François Bizet

Bertrand Muller
email: bertrand.muller@inra.fr

Yann Guédon
email: yann.guedon@cirad.fr

Beatriz Moreno-Ortega

Guillaume Fort

Identifying Developmental Zones in Maize Lateral Root Cell Length Profiles using Multiple Change-Point Models

Keywords: auxin mutant, lateral root diversity, multiple change-point model, piecewise linear function, principal component analysis, root apex

The identification of the limits between the cell division, elongation and mature zones in the root apex is still a matter of controversy when methods based on cellular features, molecular markers or kinematics are compared while methods based on cell length profiles have been comparatively underexplored. Segmentation models were developed to identify developmental zones within a root apex on the basis of epidermal cell length profiles. Heteroscedastic piecewise linear models were estimated for maize lateral roots of various lengths of both wild type and two mutants affected in auxin signaling (rtcs and rum-1). The outputs of these individual root analyses combined with morphological features (first root hair position and root diameter) were then globally analyzed using principal component analysis. Three zones corresponding to the division zone, the elongation zone and the mature zone were identified in most lateral roots while division zone and sometimes elongation zone were missing in arrested roots. Our results are consistent with an auxin-dependent coordination between cell flux, cell elongation and cell differentiation. The proposed segmentation models could extend our knowledge of developmental regulations in longitudinally organized plant organs such as roots, monocot leaves or internodes.

INTRODUCTION

Since the pioneering studies of [START_REF] Sachs | Über das Wachstum der Haupt und Nebenwurzeln[END_REF] and [START_REF] Darwin | The Power of Movement in Plants[END_REF], the root apex has been one of the most widely used plant organs to study cell division, cell elongation and cell differentiation which occur within successive zones [START_REF] Goodwin | Growth and differentiation in the root tip of Phleum pratense[END_REF]Erickson and Sax, 1956b). While the longitudinal cellular pattern within the root apex and the naming of the different zones are now the matter of tentative consensus views [START_REF] Barrio | Cell patterns emerge from coupled chemical and physical fields with cell proliferation dynamics: the Arabidopsis thaliana root as a study system[END_REF][START_REF] Ivanov | Longitudinal zonation pattern in plant roots: conflicts and solutions[END_REF], there is still no general agreement regarding the criteria used to define the limits between these zones [START_REF] Verbelen | The root apex of Arabidopsis thaliana consists of four distinct zones of growth activities: meristematic zone, transition zone, fast elongation zone and growth terminating zone[END_REF][START_REF] Ivanov | Longitudinal zonation pattern in plant roots: conflicts and solutions[END_REF][START_REF] Bizet | Length and activity of the root apical meristem revealed in vivo by infrared imaging[END_REF]. Historically, the shootward limit of the division zone (also called the root apical meristem, according to [START_REF] Barrio | Cell patterns emerge from coupled chemical and physical fields with cell proliferation dynamics: the Arabidopsis thaliana root as a study system[END_REF][START_REF] Ivanov | Longitudinal zonation pattern in plant roots: conflicts and solutions[END_REF], where cells divide at various rates and in different proportions) was identified by the presence/absence of mitotic figures in longitudinal sections [START_REF] Clowes | Apical meristems of roots[END_REF][START_REF] Hejnowicz | Growth and cell division in the apical meristem of wheat roots[END_REF]. By the turn of the last century, molecular markers have revolutionized the histology and, regarding cell division, cyclins which show marked overexpression at precise time points during the cell cycle have been used [START_REF] Ferreira | Three discrete classes of Arabidopsis cyclins are expressed during different intervals of the cell cycle[END_REF][START_REF] West | Cell cycle modulation in the response of the primary root of Arabidopsis to salt stress[END_REF]. Such type of discrete labeling leads to a probabilistic pattern and this has allowed the identification of a transition zone in the shootward region of the root apical meristem. In this zone, a progressive decrease of the occurrence of cell division is observed while cells acquire the capacity to elongate through vacuolization [START_REF] Baluška | Specific patterns of cortical and endoplasmic microtubules associated with cell growth and tissue differentiation in roots of maize (Zea mays L.)[END_REF] and cortical microtubule reorganization [START_REF] Baluška | Specialized zones of development in roots: view from the cellular level[END_REF][START_REF] Baskin | Regulation of growth anisotropy in well-watered and water-stressed maize roots. II. Role of cortical microtubules and cellulose microfibrils[END_REF]. After the transition zone, cells move to a rapid elongation zone and, to our knowledge, there is no consensus molecular marker for this zone although some members of the expansion gene family show tight association of their expression with elongation rate in monocot leaves [START_REF] Muller | Association of specific expansins with growth in maize leaves is maintained under environmental, genetic, and developmental sources of variation[END_REF] or internodes [START_REF] Lee | Expression of β-expansins is correlated with internodal elongation in deepwater rice[END_REF]. Growth cessation at the shootward limit of the elongation zone has been associated with cell wall stiffening [START_REF] Tomos | Biophysical and biochemical control of cell expansion in roots and leaves[END_REF], peroxidase activity (though more convincingly in aerial organs, see [START_REF] Macadam | Peroxidase activity in the leaf elongation zone of tall fescue. II. Spatial distribution of apoplastic peroxidase activity in genotypes differing in length of elongation zone[END_REF] and the burst of reactive oxygen species [START_REF] Dunand | Distribution of superoxide and hydrogen peroxide in Arabidopsis root and their influence on root development: possible interaction with peroxidases[END_REF] but none of these events were used as marker to locate this limit. Moreover, the formation of a root hair bulge has often been taken as a marker of the switch from elongation to differentiation although this tends to occur before the end of the elongation zone [START_REF] Ma | Regulation of root elongation under phosphorus stress involves changes in ethylene responsiveness[END_REF][START_REF] Le | Position and cell type-dependent microtubule reorientation characterizes the early response of the Arabidopsis root epidermis to ethylene[END_REF].

Alternative to cellular features or molecular markers are kinematic studies [START_REF] Sharp | Growth of the maize primary root at low water potentials. I. Spatial distribution of expansive growth[END_REF][START_REF] Muller | Spatial distributions of tissue expansion and cell division rates are related to irradiance and to sugar content in the growing zone of maize roots[END_REF][START_REF] Walter | Spatio-temporal dynamics of expansion growth in roots: automatic quantification of diurnal course and temperature response by digital image sequence processing[END_REF]. They are based on the combination of non-destructive observations of the spatial distribution of local growth within the root apex with the cell length profile in the same zone (Erickson and Sax, 1956b). These techniques quantify the cell division rate and locate the shootward limit of the division zone by using an analogous of the continuity equation used in fluid mechanics here applied to local cell density (Erickson and Sax, 1956b;[START_REF] Beemster | Analysis of cell division and elongation underlying the developmental acceleration of root growth in Arabidopsis thaliana[END_REF][START_REF] Muller | Spatial distributions of tissue expansion and cell division rates are related to irradiance and to sugar content in the growing zone of maize roots[END_REF]. However, averaging local growth profiles for several roots was identified as a source of bias, leading to smooth rapid individual variations and probably to overestimate the size of the division zone (van der [START_REF] Van Der Weele | A new algorithm for computational image analysis of deformable motion at high spatial and temporal resolution applied to root growth. Roughly uniform elongation in the meristem and also, after an abrupt acceleration, in the elongation zone[END_REF]. Moreover, when growth is non-stationary, this technique requires the incorporation of a time-dependency term [START_REF] Silk | Steady form from changing cells[END_REF][START_REF] Beemster | Analysis of cell division and elongation underlying the developmental acceleration of root growth in Arabidopsis thaliana[END_REF] which further adds uncertainties.

Developmental zones can also be identified in root apices on the basis of cell length profiles alone. Meristematic cells are short in length. The exit from the cell cycle and the entry into the elongation zone are characterized by a rapid increase in cell length while the end of the elongation zone is expected to correspond to cell length reaching a plateau. Different methods have been used to determine meristem length based on cell length profiles including expert visual methods [START_REF] Casamitjana-Martinez | Root-specific CLE19 overexpression and the sol1/2 suppressors implicate a CLV-like pathway in the control of Arabidopsis root meristem maintenance[END_REF][START_REF] Mouchel | Natural genetic variation in Arabidopsis identifies BREVIS RADIX, a novel regulator of cell proliferation and elongation in the root[END_REF], empirical thresholds on cell length [START_REF] Beemster | Variation in growth rate between Arabidopsis ecotypes is correlated with cell division and A-type cyclin-dependent kinase activity[END_REF] and geometrical approaches to detect change points [START_REF] French | Identifying biological landmarks using a novel cell measuring image analysis tool: cell-o-Tape[END_REF]. These methods are well suited to simple root structures such as in Arabidopsis thaliana where only a few cells per files are present and abrupt changes in length can be easily detected. It becomes more difficult when larger, more complex roots, with many cell files are considered or when not all the cells can be visible from the quiescent center, hampering the use of methods based on cell indices [START_REF] French | Identifying biological landmarks using a novel cell measuring image analysis tool: cell-o-Tape[END_REF]. Recently, a method was proposed that delineates complex root meristems based on a statistically-defined cell length threshold [START_REF] Bizet | Length and activity of the root apical meristem revealed in vivo by infrared imaging[END_REF]. This method proved to be robust although it does not allow to detect patterns within the meristem (such as gradual increase or decrease in cell length or change points) whereas these can be useful information when analyzing the impact of environmental stresses or genetic effects. Moreover, none of these methods were designed to identify the limit between the elongation zone and the mature zone. Very few attempts have been made in this direction because of the difficulty to deal with the large standard deviation of cell length around this limit [START_REF] Silk | Growth patterns inferred from anatomical records: empirical tests using longisections of roots of Zea mays L[END_REF].

