? We developed several working prototypes to demonstrate possible applications.

We fabricated a stretchable awareness sleeve that fits onto various objects, such as a coffee cup (see Figure 1b), to provide notifications about incoming calls and social media events. The sleeve is equipped with electroluminescent displays that provide visual notifications; and touch sensors that let the user redirect notifications to specified computer applications. We also fabricated an ultra-thin Stretchi designed to be worn on the back of the hand; it is stretchable to allow the hand to move (see Figures 1d and5). The interface provides illuminated control widgets for music player: play, skip, and pause touch buttons, as well as a touch slider to control volume.

Finally, we experimented with Stretchis that integrate sensing into stretchable fabric. For example, Figure 6 combines printed conductive ink with rigid electronic components to create a Stretchi sport headband notification system. This Stretchi includes a rigid vibration motor for sending private notifications to the wearer, who can touch the band to display them on the screen of a mobile phone. The stretchable headband is easy to slip on and can accommodate various head sizes and hair styles.

TECHNICAL EVALUATION

We ran a series of tests to evaluate the technical behavior, reliability, and robustness of Stretchi sensors and displays.

Capacitive Touch and Proximity Sensing

We conducted an experiment to evaluate the touch and proximity sensing behavior of a Stretchi under varying levels of strain, where the interface was stretched up to 120%. For comparison, Weigel et al. [START_REF] Weigel | Flexible, stretchable and visually customizable on-body touch sensors for mobile computing[END_REF] tested the iSkin sensor with an upper limit of 30% strain.

Apparatus. We printed a 15 × 15 mm rectangular pattern of PEDOT:PSS on a silicon sheet, following the described fabrication approach. We connected the sensor with a printed 1 mm wide conductive line. We took measurements with an Arduino controller.

Participants. 12 volunteers (six women and six men), 21 to 30 years old, participated. All tested the same sensor.

Procedure. For each trial, the participant approached the sensor with the index finger from an angle of 90 • at seven discrete steps, from a distance of 6 cm to a distance of 0 cm, which corresponds to direct touch. For each distance, they were asked to rest their finger for three seconds. During this time, we used the Arduino Capacitive Sensing Library to record an average of 63 (SD = 10) measurements of the sensor's capacitance value.

This process was repeated five times, with the sensor stretched at five different levels: 0%, 30%, 60%, 90% and 120% of its natural length. The experimenter attached the sensor between two clamps and progressively stretched it from 0% to 120%.

Results. Figure 7 summarizes the results. We take the median of all measurements for each unique combination of participant, strain level, and distance. The key finding is that the sensor is extremely resilient to strain: For both touch and proximity sensing, the sensing capabilities of the sensor remain intact.

The high variance of values observed for touch sensing can be largely explained by the variability in the way participants touched the sensor, for example, touching hard versus gently; or touching the center versus the edges of the sensor. However, the transition from 1 cm distance to touch always increased the measured value by at least 453% (by a mean factor of 24). We could thus easily differentiate touch and proximity events, with perfect (100%) recognition accuracy.

The behavior of proximity sensing is consistent across all strain levels (Figure 7b). Proximity sensing is clearest at around 3 cm from the sensor. We could further increase the sensor's sensitivity by applying a larger area of conductive ink. Even though all 12 participants tested the same sensor over two different days, we did not observe any degradation of the sensing values over time.

Luminosity of Displays

We also tested how the brightness of a printed display is affected by different strain levels.

Apparatus and Procedure. We fabricated a rectangular 40× 20 mm display. We used the controllable head of a CNC milling machine (with an x and y resolution of 10 µm/step) to stretch the display (see Figure 8). We applied strains from 0% to 50% and back again to 0% (Figure 7b). We repeated this process 10 times. We used a PYL-ELI-ISC inverter with 12 V DC input and powered the display with 150 V AC at 1.3 kHz frequency. We used a photodiode to measure variations of the light emitted by the display. More precise measures of display luminance require specialized measurement tools; however, this evaluation offers an initial assessment of the reliability of printed Stretchi displays.

