Because of increasing global urbanization and its immediate consequences, including changes in patterns of food demand, circulation and land use, the next century will witness a major increase in the extent of paved roads built worldwide. To model the effects of this increase, it is crucial to understand whether possible self-organized patterns are inherent in the global road network structure. Here, we use the largest updated database comprising all major roads on the Earth, together with global urban and cropland inventories, to suggest that road length distributions within croplands are indistinguishable from urban ones, once rescaled to account for the difference in mean road length. Such similarity extends to road length distributions within urban or agricultural domains of a given area. We find two distinct regimes for the scaling of the mean road length with the associated area, holding in general at small and at large values of the latter. In suitably large urban and cropland domains, we find that mean and total road lengths increase linearly with their domain area, differently from earlier suggestions. Scaling regimes suggest that simple and universal mechanisms regulate urban and cropland road expansion at the global scale. As such, our findings bear implications for global road infrastructure growth based on land-use change and for planning policies sustaining urban expansions. Today, the road system grooves and fragments the 19 million hectare surface of the Earth with more than 14 million km of paved surface. At the backbone of human colonization, road expansion embodies a complex blend of economic growth and often unsustainable development . From anStatistical laws governing road networks have been extensively explored ,12. Such2.The GRN has been extracted from commercial vectorial maps of major roads on the Earth that are used mostly for navigation and cartography . The GRN contains the major roads network for 2015 organized in four hierarchies: primary roads with limited access (H1), primary roads with non-limited access (H2), secondary roads (H3) and local roads (H4). The GRN does not contain local urban roads but only the major ones; therefore, the GRN can be used to analyse the major road infrastructure but not the urban morphology related to urban block size and form. The total road length deduced from the GRN database (14 522 470 km in 2015) is much larger than that estimated from a dataset recently used for global road environmental impact estimation (gROADS) , which aprimal road network [l). To extract the final GRN network, we removed street junctions connecting only two roads and split each link, regardless of hierarchy, at any intersection with three or more roads. Defining three layers of major land uses—urban, cropland and seminatural—we labelled each road with the land use it belongs to. These three land-use classes, corresponding to the three main land uses on the Earth, are extensively used in the global land-use literature [U) if they totally belong to urban areas, cropland (C) if they totally or partially overlap a cropland area and seminatural (Sn) if they are free of any agricultural or urban land use . Figure 1a shows a global overview of the GRN and the three classes. It is important to note that the adopted labelling methodology allows us to avoid multiple land-use associations and treat croplands differently, as a single road segment can connect more than one cropland unit. A detailed description of the dataset along with an atlas of detailed visualizations and spatial analysis methodologies are provided in Material and methods and electronic supplementary material. Key to our spatial inventory at the global scale [Starting from the original database, we produced the GRN as a  network , in whicterature . We extrterature  and cropterature . We thenal scale ,25, we sal scale , data frl) of road segments [l〉=4.8 km (with standard deviation (s.d.) 8.9 km).We here focused on a seemingly simple yet fundamental road network feature: the lengths . The mean road lengths in the different land-use classes are 〈l〉U=1.2 km (s.d. 1.3 km), 〈l〉C=7.4 km (s.d. 9.0 km) and 〈l〉Sn=7.0 km (s.d. 12.0 km). The large standard deviation values highlight an important feature of road length distributions: these distributions are heavy-tailed and potentially reminiscent of power laws [C and Sn roads are generally longer those U roads. Indeed, the distribution of U road lengths is very different from that of C and Sn segment lengths. Expectedly, cities encompass shorter streets than agricultural or seminatural areas. But apart from these different mean lengths, how fundamentally different are these distributions? Are they possibly rescaled versions of the same universal distribution, obtained by varying only the mean road length? To test this hypothesis, we examined whether road length distributions can be described by a finite-size scaling form [l〉 is the mean road length in the land-use class of interest and the function F(⋅) is identical across classes. Normalization of the distribution requires that the exponents γ and α must satisfy α=1/(2−γ) [F(x) must satisfy appropriate limiting behaviours as x→0 and p(l) by data collapse [lγp(l) versus l/〈l〉α for each land-use class separately and varying α until the curves describing each class collapse onto the same curve, thus providing a plot of the scaling function F. Figure 1c shows the plot of p(l) for each land use and figure 1d shows the resulting plot with α=1 and γ=1 . We find that road length distributions associated with urban and cropland classes collapse onto nearly the same curve, whereas the road length distribution associated with seminatural areas fails to do so. Different definitions of urban boundaries and different methods for the classification of roads crossing land-use boundaries led to indistinguishable results . Therefore, road length distributions in urban and cropland areas share the same fundamental structure at the global scale, despite large differences in the mean road length.The wer laws , and thuing form –34:2.1pcollapse , i.e. bya,d), finding different scaling behaviours in urban patches compared to cropland ones as explained below.Equation  describefigure 2l | A〉U within an urban patch of area A increases sublinearly with A, 〈l | A〉U∝Aδ with δ=0.41±0.02 (mean ± s.e. estimated via least squares fit of log-transformed data) for areas below AUT≃4×107 m2, above which it remains relatively constant . The distribution pU(l | A) of urban road lengths conditional on the urban patch area A (b) appears to be well described by the scaling form:GU is a scaling function with suitable properties [a,b). For each patch, we computed the total road length L and found that the mean total road length in urban patches increases sublinearly with A, 〈L〉U∝Aβ with β=0.62±0.01  below AUT, above which it becomes effectively linear (β=1.06±0.02) (c). As 〈l | A〉U is relatively constant above AUT, the linear scaling of 〈L〉U implies that the mean total number of roads increases linearly with A above AUT.We found that the mean road length 〈constant b, inset.h area A b appearsoperties , as veri06±0.02) c. As 〈l l | A〉C associated with a cropland patch of area A displays a triphasic behaviour . Initially, 〈l|A〉C increases until A≃105 m2, although very few crop patches are found below this scale and we will thus neglect these data in our discussion. Then, 〈l | A〉C decreases until ACT≃109 m2 and remains relatively constant above ACT. The distribution pC(l | A) of cropland road lengths conditional on the cropland patch area A (d) appears to be well described by the scaling form:GC is a scaling function with suitable properties [figure 2d,e. Analogous to the distribution of urban road lengths, the distribution pC(l | A) is invariant for all cropland areas larger than ACT. Conversely, the mean total road length in cropland patches (figure 2f) is approximately constant below ACT and increases linearly above this threshold , implying that the mean total number of cropland roads also increases linearly with A above ACT. We thus found that the average road length versus cropland patch area is well described by 〈l|A〉C with ACT=(7.4±0.5)×107 m2.Road length statistics in cropland patches also display multiple scaling regimes, although with different behaviour compared to urban areas. Specifically, the mean road length 〈ehaviour e, inset.h area A d appearsoperties , as demoGC and GU related? Superimposing the two scaling functions  suggests that GC≃GU, such that equations HH into 10 log-binned intervals (Ak) to account for variable numerosity of the samples [l}Ak) belonging to the faces whose areas are included in the kth face area bin  and computed the relative proportion p(l | A)=p(l | Ak) that measures the probability to find a road of a given length in the area bin Ak)(. We then tested whether the curves lγp(l | A) plotted against l/〈l | A〉α (where 〈l | A〉 is the average road length in the set {l}k) for each of the area bins collapse onto the same curve, i.e. whether p(l | A) displays finite-size scaling . By using the full dataset, a satisfactory collapse has been found for α=1.1 and γ=1.1 as shown in figure 4d,e. Such collapse indicates a universal scaling curve that regulates the fragmentation of the road network at all scales of observation.The scaling structure of a road network can be also investigated by observing the recursive fragmentation dependent on the hierarchy of roads. The GRN subdivides the surface of the Earth into non-overlapping regions called segments ). We def samples . We coll3.α=γ=1)F(⋅) is independent of the land-use class, with land use instead impacting the mean road length 〈l〉. At finer scales, we investigated the dependence of road length distributions on the area of the homogeneous patch under study, separately for each land-use class. We found that road length statistics conditional on patch area closely match a universal scaling law of the formG(⋅) is independent of the land-use class (i.e. G=GU=GC). That is, land use affects p(l | A) solely through the scaling of the mean road length 〈l | A〉 with patch area A.Our study provides evidence for general statistical laws describing global road lengths conditional on land use. At the global scale, urban and cropland roads share a universal distribution after rescaling that satisfies  and pC(l | A) are invariant for all cities and cropland patches larger than such a threshold. In particular, because F and G are quite similar in shape and nearly coincide if one neglects small urban and cropland patches, i.e. below the critical sizes AUT and ACT . The observation of nearly universal probability distributions of urban and cropland road lengths suggests that the processes that govern road expansion are to a large extent inevitable, regardless of climate, topography and social and economic constraints, echoing general results on self-organized patterns [Equation  has intepatterns ,38. Proppatterns .These results suggest that local conditions, such as the socio-economic development or the demography of a specific region, may simply accelerate or delay the development of road infrastructure, affecting the characteristic length scale of the road length distribution but not its scaling form. Significantly, we find that road length distributions belonging to seminatural areas do not collapse onto the same distribution, even after rescaling. We argue that such diversity stems from the diverse purpose of roads in natural areas which are not built specifically for direct access to the land and are therefore regulated by diverse, site-specific processes and possibly far from optimized. The linkage of optimality and self-organization is known to induce a variety of scaling phenomena, as shown by studies on network structures