A generic segmentation method, (i) based on cell length profiles, (ii) relying on minimum a priori biological assumptions, (iii) applicable to a large diversity of root growth dynamics, and (iv) able to detect limits both between the division zone and the elongation zone and (v) between the elongation zone and the mature zone is thus missing. The aim of this study was to propose such a generic method for identifying root developmental zones in cell length profiles. We used heteroscedastic piecewise Gaussian linear models (i.e., with a residual variance specific to each developmental zone; [START_REF] Hawkins | Point estimation of the parameters of a piecewise regression model[END_REF]. These specific multiple change-point models are distinct from segmented regression or broken-line models [START_REF] Muggeo | Estimating regression models with unknown breakpoints[END_REF] which are constrained to be homoscedastic (i.e., with a residual variance common to the different developmental zones). This assumption appeared to be unrealistic given the large changes in variance along root cell length profiles [START_REF] Goodwin | Growth and differentiation in the root tip of Phleum pratense[END_REF][START_REF] Pritchard | Biophysics of the inhibition of the growth of maize roots by lowered temperature[END_REF]. As a biological material, we choose maize lateral roots showing a large diversity of lengths and diameters, likely corresponding to various growth dynamics with acceleration, deceleration and rapid growth arrest as reported in other species [START_REF] Freixes | Root elongation and branching is related to local hexose concentration in Arabidopsis thaliana seedlings[END_REF]. These dynamics were expected to correspond to meristem enlargement, shrinkage or exhaustion, respectively [START_REF] Dubrovsky | Water deficit accelerates determinate developmental program of the primary root and does not affect lateral root initiation in a Sonoran Desert cactus (Pachycereus pringlei, Cactaceae)[END_REF][START_REF] Sánchez-Calderón | Phosphate starvation induces a determinate developmental program in the roots of Arabidopsis thaliana[END_REF]. In order to increase the sources of variability in our lateral root sample, and given the known impact of auxin on the establishment and the maintenance of the root meristem [START_REF] Pacifici | Plant hormone cross-talk: the pivot of root growth[END_REF] and on the balance between cell division and differentiation [START_REF] Dello Ioio | A genetic framework for the control of cell division and differentiation in the root meristem[END_REF], we used two independent maize mutants altered in auxin signaling: rtcs codes for a LOB-domain transcription factor and carries auxin-responsive elements in its promoter [START_REF] Taramino | The maize (Zea mays L.) RTCS gene encodes a LOB domain protein that is a key regulator of embryonic seminal and post-embryonic shootborne root initiation[END_REF] while rum1 codes for an AUX/IAA protein [START_REF] Woll | Isolation, characterization, and pericycle-specific transcriptome analyses of the novel maize lateral and seminal root initiation mutant rum1[END_REF]. Both mutants are also defective in seminal root formation. The objectives of this work were thus twofold: (i) design statistical models for identifying development zones in cell length profiles observed in root apices and (ii) on this basis, identify emerging properties in particular in terms of coupling/uncoupling between cell division, elongation and differentiation processes and characterize the intrinsic modulation of the root developmental pattern for a large diversity of lateral roots as well as the impact of a perturbation in auxin signaling.

MATERIALS AND METHODS

Plant Material, Growth Conditions, and Lateral Root Apex Sampling

Maize (Zea mays L.) seeds of the hybrid B73xUH007 (referred to as wild type in the following) used in this study were produced within the European FP7 project EURoot (http://www.euroot.eu). Seeds of rtcs [START_REF] Taramino | The maize (Zea mays L.) RTCS gene encodes a LOB domain protein that is a key regulator of embryonic seminal and post-embryonic shootborne root initiation[END_REF] and rum-1 [START_REF] Woll | Isolation, characterization, and pericycle-specific transcriptome analyses of the novel maize lateral and seminal root initiation mutant rum1[END_REF] maize mutants, both in the B73 background, were provided by Frank Hochholdinger (University of Bonn, Germany). Germinated seeds were transferred upon emergence of the radicle on the top of 70 × 40 cm rhizotrons adapted from [START_REF] Neufeld | A rootbox for quantitative observations on intact entire root systems[END_REF]. Root systems were allowed to develop between a layer of cellulose acetate tissue in contact with nutrient and water rich compost and a slide of plexiglass. Rhizotrons were installed into 1 m 2 growth chambers under controlled conditions (24/20 • C day/night temperature, 14 h photoperiod, 1 kPa VPD, and PPFD of 300 µmol photons/m 2 /s).

After 2 weeks, a selection of approximately 1 cm long root apices from 42 lateral roots encompassing the diversity of roots present along the primary root of several replicate plants was harvested. The lateral roots were sampled within 3 categories based on (i) the root length relative to its neighbors (Figure 1D), and (ii) the proximity of the first root hair to the root tip (Figures 1E-G) which is known to be a good indicator of the elongation rate [START_REF] Watt | Soil strength and rate of root elongation alter the accumulation of Pseudomonas spp. and other bacteria in the rhizosphere of wheat[END_REF][START_REF] Pagès | Estimating root elongation rates from morphological measurements of the root tip[END_REF], being more or less closely associated with the end of the elongation zone [START_REF] Le | In the early response of Arabidopsis roots to ethylene, cell elongation is up-and downregulated and uncoupled from differentiation[END_REF]. We named these roots A, B and C with type A roots being long, type C roots being very short (<1 cm) and type B roots being intermediate in length. This typology matches that recently identified in pearl millet on the basis of anatomical features [START_REF] Passot | Characterization of pearl millet root architecture and anatomy reveals three types of lateral roots[END_REF]. Using independent experiments (not described here), we evaluated the growth patterns of these roots and found that type A root growth rate was high (8-17 mm day -1 ) and increases markedly the first days of growth by contrast with type B and C roots which show slow and fast deceleration, ending up in growth arrest in (on average) 6 and 3 days respectively. In the two mutants, roots of similar types were found although long roots were globally shorter than in the wild type. In addition, vigorous lateral roots emerging from curvatures of the primary roots induced by the alteration of gravitropism were categorized as A ′ . Each of these root types were present in all individual plants and our sampling protocol aimed at capturing the largest diversity, including that due to plant vigor.

Image Analysis and Acquisition of Lateral Root Cell Length Profiles and Morphological Properties

Root apices were placed in a fixative solution of 1:3 vol/vol acetic acid: 70% ethanol and stored at 4 • C. After 2 days, the fixed material was moved to a clearing solution of chloral hydrate (200 g chloral hydrate in 20 ml glycerol and 30 ml water) for at least 4 h [START_REF] Wuyts | High-contrast three-dimensional imaging of the Arabidopsis leaf enables the analysis of cell dimensions in the epidermis and mesophyll[END_REF]. Roots were mounted in the same solution and imaged within a week. Root apices were observed using an optic microscope (Olympus BX61 TRF, Japan) under natural fluorescence conditions using UV illumination (360-370 nm) to allow observation of cell walls in epidermal root cells. Individual root apices (Figure 1) were imaged at 10 x magnification by gathering 2-3 contiguous images, until the zone where root hair development was observed.

All image processing and data extraction were performed manually using the ImageJ image analysis software (Rasband WS. U.S. National Institutes of Health, Bethesda, MD, USA). The contrast of the microscopy images was first adjusted to make the cell files appear as clearly as possible. Cell lengths from all clearly visible files (usually the central 3-4 cell files) of the epidermal tissue were manually measured for each root as the distance between two consecutive transverse cell walls. Cell length sampling started at the root cap junction and spanned shootward, as long as the quality of the image allowed it, and in all cases far beyond the occurrence of the first root hair. Each cell was assigned to a longitudinal position equal to its orthogonal projection onto a virtual line passing through the middle of the root, taking the root cap junction as the origin. The onset of root hair formation was estimated for each root by locating the most rootward cell showing an incipient root hair bulge. Lateral root diameter was evaluated at two positions beyond the first root hair. Two hours were generally sufficient to process a typical root sample. After exploration of cell length profiles, 36 lateral roots were retained for further analyses (18 wild-type, 8 rtcs and 10 rum-1 individuals), the 6 others being rejected because of a too sparse sampling of cells. An average of 160 cell lengths was measured for each selected root, with a minimum of 52 and a maximum of 267.

Multiple Change-Point Models for Identifying Development Zones in Lateral Root Cell Length Profiles

Definition of Heteroscedastic Piecewise Gaussian Linear Models and Gaussian Change in the Variance Models Multiple change-point models were used to delimit developmental zones within a cell length series x of length T. We made the assumption of heteroscedastic piecewise Gaussian linear models where the within-zone parameters were the intercept, slope and residual variance. The heteroscedasticity assumption (a residual variance different in each zone) was suggested by the data characteristics and validated a posteriori.

We adopted a retrospective or off-line inference approach whose objective was to infer the number of developmental zones J, the positions of the J -1 change points τ 1 , . . . , τ J-1 (with the convention τ 0 = 1 and τ J = T + 1), the J within-zone intercepts α j , slopes β j and residual variance σ 2 j . For the selection of the number of developmental zones, we used the slope heuristic proposed by [START_REF] Guédon | Slope heuristics for multiple change-point models[END_REF]. The principle of this kind of penalized likelihood criterion consists in making a trade-off between an adequate fitting of the model to the data and a reasonable number of parameters to be estimated.

Once the number of developmental zones J had been selected, the cell length series was optimally segmented into J zones using a dynamic programming algorithm [START_REF] Auger | Algorithms for the optimal identification of segment neighborhoods[END_REF]). This optimal segmentation (i.e., the most probable segmentation among all the possible segmentations for a fixed number of developmental zones) defines the optimal change points and relies on the estimation of withinzone parameters. It thus defines the optimal piecewise linear function which is not assumed to be continuous at change points. This optimal piecewise linear function should only be considered as the most likely explanation of the cell length series using multiple change-point models but not as a generative model of the underlying biological mechanism. [START_REF] Guédon | Exploring the latent segmentation space for the assessment of multiple change-point models[END_REF] generalized the dynamic programming algorithm of [START_REF] Auger | Algorithms for the optimal identification of segment neighborhoods[END_REF] to compute the top N most probable segmentations. This algorithm was useful since, in some cases, a well-supported alternative segmentation was more consistent with biological assumptions than the optimal segmentation. The assessment of multiple change-point models relied on two posterior probabilities (see Methods S1 in Supplementary Material for formal definitions):

• posterior probability of the selected J-developmental-zone model, i.e. weight of the J-developmental-zone model among all the possible models. This posterior probability is an output of the slope heuristic (see Methods S1 in Supplementary Material). • posterior probability of the optimal segmentation in J developmental zones, i.e. weight of the optimal segmentation among all the possible segmentations in J developmental zones.