Results. The sensor measured relative raw brightness values at 13 (0% strain), 17 (30% strain), and 16 (50% strain). All values remained highly stable during all 10 stretching cycles.

The increase in brightness when the display is stretched is due to the fact that the voltage is applied to a thinner layer of the phosphor/PDMS mix. The estimated luminance values that correspond to these measurements range between 120 -200 cd/m 2 , which is comparable to the range of values reported by Olberding et al. [START_REF] Olberding | Printscreen: Fabricating highly customizable thin-film touch displays[END_REF]. Note that all the display layers are transparent, and the display emits light in two directions, towards both the viewing and the hidden surface of the Stretchi. The brightness could be increased by printing an additional light-reflective or white layer, under the display.

Durability of Stretchable Conductors

We further tested the extent to which stretchable conductors produced with our approach can endure a large number of stretches while still remaining conductive. We applied a total of 6000 stretches, where for each cycle of stretching, the strain increased from 0% to 50%. The 50% strain value is an upper bound for wearables and on-skin devices in normal use. For comparison, Lipomi et al. [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF] tested the durability of their stretchable conductors by applying 1000 stretches from 0% to 30%.

Apparatus and Procedure. We used the described fabrication method to print a 60×6 mm layer of PEDOT:PSS on a 1millimeter thick PDMS substrate (SORTA-CLEAR 40). The test pattern was sealed with an additional layer of PDMS. We measured an initial, 0% strain, resistance of 2.7 kΩ between the two ends of the test pattern. We then used the controllable head of a CNC milling machine (with an x and y resolution of 10 µm/step) to stretch the substrate. We applied a strain from 0% to 50% and then back again to 0%. We repeated this process 6000 times.

Results. The resistance increased non-reversibly after the first stretches: 177 kΩ after 10 stretches. As Lipomi et al. [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF] explain, this is because the first stretches introduce tiny breaking lines in the conductive layer. These lines are permanent and increase the conductor's resistance, without hindering conductivity. Subsequent stretches had a low impact on the conductor's resistance, reaching a value of less than 350 kΩ after 6000 stretches. This additional increase (< 29 Ω per stretch on average) is negligible for most capacitive sensing methods. Even so, we cannot reliably detect and measure strain, which requires different sensing technologies. An increase in the conductor's resistance can also reduce the display's brightness but does not hamper its usability in most real-world situations. Of course, we could also increase the voltage supply of the display to compensate for this loss.

Break Point. After the end of the above procedure, we stretched the same patterned substrate until conductivity reached a complete breakdown. This occurred when strain reached the level of 153%, where the resistance exceeded the value of 10 MΩ. Lipomi et al. [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF] report a higher break point for their method at 188% strain. However, they used a virgin conductive film for this test, not previously exposed to strain.

DISCUSSION AND FUTURE DIRECTIONS

Our method accommodates multiple conductive and display materials without changing the overall structure of Stretchi components. For example, we are interested in using WO 3 powder to support electrochromic [START_REF] Yan | Stretchable and wearable electrochromic devices[END_REF] as well as electroluminescence displays. Designers can also integrate more conventional electronics as separate layers within a Stretchi's' multi-layer architecture, including non-stretchable or nontransparent components in parts of the interface, e.g., to support richer sensing modalities such as pressing, bending or folding [START_REF] Gong | Printsense: A versatile sensing technique to support multimodal flexible surface interaction[END_REF][START_REF] Olberding | Foldio: Digital fabrication of interactive and shape-changing objects with foldable printed electronics[END_REF].

