In this research, vibration frequency analysis of three layered functionally graded material (FGM) cylinder-shaped shell is studied with FGM central layer and the internal and external layers are of homogenous material. Strain and curvature-displacement relations are taken from Sander’s shell theory. The shell frequency equation is obtained by employing the Rayleigh Ritz method. Influence on natural frequencies (NFs) is observed for various thickness of the middle layer. The characteristics beam functions are used to estimate the dependence of axial modal. Results are obtained for thickness to radius ratios and length to radius ratios for different edge conditions. The validity of this method is checked for numerous results in the open literature. Shell vibration analysis carried out by employing different numerical techniques like Galerkin method, Rayleigh Ritz method, different quadrature method and finite difference method. These shells are fabricated by isotropic, laminated and multi-layered materials. Functionally graded materials have been developed by applying powder technology. Functionally graded materials are utilized for various objectives because of their proper material distribution in their fabrication. They are mostly used for high pressure and heat dominant surroundings. Sharma et al.3 scrutinized behaviour of vibrations for cylinder-shaped shells by employing the Rayleigh Ritz technique for clamped-free boundary conditions. Loy et al.4 analysed the fundamental frequencies of circular shaped shells by a generalized differential quadrature method (DQM). Further Loy et al.5 investigated the vibrations of functionally graded (FG) cylindrical shells fabricated by stainless steel and nickel. They showed the effects of formations of essential constituents on the frequencies. Moreover, Pardhan et al.6 explored vibration behaviour of FG cylindrical shells fabricated by stainless steel and zirconia for different edge conditions. Zhang et al.7 scrutinized free vibrations of cylindrical shells for different edge conditions by employing a local adaptive DQM. Naeem et al.8 employed a generalized DQM for the functionally graded material cylindrical shells to investigate vibration behaviour. Pellicano9 showed the response of an isotropic cylindrical shell for linear and non-linear vibrations by employing analytical experiment method. Vibration study of FG cylindrical shells has been done by Iqbal et al.10 and the shell governing motion equations were solved by using wave propagation technique. This technique was exceptionally helpful for vibration analysis. Axial modal dependence was estimated with help of beam functions in exponential form. Li et al.11 determined free vibration analysis of three layered cylindrical shells with FG material central layer. Flugge’s shell theory was used by them. Vel12 observed free and forced vibration of cylinder-shaped shell by using the elasticity solution technique for simply - supported conditions at both ends. Lam et al.13 showed the frequency vibration behaviour of multi layered FGM cylindrical shells for different edge conditions. Arshad et al.15 studied the FGM cylindrical shell for vibration frequency analysis with simply - supported end point conditions under different volume fraction laws. They used Love’s shell theory. Rayleigh Ritz technique was employed by them to solve the problem. Further he investigated vibration characteristics of FGM cylindrical shell with the effect of different edge conditions for exponential volume fraction law. Shah et al.16 analysed vibrations of NFs for fluid filled and empty CS constructed by elastic foundation. Naeem et al.17 explored the vibration behaviour of three layered functionally graded material cylindrical shell for different edge conditions. The internal and external layers were fabricated by FG materials whereas the central layer was of isotropic material. They used the Love’s thin shell theory. Arshad et al.18 examined the vibrations of natural frequencies of bi-layered cylinder-shaped shell. One layer was fabricated by isotropic material and the other was of functionally graded material. Rayleigh Ritz technique was utilized. Shah et al.19 scrutinized the vibration behaviour of three layered FGM CS constructed by Winkler and Pasternak basis. They used wave propagation approach for the solution of the model.Vibration of FGM cylindrical shell is a widely studied area of research in theoretical and applied mechanics. Among a large number of studies on vibrations of cylindrical shells (CS) we cite a few. Arnold and War-burton20 investigated vibrations of rotating cylindrical shells composed of FG materials. Natural frequencies of cylindrical shell were studied with effects of volume fraction law and different ratios.Ahmad and NaeemR, thickness h and length L as shown in Fig. x, θ, z) is fixed at the middle surface of the cylindrical shell, where x, θ and z lie in the axial, circumferential and radial directions of the shell, and  are the displacements of the shell in x, θ and z directions respectively.Consider a cylinder-shaped shell of radius ε1, ε2, γ and K1, K2, τ represent the strains and curvatures reference surface relations respectively. Prime (′) denotes the transpose of a matrix. These relations are taken from Sanders’ shell theory and written as:S] is defined asaij denote the extensional stiffness, bij the coupling stiffness and dij the bending stiffness. . They are defined as:The strain energy for a CS is represented by et al.5For isotropic materials E and bij coupling stiffness turn to zero for homogenous CS and ≠0 for FGM cylindrical shells and values of bij depend on the material distribution. Also bij become negative and positive due to irregularity of material properties at the mid plan. Here Young’s modulus represented by With the help of expression  and 5),,5), \docBy putting these expressions  and 4)  4) in thShell kinetic energy is symbolized by Ì and is stated as:t designates the time. Mass density is represented by ρ and ρt denotes the mass density for each unit length and it is expressed as:Here variable The Lagrange energy functional denoted by xm, ym and zm represent the amplitudes of vibration in the x, θ and z direction respectively, the axial and circumferential wave numbers of mode shapes are denoted by m and n respectively, ω signifies the angular vibration frequency of the shell wave. U(x), V(x), and W(x), denotes the axial model dependence in the longitudinal, circumferential and transverse directions respectively. Here we take φ(x) represents the axial function which satisfies the geometric edge conditions.The Rayleigh-Ritz procedure is used to achieve the natural frequencies of cylindrical shell. Now the displacement fields are presummed by the following relations:φ μm signify the roots of some transcendental equations and σm are parameters which depend on the values of μm.Here values of For generalization of this problem following non-dimensional parameters are used.Now expression  is altermax, Ìmax and After substituting expression (3.4) into the expressions for xm, ym and zm as follows,Rayleigh-Ritz procedure is employed to get the eigenvalue form problem of the shell frequency equation. The Lagrangian energy functional The obtained equations by arrangements of terms are written in matrix form as14 of the shell middle layer constructed from two constituents using trigonometric volume fraction law (VFL) are given by the following relations:In present study a cylindrical shell is considered constructed from three layers, the internal and external layers are fabricated by isotropic material while the central layer is constructed from FG materials nickel and stainless steel. The volume fractionsVf1+Vf2 = 1, where h is the shell thickness and υ denotes the power law exponent. It is presumed that each layer is of thickness h/3. Following are the material parameters: These relations satisfy the VFL i.e.z = −h/6, Efgm = E2, ρfgm = ρ2 and the material properties at z = h/6 becomes:From expression  at z = −z = −h/6 and the properties of material are combination of stainless steel and nickel at z = +h/6. The stiffness moduli are modified as:i = 1, 2, 6 and (iso) represents the internal and external isotropic layers and FGM represents the central functionally graded material layer.Thus the shell is consisted of purely stainless steel at ς- ς) and clamped-free  with those available in Loy & Lam4 for ς-Results for an isotropic cylindrical shell with following edge conditions, simply supported-simply supported  are decreased for type-I and increased for type-II less than 1% when power exponent law increased from υ = 1–20 for n = 1–5. Hence natural frequencies are affected by the configuration of the essential materials in the three layered CS.Tables ς-ς, ς-ς-Figures entclass1pt{minimaς-ς and f-f boundary conditions 105%, 84% and 95% for ς-ς-Figures L/R ratios. When L/R ratios increased 100%, 200% and 500% then natural frequencies decreased 72%, 87% and 95% respectively for n = 1. Frequencies increased with the increased of h/R ratios. Thickness to radius ratios has significant effect on natural frequencies (Hz).In present study, frequency analysis of three layered FGM cylinder shaped shell is done for different thickness of the shell middle layer. Strain and curvature displacement relationships are adopted from Sander’s theory. To solve the current problem Rayleigh Ritz method is employed. Natural frequencies are examined for six edge conditions. It is noticed that Natural frequencies becomes minimum with the increase in thickness of the shell FGM middle layer. These also decreased with the increased of Appendix