These two posterior probabilities reflect the hierarchical nature of the inference with two successive steps: (i) selection of the number of developmental zones using the slope heuristic considering all the possible segmentations in J developmental zones for J = 1, . . . , J max and (ii) computation of the optimal segmentation in the number of developmental zones previously selected.

We used different diagnostic tools [START_REF] Guédon | Exploring the latent segmentation space for the assessment of multiple change-point models[END_REF] to assess the assumption of the segmentation in developmental zones and in particular two types of posterior probability profiles that summarize all the possible segmentations for a fixed number of developmental zones: posterior zone probability profiles and posterior segmentation probability profiles. It is often of interest to quantify the uncertainty concerning change-point position. To this end, we computed uncertainty interval for each change point using the smoothing algorithm proposed by [START_REF] Guédon | Exploring the latent segmentation space for the assessment of multiple change-point models[END_REF].

All these quantities used for diagnostic are formally defined in Methods S1 in Supplementary Material.

We conducted a residual analysis to decipher the weights of the change in slope and of the change in residual variance in the determination of change points. To this end, we computed the residual series by subtracting the piecewise linear function corresponding to the selected segmentation from the measured series. We then estimated a Gaussian change in the variance model applying the methodology previously described for heteroscedastic piecewise Gaussian linear models. In a Gaussian change in the variance model, we suppose that J -1 change points τ 1 < • • • < τ J-1 exist such that the mean is assumed to be constant and the variance is constant between two successive change points:

if τ j ≤ t < τ j+1 , E(X t ) = α, Var(X t ) = σ j 2 .
In our context of residual analysis, the estimated mean was always very close to 0. Details on the statistical methods for multiple change-point models are given in Methods S1 in Supplementary Material.

Illustration of the Application of Multiple Change-Point Models on Selected Maize Lateral Root Apices

Three successive zones are expected along the apex of growing roots starting from the tip: the root apical meristem (called division zone and abbreviated DZ hereafter), the elongation zone (EZ) and the mature zone (MZ). We assumed that the division zone was characterized by small cells, the elongation zone by cells of gradually increasing length and the mature zone by large cells.

In our modeling framework, the limit between two successive zones corresponds to a marked change in slope and in residual standard deviation. The example presented in Figure 2 illustrates a typical type A lateral root where the DZ-EZ and EZ-MZ limits correspond to changes both in slope and in residual standard deviation. The residual analysis (Figure 3) highlights the role played by the change in residual standard deviation for defining limits between consecutive zones since the uncertainty intervals for the DZ-EZ and EZ-MZ limits given by the piecewise linear model estimated on the basis of the measured series and the change in the variance model estimated on the basis of the residual series are very close.

In a few cases, the optimal piecewise linear function did not fit our biological assumptions. The type A lateral root presented in Figure 4 illustrates such a case where the optimal piecewise linear function deduced from the optimal 2-segment model can be interpreted as a division zone followed by a mature zone according to our biological assumptions. The missing elongation zone could not be identified in the optimal 3-segment piecewise linear function (the slope in the elongation zone is not significantly different from zero) but rather in the well-supported alternative 3-segment piecewise linear function corresponding to the second segmentation. In this example, the difficulty comes from the rather sparse sampling of cells in the mature zone (16 cells in the retained segmentation instead of 62 in the example shown in Figure 2) in conjunction with the high MZ residual standard deviation. As a consequence, the uncertainty interval for the EZ-MZ limit is large. The example presented in Figure S1 illustrates a similar situation but where the optimal 3-segment piecewise linear function is consistent with our biological assumptions.

The example presented in Figure 5 illustrates the case of a type A lateral root for which the determination of the EZ-MZ limit is rather uncertain. The optimal limit at 1,132 µm is at the shootward end of the uncertainty interval. This limit entails a jump of -44.7 µm between the two linear functions. We thus retained the limit at 689 µm corresponding to the third segmentation which only entails a jump of 11.1 µm. It is also consistent with the residual analysis since the optimal segmentation of the residual series deduced either from the optimal segmentation or from the third segmentation has the same limits as the third segmentation of the measured series. The example presented in Figure S2 illustrates a similar situation in the case of a type B lateral root without a division zone for which the second segmentation is far more consistent with the approximate continuity of the selected piecewise linear function.

These examples illustrate the strategy we adopted for selecting piecewise linear functions combining the inference of multiple change-point models with biological assumptions. We first computed the optimal piecewise linear function for the number of developmental zones given by the slope heuristic. We then identified the division zone, the elongation zone and the mature zone and checked their characteristics according to our biological assumptions (knowing that the division zone or the division and the elongation zones can be absent in type B or C roots). If two consecutive zones were merged (e.g., the elongation and the mature zones for the example presented in Figure 4), we explored the well-supported (in terms of posterior probabilities) segmentations with one more zone. If the optimal piecewise linear function was strongly inconsistent regarding the approximate continuity assumption (the presence of a huge discontinuity was rather inconsistent with the biological assumption of a gradual increase in cell length), we explored well-supported alternative segmentations. No other biological assumptions (e.g., position of the EZ-MZ limit with respect to the first root hair position) were used for selecting piecewise linear functions.

RESULTS

We assumed that the developmental pattern was common to the studied lateral roots even if the most rootward developmental zones (i.e., the division zone or the division and the elongation zones) were absent in some roots. The analysis of this developmental pattern broke down in two steps:

1. Identification and characterization of the successive developmental zones along each lateral root. For this individual analysis, we focused in particular on the selection of the number of developmental zones, on the roles played by the change in slope and the change in residual standard deviation in the determination of the limits between these zones and on the uncertainty concerning these limits. 2. Comparison of the developmental zones of the lateral roots in order to identify commonalities and differences between these zonations.

Selection of the Number of Developmental Zones

We retained the number of developmental zones given by the slope heuristic and the optimal segmentation in this number of developmental zones for 26 individuals among 36 (Tables 123for the wild type and the rtcs and rum-1 mutant respectively). This includes the three 4-zone individuals For each lateral root (ordered in decreasing length of the division zone), the piecewise linear model is described in the first row and the Gaussian change in the variance model estimated on the basis of the residual series is described in the second row. For each multiple change-point model, the standard deviations (s.d.) estimated for each zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ)-, the limits between zones with associated 0.05-uncertainty intervals, the first root hair position (all in µm), the selected segmentation posterior probability -an asterisk indicates that the segmentation is the optimal one-, the selected model posterior probability -an asterisk indicates that the model is the one given by the slope heuristic (SH)-, and the number of zones given by the slope heuristic are given.

presented in Tables S1, S2 (see below for the interpretation of this pattern). Figure 2 shows a typical 3-zone individual while Figure 6 shows a typical 2-zone (elongation and mature zones) individual and a typical single-zone (mature zone only) individual both categorized as type C and illustrating the diversity of cell length pattern in arrested roots. For two individuals, we retained a well-supported alternative model with one more developmental zone than the model selected by the slope heuristic and for two other individuals (see Figure 5, Figure S2), we retained a well-supported alternative segmentation in the number of developmental zones given by the slope heuristic. For the 6 remaining individuals, we retained a model with one more developmental zone than the model selected by the slope heuristic and the optimal segmentation in this number of developmental zones (see Figure S1) except for one individual (see Figure 4) for which we retained For each lateral root (ordered in decreasing length of the division zone), the piecewise linear model is described in the first row and the Gaussian change in the variance model estimated on the basis of the residual series is described in the second row. For each multiple change-point model, the standard deviations (s.d.) estimated for each zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ)-, the limits between zones with associated 0.05-uncertainty intervals, the first root hair position (all in µm), the selected segmentation posterior probability -an asterisk indicates that the segmentation is the optimal one-, the selected model posterior probability -an asterisk indicates that the model is the one given by the slope heuristic (SH)-, and the number of zones given by the slope heuristic are given. For each lateral root (ordered in decreasing length of the division zone), the piecewise linear model is described in the first row and the Gaussian change in the variance model estimated on the basis of the residual series is described in the second row. For each multiple change-point model, the standard deviation (s.d.) estimated for each zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ)-, the limits between zones with associated 0.05-uncertainty intervals, the first root hair position (all in µm), the selected segmentation posterior probability -an asterisk indicates that the segmentation is the optimal one-, the selected model posterior probability -an asterisk indicates that the model is the one given by the slope heuristic (SH)-, and the number of zones given by the slope heuristic are given. a well-supported alternative segmentation. All these choices of alternative model or segmentation were supported by the biological assumptions stated before regarding the succession of developmental zones and their main properties in terms of cell length. It should be noted that among the 10 individuals for which we did not retain the number of developmental zones given by the slope heuristic and the optimal segmentation in this number of developmental zones, 7 were characterized by a rather sparse sampling of cells within the mature zone (wild-type A8, rtcs A3, A2, A ′ 39, B15, rum-1 A7, A ′ 40); see Tables 456.

For three wild-type individuals, 4 zones were identified where the first two zones correspond to a split of the division zone (Table 1,Tables S1,S2) that can be interpreted as a proliferation zone followed by a transition zone. The segmentations in 3 and 4 zones are nested or almost nested in the case of A13 (Figure S3). The limit between the two successive zones within the division zone corresponded mainly to a change in slope with a negative slope in the first zone followed by a positive slope or a slope non-significantly different from zero in the case of B32 in the second zone (Tables S1,S2). When the residual series was extracted using the 4-segment piecewise linear function, the first two zones could only be identified in B32 but not in the two other individuals for which they were merged consistently with the similar residual standard deviations estimated for these two zones in A13 and B33 (Table S1).