derived from selection principles ,39–42, iL) scales sublinearly with the total population of a city (P) within its boundary, i.e. L∝Pη with 0.7<η<0.9 [figure 2c, an urban connected component of 109 m2 and ten components of 108 m2 together require approximately the same total length of major roads. This holds for all urban patches bigger than AUT≃4×107 m2, thus including all major cities. This result is consistent with other recent studies performed for a set of cities in Europe and the USA [L and population may have been an artefact of census-based data, which are built regardless of land use within the political boundary of a given region. Therefore, it seems plausible that roads from different land-use classes, with correspondingly different characteristic lengths, may have been mixed. Moreover, census areas are defined differently among various countries, thus precluding cross-country comparative analyses. Indeed, it has been shown that different methods to define city boundaries, based on tuning urban population density, could drastically affect the scaling of urban measurements [Urban scaling approaches have suggested that larger cities require less infrastructure than bigger ones. Indeed, if the total road length . As it has been estimated that by 2030 the total urban area, AUtot, could potentially grow by 200% [Ltot of 3 485 400 km of new paved urban roads.Although the above implications are in need of deeper scrutiny, which should include scaling analyses of different urban measurables  require the same investment in road paving as n smaller cropland patches of area A/n, provided that A/n>ACT. From this perspective, larger cropland patches are predicted to be as efficient in terms of road infrastructure requirements as smaller ones (within the linear regime highlighted in figure 2d). By contrast, cropland areas smaller than ACT require a relatively greater length of roads . Such cropland patches are typically composed of several small and scattered cropland units located in wild or mountainous areas. A typical example is given by scattered agricultural plots in forest areas ; in such configurations, long roads serve tiny cropland units, indicating potentially higher environmental impacts of sparse agricultural expansion. Meanwhile, cropland patches around urban areas are generally more compact and more fragmented, in the linear regime. Because cropland roads can connect more than one cropland patch, however, the proportion NC ∝ACtot∝LCtot would overestimate LCtot. Moreover, by considering the competition among cropland patches and seminatural areas and the scarcity of residual free land to colonize [Ltot. Such a hypothesis is corroborated by the similarity of the distributions p(l)C and p(l)Sn shown in figure 1c, which implies a similar level of fragmentation of seminatural areas. An important distinction arises therefore between urban and cropland areas in that the site-specific degree of reuse of existing infrastructure in cropland areas prevents the same kind of strict predictability allowed for urban areas.Our approach also suggests that cropland patches smaller than colonize , future 4.4.1.The general idea here is to transfer the land-use information, represented by continuous values, onto the road network, which is represented by lines vectorial geometry. To do so, a series of spatial analysis operations have been performed, mainly in ArchMap, Qgis and Python environments. An illustration of the data preparation schema is presented in the electronic supplementary material. Data preparation includes the extraction of the urban footprints, a preparation of the road network and a join phase between land uses and the road network. Below we report details of each phase; in the electronic supplementary material, we report detailed visualization of the final road network.4.2.https://www.ngdc.noaa.gov). The NTL data contain continuous values from D=1 to D=63, representing the intensity of stable light in a grid of 30 arc seconds (approx. 1 km). DMSP-OLS have been widely used to characterize urban footprint and urban evolution at the global scale [D>28. We tested other classification methods as well as different numbers of classes; but given the sharp separation between highly and poorly illuminated places, different values of D did not affect the main results of the scaling analysis . Using the same procedure, we classified urban footprints from 1997 to 2012.Urban footprint data have been extracted from inter-calibrated night-time light (DMSP-OLS) remote imagery using original images and data processing by NOAA’s National Geophysical Data Center and DMSP data collected by the US Air Force Weather Agency . Detailed visualizations of the cropland mask are provided in the electronic supplementary material.The 1 km global IIASA-IFPRI  cropland4.4.https://developer.garmin.com/datasets/digital-atlas/. The topology of the road network has been corrected using Archmap software and ad hoc Python scripts for the purpose of joining connected roads at junctions with only two roads and to remove small road links representing highway ramps and crossroads intersections which are not representative of any fragmentation process and are potential noise for the statistical analysis. Roads shorter than 100 m cover only ≈0.03% of the total road length; these roads, as confirmed by an extensive and scrupulous inspection of the GRN dataset, appear to be highway ramps or road segments for large road junctions and were therefore excluded from our analyses. Coupling the GRN with two global land-use inventories, roads have been classified into three categories: urban roads, i.e. roads that entirely belong to urban areas; cropland roads, i.e. road that intersect or belong to cropland areas; and seminatural roads, i.e. roads that are completely free of direct urban or cropland use . Data of road network are released under licence agreement with DeLorme .Geo-located vectorial road data are usually produced for GPS navigation and cartography. Our dataset was produced by and acquired from DeLorme . These data represent a complete and updated dataset of primary and secondary roads at the global scale. These data have commercial restrictions; additional information and request for data samples can be addressed to the contact details at 4.5.p(l)=lγ−F(l/〈l〉α) requires that the exponents γ and α satisfy α=1/(2−γ) [F(x) must satisfy appropriate limiting behaviours for x→0 and p(l) by data collapse [lγp(l) versus l/〈l〉α for each land-use class separately and varying α until the curves describing each class collapse onto the same curve, thus providing a plot of the scaling function F(⋅). We similarly verified our scaling ansätze .Normalization of the distribution =1/(2−γ) . Furtherx→∞ ) o) op(l)=lquations d, inset.4.6.F and G is mediated by the distribution of patch areas p(A), via A<AUT, we findl|A〉 on A that holds approximately for A>AUT. Equation  with the distribution of urban road lengths conditional on the urban area being A>108 m2 . Figure 3b shows that the two distributions overlap almost completely, implying that equation (figure 2e). This unintuitive result appears to be due to long roads fragmenting wild areas for agricultural purposes. Therefore, the tail of the ensemble distribution of cropland road lengths (green solid line in figure 1c and green dashed line in figure 3b) is ‘fatter’ than the tail of the cropland road length distribution conditional on cropland area A>109 m2 .The relationship between the scaling functions Equation  for A>AU


We report on a wavelet based space-scale decomposition method for analyzing the response of living muscle precursor cells (C2C12 myoblasts and myotubes) upon sharp indentation with an AFM cantilever and quantifying their aptitude to sustain such a local shear strain. Beyond global mechanical parameters which are currently used as markers of cell contractility, we emphasize the necessity of characterizing more closely the local fluctuations of the shear relaxation modulus as they carry important clues about the mechanisms of cytoskeleton strain release. Rupture events encountered during fixed velocity shear strain are interpreted as local disruptions of the actin cytoskeleton structures, the strongest (brittle) ones being produced by the tighter and stiffer stress fibers or actin agglomerates. These local strain induced failures are important characteristics of the resilience of these cells, and their aptitude to maintain their shape via a quick recovery from local strains. This study focuses on the perinuclear region because it can be considered as a master mechanical organizing center of these muscle precursor cells. Using this wavelet-based method, we combine the global and local approaches for a comparative analysis of the mechanical parameters of normal myoblasts, myotubes and myoblasts treated with actomyosin cytoskeleton disruptive agents . Among these three filament networks, the actin filaments are involved in many mechanical processes such as cellular reshaping, locomotion, substrate adhesion, phagocytosis and plasma membrane compartmentalization3, they henceforth have been assigned the role of active CSK organizer. Actin polymerization and actomyosin dynamics produce the driving motile force of eukaryotic cells 4, they are both driven by ATP. Actin dynamics is tightly regulated in time and space by a considerable number of actin binding proteins (ABPs). Genetic defects and abnormal expression of ABPs are often related to congenital and acquired human diseases confirming their critical role in actin CSK dynamical regulation6. MFs are tracks for their ATP-driven myosin molecular motors. Among myosins, non-muscle myosins II (NMM II) are the principal actin CSK regulatory proteins7; they have an important role in cell shaping and motility8. The actomyosin apparatus acts as a mechanical tensor in the mechanical coupling of the CSK to the extra cellular matrix (ECM) via focal adhesions (FAs), in mechanotransduction of external stresses to the nucleus9, and in exertion of resistance against forces3. In particular, ventral stress fibers have a key role in mechanosensing10 and can be classified in (i) peripheral stress fibers running along the edges of adherent cells, and (ii) perinuclear stress fibers drapped over the nucleus11. Perinuclear caps have a protective and mechanical confining role for the underlying nuclei. Given that the nuclear membranes and their adjacent lamina network are very sensitive to disruptions and deformations, perinuclear caps are the guardians of their mechanical stability, ensuring a correct chromatin organization and assisting the cell cycle timing and nuclear machineries involving DNA12. Soft perinuclear zones withstanding rather large deformations without CSK rupture confer to the cell a ductility upon deformation and assist its shape recovery. Conversely, highly tensed perinuclear zones propitious to localized failures (brittle) by disruption of cross-linked CSK domains, impede a complete shape recovery after deformation. To distinguish and quantify these two situations, we took, as cell models, muscle precursor cells, namely myoblasts (C2C12) and their differentiated form in myotubes, and we tested their proneness to ductile or brittle failures in normal and altered growth media.Living cells are active mechanical machines which can withstand forces and deformations and can adapt quite rapidly to their mechanical environment. This malleability is mediated by three major cytoskeleton (CSK) filament networks, namely microtubules (MTs), actin filaments (F-actin), and intermediate filaments (IFs)13. C2C12 myoblasts can also be differentiated into adipocytes or osteoblasts when stimulated with suited nuclear transcription factors and other molecular cues15. When forced to adhesion on solid surfaces, myoblasts exhibit the characteristic spindle-shaped morphology, typical of mesenchymal cell lineage . Their morphology differs notably from the one of their differentiated myotubes which adopt elongated tubular shapes . Note that a small percentage of spindle-shaped unfused myoblasts can still be found among myotubes after 5 days of differentiation in DM. Adherent cells such as C2C12 myoblasts are mechanosensitive which means that the cell substrate for adhesion has a critical impact not only on their response to different stimuli, but also on their development, differentiation, disease, and regeneration16. While pulling on their sticky environment, these cells are capable to probe its elasticity and to a certain extent adapt their own elasticity to their environment16. C2C12 myoblasts can moreover adjust their myogenic differentiation to the substrate stiffness18. For the majority of tested cells requiring an anchoring on a surface, softer substrates lead to higher cell motility while stiffer substrates normally produce greater spreading and cell contractility20. Collagen gel coated surfaces are an exception in this context since, although classified as soft surfaces, they may still lead to maximal cell spreading21.C2C12 myoblast cells are immortalized cells derived from mouse satellite cells that can be switched to differentiation into myotubes by replacing their proliferation growth factor rich medium (GM) by a growth factor deprived medium (DM). After a few (~5) days in DM, confluent differentiated myoblasts fuse spontaneously and form syncitia of multinucleate myotubesage Fig. Top. Thepes Fig. Bottom. 