Background: While primary progressive aphasia (PPA) is associated with frontotemporal lobar degeneration (FTLD) pathology due to tau or TDP, clinical-pathological studies also demonstrate many cases have Alzheimer's disease (AD) pathology. The logopenic variant of PPA (lvPPA) is most often associated with AD pathology, but this has proven to be the least reliable PPA to diagnose using published clinical criteria. In this study, we used cerebrospinal fluid (CSF) analytes to identify patients with likely AD pathology, and relate phenotypic features of lvPPA to CSF.Methods: We studied 46 PPA patients who had available CSF analytes, including 26 with a clinical diagnosis of lvPPA, 9 with non-fluent/agrammatic variant (naPPA), and 11 with semantic variant (svPPA). We identified patients with likely AD pathology using amyloid-beta 1–42 (Aβ1−42) < 192 pg/ml and assessed MRI gray matter atrophy in these patients.Results: We found that 23 (49%) of 46 PPA patients have a low CSF Aβ1−42 level consistent with AD pathology. Twenty-one (91%) of 23 patients had a lvPPA phenotype, and 18 (79%) of 23 cases with an elevated CSF Aβ1−42 level did not have a lvPPA phenotype. Patients with a lvPPA phenotype demonstrated GM atrophy in the left lateral temporal lobe, and this was also seen in those with a CSF Aβ1−42 level < 192 pg/ml.Conclusion: The lvPPA clinical phenotype may be a useful screen for CSF analytes that are a surrogate for likely AD pathology, and may help establish eligibility of these patients for disease-modifying treatment trials. Primary progressive aphasia (PPA) refers to a syndrome of declining language ability that results from a neurodegenerative disease. Three variants of PPA have been identified: non-fluent/agrammatic (naPPA), semantic (svPPA), and logopenic (lvPPA) . It is v1−42) to identify PPA patients with likely AD pathology, and assessed whether this corresponds to PPA patients with a lvPPA phenotype. Several previous studies have examined Aß1−42 in PPA. In one study, patients with a clinical diagnosis of lvPPA had lower Aß1−42 levels than controls and naPPA patients  analytes have been shown to serve as a sensitive biomarker for AD pathology , this stpatients . In a lapatients . Here wen = 26, svPPA: n = 11, naPPA: n = 9) (1−42 level that was < 192 pg/mL and AD pathology. One autopsy case with FTLD-Tau pathology had a CSF of Aß1−42 level that was >192 pg/mL and did not have a lvPPA phenotype. Some of these cases have participated in other CSF biomarker studies.Patients included for this study were seen in the Department of Neurology out-patient clinic of the Penn FTD Center and data were retrieved from The Integrated Neurodegenerative Disease Database (INDD)  at the U: n = 9) . All patWe assessed the neuropsychological profile for each subtype of PPA . We evalAll procedures, including CSF collection and MRI, involved participation in an informed consent procedure, and were performed in accordance with the Helsinki Agreement and the rules of the Institutional Review Board at the University of Pennsylvania.1−42, p-tau, and t-tau levels, as previously described (CSF samples were obtained by routine lumbar puncture according to standard operating procedures of the Alzheimer's Disease Neuroimaging Initiative (ADNI) . In brieescribed , and a sescribed .1−42 level, and examined this cut-point in our cohort. The variables utilized in this analysis were the categorical variable “clinical phenotype” and the continuous variable “CSF Aß1−42 level.” A receiver operating characteristic (ROC) curve analysis used sensitivity and specificity to define the cut-point between these groups, and provided the area under the curve (AUC). In both the analysis using the predetermined cut-point and the empirically determined cut-point in our cohort, we tabulated the frequency of patients with lvPPA compared to PPA patients with another phenotype in the two CSF-determined groups, and used chi-squared analyses to assess whether there was a statistically significant difference between phenotypes within and between CSF-defined groups.In the first analysis, we grouped patients according to the predefined cut-point of 192 pg/ml. Since the cut-point based on an AD phenotype may not generalize to non-amnestic cases with AD pathology, we also implemented a second analytic approach. Here we used a k-means cluster analysis with a two-group solution to identify the groups among all PPA patients in our cohort according to their CSF Aßp > 0.1). MRI exclusion criteria included poor-quality MRI at visual inspection . Briefly, participants underwent a structural T1-weighted MPRAGE MRI acquired from a SIEMENS 3.0T Trio scanner with an eight-channel coil using the following parameters: repetition time (TR) = 1,620 ms; echo time (TE) = 3 ms; 160 1.0 mm slices; flip angle = 15°; matrix = 192 × 256; and in-plane resolution = 0.9766 × 0.9766 mm. T1 MRI images were preprocessed using antsCorticalThickness  with 10,000 permutations that is equivalent to an analysis protecting for multiple comparisons. We report clusters that survived p < 0.01 with a minimum of 150 adjacent voxels. We report two sets of t-tests: First, we examined each patient group  relative to controls; and second, we examined each CSF-subgroup relative to controls .High resolution T1-weighted MRI scans were available for 20 lvPPA, 10 naPPA, and 11 svPPA, and we compared these PPA patients with 69 demographically matched control participants. The PPA patients with MRI matched the clinical and demographic characteristics of those without MRI  of these cases had an lvPPA phenotype, revealing significantly more cases of clinically diagnosed lvPPA than non-lvPPA among PPA patients with a lower CSF Aß1−42 level (p < 0.001). The sensitivity for low CSF Aß1−42 level to identify lvPPA compared to non-lvPPA is 91%; the specificity is 89%; the positive predictive value is 87%; and the negative predictive value is 92%. Of the 2 non-lvPPA cases with a lower Aß1−42 level, one had an svPPA phenotype (CSF Aß1−42 level = 167 pg/ml) and the other had a naPPA phenotype (CSF Aß1−42 level = 183 pg/ml). The svPPA patient with CSF Aß1−42 level < 192 pg/mL had an age at onset of 68. Not only was this well-above the mean age of onset of lvPPA patients with CSF Aß1−42 < 192 pg/mL, but it was also well above the mean age at onset of the svPPA patients with CSF Aß1−42 level > 192 pg/mL , suggesting the possibility of AD co-pathology. Duration of disease at the time of obtaining CSF was shorter in this svPPA patient (1 year) and longer in this naPPA patient (7 years), and differed from the mean disease duration of lvPPA patients with CSF Aß1−42 < 192 pg/mL  and from that of PPA patients with CSF Aß1−42 level >192 pg/mL . The svPPA and naPPA patients with CSF Aß1−42 < 192 pg/mL levels did not differ from lvPPA patients with CSF Aß1−42 < 192 pg/mL, and did not differ from PPA patients with CSF Aß1−42 level >192 pg/mL with regards to education level.A summary of CSF analyte values is provided in 1−42 level < 192 pg/mL, significantly greater than the number of lvPPA cases with CSF >192 pg/mL (p < 0.001). The sensitivity for lvPPA to identify low CSF Aß1−42 level compared to elevated CSF Aß1−42 level is 81%; the specificity is 75%; the positive predictive value is 91%; and the negative predictive value is 78%. The mean CSF Aß1−42 level of the 5 lvPPA cases with CSF >192 pg/mL was 317 pg/mL (SD = 186). The lvPPA patients with CSF Aß1−42 < 192 pg/mL did not differ significantly from those with CSF Aß1−42 level >192 pg/mL with regards to education level and disease duration at the time that CSF was obtained. The age at onset, however, differed significantly , with the 5 lvPPA patients with an elevated CSF Aß1−42 level having an older age at onset  than lvPPA cases with CSF Aß1−42 < 192 pg/ml .Twenty-one (80.8%) of 26 cases with a clinical diagnosis of lvPPA had a CSF Aß1−42 level in non-amnestic AD with early-onset disease may differ from the CSF Aß1−42 level associated with later-onset amnestic AD, we also used a cluster analysis to partition the entire cohort of PPA patients (n = 46) according to the CSF Aß1−42 level. One cluster included PPA patients with a lower Aß1−42 level , and the second cluster included PPA patients with a higher Aß1−42 level . A ROC curve analysis in this sample defined a cutpoint at 204.2 pg/mL, yielding 91% sensitivity, 89% specificity, and an area under the curve = 0.914. Independent samples t-test showed that the Aß1−42 level of the cohort with likely AD pathology to be significantly lower than that of the cohort less likely to have AD pathology .Since CSF Aß1−42 level and a high Aß1−42 level. This demonstrated distinct areas of atrophy that stratified the two groups  is present in the CSF of many PPA patients, suggesting that these PPA patients may have AD pathology (1−42 level in the range consistent with likely AD pathology have a clinical phenotype of lvPPA.We found that a low Aβathology . It is pathology . In our athology , which i1−42 level had lvPPA, this phenotype alone cannot be used reliably to identify patients with likely AD pathology. For example, in our cohort we found two patients with a low CSF Aß1−42 level who did not have a lvPPA phenotype: One patient had svPPA and another had naPPA. Without autopsy evidence, we can only speculate about the basis for this discrepancy. One possibility is that these cases in fact have AD pathology but in an anatomic distribution more consistent with these non-lvPPA phenotypes. Several patients with atypical presentations of AD pathology have been reported with svPPA or naPPA phenotypes (While a large majority of cases with a low CSF Aßenotypes , 25. A senotypes , and we enotypes .1−42. The five lvPPA cases with elevated CSF Aß1−42 levels had an age of onset that was older than that of the lvPPA patients with lower CSF Aß1−42 levels. With the caveat that we did not have a pathologic diagnosis in our cases, our findings are consistent with the claim that a clinical evaluation for lvPPA can be an inexpensive way to screen PPA