Results from advanced materials research [START_REF] Amjadi | Highly stretchable and sensitive strain sensor based on silver nanowireelastomer nanocomposite[END_REF] show that conductors based on silver nanowire (AgNW) networks can be embedded between PDMS layers and produce highly stretchable sensors (up to 70%). AgNW networks are also transparent but have additional interesting properties, i.e. they are more conductive than PEDOT:PSS and can act as strain sensors [START_REF] Amjadi | Highly stretchable and sensitive strain sensor based on silver nanowireelastomer nanocomposite[END_REF]. Sensing strain is particularly challenging, especially since the change in resistance of PEDOT:PSS conductors is not fully recoverable after stretching. In the future, we plan to produce strain sensors based on AgNW networks and integrate them into our fabrication process. We would also like to extend our approach to accommodate ionic [START_REF] Wang | Extremely stretchable electroluminescent devices with ionic conductors[END_REF][START_REF] Yang | Electroluminescence of giant stretchability[END_REF] in addition to electronic conductors.

CONCLUSION

We present an inexpensive method for fabricating Stretchis, highly stretchable interfaces that combine sensing and display capabilities. This method enables designers and novice makers to embed transparent conductors and electroluminescent displays into stretchable PDMS substrates. Inspired by state-of-the-art research on flexible electronic materials [START_REF] Lipomi | Electronic properties of transparent conductive films of pedot:pss on stretchable substrates[END_REF][START_REF] Lu | Rapid prototyping for soft-matter electronics[END_REF][START_REF] Wang | Highly stretchable and self-deformable alternating current electroluminescent devices[END_REF], we introduce new fabrication techniques that do not require specialized treatments or expensive equipment. In particular, we show how to prototype stretchable user interfaces by using standard design software and screen-printing techniques. Despite the use of inexpensive equipment, our results demonstrate that we can produce durable and highly stretchable sensors and displays that remain functional under strain levels of greater than 100%. We designed three sample applications that support touch and proximity sensing, but the Stretchi fabrication approach is applicable to a variety of additional sensing technologies.

Figure 2

 2 shows representative Stretchi structures, where layers can be stacked in different ways. A Stretchi can contain several layers of the same type, such as two sensing layers dedicated to different types of sensing or different user interface components.

Figure 2 .

 2 Figure 2. Three sample variations of Stretchi multi-layer structures. The top layer is the Stretchi viewing surface.

Figure 3 .

 3 Figure 3. Printed layer of PEDOT:PSS on PDMS recorded under 20x magnification. Left: Without the binding layer, the liquid ink forms drops and the layer is not conductive. Right: A binding layer printed between PDMS and PEDOT:PSS. The ink forms a connected layer and is conductive. The screen printing mask pattern is visible.

Figure 4 .

 4 Figure 4. Schematic of a stretchable display printed directly on a PDMS substrate. A phosphor/PDMS layer is sandwiched between two transparent conductive layers (electrodes) of PEDOT:PSS. Binding layers allow for printing the conducive layers on PDMS.

Figure 5 .

 5 Figure 5. Stretchi fabrication process: (a) The designer uses standard design software to create a virtual layer corresponding to each physical Stretchi layer. (b) The shapes in each layer are transferred to a screen printing net, using UV lithography, and then printed onto the stretchable PDMS substrate. (c) Each printed layer is cured or heat-dried before the subsequent layer is printed on top. (d) The designer adds power and (e) tests the Stretchi.

1 2Figure 6 .

 16 Figure 6. A Stretchi integrated into a stretchable headband. The Stretchi includes a (1) rigid vibration motor that sends notifications to the wearer and a (2) stretchable touch sensor to support touch interaction.

Figure 7 .

 7 Figure 7. Experimental results for touch (a) and proximity sensing (b) under variable strain (0% -120%). Sensor values are relative[START_REF] Badger | [END_REF], where no sensing corresponds to values close to 0. We show between-participant standard deviations for 12 participants.

Figure 8 .

 8 Figure 8. A CNC milling machine strains the Stretchi display in order to measure its brightness. Left: 0% stretch. Right: 50% stretch.

ACKNOWLEDGMENTS

This research is funded by the European Research Council grant CREATIV (n o 321135). We thank Niels Mourette for his valuable technical help.