24. We chose a nano indentation technique 32 to probe the mechanics of living myoblasts for its high spatial resolution and sensitivity. With this technique, the elastic (Young) modulus E of living cells was previously found to range from a few hundreds of Pa to hundreds of kPa. Varying the shape of the indentation probe tip can yield quite different Young modulus estimations33. Sharper tips  produce a greater and more localized shearing and hence lead to higher Young’s modulus than spherical tips34. They are better suited to probe local  mechanical properties36 and to investigate local perturbations including disruptions of the CSK network. Spherical tips are instead used to estimate more global cell mechanical properties33. The Young’s modulus of muscle cells increases from myoblasts (the softest) to smooth, skeletal and cardiac muscles (the stiffest). This variability reflects also their strong adaptability to mechanical constraints and the variety of their in vivo organic functions. Actually, while myoblasts rather need high motility and deformability to migrate through the muscle tissue, cardiac or skeletal muscle cells instead require higher resistance against mechanical stress. Sharp  indenters are better suited for the characterization of the spatial inhomogeneity of cell mechanics, and more precisely of their stress fiber resistance to deformation. This explains that we selected very sharp AFM tips  for the present study. Beyond a global characterization of the elasticity and viscosity of muscle precursor cells which can be estimated either from the work integrals of approach and retract force curves37 (as done here), or from cantilever position modulation experiments40, our study also focuses on local disruption events which have seldomly been evoked in such a context. Whereas the study of fracture mechanics and crack propagation in solid materials was developed during the beginning of the twentieth century41, these concepts were applied only recently to living organisms at large scales, e.g. bone fractures43, and there are still a few evidences of such phenomena in living cells44. A fracture occurs inside a strained material to release locally the strain, leading ultimately to a plastic irreversible deformation of the material (nonlinear regime). Upon deformation, an elastic material will store the elastic strain energy, up to the point of failure where this energy will be dissipated by plastic deformation or surface energy. In the case of living cells, this strain energy can also be released by viscous effects. As far as soft material mechanical failures are concerned, a lot of works have been performed on soft glassy materials or biopolymer gels. These failures originate from nonlinear elastoplastic or viscoplastic disassembly of cross-linked networks47. The global viscoplasticity of cells under large deformations has been studied by micron-scale stretching devices48. At these sub-micron scales, it appears that the destabilization of microtubules is more important and plays a major role in the viscoplasticity of cells. At nanometer-scales, and with a highly sensitive technique such as AFM, the local CSK actin filament network cohesion and its prestress determine the cell elasticity and its resilience to deformation49. The amplitude and strength of the local rupture events are not detectable by instrumental techniques performed at the level of the whole cell; this may explain why these behaviors were not much reported in previous works. In our experiments, we focused on the perinuclear region not only because it is the thickest part of an adherent cell but also because this area can be considered as a master mechanical organizing center. More precisely, we addressed the perinuclear actin cap firmness as a criteria for cell contractility and wellness of their actomyosin machinery. The muscle cell precursors were the best candidates to perform such a demonstration.Single cell approaches have been widely developed in the past two decades to evaluate the mechanical properties of cells and their interplay with the extracellular environment, both in physiological and pathological conditions50  until a defined cantilever deflection value (i.e. a set-point force) was reached (position Zsp)  back to its starting Z position  which is related to the second derivative of the force indentation curves (FICs) (see the Section Methods and the Supplementary Information)54. More details on the cantilever calibration and the FICs can be found in the Supplementary Information. From the loading and unloading FICs, different parameters could be retrieved, such as the dissipation loss Dl  the cell behaves like a strain-softening material. Inversely, if Gg > Gi  the cell behaves like a strain-hardening material.AFM is a versatile technique which has been specifically used to collect the mechanical parameters of living adherent C2C12 myoblasts and myotubes50 Fig. . We used50 Fig.  was move Dl (Eq. ), the in(Gi (Eq. ), the gl(Gg (Eq. ). Rapidlw55  or stretching and activation of nonselective cation channels (Ca2+ ion influx through the channels). This Ca2+ ion influx can occur both on the zone pinched by the cantilever tip or at some distance, corresponding to increased membrane tension. The typical time for a force-induced calcium influx was shown56 to be tCa ~ 2 s. Importantly, this accumulation of free Ca2+ ions was shown to precede a perinuclear actin assembly. When the initial stretch was not maintained, the increase of perinuclear actin indeed reached a maximum within about tactin ~ 20–30 s. The remodeling of the actin cytoskeleton around the nucleus was therefore found56 to occur over time intervals longer than tCa. This remodeling may involve specific actin binding proteins such as formins (INF2) that will reinforce mechanical protection of the nucleus by the perinuclear actin network57. If the reinforcement of the nuclear actin rim is triggered by Ca2+ ion influx, this process will likely lead to an apparent shear thickening of the force-indentation curves and to a possible amplification of the observed rupture events.By indenting the cells with sharp pyramidal tips, their extracellular membrane is also locally sheared and CaGi, Gg, Dl and Wg/Wi (See the Section Methods). The red plots were constructed from a set of 54 C2C12 myoblasts, with 30 FICs being recorded on each cell. The blue plots correspond to C2C12 myotubes, again with 30 FICs recorded on each cell. The two histograms of log10(Gi) and log10(Gg)  and their medians are more representative of the cell mechanical deformability; the myotubes behave more closely to strain-softening materials (51% of the FICs have Wg/Wi < 1), whereas the myoblasts behave more likely as strain-hardening materials (21.5% only of the FICs have Wg/Wi < 1).Figure Gg) Fig.  show thabes Fig. . As far pes Fig. , with a 60, the passive fluctuations being overwhelmed by the active fluctuations in the low frequency range (< a few tens of Hz)63. ATP deprived solutions  were also shown to stiffen differentiated muscle fibers into a state similar to “rigour mortis” via the blockage of the muscle myosin II on actin filaments65. Without ATP, cross-linking of actin microfilaments by ADP-non muscle myosin II (ADP-NMMII) “freezes” the actomyosin network66. A complete ATP depletion (see the Section Methods for further details) can cause a 100-fold increase of the elastic modulus of the actomyosin network66. Interestingly, ATP depleted cells loose their typical intracellular actin organization but they retain their initial morphology (prior to depletion)69. Contradictory experiments also showed that ATP depleted cells could behave as a softer material71. Blebbistatin inhibits completely the NMMII motor activity by slowing down the phosphate release after ATP hydrolysis, thus setting NMMII in a weak actin-binding state and disassembling the stress fibers and the focal adhesions72. Fluorescence staining of the MT and MF corroborated these reported observations  and ATP-driven (active) fluctuations on living cell rheologyGg〉 = 0.79 ± 0.10 kPa , and there is clearly a shift of the Dl histogram to larger values, with only a small percentage (<25%) of the FICs with a mean Dl value close to 0.4, typical of normal myoblasts, and a larger percentage (>50%) with Dl values above 0.5. Their ratio 〈Wg/Wi〉 is also very interesting since it increases dramatically to 3.7 ± 0.7, the fraction of FICs with Wg/Wi < 1 dropping down to 2%. ATP depleted myoblasts are much softer, they dissipate more energy and surprisingly they behave as strain-hardening materials.The global mechanical parameter distributions for ATP depleted myoblasts are reported in Fig. kPa Fig.  and 〈Gi〉kPa Fig. . These cGg〉 = 0.79 ± 0.10 kPa and 〈Gi〉 = 0.19 ± 0.06 kPa. But the distribution of dissipative loss Dl is quite different, since it decreases markedly to very low values (〈Dl〉 = 0.27 ± 0.12), more than half of the FICs presenting Dl values around 0.2   for the same set of cells as considered in Fig. Dl or Wg/Wi with a colour dot, its error with a vertical line and for each interval we added a circle with a size proportional to the percent of FICs corresponding to this interval (in Gi or Gg). Considering the most probable values of Dl and Wg/Wi (highlighted by the largest circles), we observe that, for all cell types, Dl decreases very slightly with Gi < 2 kPa), and that these plateaus match the Gi values of the histogram maxima in Fig. Dl vs Dl vs λ ~ −0.2, meaning that these two quantities are strongly correlated for the myotubes . Indeed Dl with a characteristic decay factor |λ| = 0.2.To elaborate on possible cross-correlations of the global mechanical parameters, we plotted (logarithmic representation) in Fig. 0Gi Fig.  .Another amazing alignment is found on the kPa Fig. , and thiGg) Fig. , we then0.1 Fig. . This poues Fig. . This di/Wi Fig.  and \docGg) Fig.  are far Wg/Wi vs Gi for small values of Gi  was very attractive because it suggested that we could build a general model for the mechanical response to strain of these cells. But we failed to extract a simple relation between the dissipation loss Dl and the global relaxation modulus Gg for myoblast cells, whereas we got a very nice exponential decay for the myotubes. This is an indication that we are missing some information that cannot be extracted from these large scale mechanical parameters. We therefore decided to focus on the local variations (fluctuations) of the FICs and we observed that in some situations  rupture events, as confirmed in the cross-correlation plot of Dl  Fig.  and \docEd) Fig.  for myot Dl Fig. . Those c> 1 Fig. .Figure 8G1, S and G2 phases. The myotubes are differentiated cells that no longer divide and these perinuclear tensed stress fibers are not needed. When indenting myoblasts, we have observed that the cell stiffness was reinforced upon indentation (strain-stiffening), as the signature of a possible local reinforcement of the perinuclear zones by actin stress fibers. Another important concomitance was the fact that those strain-stiffening cells were also those with higher energy disruption events and higher dissipative loss. These disruption events could then be directly related to local brittle failures of the actin stress fibers. Conversely, these high energy failures almost vanished for myotubes because these differentiated cells have a very different actin network organization. These multinucleated elongated cells were obtained by the fusion of bipolar elongated shape myoblasts (plasma membrane fusion). This required not only non-muscle myosin II and actin dynamics but also cadherins, integrins and DOCK1 , all via the activation of Rac174. Typical bipolar-shaped cells committed to membrane fusion and thus to myogenic differentiation are pointed out by arrows in Fig. Comparing the different morphologies of myoblasts and myotubes and more precisely their actin CSK networks from fluorescence microscopy images has helped us unravel the origin of the two groups of failure events detected in this study Fig. –S3. One μm/s), the cell indentation takes only a few seconds (3–4 s) which is at least one order of magnitude shorter than the relevant time scales for nonmuscle myosin II actin CSK remodeling in coordination with the two other CSK networks. To check whether actomyosin dynamics may have a role in these FIC disruption events and to which extent they may impact the systematic loss of mechanical work (Dl) recorded upon indentation of living myoblasts, we inhibited ATP synthesis. The mechanical response of ATP depleted cells revealed that they were globally softer but that they dissipated more mechanical work, with a greater propensity to strain-stiffening (〈Wg/Wi〉 > 3). We were puzzled to observe that the distributions of the force drop and energy values of their FIC disruption events were very similar to normal myoblasts, as if these cells would have kept locally the actin network cross-linked architecture. ATP depletion affects dramatically the actin network during the early hours after drug administration69. F-actin depolymerizes after ~30 minutes, before apical actin ring loss, and actin aggregates appear in the perinuclear regions after ~60 minutes69. Disruption of the well-organized F-actin network (stress fibers and actin cortex) and secondary formation of ectopic actin aggregates  were observed in different cell lines after ATP depletion75. Microtubules have a persistence length ranging from 5 to 100 μm in vitro, which makes them substantially stiffer than other cytoskeletal filaments (when considered as isolated filamentous structures). High resolution microscopy techniques have highlighted the existence of smaller persistence length filaments that would be due to non-thermal force bending, suggesting the interplay of forces exerted by active molecular motors or passive cross-linkers of the actin CSK network and/or the actin cell cortex76. ATP depletion does not impact the microtubule polymerization which is GTP dependent, these filaments remain intact, and the microtubule CSK network is unchanged. We observed that both Gg and Gi of the C2C12 myoblasts decrease upon ATP depletion . Finally, the fact that blebbistatin treated myoblasts behaved much softer and less dissipative than the three other tested examples of muscle cells, with quite undetectable disruption events, was an additional indication of the presence of tightly cross-linked structures in the myoblast actin CSK . Fluorescence microscopy images of blebbistatin treated myoblasts provided a visual confirmation of their complete loss of mechanical actomyosin contractility with no evidence of perinuclear stress fibers ) in a jelly state reminiscent of the initial spread adherent morphologyTo conclude, we proposed in this study an original method for analyzing the temporal response of living muscle precursor cells upon sharp shearing indentation, and quantifying their aptitude to sustain such a local stress. Rupture events encountered during the force-indentation curves were related to local disruptions of actin cytoskeleton structures, the strongest ones being produced by the tighter and stiffer structures (stress fibers or actin aggregates). This local malleability and susceptibility to actin CSK failures are an important aspect of living cell dynamics, shape maintenance and quick recovery from local stresses. They also play a major role in their quick adaptation to various external environments, such as the differentiation of myoblasts and their fusion into myotubes for example.μg/ml streptomycin, Gibco, Thermo Fisher Scientific) and 10 mM HEPES (Gibco). GM was stored at 4 °C up to 1 month; fresh GM was replaced every 2 days. The myoblasts were maintained in a 5% CO2 atmosphere at 37 °C inside 90 mm diameter petri dishes until ~70% confluency. To avoid that 60–70% confluent myoblasts differentiate spontaneously, the dishes were washed with preheated PBS and cells were detached from the dish bottom with 0.25% trypsin-EDTA (Gibco) for 3 minutes at 37 °C. Then the cells were harvested and either re-plated at lower concentration or frozen-stored. Prior to AFM FIC collection, C2C12 myoblasts (~1.5 105 cells from passages 10 to 14) were seeded on collagen coated petri dishes CCPDs (35 mm diameter) in GM and kept at least 24 hours at 37 °C with 5% CO2. Then the medium was changed and replaced by 2 ml of GM and the petri dish was transferred to the AFM. Each sample was used within 2 to 3 hours and discarded afterwards.C2C12 myoblasts (ATCC CRL-1772) were cultured in a growth medium (GM) composed of high glucose  Dulbecco’s Modified Eagle Medium with L-glutamine  supplemented with 20% fetal bovine serum , 1% penicillin-streptomycin antibiotics  Dulbecco’s Modified Eagle Medium with L-glutamine , supplemented with 2% Donor Horse Serum , 1% penicillin-streptomycin antibiotics  and 10 mM HEPES (Gibco). The myoblasts were maintained in DM at 37 °C with 5% CO2 for at least 5 days, renewing the medium every 2 days. Different culture supports were used  to compare their impact on C2C12 cell myogenic differentiation. C2C12 myoblast differentiation was monitored by DIC and time-lapse video recording. Multinucleated myotubes were observed with fluorescence confocal microscopy and probed by AFM at their 5th day of differentiation. A fusion index, defined as the percentage of nuclei contained in myotubes compared to the total number of nuclei observed in each image field, was used to quantify the level of myotube formation in different culture conditions.Confluent (60%) C2C12 myoblasts on CCPDs were induced to differentiation, replacing GM by a differentiation medium (DM)C2C12 myoblasts grown on CCPDs (35 mm diameter) were rinsed twice with preheated PBS and fixed with freshly made 4% paraformaldehyde  in PBS for 20 minutes at room temperature (RT) (24 °C). Then the sample was maintained in 2 ml PBS before AFM measurements which lasted ~4 hours at RT.78, namely the mitochondrial electron transport chain (ETC) complex III with antimycin A (AMA)79 and glycolysis with 2-deoxy-D-glucose (2-DG)80. ATP depletion buffer composition: 140 mM NaCl, 5 mM KCl, 1 mM MgCl2, 2 mM CaCl2, 10 mM HEPES, 6 mM 2-DG (Sigma), 5 μM AMA (Sigma). The filtered solution was adjusted to pH 7.4 and stored at −20 °C. CCPDs with adherent C2C12 myoblasts were first rinsed twice with preheated PBS and filled with freshly thawed ATP depletion buffer (2 ml). 15 minutes were necessary for the cellular ATP concentration to decrease below 7% of its initial concentration81. AFM FIC capture was performed on single cells, inside the ATP-depletion buffer at RT for 2–3 hours, before discarding the CCPD.Living cells rely on a combination of oxidative and glycolytic energy metabolism for ATP production. For a complete ATP depletion, both pathways must be inhibited82. Aliquots of 100 μM blebbistatin dissolved in dimethylsufoxide solution (1% DMSO in GM) were stored at −20 °C. GM was replaced with the same amount (2 ml) of preheated GM containing blebbistatin (50 μM) for 20 minutes83. AFM experiments on C2C12 cells inside GM-blebbistatin were performed in ~2–3 hours at RT.C2C12 myoblasts were treated with (S)-(-)-Blebbistatin (Santa Cruz Biotechnology) to inhibit the activity of the non-muscle myosin IIin vivo extracellular matrix (ECM). Type I collagen aqueous solution (3 mg/ml) from bovine skin and tendon BioReagent (Sigma Aldrich) was diluted in ultra-pure water to get 100 μg/ml. Petri dishes were incubated three hours with this solution (8 μg/cm2) at 37 °C to allow proteins to bind, dried overnight at RT under clean atmosphere and rinsed with Dulbecco’s Phosphate Buffer Saline  before use.We tested different surface treatments: gold coating, standard (SPD) and type I collagen coated petri dishes (CCPD), and we observed that CCPDs lead to greater C2C12 myoblast lengths, widths and areas. In this study, we chose type I collagen for myoblast adhesion because this protein is one of the major insoluble fibrous protein found in the X-Y Motor Precision Stage (JPK) with 20×20 mm motorised stage, a vibration isolation table (Melles Griot), a foam-based acoustic isolation system and a white light LED illumination . The AFM Z-piezotransducer with movement range of 100 μm was controlled by a closed loop feedback system with sub-nanometric precision. Proportional gain (P gain) was set at 20 and integral gain (I gain) at 0.002. The calibration of the AFM probes was performed via the thermal noise method84. The vertical deflection (ΔD) (nm) of the cantilever is proportional to the force applied to the sample; it can be converted to a tip-sample interaction force ΔF (nN) knowing the stiffness of the cantilever k (N/m) through the Hooke’s law: ΔF = kΔD and the sensitivity of the photodiode quadrant.A CellHesion 200 Atomic Force Microscope  coupled to a transmission inverted microscope and a CCD camera was used for nano-indentation experiments. The apparatus was equipped with a μm/s) on two myoblasts and two myotubes. For each speed, 10 FICs were collected with a force set point of 9 ± 1 nN. Between each 10 FIC recording series, the sampling rate was adjusted to collect similar sample size (~20000) for each FIC. To reconstruct the mechanical parameter histograms, a protocol for cell indentation was elaborated: for each cell, 30 successive FICs were recorded on the perinuclear zone. The force set-point, the indentation velocity, the cantilever displacement range ΔZ and the sampling rate were fixed to 8 ± 2 nN, 1 μm/s, 6 ± 2 μm and 3.5 kHz respectively.Force indentation curves (FICs) were recorded with sharp tip triangular (SNL) or rectangular cantilevers (qp-CONT) (see Supplementary Information) and targeted on perinuclear zones. To evaluate the impact of both the hydrodynamic drag and the damping of adherent cell layers on FIC shapes, four set of FICs were recorded at different scan velocities  corresponding to 2Fd was used. The second step was to filter the FICs to get rid of the background noise and, when needed, to compute the successive derivatives of the FICs. These two tasks were achieved simultaneously using the continuous wavelet transform89. Details can be found in the Supplementary Information.The first step was to eliminate the drift of the FICs resulting from either a mis-alignment of the laser photodiode beam on the tip of the cantilever or an hydrodynamic drag produced by the liquid surrounding the cantilever. The elimination of the drift coming for optical misalignment was performed from a parametrization of the sum of the loading and unloading curves, out of contact with the sample, assuming that the hydrodynamic drag inflects similarly the loading and the unloading curves. To parametrize and compensate the hydrodynamic drag force 91 have led to general hereditary integral representation of stress-strain relationships for the indentation of linear viscoelastic materials by axisymmetric indenters:G(t) is the stress relaxation modulus, ν the Poisson ratio, θ the cantilever tip half-angle, F is the loading force, h = Z − Zc describes the displacement of the indenter, and n is a positive integer which depends on the shape of the indenter. The stress relaxation modulus G(t) retains the memory of the deformation. For a pyramidal indenter tip, we have n = 1 and C1 = tan θ/π, where θ is the nominal tip half-angle:Viscoelasticity theories developed in the second half of the twentieth centuryV0, dZ = V0 dt, and the stress relaxation modulus G can be rewritten as:G with Z keeps the memory of the whole deformation. For pyramidal  tips, Eq.  (see Supplementary Eqs (G(Z) were computed from wavelet transforms of the FICs at a smoothing scale s92:The first- and second-order derivatives of tary Eqs  and henc91Gg. Gg is the prefactor of the parabola that crosses both the contact point and the final set point Zsp of the FIC  is a parabola, Z  in this study, we were rather sensing spatial and gradual variations of the cell rigidity when getting closer to the nucleus.If the cells were homogeneous elastic balls, the FICs should be pure parabolan Z Fig. . CompariWl (loading FIC) is not recovered in the output work Wu upon strain release (unloading FIC)94. This dissipation of mechanical work is described by the ratio Dl:Loading and unloading FICs collected from living cells are rarely superimposed, reflecting that a fraction of the input work Dl tends to zero for a purely elastic material and to 1 for a purely viscous material. The work integrals corresponding to the parabolic interpolations Gg and Gi are also computed accordingly:Wl, Wu, Wg and Wi) during 30 successive load-unload scans are compared in Fig. Wl/Wg > 1 and Wl/Wi < 1; they both never cross the value 1 during the 30 successive load-unload scans. When these two ratios collapse to 1, a pure elastic ball response is found, this occurs for FIC index = 4 only in this example. We also note that while repeating the cell deformation (after 20 load-unload scans), the ratios tend to be more distant from 1 than initially, meaning that the cell response gets farther from a simple elastic ball with time. For the myoblast of Fig. Wl/Wg < 1 and Wl/Wi > 1, and as before these two ratio curves do not cross during the 30 load-unload scans. Surprisingly, whereas Wl/Wg is quite constant, Wl/Wi is very irregular and increases to values around 3 after 8 load-unload scans. This evolution can be explained by the fact that, underneath a softer cell cortex, these indentations progressively unveil a stiffer cell subdomain that ultimately yields under the constraint . The dissipation loss Dl is only slightly smaller for the second myoblast  = Fu(Zsp), Dl can be simply expressed as a function of the ratio of the exponents αl and βu:The temporal evolution of the different work integrals /dZ2, we took the local minima Gm as searching criteria /dZ2 with a peak prominence ≥1 MPa. Gm and GM are marked with black symbols in Fig. Gm and GM, we could then detect the beginning and the end of the disruption events, represented with blue symbols in Fig. Zd. Finally, the force drop Fd was corrected, taking into account the increasing trend of the FIC after the rupture event. A linear interpolation of the FIC in a small interval (~20 nm) beyond the local minima of the FIC gave the best interpolation. Nonlinear interpolations of the FIC did not work better. We defined the energy released during this disruption event as:The protocol that we have elaborated to track singular events in FICs is shown in Fig.  nm (Eq. ). Noticiria Fig. . We defiSupplementary information file


Arabidopsis thaliana sepal development, fluctuations in the concentration of the transcription factor ATML1 pattern a field of identical epidermal cells to differentiate into giant cells interspersed between smaller cells. We find that ATML1 is expressed in all epidermal cells. However, its level fluctuates in each of these cells. If ATML1 levels surpass a threshold during the G2 phase of the cell cycle, the cell will likely enter a state of endoreduplication and become giant. Otherwise, the cell divides. Our results demonstrate a fluctuation-driven patterning mechanism for how cell fate decisions can be initiated through a random yet tightly regulated process.Multicellular development produces patterns of specialized cell types. Yet, it is often unclear how individual cells within a field of identical cells initiate the patterning process. Using live imaging, quantitative image analyses and modeling, we show that during DOI:http://dx.doi.org/10.7554/eLife.19131.001 Plant and animal organs contain several types of cells that perform different roles. As a plant or animal develops, these different cell types can form intricate patterns. To start the pattern, a few cells within a group of identical cells must somehow become different from their neighbors. Some patterns of cells are very well organized and easily reproduced. However, sometimes cells spontaneously become different from their neighbors, producing a less ordered pattern.Arabidopsis , a scattered pattern of giant cells and small cells spontaneously forms within a part of the developing flower called the sepal. A protein called ATML1 is a key regulator in the formation of giant cells, but because it is found in both giant cells and small cells, it is not clear how this regulation works.In a plant called Mathematical models of this process suggest that identical cells could initially acquire subtle differences, potentially from random fluctuations in the activity of key regulatory molecules, to start the patterning process. Meyer, Teles, Formosa-Jordan et al. used a combination of microscopy, image analysis and mathematical modeling to investigate how the level of ATML1 fluctuates in cells to give rise to the pattern within the sepal. The experiments show that early in the development of the sepal, the levels of ATML1 fluctuate up and down in every sepal cell. If ATML1 reaches a high level specifically when a cell is preparing to divide, that cell will decide to become a giant cell, whereas if the level of ATML1 is low at this point, then the cell will divide and remain small.Overall, the findings of Meyer, Teles, Formosa-Jordan et al. demonstrate that fluctuations of key regulators while cells are preparing to divide are important for creating patterns during development. A future challenge is to examine whether other tissues in plants, or tissues in other organisms, use a similar mechanism to generate patterns of cells.DOI:http://dx.doi.org/10.7554/eLife.19131.002 Wolpert’s French flag model proposes that a group of identical cells differentiate into different cell types based on threshold concentrations of a morphogen gradient  promoter  and transformed it into atml1–3 mutant plants  and AP2-2XYpet (pAP2::AP2-2XYpet), and the SEC24A transcriptional reporter (pSEC24A::H2B-GFP). VIP1 is a mechano-sensitive transcription factor that localizes to the nucleus upon hypo-osmotic treatment  in order to perform segmentation of each individual nucleus  was initially performed by applying the FeatureJ ImageJ plugin (http://www.imagescience.org/meijering/software/featurej/) to each denoised image, facilitating the subsequent registration step. Pairing of individual nuclei in two consecutive time points was computed by registering pairs of successive images , nuclei with an area of ≥35 µm2 with an eccentricity of ≤0.7 were classified as 4C (G2) and nuclei with an area of >35 µm2 with an eccentricity of >0.7 were classified as 8C (endoreduplicating). Nuclei that bordered these area thresholds were manually checked to ensure that they were correctly classified, with a small number of incorrectly classified nuclei being reclassified. Manual correction was based on additional knowledge from the live imaging time series and visualization in 3D . Importantly, information of ATML1 concentration values was not used for ploidy classification at any stage. Additionally, independent manual correction of ploidy classification by different researchers produced highly similar results. For other genotypes , area and eccentricity threshold values were slightly adjusted in order to account for changes in segmentation parameters . Flowers that have a broader distribution of giant and small cells tended to have slightly inflated segmented masks in order to increase the number of nuclei successfully segmented through the entire time course. The inflation of the segmented masks leads to slightly increased nuclear area, which we accounted for when we defined the thresholds.Building on these results which show that area thresholds can be used to effectively separate cell cycle stages, we defined a set of area and eccentricity thresholds to classify cells into different ploidies  to our lgo-2 mutant, which exhibits a loss-of-giant cell phenotype , is required for giant cell formation; LGO triggers endoreduplication once giant cell fate has been established . To verihenotype . Plants lgo-2 mutant, which fail to endoreduplicate in early stage sepals. In this scenario, we would expect the same number of lgo-2 nuclei to surpass the ATML1 threshold in G2 as in wild type. Cells that pass the threshold would still divide because they are unable to endoreduplicate. To test this, we live imaged our mCitrine-ATML1 reporter in the lgo-2 mutant background . We therefore assigned ATML1 to activate a downstream target that inhibits cell division and promotes entry into endoreduplication. Only if the downstream target passes its own specific threshold in G2, does it successfully drive a cell to endoreduplicate to form a giant cell . Hence, In the model, different ploidy and cell division checkpoints were determined using a linearly increasing timer variable, which represents the cell cycle. The timer resets at every cell division checkpoint with a small amount of noise .Video 11The model qualitatively reproduced our experimental data and led to a scattered pattern of giant cells in a growing tissue . SpecifiTo test whether our model could recapitulate G2-mediated giant cell fate specification, we performed a ROC analysis on the simulated time traces, mimicking the analysis performed on the experimental data . ConsistATML1 dependent and basal production rates, whilst keeping the number of predicted giant cells close to experimental values . With no feedback or low feedback strengths, we could qualitatively match the experimental ROC analysis  and 4C (G2) . These rWe then asked whether our model could qualitatively reproduce the ATML1 dosage phenotypes we had observed with our genetic dosage series. We found that changing the basal ATML1 production rate was sufficient to gradually increase the total amount of the ATML1 in the modeled tissue, and accordingly, the fraction of giant cells in the sepal . These rHence, our model shows