patients for those who may be eligible for participation in disease-modifying therapies targeting the misfolded proteins contributing to AD pathology. Additional biomarker data would be helpful to confirm the diagnosis of PPA associated with likely AD such as amyloid-PET, although it should be noted that amyloid-PET is also associated with false-positive and false-negative findings (Another important consideration is that 26 cases in our cohort had a lvPPA phenotype, but only 21 of these cases with a lvPPA phenotype had low CSF Aßfindings .1−42 level cohort by showing nearly identical areas of reduced GMP in the lvPPA cohort and the low Aβ1−42 level cohort (1−42 cohort, including the left middle-superior temporal gyrus, left parietal region, and left precuneus. This anatomic distribution of atrophy is a subset of regions typically affected in clinical AD. Importantly, our cohort did not have significant medial temporal lobe atrophy, emphasizing that these patients had PPA and not a language-dominant variant of AD. Despite the small cohort of PPA patients with elevated CSF Aβ1−42 consistent with a non-AD form of PPA, the pattern of atrophy in this group is distinctly different from that of the low CSF Aβ1−42 cohort with lvPPA. The areas of atrophy found in the group with elevated CSF Aβ1−42 include the left anterior temporal region, and the left inferior frontal-insula region. These are areas associated with svPPA and naPPA, respectively (The MRI analysis was consistent with the finding that the lvPPA phenotype is most prominent in the low Aβl cohort . The patl cohort , 26, 28.l cohort  and audil cohort  that areectively .1−42, t-tau, and p-tau181 in a small cohort of PPA patients, AD patients, and healthy controls (181/Aß1−42 ratio allowed separating AD and non-AD patients, although there was no available converging evidence such as imaging or autopsy to support this finding. Another study also found lower levels of CSF Aβ1−42 in clinically-diagnosed lvPPA compared to other PPA patients, discriminating between lvPPA and naPPA/svPPA with 86% sensitivity and 69% specificity (Other groups have explored the utility of using CSF analytes to differentiate PPA phenotypes. One study evaluated CSF levels of Aßcontrols . They focificity . In a smcificity . In a lacificity . An impocificity , 31. In cificity , for exacificity . Serum ncificity , and othcificity  [also secificity , 33. Con1−42 is not uncommon in patients with PPA, and that there is a statistical association between a low CSF Aβ1−42 level and the lvPPA phenotype. The link between lvPPA phenotype and a surrogate marker of AD pathology was further supported by MRI imaging. The potential use of the lvPPA clinical phenotype to screen for CSF analytes as a surrogate for likely AD pathology may help establish eligibility of these patients for disease-modifying treatment trials.Several caveats should be kept in mind when considering our results. First, a relatively small number of patients participated in our study, although PPA is a relatively rare condition, and we adhered narrowly to the published criteria for PPA to determine the usefulness of AD CSF analytes within the scope of these criteria. Other CSF analytes may be informative in PPA and have been reported in some patients from this cohort elsewhere , and addThis study was carried out in accordance with the recommendations of the Institutional Review Board of the University of Pennsylvania with written informed consent from all subjects and witnessed by a responsible caregiver. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the Institutional Review Board of the University of Pennsylvania.AH, CJ, MU, DI, and MG data collection. CN, CJ, CM, and KC data analysis. CN, CM, DI, KC, and MG manuscript writing and editing.MG is an Associate Editor of Neurology, participates in clinical trials sponsored by Biogen, Alector, and Eisai, and receives support for his role as a consultant for Biogen, UCB, Ionis and Bracco. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The rheological and anti-ultraviolet (anti-UV) properties were investigated and compared to filler-free PBS as well as LDH Mg2Al/nitrate as references. Both organo-modified LDHs exhibited a remarkable chain-extension effect for PBS with an outstanding increase in the zero-shear viscosity η0 for PBS–Mg2Al/PHE (two order of magnitude increase as compared to filler-free PBS). These results were compared to data found in the literature. Moreover, HIS and PHE anions embedded into the LDH structure can successfully prevent the chain scission reactions that usually occur during photo-ageing of PBS under UV radiation exposure. This highlights the outstanding performance of the LDH hybrid materials, and in particular, their application as a polymer chain extender and UV stabilizer for PBS, which can likely be extended to other biodegradable polymers.Polybutylene succinate (PBS) nanocomposite materials were prepared using a melt compounding process. The Mg Polymer nanocomposite materials have been one of the leading scientific topics over the past decades –4. BecauA classical approach to overcome these problems is to add a stabilizing agent that may act as an anti-moisture and UV stabilizer. The small organic molecules used –9 are tyToday inorganic containers are thought to play the dual role of embedding a specific agent to avoid its ingress into a polymer as well as providing the complementary properties as a gas barrier and mechanical reinforcement for the polymer. Among the candidates, layered double hydroxides (LDH) appear to be a promising choice in endowing multiple properties to the polymer. This due to their versatility in terms of chemical composition and the relatively straightforward preparation involving soft chemistry routes.II1−xMIIIx–(OH)2](An−)x/n·mH2O, where MII and MIII are di- and trivalent cations respectively and An− is an anion. The presence of trivalent cations results in a positive charge on the layers, which has to be balanced by diverse inorganic or organic anions intercalated between the layers ·2H2O were prepared by a co-precipitation method according to a similar procedure as described by Totaro et al. [3)2·6H2O and Al(NO3)3·9H2O (molar ratio 2/1) in milli-Q water (100 mL) was added dropwise to the vigorously stirred water solution of organic guest (50 mL) for 3.5 h and the pH was kept constant (10.0 ± 0.1) with the addition of NaOH solution. When the addition of salts was finished, the reaction mixture was aged for 3.5 h at room temperature. Both the reaction and ageing processes were performed under nitrogen atmosphere to avoid contamination by carbonate. The final products were centrifuged, washed several times with deionized water and vacuum dried at 40 °C for 24 h. Next, the solid LDH fillers were ground using a Retsch CryoMill machine and sieved to obtain a fraction with particles of diameter less than 50 µm.The LDH hybrids with general formula Mgo et al. , for posThe PBS–LDH nanocomposites were obtained using a twin screw extruder Hakke MINILAB microcompounder (Thermo Electron Corporation). The melt extrusion process was performed at 120 °C with a roller speed of 100 rpm over 5 min. A LDH mass loading of 5 wt % was used as it appeared to be the right compromise to be detected by XRD, as formerly demonstrated . This waPBS–LDH nanocomposite films with thickness of about 90–100 µm were prepared by compression moulding between two Teflon sheets under 100 bar at 120 °C for one minute.PBS and PBS–LDH nanocomposite films were placed on a rotating carousel in a SEPAP 12/24 chamber from ATLAS equipped with four 400 W mercury lamps with spectral rays above 300 nm. The accelerated photodegradation was studied at 60 °C and under air atmosphere.−1 were collected.FTIR spectra of the LDH fillers were recorded using a Nicolet 380 FTIR spectrometer (DTGS detector) equipped with an attenuated total reflection single reflection diamond from Specac, and 32 scans and at a resolution of 4 cmLDH fillers and PBS–LDH nanocomposites were characterized by XRD using a Philips X-Pert Pro diffractometer with Cu Kα radiation. Data were collected at room temperature from 2θ = 2.0–70.0° with a step size of 0.03° and a counting time of 10 s per step.UV–vis spectra of the LDH substances and polymer nanocomposites were obtained using a Shimadzu UV-2101 PC spectrophotometer. Kubelka–Munk theory was applied to the LDH filler data to transform the diffuse reflectance spectra into absorption spectra.