Approximate Continuity of the Selected Piecewise Linear Functions

Contrary to segmented regression models [START_REF] Muggeo | Estimating regression models with unknown breakpoints[END_REF], the piecewise linear functions are not constrained to be continuous in the framework of multiple change-point models. We thus computed the rootward and shootward confidence intervals at each limit between two consecutive developmental zones (e.g., DZ and EZ confidence intervals at the DZ-EZ limit) in order to assess the approximate continuity of the piecewise linear function selected for each individual. The piecewise linear functions are most often approximately continuous for the mutants with overlap between confidence intervals for 13 limits among 14 for rtcs (Table 8) and for 15 limits among 17 for rum-1 (Table 9). The situation is substantially different for the wild type with overlap between confidence intervals for 15 limits among 28, the non-overlap concerning mostly EZ-MZ limits (Table 7).

The Limits between Developmental Zones are Explained Both by a Change in Slope and in Residual Standard Deviation

We conducted a residual analysis using the residual series deduced from the selected piecewise linear function for each individual. We checked that the residual series were stationary and selected for each series a Gaussian change in the variance model using the slope heuristic (Tables 123). We found the same number of zones as for the measured cell length series for 31 individuals among 36 while this number of zones corresponds to a well-supported alternative model for 4 other individuals. Fifty four limits between zones among 59 are co-localized i.e., the uncertainty intervals for a given limit for the piecewise linear model and for the change in the variance model overlapped (Tables 123). It should also be noted that we did not detect any supplementary change point within the elongation zone in the residual series. The residual standard deviation is thus approximately stationary within the elongation zone.

Consistency of the Limit between the Elongation Zone and the Mature Zone with the First Root Hair Position

The formation of a root hair bulge is often taken as a marker of the switch from elongation to differentiation. We thus compared for each root the EZ-MZ limit with the first root hair position. For about half of the individuals (16 among 33), the EZ-MZ limit matches with the first root hair position, i.e., the first hair position falls within the uncertainty interval of the EZ-MZ limit or in EZ (Tables 123). The situation is contrasting between the wild type and the rtcs and rum-1 mutants since in the case of the mutants, the EZ-MZ limit matches with the first hair position for most of the individuals (7 among 8 for rtcs and 7 among 10 for rum-1) while this is rather the exception for the wild type (2 among 15 individuals with a least two zones). In particular, the EZ-MZ limit is far from the first hair position in the rootward direction for five wild-type lateral roots of type A.

Among the individuals for which the EZ-MZ limit does not match with the first root hair position, we focused on the six For each lateral root (ordered in decreasing length of the division zone), the parameters of the piecewise linear function (slope × 1,000 in µm/mm, correlation coefficient for each zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ), n.s. for non-significant-, limits between zones and position of the most shootward cell in µm) and the number of cells assigned to each zone are given. For each lateral root (ordered in decreasing length of the division zone), the parameters of the piecewise linear function (slope × 1,000 in µm/mm, correlation coefficient for each zone -division zone (DZ), elongation zone (EZ), and mature zone (MZ), n.s. for non-significant-, limits between zones and position of the most shootward cell in µm) and the number of cells assigned to each zone are given.

3-zone individuals for which the distance between the EZ-MZ limit and the first root hair position was the largest (wild-type A8, A9, A10, A11, A31, and rum-1 A6 with a distance between and 1,099 µm); see Table S3. These 6 individuals are also the individuals with the steepest MZ slopes (see Tables 4-6 and Figure 7) and the smallest difference between the MZ slope and the EZ slope among the 3-zone individuals and are characterized by a high overlap between the confidence intervals of the EZ and MZ slopes; see Table S3. It should be noted that for most of the other 3-zone individuals (17 among 20), there is no overlap between the confidence intervals of the EZ and MZ slopes (results not shown). For these 6 individuals, the EZ-MZ limit is thus mainly explained by a change in residual standard deviation.

Finally, the cell sampling cannot fully explain these results since for 3 of these individuals, the number of cells beyond the first hair position is above 30 (Table S3). This mismatch of the EZ-MZ limit with the first hair position for some individuals can be viewed as a consequence of the fact that this limit is only explained by a change in residual standard deviation for these individuals while for most individuals, the EZ-MZ limit is explained by a concomitant change in slope and in residual standard deviation. There is a consistent relationship between the EZ-MZ limit and the first root hair position in both the wild type and the two auxin mutants (Figure S4). This relationship is shifted in the mutants with first root hairs emerging closer to the root tip compared to the wild type. For each lateral root (ordered in decreasing length of the division zone), the parameters of the piecewise linear function (slope × 1,000 in µm/mm, correlation coefficient for each zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ), n.s. for non-significant-, limits between zones and position of the most shootward cell in µm) and the number of cells assigned to each zone are given. For each lateral root (ordered in decreasing length of the division zone), the piecewise linear function (cell lengths predicted at both ends of a zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ)-linked by an arrow, limits between zones and position of the most shootward cell in µm) with associated rootward and shootward confidence intervals at each limit between zones (between brackets) is given.

A Strong Modulation of the Developmental Pattern is Observed among Lateral Roots

As expected from the root sampling strategy, a strong modulation of the developmental pattern is observed among lateral roots (Tables 123456789). While type A roots show the longest division and elongation zones (mean DZ length of 454 µm and mean EZ length of 605 µm), type B roots show much reduced division and elongation zones (mean DZ length of 277 µm and mean EZ length of 284 µm) and type C roots show a lack of division zone and often a lack of elongation zone. Figure 6A illustrates this with a root where a shrunken and no longer active elongation zone is followed by a mature zone with irregular cell length. Figure 6B illustrates a lateral root with neither a division zone, nor an elongation zone but a mature zone with irregularly increasing cell length, possibly the trace of a progressive and irregular deceleration of the root. In both cases, growth arrest is associated with meristem exhaustion and increase in cell length in the apical region. Moreover, DZ cell length is higher in the mutants (6-18 µm; see Tables 8,9) than in the wild type (4-11 µm, see Table 7). Figure 8 illustrates the diversity of the cell length pattern by showing the piecewise linear functions estimated for each lateral root. It illustrates the ordering of the three types of lateral roots in terms of (i) length of the meristem (from long meristems for type A roots toward absent meristem for most type C roots, type B roots showing shorten meristems), (ii) cell length in MZ (from the longest cells for type A roots to the shortest cells for type C roots). The systematic increase in MZ cell length for type C roots should be noted. This is confirmed by the significant linear correlation coefficients and equivalently the slopes significantly different from zero in the mature zone (Tables 4,6). Finally, only a limited proportion of potentially growing roots have MZ slopes non-significantly different from zero: nine type A roots (all mutants) among 21 and three type B roots (two wild types and one mutant) among 7 (Tables 456). Interestingly, type B roots have the less steep MZ slopes. The variability in MZ slope is also higher for type A roots than for the other types.

Choice of the Variables Summarizing Lateral Root Development for the Meta-Analysis

In order to provide a synthetic view of the modulation of the lateral root developmental pattern, we selected a set of variables for a meta-analysis. The length of the division zone, the length of the elongation zone and the first root hair position were chosen to characterize lateral root development. These three variables are strongly correlated (correlation coefficients between 0.63 and 0.83 for 3-zone lateral roots) which can be interpreted as a longitudinal scaling of lateral root developmental zones. Concerning cell dimension variables, since the elongation zone is the most structuring zone with high estimated correlation coefficients (Tables 456), we chose the cell lengths predicted at the two ends of the linear function estimated in the elongation zone for summarizing the change in cell length along the lateral roots. These two predicted cell lengths are positively correlated (r = 0.63 for 3-zone lateral roots). There is thus also a scaling effect in the cell length along the roots. The slope within the elongation zone, which is negatively correlated with the length of this zone was also incorporated. Finally, we incorporated the first root hair position and the mean root diameter within the mature zone, in order to explore the relationships between meristem length, growing zone length, root diameter and onset of differentiation. For each lateral root (ordered in decreasing length of the division zone), the piecewise linear function (cell lengths predicted at both ends of a zone -division zone (DZ), elongation zone (EZ), and mature zone (MZ)-linked by an arrow, limits between zones, and position of the most shootward cell in µm) with associated rootward and shootward confidence intervals at each limit between zones (between brackets) is given. For each lateral root (ordered in decreasing length of the division zone), the piecewise linear function (cell lengths predicted at both ends of a zone -division zone (DZ), elongation zone (EZ) and mature zone (MZ)-linked by an arrow, limits between zones and position of the most shootward cell in µm) with associated rootward and shootward confidence intervals at each limit between zones (between brackets) is given.

Exploration of the Diversity of Lateral Roots Using Principal Components Analysis

We applied a principal components analysis (PCA) to the 26 3zone lateral roots (13 wild-type, 6 rtcs and 7 rum-1 individuals) using 5 variables extracted from the analysis of individual lateral roots using multiple change-point models (DZ length, EZ length, cell length predicted at the DZ-EZ limit, cell length predicted at the EZ-MZ limit, EZ slope) completed by two morphological variables (first root hair position, mean diameter within MZ). We incorporated as supplementary variables in PCA the slope within the division zone. This slope is either negative or non-significantly different from zero for the wild type while being positive (Table 4) or non-significantly different from zero for the rtcs mutant (Table 5). The situation of the rum-1 mutant is intermediate with both positive, negative slopes and slopes non-significantly different from zero (Table 6).

We also incorporated as supplementary variables in the PCA the residual standard deviations estimated within each zone.