that fast and relatively small stochastic fluctuations of ATML1 are sufficient to pattern giant and small cells in the sepal. ATML1 activates a downstream target, which if activated in G2, will induce endoreduplication. The dynamics of the ATML1-target network creates a soft ATML1 threshold during G2.Arabidopsis sepal development, the pattern of giant cells and small cells in the epidermis is initiated through fluctuations of the transcription factor ATML1. Using live-imaging, quantitative image analyses and mathematical modeling, we have revealed that cells in which ATML1 levels surpass a soft threshold during the G2 phase of the cell cycle have a high probability of establishing giant cell identity and entering endoreduplication. A sepal epidermal cell is only competent to respond to ATML1 fluctuations during a window of time defined by G2 stage of the cell cycle.Here, we have identified a cell-autonomous fluctuation patterning mechanism for specifying cell fate in a multicellular system . During Strikingly, our fluctuation-patterning model resembles Wolpert’s French flag model in that each individual cell makes an autonomous fate decision based on the concentration of a key developmental regulator. Our model however deviates from the French flag model because it utilizes internal fluctuations instead of a diffusible morphogen to generate concentration differences. Concentration threshold-based patterning mechanisms have been traditionally viewed as being non-robust because they are sensitive to small perturbations in concentrations. Often additional mechanisms are needed to achieve robustness . This seA few recent studies have similarly demonstrated that the cell cycle provides a window of opportunity for making cell fate decisions. However, these studies suggest that G1 is the critical phase for specification. During G1, there is a growth factor-dependent restriction point, where a cell determines whether to enter quiescence (G0) or progress through the cell cycle. Cyclin/Cyclin Dependent Kinase (CDK) activity is normally reduced during the restriction point, providing a window for cells to receive extracellular signals necessary for cell fate decisions . This haWe have found that a sepal epidermal cell’s window to differentiate occurs not in G1 but in G2, suggesting a different manner of regulation than in G1-gated determination. For instance, cell fate decisions governed by the G1 phase of the cell cycle must often receive an extracellular signal to activate prodifferentiation genes instead of going into G0 quiescence. In contrast, our model suggests ATML1 fluctuations could be sufficient to pattern the sepal without a need for an extracellular signal. Alternatively, ATML1 could be priming the cell to receive a signal during the G2 phase of the cell cycle. We have previously reported that ACR4 . A soft but still reliable threshold can emerge when a target follows the dynamics of its upstream regulator. Indeed, our experimental data shows that the ATML1 threshold is soft, but robust across different plants.We have described a cell-autonomous fluctuation-driven patterning mechanism, where fluctuations of the transcription factor ATML1 must reach a concentration threshold during the G2 stage of the cell cycle to regulate cell fate decisions. This overall demonstrates that stochastic processes can be important for creating spatial patterns necessary for reproducible tissue development.pSEC24A::H2B-GFP which was in Landsberg erecta (Columbia (Col) plants were used as the wild-type accession for all genotypes except g erecta .atml1–3 (SALK_033408); exhibits a lack of giant cell phenotype. The atml1–3 mutation is a dosage dependent mutation that contains a T-DNA insertion in the homeodomain. The atml1–3 mutation can be PCR genotyped by amplifying with oAR272 (CAGGCAGAAGAAAATCGAGAT), oAR273 (GAAACCAGTGTGGCTATTGTT) and LBb1 (GCGTGGACCGCTTGCTGCAACT).lgo-2 (SALK_039905); exhibits a lack of giant cell phenotype. The lgo-2 mutation is a recessive mutation, containing a T-DNA insertion. The lgo-2 mutation can be PCR genotyped by amplifying with oAR284 (CTTCCCTCTCACTTCTCCAA), oAR285 (CCGAACACCAACAGATAATT), and JMLB2 (TTGGGTGATGGTTCACGTAGTGGG) (TAGTGGG) .dek1–4 plants do not form giant cells. The dek1–4 mutation can be PCR genotyped by amplifying with oAR448 (TGTTGGTGGAACAGACTATGTGAATTCA) and oAR449 (TGAAGACTGAAAGGACAAAAGGTGC) with a 60°C annealing temperature followed by a 4 hr product digest using BsaAI.acr4–24 plants have a severe reduction in the number of giant cells that form. The acr4–24 mutation can be PCR genotyped by amplifying with oAR302 (ATAGAAGTCCCTGTGAGAACTGCG) and oAR303 (TATGATCATAGTGCGGTCTGTTGG) with a 60°C annealing temperature followed by a 4 hr product digest using HhaI.pAP2::AP2-2XYpet plants were provided by Jeff Long (eff Long .pVIP1::VIP-mCitrine plants were provided by the ABRC (CS36991) (CS36991) .proRPS5A-ATML1/pER8 and proATML1-nls-3xGFP) and Keiko Torii (pKMP151 line #134) (ATML1 estradiol inducible lines were provided by Shinobu Takada (ne #134) .All plants used for this analysis were grown in Percival growth chambers with 24 hr light conditions at 22°C to minimize any diurnal effect on plants.ATML1, AT4G21750; giant cell enhancer trap marker, YJ158; small cell enhancer trap marker, CS70134; LGO, AT3G10525; atml1–3, CS68906, SALK_033408; lgo-2, CS69160, SALK_039905; pPDF1::GFP-ATML1, GIL91–4; pPDF1::FLAG-ATML1, GIL90–5; SEC24, AT3G07100; pVIP1::VIP-mCitrine, CS36991; CER5, AT1G51500; FDH, AT2G26250; PDF2, AT4G04890; PDF1, AT2G42840.PDF1::FLAG-ATML1 plants, which exhibit an all ectopic giant cell phenotype to Columbia plants, resulting in F1 plants that were hemizygous for the PDF1::FLAG-ATML1 transgene (PDF1::FLAG-ATML1/+). To lower amounts of ectopic ATML1 even further, PDF1::FLAG-ATML1/+ plants were crossed into the atml1–3 mutant background. Using genetic segregation and PCR genotyping, plants containing the PDF1::FLAG-ATML1 transgene in an atml1–3 mutant background were recovered and analyzed . Next, to look at the effects of atml1–3 heterozygotes, the Columbia plants were crossed with atml1–3 mutants. The resulting F1 plants were analyzed.To create genetically altered lines of ATML1 for our dosage series, we first crossed PDF1::FLAG-ATML1 plants confer giant cell identity, PDF1::FLAG-ATML1 plants were crossed with plants expressing the giant and small cell marker , lgo;pHM44 pATML1::mCitrine-ATML1  or GIL91–4 pPDF1::GFP-ATML1  were imaged either every 8 hr or every hour using a Zeiss 710 laser scanning confocal microscope with a 20x water-immersion objective . Before imaging, plant inflorescences were dissected down to early stage flowers and meristems and then taped onto slides. Dissected inflorescences were then stained with PI and mounted with a cover slip and imaged. Inflorescences were unmounted, dried, and plants were placed upright in the growth chamber for 8 hr before remounting and imaging. The resulting images were 3D cropped with ImageJ  sepals with a Zeiss 710 confocal microscope. mCitrine was excited with a 514 nm laser and emission was collected with a 519–621 filter. Imaged sepals were semi-automated image processed using a MATLAB module, which has been previously published , a 6160 bp fragment upstream of the ATML1 protein coding region was PCR amplified using oHM23 (ACCGACAATGTATGAATGTACTCT) and oHM24 (cggtaccggcgcgccGATGATGATGGATGCCTATCAATTT) and cloned into a pGEM-T Easy vector to create pHM20. Additionally, a 992 bp region downstream of the ATML1 protein coding region was PCR amplified using oHM25 (cggtaccTCGATGTTTTCGGGTAAGCTTTTT) and oHM26 (TTTGATGACTTGGTCTCCATAATTTC) and cloned into pGEM-T easy to create pHM21. pHM21 was cut with SacII and KpnI and cloned into pHM20 to make pHM22. A gateway cassette from pXQ (AscI-GW-KpnI in pGEM-T easy) was cut with AscI and KpnI and cloned into pHM22 to make pHM23 (pATML1::GW:ATML1 3’UTR). Next, mCitrine was PCR amplified using oHM42 (CACCAAAATGGTGAGCAAGGGCGAGGAGCTG) and oHM39 (atACTAGTGGCCGCTGCCGCAGCGGCAGCCGCAGCTGCTCCGGACTTGTAC) and cloned into pENTR/D-TOPO vector to make pHM30. ATML1 was PCR amplified using oHM40 (tcggcgcgccCACCCTTTTAGGCTCCGTCGCAGGCCAGAGCGGCT) and oHM41 (ccactagtATGTATCATCCAAACATGTTCGAATCTCATC) and cloned into pGEM-T easy to make pHM28. ATML1 was cut using SpeI and AscI and cloned into pHM28 to make pHM25 (pENTR mCitrine-ATML1). LR reaction between pHM25 and pHM43 to make pHM44 (pATML1::mCitrine-ATML1). The atml1–3 rescue line was generated by transforming the pHM44 pATML1::mCitrine-ATML1 transgene into atml1–3 mutants using Agrobacterium-mediated floral dipping methods  was used as a reference gene to normalize gene expression. Furthermore, three technical replicates were used to ensure the validity of each biological replicate.To perform qPCR, 3–4 inflorescences were collected per sample and total RNA was extracted using RNeasy Plant Mini Kit . Next, 1 microgram of total RNA was DNAse treated with amplification grade DNAse I  and reverse transcribed using Superscript II reverse transcriptase (Invitrogen) with oligo dT primers. Real-time PCR was performed using 480 SYBR Green I Master  on a Roche qPCR primers:ATML1 (flanks atml1–3 insertion)oHM58: GAGCTAGAGTCGTTCTTCAAGG – qPCR forward primer for ATML1 (flanks atml1–3 insertion)oHM62: GTTCTCGTGCCTCTCATGTTGTG – qPCR reverse primer for ATML1 5’UTR (upstream of start site)atml1-ATGF: GGATATACAGGCAGAAGAAAATCGAG – qPCR forward primer for endogenous ATML1 5’ UTR oAR715: CGCTGAAGCTAGTCGACTCTA – qPCR forward primer for induced ATML1 (just downstream of the start codon in both the endogenous and induced transcripts).oAR716: TTCTCCATGGTGACTTCTGCG – qPCR reverse primer for CER5 . Costanza performs segmentation following the steepest descent algorithm, providing high-resolution three-dimensional segmentation of each individual sepal nucleus.Raw fluorescence intensity images were denoised using the PureDenoise ImageJ plugin , 2010, ohttp://imagej.net/FeatureJ) for edge detection by applying the Canny method . To track the cell nuclei between two successive nuclei segmentations, t Nand t+ΔtN , the block matching framework . The registered floating image and the reference image were merged with different colors into a double channel image in ImageJ . Using ALT , registration was not always successful for all nuclei. Incorrectly tracked nuclei were manually corrected using the MorphoGraphX parent labels tools, making use of the ALT-generated optimal pairing tables, describing the mother/daughter relations between time points.Denoised images were processed using the FeatureJ ImageJ plugin  functions and scripts was developed to quantify signal intensity, as well as size and shape properties of individual nuclei from sets of confocal microscopy images processed as described above . We did perfcurve function of the Statistics and Machine Learning Toolbox in MATLAB  or endoreduplicating cells (lgo-2 mutant flowers), a common threshold was inferred from the wild-type pATML1::mCitrine-ATML1 flowers  and false positive rate (FPR) when classifying small versus giant cells, was taken as the threshold ATML1 concentration required for triggering giant fate decision in individual cells. For sepals where such a threshold could not be inferred, whether due to the absence of sufficient numbers of dividing  with the previously inferred common threshold. A lineage was considered to be a giant cell lineage if the ATML1 concentration of a given nucleus surpassed the common threshold concentration during 4C at any point of the time course. If this event never occurred, the lineage was considered to correspond to a small cell lineage.In the i, whose variables are [ATML1]i and [Target]i respectively, readsWe implemented a stochastic computational model for ATML1 mediated giant cell fate decisions in a 2D idealized