−1 and excitation wavelength of 280 nm.The fluorescence spectra of PBS–LDH nanocomposites were recorded with a Perkin-Elmer LS-55 luminescence spectrophotometer equipped with a front surface accessory and pulsed xenon excitation source. The emission signal was collected with the monochromater from 200 to 600 nm at scan rate of 600 nm min−1, then kept at high temperature for 2 min, and then cooled down to −60 °C at 10 °C min−1. After that, the thermal history of the samples was deleted and a second scan profile was performed by heating from −60 °C to 150 °C at 10 °C min−1. The glass transition temperature (Tg), the melting temperature (Tm) and the enthalpy of fusion (∆Hm) were measured from the second scan. Tg was taken as the midpoint of the heat capacity increment associated with the glass-to-rubber transition. The crystallization temperature (Tc) and the enthalpy of crystallization (∆Hc) were measured during the cooling scan.The thermal properties of the LDH fillers and polymer composites were characterized by differential scanning calorimetry (DSC) using a Perkin-Elmer DSC6 apparatus. The analyses were carried out under nitrogen, and firstly, the samples (≈10 mg) were heated from 40–150 °C at 20 °C min−1) using a Perkin Elmer TGA7 apparatus. The temperature range of 50–850 °C and heating rate 10 °C min−1 was applied. The onset degradation temperature (Tonset) was taken from the intersection of the tangent of the initial point and the inflection points. The 10% mass loss temperature (T10D) was also measured.Thermogravimetric analysis (TGA) of LDH fillers and composites was performed in air atmosphere , loss modulus (G”) and tan δ (ratio of G” and G’) were monitored automatically against frequency.The melt rheological properties of the polymer composites were measured at 120 °C using a dynamic mechanical spectrometer (ARES Rheometric Scientific T&A Instruments) equipped with two parallel plate holders of 8 mm in diameter. The measurements were performed in oscillatory frequency sweep mode with the range of frequency sweeps from 0.1 to 100 rad s−1, frequency 3 Hz, strain 0.01%).The dynamic mechanical thermal properties were measured using a Rheometric Scientific DMTA IV dynamic mechanic thermoanalysis (DMTA) instrument with a dual cantilever testing geometry. Test samples were prepared by injection moulding at 140 °C using a Minimix Molder, obtaining small-sized bars (33 × 8 × 2 mm). Such samples were scanned from −150 °C to 80 °C  and L-phenylalanine (PHE) intercalated between metal cation layers were prepared by the co-precipitation method.The organo-modified MgThe synthesis and characterization of LDH–phenylalanine compounds has already been reported by Aisawa et al. . However2Al/nitrate, as a reference sample, displays an X-ray diffraction pattern typical of lamellar compounds with a series of strong basal 00l reflections at low angles (R3m), generally observed in LDH systems. The interlayer distance can be determined in a straightforward way from the position of the first (00l) reflection indexed as (003), leading here to a value of ≈8.79 Å as expected for a nitrate-containing LDH. The position of the (110) reflection at high angles, near 2θ = 60° for Cu Kα radiation, allows the value of the lattice parameter a to be determined since a = 2d110. Its value reflects the radii of the cations and the M2+/M3+ ratio within the hydroxide layers. The value obtained here is ≈3.03 Å and suggests a Mg2+/Al3+ molar ratio slightly lower than 2 [Mgw angles . In the r than 2 .l) reflections: one attributed to the desired Mg2Al LDH hybrid phase and the other to a LDH/nitrate phase. In hkl) were assigned to each reflection (assuming 3R polytypes) with positions marked by ticks under the peaks. The presence of this nitrate phase, also evidenced by FTIR analysis (see below), can be explained by either the small excess of the organic anion used during the synthesis or the large excess of nitrate anions in the reaction medium introduced by the reactants in the form of nitrate salts. The diffraction pattern obtained for LDH/HIS is poorly defined with only one (00l) reflection visible, from which the interlayer distance was estimated as ≈12.2 Å (l) reflections are observed and their positions are consistent with an interlayer distance of ≈18.4 Å  leads us to propose a monolayer arrangement for LDH/HIS where the main plane of the molecule is oriented nearly parallel to the hydroxide layer . This interlayer arrangement suggests an important confinement of histidine molecules between LDH layers. This may explain the presence of co-intercalated nitrate ions, which decreases the constrained accommodation that would have resulted in the close proximity of the HIS molecules to one other.For the synthesis performed in the presence of histidine and phenylalanine, we observed two series of  overlap in the range of 1700–1480 cm−1 and the lattice vibration of M–O in the platelet structure can be depicted at a low wavenumber (800–650 cm−1).FTIR spectroscopy was applied to confirm the intercalation of HIS and PHE in the hybrid structures. The FTIR spectra of LDH/HIS and LDH/PHE in comparison to pristine amino acids are displayed in − associated with metal cations Al3+ and Mg2+. Finally, the third weigh loss at >400 °C can be observed for the decomposition of amino acid or nitrate anions and the residual mass at 800 °C can be considered as the mass residue of Mg2AlO7/2. Evidently these residual mass losses are not equal because of the difference in molecular weight of the guest organic species as well as the different hydration rate.The experimental chemical composition of all synthesized LDHs has been estimated based on TGA analysis and collected in 2Al LDHs should be effective as potential UV stabilizers.The UV-absorbing properties of the LDH structure with amino acids are presented in 2Al LDHs and Mg2Al LDH nitrate were prepared by melt extrusion. The XRD patterns are presented in PBS nanocomposites with 5 wt % of amino-acid-modified MgThe diffraction peaks located at 2θ = 18–30°, which are characteristic for the crystallinity of PBS, seem to not be affected by the addition of the LDH hybrid materials. There is no reflection coming from pristine LDHs modified by amino acids. In contrast, the PBS–LDH nitrate composite exhibits an initial basal spacing of 0.84 nm. In the case of PBS composites with LDH/PHE or LDH/HIS, the absence of harmonic peaks from the layered filler structure can be explained by their lower crystallinity or by a partial exfoliation occurring during thermal processing. For PBS–LDH/HIS, there is an additional peak at 2θ = 18° that is difficult to assign. Its associated small full width at half maximum (FWHM) may be attributed to rather structurally well ordered organic species. Tentatively it may be ascribed to some degradative effect coming from the imidazole cycle since there is no such effect for the other PBS composites. It is well known that imidazole cycle is prone to coordinate to metal ions as in metallo-proteins . The temTonset and T10D (temperature at which the polymer loses 10 wt %) measured for all nanocomposites are consistently lower than for PBS  to 83 and 82 °C for PBS–LDH/HIS and PBS–LDH/PHE, respectively. Small differences are also observed in Tg, but without a regular trend. However, Tg will be discussed further in the next paragraph, because DMTA is a more sensitive method to detect this. TGA and DSC traces for all nanocomposites are shown in the The introduction of LDH fillers into PBS does not improve the thermal stability of the polymer and the decomposition process proceeds mainly in the range 300–450 °C. The determined  for PBS . This trx-axis (η′ at η″ = 0), the Newtonian zero-shear viscosity η0 can be calculated using The effect of synthesized amino acid LDH fillers on the chain extending and molecular weight evolution was measured using melt rheology. The rheological data were plotted in Cole–Cole plots – a model curve used to predict the variation in complex viscosity components, where the imaginary viscosity (η”) versus real viscosity (η’) is plotted as a circle arc in the complex plane. This representation is very helpful in the analysis of polymer and polymer composites –26. By f2Al LDH filler on the PBS chain extension is presented as η”–η’ Cole–Cole plots in 0 of 7174 Pa s which is almost 80 times higher than for PBS.The effect of 5 wt % MgThe role of layered double hydroxides modified with 3-(4-hydroxyphenyl)propionic acid and amino acids such as tyrosine, tryptophan as PBS chain-extenders has been known and described previously, however the results were not so spectacular ,17,23. Ta = 3.4 in 2Al/PHE and filler-free PBS is of about 400 times (Mw(PBS–Mg2Al/PHE)/Mw(PBS) = 397, resulting in a similar change in the average repeat units under melt state polymer rheology), thus underlining once again the outstanding effect of the filler in networking PBS chains. This occurs without a transition to a gel-like structure, which would be deleterious for the PBS processability.With a value of 2Al or Zn2Al LDH cations and with different anions from carboxylic and dicarboxylic acids, ascorbic acid and amino acids) were incorporated to PBS polymer by both methods: in situ polymerization and melt extrusion. Depending on the type of organic molecules, chain extension or plasticizing effects were observed.0 from 40–50 Pa s for unmodified PBS 400, 430, 450 and 790 Pa s for Mg2Al/HPP, Mg2Al/stearate, Mg2Al/TRP and Zn2Al/HPP, respectively (when used at 2.5–3 wt %).In the case of in situ polymerization, an increase in the Newtonian zero-shear viscosity of nearly 20–28 times was observed when LDHs with embedded PBS oligomers were used, whereas the smaller molecules increased η2Al/TYR and Zn2Al/cinnamic LDHs (5 wt %), from 115 to 245 and 360 Pa s respectively, while in other cases the differences were smaller. Our present results are here spectacular, since an increase of 80 times for η0 is observed, from 90 Pa s (unmodified PBS) up to more than 7,000 Pa s for Mg2Al/PHE LDH (5 wt %), respectively. Moreover, the PBS nanocomposite composition displays the desired UV-stabilizing effect (described below), which was described previously but only for PBS composites with Zn2Al/amino acid LDHs [2Al) that is known to present some UV screening due to Zn2+ cations. However, hydrotalcite-type Mg2Al (as used in this work) has yet to be investigated.The effectiveness of the LDHs was lower when they were added to the PBS in the melt extrusion process, most likely because LDH fillers were less dispersed in this case compared to the in situ polymerization approach. The best results were obtained for Zncid LDHs . This woE′) and tan δ, which is the ratio of the loss modulus to the storage modulus  up to RT, respectively. More detailed, at room temperature, the increase in E’ is 26% for PBS–LDH/PHE, 20% for PBS–LDH/HIS and 17% for PBS–LDH/nitrate. The larger storage modulus with respect to PBS for such composites indicates a mechanical reinforcement of the matrix due to a better interfacial interaction filler/polymer achieved. Such results are consistent with rheology experiments. For all samples, the glass transition temperature is recognized by the large decrease in the storage