The cell length predicted at the DZ-EZ limit is strongly correlated with the residual standard deviations estimated in DZ and EZ (r = 0.8 and r = 0.81 respectively) and the cell length predicted at the EZ-MZ limit is strongly correlated with the residual standard deviation estimated in each zone (r = 0.58, r = 0.77 and r = 0.58 for DZ, EZ and MZ respectively). The first axis accounting for 49% of variance corresponds to the longitudinal variables (mainly DZ and EZ lengths but also first root hair position) while the second axis accounting for 29% of variance corresponds to the cell length variables (cell lengths predicted at the DZ-EZ and the EZ-MZ limits), longitudinal variables and cell length variables being uncorrelated (Figure 9A). All these five variables are highly structuring; see the distances of their projections to the correlation circle. The residual standard deviations within DZ and EZ incorporated as supplementary variables are highly related to the second axis and thus with cell length variables. The EZ slope is less affected by the difference in cell length within EZ (cell length predicted at the EZ-MZ limit -cell length predicted at the DZ-EZ limit) than by the EZ length. Hence, the EZ slope increases when the EZ length decreases; see Figure 9A. Roots from all three genotypes are spread along the first axis but with type B roots being clearly shifted to the left consistently with their smaller DZ and EZ lengths (Figure 9B). Moreover, rtcs and rum-1 individuals

DISCUSSION

Successive Developmental Zones in the Root Apex Are Well Characterized by Piecewise Linear Functions Heretoscedastic piecewise linear models with at most 4 developmental zones were selected for each lateral root with minimum a priori biological assumptions. This result is consistent with the expectation of 3-4 zones for elongating roots: a root apical meristem [START_REF] Ivanov | Longitudinal zonation pattern in plant roots: conflicts and solutions[END_REF] possibly including a transition zone, an elongation zone and a mature zone. For non-elongating roots such as type C roots, the lack of both a division zone and an elongation zone is also consistent with the biological knowledge about determinate roots [START_REF] Dubrovsky | Water deficit accelerates determinate developmental program of the primary root and does not affect lateral root initiation in a Sonoran Desert cactus (Pachycereus pringlei, Cactaceae)[END_REF][START_REF] Sánchez-Calderón | Phosphate starvation induces a determinate developmental program in the roots of Arabidopsis thaliana[END_REF]. These are clear elements of validation of the piecewise linear model assumption but to which extent the linearity assumption matches our biological knowledge within each zone?

Cell length in the apical meristem was well approximated by a single linear function. For only 3 roots among 26, the optimal model induced a split of the apical meristem into 2 zones, the first with a negative slope and the second with a positive or nil slope. Such a split is consistent with the concept of a transition zone where cells progressively leave the cycle while local elongation rate has not yet changed [START_REF] Dello Ioio | A genetic framework for the control of cell division and differentiation in the root meristem[END_REF][START_REF] Baluška | Root apex transition zone: a signalling-response nexus in the root[END_REF]. Because cell length is the result of an equilibrium between cell division rate and local tissue elongation rate [START_REF] Green | Growth and cell pattern formation on an axis: critique of concepts, terminology, and modes of study[END_REF], a split into 2 zones with negative and positive slopes is consistent with relative elongation rate being constant throughout the meristem (van der [START_REF] Van Der Weele | A new algorithm for computational image analysis of deformable motion at high spatial and temporal resolution applied to root growth. Roughly uniform elongation in the meristem and also, after an abrupt acceleration, in the elongation zone[END_REF] but lower and higher than cell division rate in the two domains respectively [START_REF] Ivanov | What the distribution of cell lengths in the root meristem does and does not reveal about cell division[END_REF] due to more or less cells being engaged in the cycle. The fact that a transition zone was identified in only 3 roots could be due to the short size of this zone [START_REF] Pacifici | Plant hormone cross-talk: the pivot of root growth[END_REF] in conjunction with changes in slope and in residual standard deviation of small amplitude.

Beyond these 3 roots, DZ slopes were not systematically zero but rather essentially negative in the wild type while being positive in the rtcs mutant and mixed in the rum-1 mutant. As stated above, a negative slope indicates a proliferative activity being higher than local tissue expansion leading to apparent decrease in cell size (Erickson and Sax, 1956a;[START_REF] Green | Growth and cell pattern formation on an axis: critique of concepts, terminology, and modes of study[END_REF][START_REF] Ivanov | What the distribution of cell lengths in the root meristem does and does not reveal about cell division[END_REF]. Our results thus suggest that: (i) elongation and division are rarely in perfect equilibrium in the meristem; (ii) in roots from wild-type plants, this equilibrium is in favor of cell division as cells move away from the tip; (iii) deceleration or slow growth is associated with a shortening of the meristem, as classically reported to account for changes in elongation rate of roots exposed to various stresses [START_REF] Barlow | The response of the primary root meristem of Zea mays L. to various periods of cold[END_REF][START_REF] Ryan | Aluminium toxicity in roots: an investigation of spatial sensitivity and the role of the root cap[END_REF][START_REF] West | Cell cycle modulation in the response of the primary root of Arabidopsis to salt stress[END_REF], but not with changes in meristematic cell length; (iv) auxin strongly interferes with the balance between cell division and tissue expansion, at the benefit of the later in the meristem, consistently with the knowledge about the role of this hormone [START_REF] Pacifici | Plant hormone cross-talk: the pivot of root growth[END_REF]. The outcomes of our model thus essentially fit what is known about the functioning of the root apical meristem and the balance between division and elongation.

Within the elongation zone, the linearity was clear as shown by the high correlation coefficient estimated for most of the lateral roots and the absence of split of this zone. Because the elongation zone corresponds to a zone without cell division, a linear increase in cell length suggests, if elongation of the root is stationary, a linearly increasing local elongation rate and thus a constant relative elongation rate within the elongation zone [START_REF] Silk | Growth patterns inferred from anatomical records: empirical tests using longisections of roots of Zea mays L[END_REF]. This is what is found in Arabidopsis roots when biases due to averaging among individual roots are eliminated [START_REF] Van Der Weele | A new algorithm for computational image analysis of deformable motion at high spatial and temporal resolution applied to root growth. Roughly uniform elongation in the meristem and also, after an abrupt acceleration, in the elongation zone[END_REF]. However, the growth of most of our lateral roots was likely non-stationary. Type C roots were already arrested, type B roots were slowly decelerating and type A roots were mostly accelerating with potentially large variability among roots. Our results suggest that even under non-stationary growth regimes, the linear model appears as the most straightforward way to represent the elongation zone in the root apex.

More unexpected was the identification of mature zones often displaying significantly positive slopes as one would not expect from a "classical" mature zone in which cells no longer elongate. A first hypothesis is that this results from nonstationary growth rates, in particular in type A roots. These roots were likely accelerating according to our observations and we know that accelerating roots show increasing mature cell length-e.g., in Arabidopsis [START_REF] Beemster | Analysis of cell division and elongation underlying the developmental acceleration of root growth in Arabidopsis thaliana[END_REF] and in maize [START_REF] Muller | Spatial distributions of tissue expansion and cell division rates are related to irradiance and to sugar content in the growing zone of maize roots[END_REF]. A possibility is that in these roots, we were observing a transition toward a new growth regime with an elongation zone that progressively increases in length, a first root hair located at an increasing distance from the tip and cells in the mature zone with increasing lengths as classically observed [START_REF] Beemster | Analysis of cell division and elongation underlying the developmental acceleration of root growth in Arabidopsis thaliana[END_REF][START_REF] Beemster | Variation in growth rate between Arabidopsis ecotypes is correlated with cell division and A-type cyclin-dependent kinase activity[END_REF]. Notably, arrested roots also show positive slopes in the mature zone, a result which suggests the opposite situation for decelerating roots, with a progressive decrease of mature cell length as roots decelerate. A second hypothesis is that the mature zone, as identified in our study, is partly a zone with growth deceleration. This hypothesis fits with most classically reported velocity patterns showing progressive deceleration toward the end of the elongation zone in a zone that can occupy a significant length after the zone with fast elongation (see [START_REF] Sharp | Growth of the maize primary root at low water potentials. I. Spatial distribution of expansive growth[END_REF]in maize, Beemster and Baskin, 1998in Arabidopsis, Bizet et al., 2015 in poplar;Yamaguchi et al., 2010 in soybean). This zone would be longer and therefore its proportion in the mature zone higher for fast growing roots which would explain why MZ slopes in type A roots are globally higher than in type B roots.

Overall, there is a consistent match between the EZ-MZ limit and the first root hair bulge. Because the first root hair bulge is reported to occur rather before the end of elongation, at least in Arabidopsis [START_REF] Le | In the early response of Arabidopsis roots to ethylene, cell elongation is up-and downregulated and uncoupled from differentiation[END_REF][START_REF] Le | Position and cell type-dependent microtubule reorientation characterizes the early response of the Arabidopsis root epidermis to ethylene[END_REF][START_REF] Ma | Regulation of root elongation under phosphorus stress involves changes in ethylene responsiveness[END_REF], this further strengthens the second interpretation for the positive slopes detected in MZ using our method. In addition, a distinction could clearly be seen between the wild type and the auxin signaling mutants in agreement with auxin playing also a rolelikely in interaction with ethylene; see [START_REF] Ivanchenko | Ethyleneauxin interactions regulate lateral root initiation and emergence in Arabidopsis thaliana[END_REF] and [START_REF] Cho | Regulation of root hair initiation and expansin gene expression in Arabidopsis[END_REF]-in coupling cell elongation and differentiation.

What Can be Biologically Deduced from the Identification of Developmental Zones in Roots with Non-Stationary Growth or in Arrested Roots?

Whereas cell length patterns can be interpreted in a straightforward way in roots with stationary growth [START_REF] Silk | Growth patterns inferred from anatomical records: empirical tests using longisections of roots of Zea mays L[END_REF][START_REF] Fiorani | Quantitative analyses of cell division in plants[END_REF], it becomes more challenging when accelerating, decelerating or arrested roots are considered in particular in the absence of the growth rate profile along the apex. We suggest that the cell length pattern depends on the different time constants for growth rate change, on the one hand, and for cell crossing the apex, on the other hand. Based on Arabidopsis and maize data, we know that cells cross the root meristem in 50-100 h while they cross the elongation zone in a much shorter time (3-10 h; see Muller et al., 1998 for maize data andBeemster and[START_REF] Beemster | Analysis of cell division and elongation underlying the developmental acceleration of root growth in Arabidopsis thaliana[END_REF][START_REF] Verbelen | The root apex of Arabidopsis thaliana consists of four distinct zones of growth activities: meristematic zone, transition zone, fast elongation zone and growth terminating zone[END_REF] for Arabidopsis data). If we assume that time constant for growth rate changes (i.e., time needed for growth to change by + or -50%) for our maize lateral roots is 1-4 days (type C and B roots typically stop growing within 2-6 days, whereas type A roots never accelerate by more than 50% in 3 days, unpublished data), non-stationary growth would not be an issue when dealing with patterns in the elongation zone or in the mature zone. By contrast, this suggests that a snapshot of a meristem in a root with non-stationary growth gathers cells produced while the root was growing at different rates. The shrinkage of the meristem in type B roots prior to their growth arrest and the lack of meristem in most type C roots fit our knowledge of the functioning of determinate roots [START_REF] Dubrovsky | Water deficit accelerates determinate developmental program of the primary root and does not affect lateral root initiation in a Sonoran Desert cactus (Pachycereus pringlei, Cactaceae)[END_REF][START_REF] Sánchez-Calderón | Phosphate starvation induces a determinate developmental program in the roots of Arabidopsis thaliana[END_REF]. Type B and C roots would then behave as determinate roots with rapid (type C) or more progressive (type B) meristem shrinkage followed by meristem exhaustion (type C). This pattern of growth cessation in lateral roots thus differs from growth cessation in both monocot and dicot leaves in which, after the end of cell division at the base, all cells elongate reaching the same final length as those of the mature zone [START_REF] Granier | Spatial and temporal analyses of expansion and cell cycle in sunflower leaves. A common pattern of development for all zones of a leaf and different leaves of a plant[END_REF][START_REF] Muller | The elongation rate at the base of a maize leaf shows an invariant pattern during both the steady-state elongation and the establishment of the elongation zone[END_REF][START_REF] Parent | Spatial and temporal analysis of non-steady elongation of rice leaves[END_REF]. Despite non-stationary growth, other remarkable results could be observed at the population level. For instance, there was a positive correlation between DZ length and EZ length suggesting that meristem shrinkage occurs in pace with a shortening of the root growing zone.