growing tissue. The model has a core simplified ATML1 regulatory network that can prevent cell division, driving cell endoreduplication. We modeled ATML1 cell concentration dynamics as a basally produced protein that self-activates and is linearly degraded . ATML1 eAP is a basal ATML1 production rate, XV is the maximal ATML1-dependent production rate for the X (either ATML1 or Target concentration) variable, XK is the ATML1 concentration at which the ATML1-dependent production rate has its half-maximal value, X nis the Hill coefficient and XG is the linear degradation rate for the X variable. For simplicity, we will refer to AV as the ATML1 auto-induction rate, so no feedback is considered when AV = 0.where A cell is defined by a set of vertices in 2D and we set the tissue to grow exponentially and anisotropically by moving vertices outwards from the center of mass of the tissue. Hence, all cells grow anisotropically, and they divide according to a timer variable present in each cell. We implemented dilution of ATML1 and its target variables due to growth. During sepal development, nuclear and cell area of epidermal cells are correlated . We usediTimer is a variable in cell i, and CP is the basal timer production rate. The timer resetting was performed at each time step according to the following equation:where iU is a uniform randomly distributed number in the interval  levels higher than a specific threshold TΘ during the 4C stage. In that case, endoreduplication occurs, and cells reset their timer when they reach the C,DΘ threshold, but keeping its ploidy to 4C. We imposed that 4C cells having endoreduplicated once cannot undergo cell division anymore. As a consequence, these cells will increase their ploidy every time they pass the timer threshold C,SΘ representing entry into S phase.Cell ploidy was modeled as a discrete variable dependent on the timer and cell division, which also depends on the ATML1 network. Specifically, cell ploidy increases from 2C to 4C when the timer reaches a threshold Our experimental data shows that the nuclear area scales with the DNA content and ploidy in the cell –2. PreviX in cell i, the resulting stochastic equations would readDynamic stochasticity was introduced in the ATML1, Target and Timer variables by extending its deterministic dynamics to its Langevin form . In partxi+F and xi–F are positive functions that represent the birth and death processes for the species X in cell i. Hence, we take into account stochasticity coming from production and degradation of the modeled species. i(t)ε is a normalized cell area; we assume i(t)ε=0Ei(t),E where 0E is an effective cell area used to normalize noise, and i(t)E is the area of cell i in arbitrary units. Xiη is a random Gaussian variable with zero mean that fulfills Xi(t)ηX’j(t’) ⟩=δ(t-t’)δXXδij⟨η, where i and j are cell indices, X and X’ the modeled variables, XXδ and ijδ are Kronecker deltas and δ(t-t’) is the Dirac delta. Note that, as the standard chemical Langevin equation , and MinE and MaxE the minimal and maximal areas of the cells at the start of the simulation. This made larger cells being initiated at more advanced stages of the cell cycle, and hence, being more likely to divide. Ploidies were initially set to either 2C or 4C, depending on whether the initial timer values set by C,SΘ.being C,SΘ and C,DΘ respectively) were assigned so that we could recover 2C and 4C percentages of cells in atml1–3 mutants (AP) and linearly spaced values of the ATML1 auto-induction production rate (AV). Specifically, we performed simulations on 121 logarithmically spaced AP values between 0 and 2.2, and 11 linearly spaced AV values between 0 and 2.5. From these parameter scans, AP and AV parameters were chosen for the simulations shown in AP and AV parameter values for representing the wild-type sepal in We assigned different parameter values based on experimental evidence when available. Threshold values of the timer for the synthesis phase and division checkpoint .The computational implementation of the model was performed through the open source C++ Organism package, (Receiver operator characteristics (ROC) analysis section and ROC analysis was also applied to the ATML1 concentration maxima across the different simulated lineages, by following a similar procedure as for the experimental data  operating point of the ROC curve. Specifically, we used 30 different random subsamples of the small cell population with as much cells as the pool of giant cells, so that the total cost of misclassification of positive and negative cases for the threshold determination would remain equivalent and similar to the experimental analysis. As a result, the computed soft threshold A*Θ was defined as the mean of the 30 different optimal thresholds found using random subsamples. This subsampling method, when applied to the target maxima throughout 4C time courses, could accurately predict the hard threshold of the target variable imposed in the simulations TΘ, which we denote by T*Θ .Thank you for submitting your article "Fluctuations of the transcription factor ATML1 generate the pattern of giant cells in the The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission. Overall, the experimentalists thought the work was well done, but there were several additional controls and tests of the model that were needed to make this a truly solid story. The modeling was evaluated by another expert who again finds the work generally strong but offered some cautions in interpretations. All felt that the authors should be commended for the very clear writing and explanation of complex data in an accessible way.Summary:Arabidopsis sepal, an organ with a stereotyped size and shape and an epidermis consisting of several cell types, the authors use live cell imaging and modeling to investigate how the pattern of giant cells (GC) arises. The authors provide evidence that stochastic gene expression, a widespread phenomenon that arises from the nature of gene expression as it responds to intrinsic and extrinsic sources of variation, enables equivalent cells to adopt fates different from some of their neighbors, dependent upon cell-autonomous accumulation of a particular key regulator. In this manuscript the authors present a case for ATML1 being such a factor, whose activity in promoting GC fate is determined by stochastic levels that exceed a threshold.During development, patterns emerge in tissues and organs. The underlying basis for these patterns is a topic of broad interest in the community. Using the Essential revisions:1) Better test of the need for ATML1 in G2High ATML1 levels are moderately predictive of GC identity . This is OK, but it makes it all the more important to add in something that suggests the ATML1 level is a major component driving (and not simply reflecting) the fate.Two possible ways to address this are: (1) The best way to do this would be to increase levels of ATML1 only during G2 and only during G1 and test whether these manipulations fit their model. I don't know all the tools available-in other systems there are G1, S and G2/M specific promoters and there are degradation elements linked to the cell cycle. Could these be employed? (2) If the first test is technically impossible, then another part of their model is that there is a feedback of ATML1 on its own expression. The ATML1 site from their ATML1 or PDF1 promoters could be removed to see whether this makes an important contribution.2) Provide more convincing evidence that correlation of expression patterns (bursts in G2) aren't more general feature of TFs in these cells.pSEC24A::H2B-GFP abundance as a control for whether the relative broad variation in pATML1::mCitrine-ATML1 abundance in the developing sepal is a general phenomenon. H2B is quite different from ATML1 in that it is incorporated in the nucleosomes and likely much more abundant compared to regular transcription factors. Also a different fluorescent probe is used. To prove that varying expression of ATML1 in the sepal epidermis is not a common feature among transcription factors, it would be better to use an unrelated mCitrine-TF fusion as control.Reviewers were concerned by the use of 3) Some model interpretations are a bit overstretched and should be reconsidered.For instance, the assertion in the fifth paragraph of the subsection “A model with stochastic fluctuations of ATML1 reproduces giant cell patterning”, that the model correctly recapitulates G2-mediated giant cell fate specification and that the lower AUC values recovered in 2C stages than in 4C in the model indicate 'that high ATML1 levels during the G2 phase of the cell cycle are important for giant cell fate commitment' is not really justified . Since a hard threshold for the 'Target' gene to start endoreduplication is hard-coded specifically in the G2 phase of the model, no other outcome could have been expected. Instead, I think the major contribution of the model is that it shows that stochasticity in ATML1 expression alone, associated with a hard threshold on ATML1 levels to induce giant cell formation, is not sufficient to explain the observations, and that it predicts instead that a hard G2-associated threshold on another stochastically influenced downstream factor, causing the ATML1 threshold to become soft, is needed to explain the imperfect relationship between high ATML1 expression in G2 and the induction of giant cell formation. It seems that additionally this downstream factor needs to have a higher degradation rate so that the 'Target' follows the dynamics of ATML1 , and that the auto-induction rate of ATML1 should not be too strong. These are in my view the real predictions of the model, the fact that e.g. overexpression of ATML1 in the model would lead to an ectopic giant cell phenotype can already be deduced from the model form without running simulations.Arabidopsis sepal" for further consideration at eLife. Your revised article has been favorably evaluated by Christian Hardtke as the Senior Editor, and a Reviewing Editor.Thank you for resubmitting your work entitled "Fluctuations of the transcription factor ATML1 generate the pattern of giant cells in the The manuscript has been significantly improved in many places, but there is still one issue to address.Because you cannot supply ATML1 at a specific phase in the cell cycle, nor delete promoter elements, but instead rely on following endogenous ATML1 transcripts after induction to infer a feedback loop, this last experiment must be very carefully done and interpreted.A concern is whether the technical set up (and point at which ATML1 is monitored) has the capacity to report a "strong" feedback and indeed, what level of transcriptional up-regulation would be considered strong feedback. How can the experiment be calibrated? You will either need to cite evidence from similar experiments (same induction system and timing and tissue for monitoring) that put the 1.7 fold increase in context, or do this type of control experiment yourselves. Essential revisions:1) Better test of the need for ATML1 in G2High ATML1 levels are moderately predictive of GC identity . This is OK, but it makes it all the more important to add in something that suggests the ATML1 level is a major component driving (and not simply reflecting) the fate.Two possible ways to address this are: (1) The best way to do this would be to increase levels of ATML1 only during G2 and only during G1 and test whether these manipulations fit their model. I don't know all the tools available-in other systems there are G1, S and G2/M specific promoters and there are degradation elements linked to the cell cycle. Could these be employed? (2) If the first test is technically impossible, then another part of their model is that there is a feedback of ATML1 on its own expression. The ATML1 site from their ATML1 or PDF1 promoters could be removed to see whether this makes an important contribution.We will address each of the three points raised here.Arabidopsis .First, we agree that expressing ATML1 specifically during different stages of the cell cycle would be a great experiment; however, we also agree that it is technically impossible at this time. Overexpression of ATML1 in internal tissues is seedling lethal . Since cell cycle genes are expressed in dividing cells in all cell layers at all stages of development, expressing ATML1 under one of these promoters would cause lethality. In addition, while there are well characterized combinations of promoters and degradation elements that can give rise to expression in S through early G2 and late G2 through M phase, there are currently no equivalently characterized promoters and degradation elements for G1 in Second, we tested the feedback of ATML1 on its own expression using an ATML1 inducible system previously published . We are doing this in place of the experiment proposed (mutating the ATML1 binding site in the ATML1 promoter) because while one ATML1 binding site has been characterized in the ATML1 promoter , there may be additional ATML1 binding sites within the promoter according to our searches with the ATML1 consensus binding sequences . This would make the deletion experiment difficult to interpret. Using the inducible system, we found that ATML1 works in a weak positive feedback loop as predicted by the model; endogenous ATML1 expression increased by 1.7 fold when ATML1 was induced with estradiol for 24 hours in inflorescences .