modulus and by the corresponding peak maximum in tan δ. The values extrapolated  highlight a slight increase with respect to the homopolymer, and therefore, the filler can be assumed to affect the mobility of the chains as a nucleating agent.The temperature and frequency-dependent mechanical relaxation data for the composites were recorded via the storage modulus  and the overlap of PBS impede any classical spectroscopic characterization. Nevertheless, considering the tethered molecular backbone of PHE as well as the Cole–Cole response indicative of a chain-extension effect rather than a gel-like transition, this may indicate the presence of a jamming structure due to mobility hindrance of PBS chains as observed for other LDH composites with polystyrene  rather t2Al/HIS , which may be affected under UV radiation of the polymer chains.The worst behaviour in terms of UV stability for Mg fillers ) may be 2Al LDHs are presented in 2Al/nitrate LDH shows lower transmittance, as described above. The evolution of the UV–vis transmittance of PBS and PBS nanocomposites was measured during photodegradation at 60 °C (UV–vis transmittance spectra of PBS and the nanocomposites with 5 wt % of MgSimilar trends have been also registered using fluorescence spectroscopy . The int2Al/TYR LDH and is described in the literature [A similar phenomenon was observed for the PBS composite with Znterature . It can 2Al layered double hydroxides and subsequently dispersed in PBS by melt blending. The XRD analysis indicated the presence of nitrate phase in the LDH structure, which can be explained by the large excess of nitrate anions in the reaction medium during co-precipitation. For this reason, the LDH framework with nitrate anions was also synthesized and applied for PBS as a reference.Two amino acids, L-phenylalanine and L-histidine, have been interleaved into MgThe best results were obtained in the case of the LDH/PHE filler. By the use of melt rheology, an outstanding chain-extending effect was observed, with an increase of the Newtonian zero-shear viscosity of almost 90 times in comparison to filler-free PBS samples.Moreover, the synthesized organo-modified LDHs were found to be effective as UV-stabilizers since they successfully prevent the chain scission reactions which usually occur during photo-ageing of PBS. Especially in the case of LDH/PHE, the UV-stabilizing effect is quite sustainable over time, thus making such a filler composition a good alternative in the design of polymer composites with these specific properties. These properties are important from the point of view of processability as well as for their prolonged shelf life and extended use.File 1Additional experimental data and experimental schemes.


During embryogenesis cells make fate decisions within complex tissue environments. The levels and dynamics of transcription factor expression regulate these decisions. Here, we use single cell live imaging of an endogenous HES5 reporter and absolute protein quantification to gain a dynamic view of neurogenesis in the embryonic mammalian spinal cord. We report that dividing neural progenitors show both aperiodic and periodic HES5 protein fluctuations. Mathematical modelling suggests that in progenitor cells the HES5 oscillator operates close to its bifurcation boundary where stochastic conversions between dynamics are possible. HES5 expression becomes more frequently periodic as cells transition to differentiation which, coupled with an overall decline in HES5 expression, creates a transient period of oscillations with higher fold expression change. This increases the decoding capacity of HES5 oscillations and correlates with interneuron versus motor neuron cell fate. Thus, HES5 undergoes complex changes in gene expression dynamics as cells differentiate. The single-cell gene expression changes during spinal cord formation and neurogenesis in mice are unclear. Here, the authors use a HES5 reporter to live image, then mathematically model, oscillations in single cells in presumed progenitors and neurons of the developing spinal cord. In the dorso-ventral (D-V) axis of the spinal cord elegant experiments have shown that fate decisions require integration of a wide range of signals over time, many in the form of morphogen gradients, resulting in downstream gene expression changes3.During embryogenesis cells balance proliferation with differentiation to make cell state transitions that lead to the formation of functional organs. This is exemplified by development of the central nervous system, which requires the balance of neural progenitor maintenance with differentiation during multiple waves of differentiation into neuronal and glial cell-types7. However advances in single-cell live imaging of gene expression have shown that it is often highly dynamic, suggesting that the control of cell state transitions is more complex10. Rather than being in an on or off state, a handful of transcription factors (TFs) have been shown to oscillate with periodicity of a few hours11. Oscillations have been long described in somitogenesis12, but are a relatively recent discovery in neurogenesis. This is because unlike somitogenesis where oscillations are synchronous within each somite, they tend to be asynchronous in neural progenitor cells (NPCs) and so required unstable reporters and single cell imaging to be discovered13. Thus, it is not only changes in gene expression levels that are important, but the short-term dynamics of gene expression can also carry important information for cell state transitions. Indeed, there is experimental and theoretical evidence that cell fate transitions may be controlled by a change in the dynamic pattern of gene expression, which could be from oscillatory to stable expression, or to oscillatory with different characteristics15.Single-cell transcriptomics have greatly enhanced our understanding of gene expression changes and networks involved in fate decisions and of the bifurcation points where decisions are made17. Like HES1, HES5 is a Notch target bHLH TF, which is highly expressed by NPCs and decreases in expression as differentiation proceeds19. Knock-out mice and over-expression studies have shown that HES5 functions to maintain the undifferentiated progenitor state through repression of proneural genes, such as Neurog2 and Atoh1 that promote neuronal differentiation22. Like HES1, HES5 has been reported to oscillate in NPCs in vitro9.In the case of the transcriptional repressor HES1, a key target of Notch signaling, it has been known that oscillatory expression is driven by transcriptional auto-repression coupled with delays, instability of mRNA and protein and non-linearity of reactions, common principles of many biological oscillators25. Other theoretical studies provide additional support for the importance of a change in dynamics by showing that gene expression networks in the D-V dimension of the spinal cord can generate multi-way switches (stable or oscillatory)26.Changes in HES1 dynamics are mediated by a change of the parameters or initial conditions of the oscillator, likely through changes in mRNA stability or protein translation under the influence of a microRNA, miR-929. Current ideas for the role of such embedded stochasticity include cases where it would be an advantage31 or conversely, an impediment for cell fate decisions33 and mechanisms to suppress noise after a fate-decision34.An additional revelation of single-cell live imaging studies is that gene expression is characterised by varying degrees of noise due to the stochastic nature of transcription35. A study using electroporation of a promoter reporter of Hes5-1 in chicken spinal cord tissue reported activation of Notch signaling throughout the progenitor cell cycle but most frequently before mitosis36. However, this approach suffered from plasmid loss and varying degrees of plasmid transfection and did not report on endogenous HES5.However, although these studies have shed new light into the problem of cell-state transitions, how cells make decisions in the context of a multicellular tissue is poorly understood. This is because both single-cell transcriptomics and live imaging data are routinely performed in single cells taken out of the tissue environment. Existing studies of oscillatory expression in the mouse brain and spinal cord lack the statistical power needed to give a comprehensive understanding of the dynamics in the tissueHes5 genetic oscillator operates close to a bifurcation point where noise can tip it from aperiodic to periodic expression. Taken together, our findings suggest that single progenitor cells in a tissue are noisy and are thus primed to enter a transient oscillatory phase as the cells differentiate. Additionally, our study shows that tissue level single-cell heterogeneity has a complex origin in both short-term and long-term dynamics and that the dynamics are decoded en route to differentiation, where they correlate with the choice of cell fate that the cells adopt.Here, we develop ex vivo slice culture of embryonic Venus::HES5 knock-in mouse spinal cord (E10.5) to study the expression dynamics of HES5 in the context of a tissue, with single cell resolution. We report that HES5 expression has a 10-fold range between cells in a single expression domain that arises from short-term fluctuations and longer-term trends of decreasing HES5. We use hierarchical clustering to define distinct clusters of single cell HES5 expression dynamics. New statistical tools show that oscillatory HES5 is more frequently observed in cells that transition towards differentiation where it is coupled with an overall decrease in HES5 expression generating larger instantaneous fold changes. Oscillatory decline of HES5 correlates with interneuron fate, suggesting the dynamics are decoded in the choice of cell fate. By contrast, dividing