How Can We Interpret the Changes in Residual Standard Deviation at the Limit between Developmental Zones?

The limits between successive developmental zones are explained in most cases by a concomitant change in slope and in residual standard deviation. There are at least two reasons for the concomitant change in slope and in residual standard deviation at the DZ-EZ limit. First, epidermal cells differentiate into two distinct types ultimately giving rise to root hair cells (trichoblasts) or non-hair cells (atrichoblasts). While in Arabidopsis, these cells are located in distinct cell files [START_REF] Dolan | Clonal relationships and cell patterning in the root epidermis of Arabidopsis[END_REF] and differences in length can be found right at the quiescent center [START_REF] Beemster | Analysis of cell division and elongation underlying the developmental acceleration of root growth in Arabidopsis thaliana[END_REF], in poaceae as in some other species [START_REF] Sinnott | Cell polarity and the differentiation of root hairs[END_REF][START_REF] Cormack | The development of root hairs in angiosperms[END_REF], trichoblasts and atrichoblasts belong to the same files and differences in length can be seen after the last division at the shootward limit of the meristem which is asymmetrical [START_REF] Cormack | The development of root hairs in angiosperms[END_REF]. Second, the number of rounds of division per cell file in a maize root meristem is very limited, probably no more than 3-4 [START_REF] Barlow | Cellular packets, cell division and morphogenesis in the primary root meristem of Zea mays L[END_REF][START_REF] Muller | Spatial distributions of tissue expansion and cell division rates are related to irradiance and to sugar content in the growing zone of maize roots[END_REF] with different cell files being asynchronous [START_REF] Barlow | Cellular packets, cell division and morphogenesis in the primary root meristem of Zea mays L[END_REF]. An abrupt increase in dispersion of cell length could thus be due to (i) some cells living the cycle and elongating while others accomplish a last mitosis due to asynchronicity, and (ii) not all cells experiencing exactly the same number of rounds of division. Experiencing 3 or 4 rounds will make a marked difference in terms of length and this difference will show up at the shootward limit of the meristem.

The change in residual standard deviation at the EZ-MZ limit is more challenging to interpret since it occurs after the completion of cell division which is an obvious source of cell length dispersion. Indeed, cells are not supposed to slide from one file to another at a given position from the apex. A possibility is that this abrupt increase is artefactual, due for instance to some cross walls being missed or by contrast attributed to the epidermis whereas they belong to lower layers. However, the concomitant change in slope and in residual standard deviation for a large majority of roots gives little support to this hypothesis. In earlier reports [START_REF] Goodwin | Growth and differentiation in the root tip of Phleum pratense[END_REF][START_REF] Pritchard | Biophysics of the inhibition of the growth of maize roots by lowered temperature[END_REF], a massive increase in epidermal cell length dispersion was already reported at the EZ-MZ limit in cereal roots. According to our interpretation, this change also occurs in parallel with growth deceleration. How these developmental events are generated and coordinated deserves further investigations.

Comparison between Segmented Regression Models and Multiple Change-Points Models

Segmented regression or broken-line models are regression models where the regression function is piecewise linear, i.e., made of straight lines connected at change points [START_REF] Muggeo | Estimating regression models with unknown breakpoints[END_REF]. But the homoscedasticity assumption of these models (a residual standard deviation common to the different developmental zones) is very unrealistic in our context. We thus adopted the framework of multiple change-point models which are latent structure models [START_REF] Guédon | Exploring the latent segmentation space for the assessment of multiple change-point models[END_REF](Guédon, , 2015a) ) meaning that the outputs of a model are not only the piecewise linear function corresponding to the optimal segmentation but also include the alternative segmentations and more generally various quantities of interest computed on the basis of all the possible segmentations. Contrary to segmented regression models, the piecewise linear functions corresponding to the selected segmentations are not constrained to be continuous in the context of multiple change-point models. This may be viewed as a shortcoming for biological interpretations but the counterparts of choosing the framework of multiple changepoints models are numerous: (i) heteroscedastic models can be managed which was mandatory in our context; (ii) the detection of change points is not constrained by the continuity assumption and the approximate continuity is potentially an emerging property of interest; (iii) in multiple-change point models, the inference concerns not only the selection of the number of developmental zones and the estimation of linear function parameters as in standard statistical models such as segmented regression models but also the latent segmentation space (e.g., alternative segmentations). This enables a detailed introspection of each cell length profile with many possibilities to assess biological assumptions.

CONCLUDING REMARKS

The proposed method could successfully handle roots with rather strong modulation of the developmental pattern such as arrested roots without division zone or without both division and elongation zones. Our method thus appears both robust and flexible for studying genetic and environmental effects on root development. It is potentially applicable at high throughput, given the possibility to work on epidermal tissues thus avoiding the tedious preparation of longitudinal sections.

Our results highlight a strong coordination of cell division and cell elongation for a large range of fast, slow growing or decelerating lateral roots. As expected, auxin signaling had a marked influence on both coordination between division and elongation in the meristem and between cell growth and differentiation. Our method could thus be used for revisiting the coordination of developmental processes among different cell files within a tissue (e.g., trichoblast, atrichoblast ...) or, using longitudinal sections or confocal microscopy, the coordination of developmental processes among different tissues (epidermis, cortex, pericycle, stele ...). This could be very useful to extend our knowledge of developmental regulations in longitudinally organized plant organs such as roots, monocot leaves or internodes.

FIGURE 1 |

 1 FIGURE 1 | Acquisition of epidermal cell length profiles, diversity of maize lateral roots and examples of cell length profiles collected in roots of different types. (A) Autofluorescence microphotography of a maize lateral root apex obtained as a composite of 3 different microscopy images (black background). Arrowheads in the (Continued)

FIGURE 2 |

 2 FIGURE 2 | Outputs of the selected piecewise linear model in the case of a typical vigorous lateral root (rtcs A1). (A) Optimal 3-segment piecewise linear function and first root hair position; (B) Posterior division zone (DZ), elongation zone (EZ), and mature zone (MZ) probabilities. The uncertainty intervals for the DZ-EZ and EZ-MZ limits are in gray.

FIGURE 3 |

 3 FIGURE 3 | Residual analysis of the lateral root (rtcs A1) presented in Figure 2. (A) Segmentation in 3 zones of the residual series using a Gaussian change in the variance model with the division zone-elongation zone (DZ-EZ) and elongation zone-mature zone (EZ-MZ) limits (solid lines: estimated on the basis of the original series; dotted lines: estimated on the basis of the residuals series); (B) Posterior DZ, EZ, and MZ probabilities. The uncertainty intervals for the DZ-EZ and EZ-MZ limits are in gray.

FIGURE 4 |

 4 FIGURE 4 | Outputs of the piecewise linear models in the case of a lateral root (rtcs A3) for which the 2-zone model selected by the slope heuristic and the optimal 3-segment piecewise linear function do not fit biological assumptions (lack of the elongation zone and null slope within the elongation zone respectively). (A) Optimal 2-segment and 3-segment piecewise linear functions, sub-optimal 3-segment piecewise linear function and first root hair position; (B) Posterior division zone (DZ), elongation zone (EZ) and mature zone (MZ) probabilities; (C) Posterior segmentation probabilities highlighting the difference between the 2nd segmentation and the optimal segmentation in 3 zones.

FIGURE 5 |

 5 FIGURE 5 | Outputs of the selected piecewise linear model in the case of a lateral root (rum-1 A ′ 40) for which the optimal 3-segment piecewise linear function does not fit biological assumptions (piecewise linear function not approximately continuous). (A) Optimal 3-segment piecewise linear function, sub-optimal 3-segment piecewise linear function corresponding to the 3rd segmentation and first root hair position; (B) Posterior division zone (DZ), elongation zone (EZ), and mature zone (MZ) probabilities; The uncertainty intervals for the DZ-EZ and EZ-MZ limits are in gray. (C) Posterior segmentation probabilities highlighting the difference between the 3rd segmentation and the optimal segmentation.

FIGURE 6 |

 6 FIGURE 6 | Outputs of the selected piecewise linear model in the case of 2-zone -elongation zone (EZ) and mature zone (MZ)-individual and a single-zone -mature zone only-individual corresponding to arrested or almost arrested roots. (A) Wild-type C28: optimal 2-segment piecewise linear function and first root hair position; (B) Wild-type C27: optimal linear function and first root hair position.

FIGURE 7 |

 7 FIGURE 7 | Outputs of the selected piecewise linear model in the case of a lateral root for which the distance between the EZ-MZ limit and the first root hair position is large and the MZ slope is steep (wild-type A31). (A) Optimal 3-segment piecewise linear function and first root hair position; (B) Posterior division zone (DZ), elongation zone (EZ) and mature zone (MZ) probabilities. The uncertainty intervals for the DZ-EZ and EZ-MZ limits are in gray.