“To test the type of feedback of ATML1 on itself, we determined whether induction of ATML1 could rapidly stimulate transcription of the endogenous ATML1 gene using qPCR . We found that ATML1 indeed acts in a weak positive feedback loop, increasing endogenous ATML1 expression by 1.7 fold within 24 hours . This reatml1 mutants lose giant cells. To further validate this point, we now have added that inducing overexpression of ATML1 produces ectopic giant cells  Provide more convincing evidence that correlation of expression patterns (bursts in G2) aren't more general feature of TFs in these cells.Reviewers were concerned by the use of pSEC24A::H2B-GFP abundance as a control for whether the relative broad variation in pATML1::mCitrine-ATML1 abundance in the developing sepal is a general phenomenon. H2B is quite different from ATML1 in that it is incorporated in the nucleosomes and likely much more abundant compared to regular transcription factors. Also a different fluorescent probe is used. To prove that varying expression of ATML1 in the sepal epidermis is not a common feature among transcription factors, it would be better to use an unrelated mCitrine-TF fusion as control.We have added VIP1-mCitrine and AP2-2XYpet controls. . VIP1 encodes a bZIP transcription factor. Although AP2 is not fused to mCitrine, we choose to include it because of its well-characterized role in floral organ identity. Both VIP1-mCitrine and AP2-2XYPet exhibit a lower CV than mCitrine-ATML1, confirming that not all transcription factors are expressed variably in developing sepals .SEC24A transcriptional reporter (pSEC24A::H2B-GFP). […] We found that mCitrine-ATML1 concentrations were approximately twice as variable as the other reporters  and AP2-2XYpet (pAP2::AP2-2XYpet), and the 3) Some model interpretations are a bit overstretched and should be reconsidered.For instance, the assertion in the fifth paragraph of the subsection “A model with stochastic fluctuations of ATML1 reproduces giant cell patterning”, that the model correctly recapitulates G2-mediated giant cell fate specification and that the lower AUC values recovered in 2C stages than in 4C in the model indicate 'that high ATML1 levels during the G2 phase of the cell cycle are important for giant cell fate commitment' is not really justified . Since a hard threshold for the 'Target' gene to start endoreduplication is hard-coded specifically in the G2 phase of the model, no other outcome could have been expected. Instead, I think the major contribution of the model is that it shows that stochasticity in ATML1 expression alone, associated with a hard threshold on ATML1 levels to induce giant cell formation, is not sufficient to explain the observations, and that it predicts instead that a hard G2-associated threshold on another stochastically influenced downstream factor, causing the ATML1 threshold to become soft, is needed to explain the imperfect relationship between high ATML1 expression in G2 and the induction of giant cell formation. It seems that additionally this downstream factor needs to have a higher degradation rate so that the 'Target' follows the dynamics of ATML1 , and that the auto-induction rate of ATML1 should not be too strong. These are in my view the real predictions of the model, the fact that e.g. overexpression of ATML1 in the model would lead to an ectopic giant cell phenotype can already be deduced from the model form without running simulations.We agree with the reviewer that some of the modeling claims of the paper should be made clearer. We have moved and adjusted the modeling section to appear at the end of the paper, which has made it clearer what is a prediction of the model. We have also provided model files (Source code 2) to allow readers to more easily simulate the model. Additionally, we have provided code files (Source Code 1) for the image analysis and quantification protocol.In discussing the reviewers’ point that ‘the assertion in the fifth paragraph of the subsection “A model with stochastic fluctuations of ATML1 reproduces giant cell patterning”, that the model correctly recapitulates G2-mediated giant cell fate specification and that the lower AUC values recovered in 2C stages than in 4C in the model indicate 'that high ATML1 levels during the G2 phase of the cell cycle are important for giant cell fate commitment' is not really justified ’.We have altered the original text:“Consistent with our experimental observations, we found lower AUC values in 2C stages than in 4C, indicating that high ATML1 levels during the G2 phase of the cell cycle are important for giant cell fate commitment .”To:“Consistent with our experimental observations, we found lower AUC values in 2C stages than in 4C. This supports our hypothesis that high ATML1 levels during the G2 phase of the cell cycle are important for giant cell fate commitment .”We agree with the reviewer that a key prediction of the model is that the ATML1 feedback strength must not be too strong. We have stressed this point in the text, and have made it clear that our simulations where we vary the feedback strength in the ATML1 circuit predict that the system cannot contain a strong positive feedback loop. We now have further experimental evidence for this prediction (In answer to point 1), showing that the ATML1 autoregulatory feedback is weak , and men“To test the type of feedback of ATML1 on itself, we determined whether induction of ATML1 could rapidly stimulate transcription of the endogenous ATML1 gene using qPCR . We found that ATML1 indeed acts in a weak positive feedback loop, increasing endogenous ATML1 expression by 1.7 fold within 24 hours . This reWe also agree with the reviewer that it is of interest that the model suggests that a fast degradation of the target can be used to recapitulate the threshold selection behavior. However, we have not ruled out other mechanisms for enabling the target to follow the dynamics of ATML1. We changed the sentence in Methods:“To ensure that the target approximately followed and mimicked the dynamics of ATML1, we proposed that the target would have a higher degradation rate than ATML1 itself.”to“To ensure that the target approximately followed and mimicked the dynamics of ATML1, we simulated a target with a higher degradation rate than ATML1 itself.”Finally, although we agree with the reviewer that ‘the fact that e.g. overexpression of ATML1 in the model would lead to an ectopic giant cell phenotype can already be deduced from the model form without running simulations’, we think thatour simulations of ATML1 over expression do show another non-trivial result. Our simulations show that it is possible to fine-tune the number of giant cells based on the levels of ATML1 expression, showing the plausibility of the threshold based patterning mechanism.The manuscript has been significantly improved in many places, but there is still one issue to address.Because you cannot supply ATML1 at a specific phase in the cell cycle, nor delete promoter elements, but instead rely on following endogenous ATML1 transcripts after induction to infer a feedback loop, this last experiment must be very carefully done and interpreted.A concern is whether the technical set up (and point at which ATML1 is monitored) has the capacity to report a "strong" feedback and indeed, what level of transcriptional up-regulation would be considered strong feedback. How can the experiment be calibrated? You will either need to cite evidence from similar experiments (same induction system and timing and tissue for monitoring) that put the 1.7 fold increase in context, or do this type of control experiment yourselves.ATML1 was sufficiently induced in inflorescences and show that with 10 µM estradiol we have about a 700-fold induction of the transgene. Our induction is close to the 1000-fold induction that Takada et al. 2013 observed in seedlings with the same transgene after 7 days on 10µM estradiol plates. To put the induction in the context of our ATML1 overexpression lines, we measured the total ATML1 transcript after induction, including both induced and endogenous. 10µM estradiol produced a 7-fold induction of total ATML1, which is above the 5-fold increase we observed in the homozygous ATML1 OX lines in which the sepals are covered in giant cells , FDH (1.6-fold), PDF2 (1.5-fold) and PDF1 (1.2-fold)  . Note thAP and AV in the model), the model dynamics changes from having a single ‘low’ fixed-point (no to weak feedback), to a bistability region (strong feedback). We note that if the feedback strength was increased further you could be left with a single ‘high’ fixed point, but we do not consider this regime in the paper as this could not fit the experimental data. By definition the parameters chosen for the strong feedback cases in the simulations were in the bistability region (or at its boundary), leading to a long-tailed or bimodal distribution of ATML1 expression .We modeled the different feedback strengths by varying the ratio between ATML1 on the transcription of the endogenous ATML1 gene in inflorescences using qPCR . We found that ATML1 induction with 10 µM estradiol lead to total ATML1 levels 7.1 times higher than the mock treated samples, and increased endogenous ATML1 expression 1.5-fold within 48 hours .To test the type of feedback of ATML1 on itself, we examined the effects of induction of 48 hours . This leme point . The resATML1 with increasing levels of estradiol further supports the case for weak feedback in the system, as endogenous ATML1 levels are not sensitive to small increases in exogenous ATML1. In the strong feedback case, sensitivity to ATML1 induction increases as the system is bistable and easily reaches the high value state. Thus, our results are consistent with weak feedback in the system.To further test the properties of the feedback, we also induced with 0.1 µM or 1 µM estradiol and achieved intermediate levels of induction and activation of downstream genes. In our strong feedback simulations, the parameters chosen are on, or close to, the bistability region in the system, leading to a long-tailed or bimodal distribution of ATML1 expression , which w