NPCs are less frequently periodic but significantly more noisy in their HES5 expression. Computational modelling with stochastic differential delay equations, parameterised using experimental values and Bayesian inference, suggest that in the spinal cord tissue environment the 9 to ensure that it is a faithful reporter of the un-tagged gene. In transverse sections of E10.5 spinal cord Venus::HES5 shows a broad ventral and a smaller dorsal domain . These findings confirm that the Venus::HES5 fusion protein is a faithful reporter of endogenous un-tagged HES5 expression  in live homozygous Venus::HES5 E10.5 embryo slices Fig. . FCS is nM. Fig. . The meanM. Fig. . These f38 by plotting the distribution of single-cell Venus::HES5 intensities from manual segmentation of nuclei in a single slice  of intensity divided by the mean intensity) of Venus::HES5 over time in single cells increased over 4, 8, 12, 14.25, and 17.25 h Fig. . By 8–1240. Newly born cells fated towards neuronal differentiation migrate basally away from the apical surface, exit the cell cycle and turn on markers of differentiation (Tuj1 and NeuN)40. We therefore sought to infer cell state by position, motility and division of cells using progenitor/neuronal immunochemistry data in ex vivo slices as reference.We hypothesise that the different clusters of Venus::HES5 expression may represent different cell-states. It is well known that proliferating NPCs (SOX1+/2+) are found apically in the ventricular zone, undergo inter-kinetic nuclear migration (INM) dividing at the apical surfaceThe average position of cells in cluster 1 was significantly closer to the ventricle than those in cluster 3 Fig. . FurtherNuclei of cells in cluster 1 moved both apically and basally, consistent with INM but had the shortest displacement as they returned apically. Meanwhile nuclei of cells in clusters 3 and 4 had a larger displacement, which was unidirectional towards the basal side Fig. . H2B::mCCells divided at the apical surface  are proliferating NPCs maintained by Notch signalling, while cells with decreasing Venus::HES5 levels over time (clusters 3 and 4) are neural cells undergoing cell state transition to differentiation. We do not know the significance of the subtle differences between clusters 1 and 2 or 3 and 4 but we suggest that the simplest interpretation of our data is that clusters 1+2 give rise to clusters 3+4. However, more complex alternatives may exist, such as subtle heterogeneity in progenitors translated linearly to neuronal progeny heterogeneity.9 but statistical analysis has not been performed. Here, we have focused on the endogenous Venus::HES5 fluorescent fusion protein because unlike luciferase, it allows single cell spatial resolution in the tissue environment by confocal microscopy. The t1/2 of Venus maturation (15 min)42 is suitably short compared to HES5 protein half-life .Previous reports show periodic HES5 expression in embryonic mouse cortical NPCs using luciferase and fluorescence imagingTo analyse oscillations, we first subtracted long-term changes in level (trend) caused by HES5 downregulation Fig. . We thenWe found that overall 41% of cells in E10.5 spinal cord ex vivo showed oscillatory Venus::HES5 expression  than dividing progenitors in clusters 1 and 2 Fig. . By contGiven that progenitor cells close to the ventricle (clusters 1 and 2) are likely to turn into the transitory and differentiating cells in clusters 3 and 4, we conclude that progenitor cells have high, dynamic and noisy Venus::HES5 expression, which evolves into a more oscillatory signal as Venus::HES5 decreases and the cells undergo differentiation. Although our observation time window is relatively short, data collected from a few cells in cluster 1 demonstrate this noisy to oscillatory transition in Venus::HES5 expression, supporting this view . ABC has advantages over commonly used point estimates because it provides a probability distribution for estimated parameters thus quantifying parameter uncertainty. We found that the experimentally measured distribution of oscillation periods and relative SD values in clusters 1 and 2  and oscillatory (high coherence) expression. It also predicts that increases in the Hill coefficient, and decreases in repression threshold and protein degradation are most likely to initiate oscillatory expression dynamics.Taken together, our modelling suggests that the HES5 oscillator in spinal cord NPCs is enabled by noise50 we did find a positive relationship between Venus::HES5 levels and noise  we investigated whether HES5 oscillations are caused by the reduction of HES5 levels. The mean levels between cells is different in clusters 1 and 2, but there was no correlation with the presence of oscillations, arguing against the protein expression level alone having a causative effect for oscillations . The small differences in peak and trough levels may be difficult to differentially decode by downstream genes. Most likely, these oscillations are a by-product of an active negative feedback loop that is required for maintaining the HES5 level around a high mean, thus repressing pro-neural genes in most apical progenitors. By contrast, oscillations in the transition to differentiation are coupled with an overall declining trend, and thus generate larger fold differences, which may be easier for downstream targets to decode. This is analogous to a ball bouncing down steps and undergoing greater instantaneous height drops (oscillatory expression) than a ball rolling down a ramp (aperiodic expression). Since HES proteins are transcriptional repressors for pro-neural genes, such as 48 and that the HES5 auto-repression network operates near a bifurcation boundary. Consequently individual cells can switch between aperiodic and periodic expression stochastically and through regulated parameter changes. Instead of considering oscillatory versus non-oscillatory cells as stable distinct subpopulations, we propose that these are readily interconverting states with considerable plasticity. The model predicts that the transition towards oscillatory behaviour is most likely regulated through changes in the Hill coefficient, repression threshold or protein degradation. Future development of this model will capture remaining features of the observed dynamics, such as the down-regulation observed during the differentiation process, as observed in cells of clusters 3 and 4, as well as other gene regulatory interactions and multi-cellular interactions.Our mathematical modelling identified that HES5 oscillations are enabled by stochastic amplification30 suggested that low HES1 molecule number leads to stochastic oscillations of HES1 through a finite number effect. By contrast, HES5 molecule number is not low, (~30–55k molecules per nucleus for HES5 versus 2–3k per nucleus for HES130). Thus other sources of noise may need to be considered, such as stochastic activation of Notch cell–cell signalling in the densely packed tissue or cell division52 and the cell cycle. Noise and stochasticity are often considered undesirable yet they may also benefit decision-making processes30. Here the benefit of noise may be to prime HES5 expression such that it is poised to become oscillatory.In our modelling, we have included the effects of intrinsic stochastic noise, since this does not introduce further model parameters and is associated with any rate process. Phillips et al.Hes oscillator which can be tuned by external signals24. The lack of statistical difference between the number of oscillating cells from homozygous and heterozygous Venus::HES5 animals suggests, that in contrast to the stochastic transcriptional bursting53, negative feedback generated oscillations can be somewhat synchronous between two alleles.The increase in cells with oscillatory HES5 in dissociated cells versus cells in tissue, is in agreement with previous suggestions of a cell-autonomous Hes1, Dll1 and Neurog2 have been shown to oscillate in NPCs. The relative timing of pulses of different genes may regulate cellular behaviour as common target genes may respond differently to in-phase or out-of-phase input pulses. Indeed, the relative phase of the Notch and Wnt-signalling oscillations in somitogenesis have been proposed to control cellular differentiation54. Imaging protein expression dynamics of multiple factors in the same cell during cell fate decisions would help to reveal the relative timing of multiplexed oscillatory gene expression.Other genes in the Notch-Delta network, such as The second main contribution of this paper is to increase the depth of our understanding of the degree and origin of cellular heterogeneity in gene expression in a tissue environment. We conclude that HES5 expression in the spinal cord is not an ergodic system, since tissue level variability cannot be explained from short-term single cell variability but through a combination of cell sub-states co-existing in the tissue  and transitions between these sub-states. A progenitor zone close to the ventricle (<50 μm) shows maximum heterogeneity in cell-states, as all four dynamic expression clusters are equally represented in this zone, but minimum cell-to-cell heterogeneity in HES5 expression levels. By contrast, in the progenitor zone further from the ventricle there is minimum heterogeneity in cell-states, as it occupied mainly by cells in clusters 3 and 4, and maximum cell-to-cell heterogeneity in HES5 expression levels, approaching a 10-fold range in HES5. Furthermore, single cells undergoing differentiation start to down-regulate Venus::HES5 at any point between 20 and 50 μm away from the ventricle indicating that cells can make the cell fate decision at any point along the apical–basal dimension of the progenitor zone. Though we could not resolve the differences between clusters 1 and 2, and clusters 3 and 