FIGURE 8 |

 8 FIGURE 8 | Piecewise linear function estimated for each lateral root (types A, B, and C in red, blue and green respectively): (A) wild type, (B) rtcs and rum-1 mutants.

FIGURE 9 |

 9 FIGURE 9 | Principal component analysis applied to the 26 3-zone lateral roots: (A) Variables factor map with solid black arrows corresponding to variables (division zone, elongation zone and mature zone abbreviated respectively as DZ, EZ, and MZ, first root hair position abbreviated as hair pos., mean diameter within the mature zone abbreviated as diameter; DZ cell length and MZ cell length are shortcuts for cell length predicted at the DZ-EZ limit and at the EZ-MZ limit respectively) used to build the principal components and dotted blue arrows corresponding to supplementary variables (residual standard deviation abbreviated as s.d.); (B) Individuals factor map with wild-type individuals in green, rtcs individuals in orange and rum-1 individuals in red (type A roots are indicated by diamonds and type B roots by circles). The genotype centroids are indicated by squares using the same colors.

TABLE 1 |

 1 Multiple change-point models estimated for wild-type lateral roots.

	Posterior probability

TABLE 2 |

 2 Multiple change-point models estimated for rtcs lateral roots.

	Posterior probability

TABLE 3 |

 3 Multiple change-point models estimated for rum-1 lateral roots.

	Posterior probability

TABLE 4 |

 4 Piecewise linear functions selected for wild-type lateral roots.

			Division zone				Elongation zone				Mature zone
		Slope Correlation No. cells DZ-EZ limit Slope Correlation No. cells EZ-MZ limit Slope Correlation No. cells End position
	A10	0.8	0.12 n.s.	132	855	59.5	0.88	67	1,494	49.6	0.78	68	2,990
	A31	-6	-0.42	78	676	50.5	0.9	95	1,368	65.9	0.73	70	3,126
	A9	-5.3	-0.35	94	587	42.4	0.91	61	1,219	43.1	0.76	67	3,557
	A8	-3.2	-0.36 n.s.	17	553	59.2	0.83	34	1,100	49.4	0.77	37	2,459
	A13	-3.6	-0.24	109	510	90.9	0.87	76	973	30.1	0.35	53	2,582
	B33	-0.2	-0.01 n.s.	103	439	201.8	0.82	36	672	15.1	0.21 n.s.	59	2,198
	B32 -16	-0.53	110	411	116.3	0.81	48	716	35.2	0.44	39	1,852
	B19	11.6	0.42	30	366	118.8	0.85	31	600	21.2	0.36	36	2,283
	A11	-6.3	-0.25 n.s.	55	328	83	0.9	77	655	74.6	0.72	50	1,898
	A12	1.3	0.05 n.s.	49	320	102.5	0.91	44	722	13.3	0.33	77	2,967
	B34	-9.6	-0.4	27	278	140.1	0.72	57	461	27.6	0.55	68	2,182
	B20	1	0.02 n.s.	48	232	201.4	0.88	52	574	15.9	0.44	34	2,775
	B35 -22.1	-0.32	44	215	183.3	0.76	44	388	3.2	0.09 n.s.	79	2,771
	C25					159	0.75	18	144	30.6	0.82	110	1,961
	C28					108.6	0.53	40	115	9.4	0.26	111	1,424
	C26									28.1	0.82	133	2,115
	C27									22.7	0.72	136	2,172
	C30									27.6	0.69	170	2,214

TABLE 5 |

 5 Piecewise linear functions selected for rtcs lateral roots.

			Division zone				Elongation zone				Mature zone
		Slope Correlation No. cells DZ-EZ limit Slope Correlation No. cells EZ-MZ limit Slope Correlation No. cells End position
	A3	11.2	0.34	67	617	83.4	0.81	29	1,385	7.7	0.05 n.s.	16	2,277
	A1	8.3	0.36	92	535	59.6	0.88	60	1,185	24.5	0.34	62	3,407
	A ′	2.3	0.15 n.s.	116	506	92.2	0.93	39	1,146	25.5	0.3 n.s.	22	1,998
	A2	27.9	0.64	68	323	157.5	0.86	29	744	10.1	0.17 n.s.	35	2,257
	A ′	20.4	0.63	42	313	82.2	0.84	51	707	31.2	0.47	40	1,514
	A ′	9.7	0.51	46	272	111.7	0.82	41	505	28.7	0.49	47	1,621
	A ′					66	0.84	37	980	23.5	0.23 n.s.	27	1,844
	B15					139.5	0.96	30	520	-9.5	-0.18 n.s.	22	1,493

TABLE 6 |

 6 Piecewise linear functions selected for rum-1 lateral roots.

			Division zone				Elongation zone				Mature zone
		Slope Correlation No. cells DZ-EZ limit Slope Correlation No. cells EZ-MZ limit Slope Correlation No. cells End position
	A5	-3.3	-0.28		157	787	65	0.96	73	2,360	21.1	0.22 n.s.	29	4,090
	A7	3	0.15 n.s.	71	456	62	0.85	55	1,123	41	0.58	13	2,109
	A ′ 41	-6	-0.17 n.s.	75	452	75	0.84	61	1,246	16.9	0.18 n.s.	16	2,329
	A4	0	0	n.s.	101	399	42.3	0.8	87	1,068	6.5	0.08 n.s.	46	1,870
	A ′ 40	19.5	0.43		122	385	139.9	0.71	41	689	13	0.2 n.s.	52	2,235
	A6	-14.7	-0.57		68	371	43.1	0.83	67	958	73.7	0.64	31	2,139
	A ′ 42	26.7	0.57		73	295	145.3	0.93	33	627	2.4	0.06 n.s.	55	2,325
	C22						11.6	0.61	73	1,510	14.4	0.34	38	2,820
	C24						9.8	0.38	62	540	13.5	0.67	81	3,247
	C23						85.2	0.85	39	732	22.6	0.64	48	2,008

TABLE 7 |

 7 Piecewise linear functions selected for wild-type lateral roots.

		Division zone			Elongation zone			Mature zone
		Linear function Confidence intervals DZ-EZ limit	Linear function	Confidence intervals	EZ-MZ limit Linear function End position
	A10	5.4 → 6	(5.5, 6.5 | 5, 9.7)	855	7.3 → 45.4	(41.8, 48.9 | 45.7, 61)	1,494	53.3 → 127.6	2,990
	A31	9.3 → 5.7	(4.8, 6.7 | 4.2, 8)	676	6.1 → 41	(38.9, 43.2 | 47.7, 70.9)	1,368	59.3 → 175.2	3,126
	A9	7.8 → 5.1	(4.2, 6 | 5, 8.3)	587	6.7 → 33.5	(31.5, 35.4 | 40.2, 65)	1,219	52.6 → 153.4	3,557
	A8	7.7 → 6.7	(6.1, 7.2 | 2.4, 12.8)	553	7.6 → 40	(36.4, 43.6 | 42.7, 60.7)	1,100	51.7 → 118.8	2,459
	A13	7.3 → 5.4	(4.7, 6.2 | 3.9, 9.2)	510	6.5 → 48.6	(45.1, 52.2 | 69.2, 108)	973	88.6 → 137	2,582
	B33	6.4 → 6.3	(5.5, 7 | 4.4, 14.8)	439	9.6 → 56.6	(49.2, 64 | 68.6, 99.3)	672	83.9 → 107.1	2,198
	B32	9.3 → 3.6	(2.5, 4.7 | 6.1, 13.8)	411	10 → 45.5	(40.7, 50.2 | 60.6, 87.1)	716	73.8 → 113.8	1,852
	B19	6.5 → 8.9	(7.8, 9.9 | 7.7, 14.6)	366	11.2 → 39	(34.9, 43 | 66.6, 107.8)	600	87.2 → 122.9	2,283
	A11	7.3 → 5.5	(4.3, 6.7 | 3.6, 6.6)	328	5.1 → 32.3	(30.3, 34.2 | 44.1, 65.7)	655	54.9 → 147.6	1,898
	A12	5.9 → 6.2	(5.3, 7.1 | 4.5, 9.7)	320	7.1 → 48.3	(44.2, 52.4 | 59, 79.2)	722	69.1 → 98.9	2,967
	B34	6.4 → 4.9	(4.1, 5.7 | 1.4, 8.3)	278	4.8 → 30.5	(26.5, 34.5 | 52.7, 72.2)	461	62.4 → 109.9	2,182
	B20	11.3 → 11.5	(10, 13 | 6.5, 16.6)	232	11.6 → 80.4	(73.5, 87.4 | 88.5, 115.9)	574	102.2 → 137.2	2,775
	B35	9.2 → 5.1	(3.3, 7 | 4.8, 13.2)	215	9 → 40.7	(35.5, 45.9 | 55.1, 71.9)	388	63.5 → 71	2,771
	C25				5.2 → 25.7	(18.2, 33.1 | 20.9, 28.4)	144	24.6 → 80.3	1,961
	C28				6.3 → 16.7	(13.9, 19.5 | 25, 33.4)	115	29.2 → 41.4	1,424
	C26							20.3 → 79.2	2,115
	C27							28.5 → 77.2	2,172
	C30							28.7 → 89	2,214