4, our findings contrast with the schematic view that cell fate is controlled deterministically at global tissue level through signalling gradients. Together with the finding that more cells show oscillations in a dissociated culture, we suggest that NPCs make stochastic fate decisions through a complex and yet unresolved integration between global and local cell–cell signalling.Our findings highlight the importance of integrating gene expression dynamics with spatio-temporal cell behaviour to understand cell state transitions in real time in a multicellular tissue.Mus musculus) experiments were performed under UK Home Office project licences (PPL70/8858) within the conditions of the Animal (Scientific Procedures) Act 1986. Animals were only handled by personal license holders. Venus::HES5 knock-in mice (ICR.Cg-Hes5<tm1(venus)Imayo>)9 were obtained from Riken Biological Resource Centre, Japan and mated with CD-1 mice for one generation before being maintained as an in-bred homozygous line. In these mice the mVenus fluorescent protein is fused to the N-terminus of endogenous HES5. R26R-H2B::mCherry mice55 were obtained as frozen embryos from Riken Centre for Life Science Technologies, Japan and C57Bl6 mice were used as surrogates. Sox1Cre:ERT2 mice (Sox1tm3(cre/ERT2)Vep56 were obtained from James Briscoe with the permission of Robin Lovell-Badge. Sox1Cre:ERT2 (NIMR:Parkes background) and R26R-H2B::mCherry (C57Bl6 background) were crossed to generate a double transgenic line (mixed background) homozygous for R26R-H2B::mCherry and heterozygous for Sox1Cre:ERT2.Animal  was performed 18 h prior to embryo dissection. Whole embryos were screened for H2B::mCherry expression using Fluar ×10/0.5NA objective on a Zeiss LSM880 confocal microscope and the trunks of positive embryos were embedded in 4% low-gelling temperature agarose (Sigma) containing 5 mg/ml glucose (Sigma). 200 μm transverse slices of the trunk around the forelimb region were obtained with the Leica VT1000S vibratome and released from the agarose. Embryo and slice manipulation was performed in phenol-red-free L-15 media (Thermo Fisher Scientific) on ice and the vibratome slicing was performed in chilled 1xPBS (Thermo Fisher Scientific).2. FCS signals were collected inside single nuclei in either the ventral region alone or both dorsal and ventral regions for tissue experiments. Venus (EYFP) fluorescence was excited with 514 nm laser light and emission collected between 517 and 570 nm. Data from individual cell nuclei was collected using 5 × 2 s runs at 0.15–0.3% laser power which gave <10% bleaching and a suitable count rate ~1 kHZ counts per molecule (CPM). To obtain molecule number, autocorrelation curves were fit to a two-component diffusion model with triplet state using the Levenberg–Marquardt algorithm in MATLAB optimisation toolbox with initial conditions assuming a ‘fast’ diffusion component 10× faster than the ‘slow’ component57. Measurements collected from cells exhibiting large spikes/drops in count rate or with low CPM (<0.5 kHz), high triplet state (>50%), or high bleaching (>10%) were excluded from the final results. Number and brightness analysis of the count rate58 showed a high correlation with molecule number obtained from autocorrelation curve fitting. The effective confocal volume had been previously determined to be 0.57 ± 0.11 fL (mean with SD) using Rhodamine 6G with known diffusion constant of 400 μm2 s−1 allowing conversion from molecule number to concentration59. Single-cell data of number of molecules in the cell nucleus was obtained by adjusting concentration to the average volumetric ratio between nuclear volume and confocal volume. Mean nuclear volume of 523 fL was estimated using H2BmCherry intensity and 3D reconstruction from z-stack images in Imaris (Bitplane).E10.5 transverse spinal cord slices heterozygous or homozygous for Venus::HES5 were stained on ice for 1.5 h with 50 μM Draq5 (Thermo Fisher Scientific) diluted in phenol-red free L-15 (Thermo Fisher Scientific) media. FCS experiments and snapshot images of whole spinal cord were carried out using a Zeiss LSM880 microscope with a C-Apochromat ×40 1.2 NA water objective on slices placed directly on a glass-bottomed dish (Greiner BioOne) kept at 37 °C and 5% COIndividual Draq5+ nuclei in a tile-scan image of a transverse slice of the whole E10.5 spinal cord were manually segmented as ellipses using ImageJ and background Venus::HES5 fluorescence (measured via an ROI drawn outside of the cells) was subtracted. A Q–Q plot was generated for the distribution of nuclear Venus::HES5 intensities from manual segmentation of a single image and the distribution of nuclear Venus::HES5 concentrations from FCS of cells throughout the E10.5 spinal cord from multiple slices and experiments. Linear regression was used to generate a calibration curve between Venus::HES5 intensity and Venus::HES5 concentration over the middle 90% of the range. The gradient of the line was used as a scaling factor and applied to the pixel intensity values in the segmented image to transform intensity to concentration.The centroids of the manually segmented cells from a quantitative expression map were used to measure distance from the ventricle and perpendicular to the D/V axis. Neighbours were ranked based on distance from the centroid of the cell of interest and the nearest neighbours were classified as the cells in the first rank  for live imaging. 1.5 mls of DMEM F-12 (Thermo Fisher Scientific) media containing 4.5 mg/ml glucose, 1× MEM non-essential amino acids (Thermo Fisher Scientific), 120 μg/ml Bovine Album Fraction V (Thermo Fisher Scientific), 55 μM 2-mercaptoethanol, 1x GlutaMAX (Thermo Fisher Scientific), 0.5× B27 and 0.5× N2 was added. Movies were acquired using Zeiss LSM880 microscope and GaAsP detectors. For slice imaging a Plan-Apochromat ×20 0.8 NA objective with a pinhole of 5 AU was used. Ten z-sections with 7.5 μm interval were acquired every 15 min for 18 h. DMSO (Sigma) or 2 μM DBZ (Tocris) was added to media immediately before imaging. For imaging dissociated cells a Fluar ×40 1.3 NA objective with a pinhole of 6.5 AU was used. Six z-sections with 3.7 μm interval were acquired every 10 min for 24–48 h.E10.5 spinal cord slices for live timelapse microscopy were placed on a 12 mm Millicell cell culture insert (MerckMillipore) in a 35 mm glass-bottomed dish (Greiner BioOne) incubated at 37 °C and 5% COSingle NPCs in E10.5 spinal cord slices were tracked in Imaris on the H2BmCherry channel using the ‘Spots’ and ‘Track over time’ function. Spot detection algorithm used background subtraction and tracking used the Brownian motion algorithm. All tracks were manually curated to ensure accurate single-cell tracking. A reference frame was applied to the movie along the D-V and apico-basal axes of the spinal cord to allow the distance from the ventricle to be calculated. Background fluorescence was subtracted prior to analysing time-lapse intensity data. To account for any photobleaching and allow comparison of intensities between movies the mean intensity of mCherry and Venus in each spot was normalised to the mean intensity of mCherry or Venus in the whole tissue. The whole tissue volume was tracked using the ‘Surfaces’ and ‘Track over time’ function.z-position in the tissue  between Venus::HES5 intensity and z-position when all cells and time-points were plotted  by subtracting the mean of the timeseries from each timepoint and dividing by the SD of the timeseries. Standardising the data enables clustering on relative expression changes rather than absolute expression levels. Cells were aligned to all start at time 0, which refers to the start of the tracking rather than the start of the movie. Standardized single cell timeseries were then subject to hierarchical clustering using Euclidean distance and Ward’s linkage in RStudio (R Project). Experiments were clustered separately and each clustergram independently identified four clusters per experiment. The elbow method to look at the variance explained as a function of number of clusters , suggested 4–6 clusters as the optimal cluster number, however 5 and 6 clusters were not favoured by silhouette method , so we chose four clusters. Cluster relationships varied between experiments thus for annotation between experiments corresponding clusters labels were determined by 1. observation of mean Venus::HES5 expression over time per cluster and 2. calculating average single-cell COV in Venus::HES5 over time for each cluster and comparing to results of clustering experiment 1  of all the normalised Venus::HES5 intensity values for a single cell in the time window was calculated. The shoulder point of Venus::HES5 was defined as a turning point in the signal that lead to a decrease of >50% of the signal.Trunks of E10.5 embryos for cryo-sectioning were fixed in 4% PFA for 1 h at 4 °C, followed by three quick washes with 1×PBS and one longer wash for 1 h at 4 °C. Embryos were equilibrated overnight in 30% sucrose (Sigma) at 4 °C before mounting in Tissue-Tek OCT (Sakura) in cryomoulds and freezing at −80 °C. 12 μm sections were cut on Leica CM3050S cryostat. E10.5 spinal cord slices cultured on Millicell inserts were fixed in 4% PFA for 4 h. For staining, tissue and sections were washed in PBS followed by permeabilisation in PBS 0.2% Triton X-100 (Sigma) and blocking with PBS 0.05% Tween20 (Sigma) + 5% BSA (Sigma). Primary and secondary antibodies were diluted in PBS 0.05% Tween20 + 5% BSA. Tissue was incubated with primary antibodies overnight at 4 °C, then washed three times for 5–10 min in PBS 0.05% Tween20, incubated with secondary antibodies and DAPI (Sigma) for 4 h at room temperature, and washed again three times in PBS-T. Sections were mounted using mowiol 4-88 (Sigma). Primary antibodies used were rabbit anti-SOX2 , mouse anti-NeuN , mouse