TABLE 8 |

 8 Piecewise linear functions selected for rtcs lateral roots.

		Division zone			Elongation zone			Mature zone
		Linear function Confidence intervals	DZ-EZ limit	Linear function	Confidence intervals	EZ-MZ limit Linear function End position
	A3	11.9 → 18.3	(15.7, 20.9 | 17.7, 34.4)	617	26 → 90.1	(77.7, 102.4 | 63.7, 139.4)	1,385	101.5 → 108.4	2,277
	A1	7.6 → 12	(10.5, 13.6 | 13.3, 18.5)	535	15.9 → 54.6	(50.8, 58.4 | 57.1, 97.3)	1,185	77.2 → 131.6	3,407
	A ′ 36	7.5 → 8.6	(7.8, 9.4 | 7.1, 13.8)	506	10.4 → 69.4	(63.8, 75 | 51.1, 86.2)	1,146	68.6 → 90.3	1,998
	A2	5.6 → 14.6	(13, 16.2 | 10.4, 24.7)	323	17.5 → 83.8	(73.4, 94.3 | 64.3, 99.5)	744	81.9 → 97.2	2,257
	A ′ 37	5.6 → 9.1	(8.3, 9.9 | 7.1, 12.8)	313	10 → 42.4	(38.2, 46.5 | 41.6, 59.2)	707	50.4 → 75.5	1,514
	A ′ 38	6.3 → 9	(8.3, 9.6 | 2.7, 6.8)	272	4.8 → 30.8	(26.3, 35.3 | 34.2, 51.7)	505	42.9 → 75	1,621
	A ′ 39				11.1 → 59.7	(53.1, 66.3 | 34.3, 74.4)	980	54.4 → 74.7	1,844
	B15				8.9 → 79.3	(73.7, 85 | 54.9, 82)	520	68.5 → 59.2	1,493

TABLE 9 |

 9 Piecewise linear functions selected for rum-1 lateral roots.

		Division zone			Elongation zone			Mature zone
		Linear function Confidence intervals	DZ-EZ limit	Linear function	Confidence intervals	EZ-MZ limit Linear function End position
	A5	10.2 → 7.7	(6.9, 8.5 | 6.6, 12.9)	787	9.8 → 112	(106.5, 117.6 | 69, 140)	2,360	104.5 → 140.9	4,090
	A7	6.7 → 7.8	(6.9, 8.6 | 4.3, 10.5)	456	7.4 → 48.8	(43.7, 53.8 | 51.7, 91.7)	1,123	71.7 → 112.2	2,109
	A ′ 41	13.7 → 11	(9.1, 12.9 | 8.4, 16.6)	452	12.5 → 72.1	(64.9, 79.2 | 27.6, 91.6)	1,246	59.6 → 78	2,329
	A4	8.9 → 8.9	(8.1, 9.8 | 8.7, 13.1)	399	10.9 → 39.2	(36.1, 42.3 | 38.5, 61)	1,068	49.8 → 55	1,870
	A ′ 40	9.9 → 17.4	(15.7, 19.1 | 12.2, 25.4)	385	18.8 → 61.3	(52.1, 70.6 | 58.4, 86.4)	689	72.4 → 92.6	2,235
	A6	12 → 6.6	(5.6, 7.6 | 4.4, 8.4)	371	6.4 → 31.6	(28.8, 34.5 | 20.1, 59)	958	39.5 → 126.5	2,139
	A ′ 42	7.3 → 15.2	(13.5, 16.9 | 8.9, 15.3)	295	12.1 → 60.3	(55.4, 65.3 | 51.1, 72.1)	627	61.6 → 65.7	2,325
	C22				13.2 → 29.2	(26.6, 31.9 | 45.5, 65)	1,510	55.2 → 74.1	2,820
	C24				15.4 → 20.7	(18.7, 22.7 | 19.5, 28.4)	540	23.9 → 60.6	3,247
	C23				8 → 53.5	(47.1, 59.8 | 34.1, 45.5)	732	39.8 → 68.7	2,008

Frontiers in Plant Science | www.frontiersin.org

October 2017 | Volume 8 | Article 1750

October 2017 | Volume 8 | Article

ACKNOWLEDGMENTS

The authors thank Laurent Laplaze for helpful comments.

FUNDING

This work was funded by the European FP7 project EURoot and BMO Ph.D. was partly funded by INRA (Environment & Agronomy division).

AUTHOR CONTRIBUTIONS

BMO collected the data, analyzed the data and contributed to the writing of the manuscript, GF collected the data, BM designed the study, analyzed the data and wrote the manuscript, YG designed and implemented the statistical models, analyzed the data and wrote the manuscript.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpls.2017. 01750/full#supplementary-material

Conflict of Interest Statement:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Psychological Evaluation of Human Choice

Behavior in Socio-Technical Systems: A Rational Process Model Approach Tim Schürmann (&) Technische Universität Darmstadt, Darmstadt, Germany schuermann@psychologie.tu-darmstadt.de

In the age of digital services and everyday smartphone usage, the issue of online privacy has gathered more and more interest for researchers, service providers and consumers. Assuming one's digital information is private is equivalent to trusting service providers to handle one's data in a certain way or ensuring protective measures against loss of privacy. When a consumer registers for an online service or installs a smartphone app, I assume an internal psychological process to relate the benefits of their decision to the risks associated with it. However, this process is considered to be subject of uncertainty. Therefore, decisions in a socio-technical environment can be viewed as decisions governed by a probabilistic amount of trust in an outcome, or, in other words, the amount of belief one holds that a hypothesis about future events will turn out to be true. Previous research on human online behavior paints a fairly bleak picture of how we handle said uncertainty. It often adopts the paradigm of the Homo Heuristicus [START_REF] Gigerenzer | Homo heuristicus: why biased minds make better inferences[END_REF], relying on computational shortcuts rather than normatively rational inference. In a scenario as complex as online privacy, it also points out how unlikely it is for users to have a complete understanding of the capabilities and motives of involved parties [START_REF] Acquisti | Privacy and rationality in individual decision making[END_REF].

However, psychological research on broader decision making processes includes evidence that humans are in fact able to combine information in a rational sense [3]. The Sampling Hypothesis [START_REF] Vul | One and done? Optimal decisions from very few samples[END_REF] may provide the grounds for unifying research on heuristic or otherwise boundedly rational decision making on one hand with a rational account on the other. It does so by approximating Bayesian inference, sampling from probability distributions over possible hypotheses or parameter values instead of using these full distributions and creating implausibly complex computations. Its application shows that specific effects like the availability heuristic can actually be considered by-products of its sampling process [START_REF] Sanborn | Bayesian brains without probabilities[END_REF]. Vul [START_REF] Vul | One and done? Optimal decisions from very few samples[END_REF] provides evidence that in many situations, sampling only a very limited number of times provides a computationally similar result to using full yet analytically intractable probability distributions. Specifically, he links the benefits of sampling to the consumption of energy and time while arriving at a decision: why make one time-and energy-consuming decision perfectly maximizing my chance of success, when I can make many "good enough" decisions that approximate optimal results in the long run? This globally optimal solution however can produce seemingly irrational local behavior. Models that utilize such approximate Bayesian inference are termed rational process models [START_REF] Sanborn | Rational approximations to rational models: alternative algorithms for category learning[END_REF].

It appears as though human subjects, while certainly limited in their cognitive resources and computational capabilities as laid out by the bounded rationality paradigm, may make use of this process: they operate by maximizing success chances and making rational choices, but on a global rather than local level. My work utilizes this type of model to investigate how humans make decisions online, and more importantly how to sensitize them to make more adequate decisions to protect their private information. A preliminary study [START_REF] Schürmann | Adoption behavior of smartphone apps gathering private data is explained by a sampling-based rational process model[END_REF] indicates that answers to these questions are not as simple as pointing to a specific heuristic approach or a systematic gap between privacy-related attitudes and behavior. When asked whether they wanted to install a travel-related smartphone app involving beneficial and non-beneficial features, subjects showed behavioral patterns that were predicted by a rational process model. Preference trade-offs for the app's features form the basis of the model prediction as a posterior distribution. Then, sampling from said individual posterior provides the model with an approximate probability of choosing to install the app. The model stochastically chooses the option with higher utility according to its probability. It therefore allows for a seemingly irrational decision on the local level when choosing the option with lower utility instead. The rate with which subjects chose their higher utility option or deviated from it was predicted by the model, with a deviation of approximately 5% between its prediction and the empirical data. This deviation is not significantly different from zero, as indicated by a Bayesian estimation of the difference parameter between the two.

The model seems to capture the process with which subjects combine preferences about features as well as their trade-off between utility maximization and cognitive resource management. It is based on subjective utility distributions, thus avoiding the assumption of complete situational knowledge proposed in previous research [START_REF] Acquisti | Privacy and rationality in individual decision making[END_REF] to arrive at a rational decision. These subjective utility distributions in turn can be learned solely based on past experience [START_REF] Srivastava | Learning what to want: context-sensitive preference learning[END_REF]. It is worth noting that heuristic or probability-weighted alternatives of the model, following a cumulative prospect theory (CPT) approach, could possibly have resulted in a decent statistical fit as well. CPT's parameter estimation [START_REF] Boos | Probabilistic inference: task dependency and individual differences of probability weighting revealed by hierarchical bayesian modeling[END_REF] would likely capture stochastic variations descriptively if it was retrospectively fitted for a specific individual and trial. It would not, however, explain the nature of the variation or the necessity of the sampling process on theoretical grounds. Meanwhile, the rational process model approach outlined here unites the idea of Bayesian computational rationality in human cognition with limitations on the algorithmic level [START_REF] Marr | Vision: A Computational Investigation into the Human Representation and Processing of Visual Information[END_REF]. Additionally, it allows for an explanation of other phenomena observed in decision making research, like probability matching.

Based on the preliminary study, I plan to first adapt the model to other interactions with socio-technical systems. Secondly, I will explore specific mechanisms of the model to apply them to privacy interventions. For example, increasing the number of samples drawn in the model increases the chance of choosing an option with higher utility, instead of sometimes choosing a lower utility option. This may be achieved by asking subjects to state their choice repeatedly. Assuming a privacy-protecting decision (not installing an app that requires permissions to access private data) is a subject's higher utility option, an intervention increasing their internal sample count should result in a higher probability of choosing that option. However, there is a chance that they favor a privacy-disclosing option. In that case, an intervention designed to increase sampling counts might reinforce the tendency to pick the disclosing option, resulting in the opposite of the intended purpose. Future work will draw inspiration from how the mechanisms of sampling in decision making work to design privacy-protecting interventions tailored to individual preferences and thereby making use of the human tendency to operate on a globally rational level of information integration. Building on these rational process mechanisms, I aim to assess and direct user trust in the interaction with socio-technical systems as well as explain stochastic deviance from their expected behavior.

Psychological Evaluation of Human Choice Behavior