anti-NKX2.2 , mouse anti-PAX7 , rabbit anti-Olig2 , mouse anti-Isl1/2 , and rabbit anti-β3-tubulin .Primary NS cells were isolated from dissected spinal cords of E10.5–11.5 embryos from Venus::HES5 knock-in mice and cultured in DMEM/F-12 containing 4.5 mg/ml glucose, 1× MEM non-essential amino acids (Thermo Fisher Scientific), 120 μg/ml Bovine Album Fraction V (Thermo Fisher Scientific), 55 μM 2-mercaptoethanol, 1× GlutaMAX (Thermo Fisher Scientific), 0.5× B27 and 0.5× N2. NS-E cells were a gift from Jennifer Nichols .Protein half-life was obtained by transfection of 3xFlag-HES5 and 3xFlag-Venus::HES5 in to NS-E cells with Lipofectamine 3000 (Thermo Fisher Scientific) as per manufacturers’ instructions. Twenty-four hours after transfection, cells were treated with 10 μM cycloheximide (Sigma) and at 0, 15, 30, 60, 120, and 240 min after treatment lysed with. Western blots were performed using 4–20% Tris–glycine acrylamide gels (NuSep), Whatman Protran nitrocellulose membrane (Sigma) and developed with Pierce ECL substrate (Thermo Fisher Scientific). Antibodies used were anti-HES5 [EPR15578]  and anti-alpha-tubulin (clone DM1A Sigma T9026). RNA half-life experiments were obtained by 10 μM actinomycin D  treatment of primary heterozygous Venus::HES5 and primary wild-type spinal cord NS cells. Samples were taken at 0, 15, 30, 45, 60, 80, 100, 120 min after treatment and RNA prepared using RNAeasy kit (Qiagen) with DNAse treatment as per manufacturers instructions. cDNA was prepared using Superscript III (Invitrogen) as per manufacturers’ instructions and qPCR for Venus, HES5 and GAPDH was performed with Taqman (Thermo Fisher Scientific) gene expression assays.Statistical tests were performed in GraphPad Prism 7. Data was tested for normality with D’Agostino–Pearson test. The relevant parametric or non-parametric test was then performed. If necessary outlier removal was performed using ROUT method (GraphPad). COV is defined as SD over the mean.t-test (parametric) or Mann–Whitney test (non-parametric). Statistical significance (p < 0.05) for 2+ datasets was tested by Kruskall–Wallis with Dunn’s multiple comparison correction. All tests were two-sided. Multiple comparison testing involved comparing all pairs of data columns. Correlations were analysed using Spearman rank correlation coefficient. Sample sizes, experiment numbers, p-values < 0.05 and correlation coefficients are reported in each figure legend.Stacked bar plots and discrete scatter plots show mean or mean ± SD where multiple independent experiments are analysed. Statistical significance between two datasets was tested with either Student's 43 to analyse timeseries of Venus::HES5 in single cells tracked in 3D fluorescence imaging. Data was de-trended to remove long-term behaviour, such as down-regulation and to recover the oscillatory signal with zero mean. We used maximum-likelihood estimation to fit the de-trended data timeseries with two competing models: a fluctuating aperiodic one (null model) and an oscillatory one . We used the LLR statistic to compare the likelihood of data being oscillatory or non-oscillatory and determined the oscillators based on a false discovery rate of 3% independently per experiment.We adapted the statistical approach developed by Phillips et al. Gaussian processes (GPs) background: GP inference is a probabilistic modelling technique that involves fitting a time-series signal y(t) in terms of a mean function, m(t) describing the moving average of signal with respect to time and a covariance function, k(τ) describing how the signal varies around the mean with respect to timeτ = |t − t′|, representing the time interval between any pair of time points . The covariance function is typically represented by a parameterised function that encapsulates modelling assumptions. The GP provides a generative model for the data, i.e. for known mean and covariance functions synthetic data may be generated by sampling from the multivariate normal distribution 60.Detrending single cell expression timeseries: Single cell protein expression timeseries contain information about dynamics at long timescales (above 10–12 h), as well as dynamics at short timescales (2–5 h). To account for this, we first model the long-term behaviour (i.e. mean function) using a squared covariance function:yd(t) = y(t) − yse(t). We used a lengthscale αse corresponding to 10 h to remove long-term dynamics while preserving short periodicity dynamics. Next we modelled the detrended data with zero mean using GP and two competing covariance models each with characteristic parameters inferred from the data. These models are described in the following section.This allows us to determine the mean function Oscillatory and Non-Oscillatory Covariance models: The detrended timeseries can be oscillatory or non-oscillatory  covariance function, which models aperiodic stochastic fluctuations, while KOUosc is a oscillatory OU covariance function, which models periodic stochastic timeseries. The parameters determining these models are:σOU, σOUosc ; related to signal amplitude by signal variance α which represents the rate at which correlations between subsequent peaks decay over time ;lengthscale β; related to periodicity by T = 2π/β.frequency In the following section, we discuss how the probability of the observed data given the periodic model is affected by technical noise and we introduce a global calibration technique that accounts for this.Global calibration of technical noise: The fluorescent detrended signal from an oscillatory cell contains periodic variations over time (modelled by the stochastic and periodic covariance model KOUosc), as well as additive technical noise: ε denotes white noise of zero mean and variance σn. The technical noise parameter σn is unknown and needs to be estimated.43 use an experimental estimation of technical noise where signal collected from empty areas of the cell culture dish (background) is used to approximate signal detection noise in bioluminescence imaging. Here we use a fluorescence reporter which suffers from variability of noise from the detector  and auto-fluorescence produced by the cells thus showing an overall increased noise level in cells than background.Phillips et al. N is the total number of cells in each dataset. An equivalent and more convenient optimisation is to maximise the joint log-likelihood function:ni representing the number of time-points in each trace and To account for this, we propose a different strategy to estimate technical noise directly from the data by observing the relationship between likelihood and signal-to-noise ratio, i.e. optimising joint likelihood. The joint likelihood function describes the probability of observing all the data per experiment under a global model:The global estimation of For single cell parameter estimation (described in the next section), we calibrated the technical noise level expressed in fluorescent intensity units at the global level and re-estimated the remaining parameters for each cell. In this way we ensure that cells have the same  amount of technical noise variance and that the remaining variability comes from the true signal. This generates a fit that is robust and avoids over-fitting. In the following, we describe the parameter estimation procedure:Maximum-likelihood estimation and LLR: Using the global calibrated level of technical noise, we further optimise GP covariance model parameters for single cell timeseries. Since we do not know a priori if single cell traces are periodic or not, we estimate parameters for both OU and OUosc models by optimising log-likelihood with respect to hyperparameters β → 0). However, a higher likelihood for OUosc compared to likelihood of OU indicates increased probability of the signal to be oscillatory. To quantify the probability for the single cell data to be oscillatory, we use the LLR statistic:ni, the following normalisation was used LLRi = LLRi/ni.The likelihood value of the periodic model does not directly indicate if the data is oscillatory or non-oscillatory since OUosc can also fit non-oscillatory data  distribution43 and controlling for the false discovery rate (FDR) per experiment. We obtain a synthetic null distribution using the generative OU (non-oscillatory) models that have been fitted to data. This allows generating a large number of synthetic traces that are non-oscillatory. To find statistically significant oscillators we set FDR to 3%. Controlling the FDR ensures that false positives are very unlikely to be considered oscillatory  and an example from data is shown in Supplementary Fig. 61 has been shown to outperform standard Lomb–Scargle periodogram  in terms of specificity and sensitivity43.Prior distribution on lengthscale: Lengthscale is a parameter of the covariance model that describes the rate at which subsequent peaks in the oscillatory signal become uncorrelated over time. In practice, we found that estimating lengthscale as a free parameter can lead to α → 0 for some of the cells  (existing in GPML toolbox) that has a sigmoidal expression around a lower bound, l and an upper bound, L:η is a parameter that controls the shape, with higher values leading to a box appearance. The shape of the prior affects the posterior distribution of lengthscales that is estimated from the data. Although both a restrictive, box-shaped prior and relaxed, smooth prior help prevent α → 0 62. The analytic signal consists of a real part identical to the data: Rezd(t) = yd(t); and an imaginary part representing the data with a π/2 phase shift, where phase shift denotes a difference in the peaks of two waves. It is generally assumed that, Rezd(t) is proportional to a cosine wave, thus Imzd(t) will be proportional to a sine wave or viceversa. The analytic signal relates to instantaneous amplitude A and phase φ by: zd = A exp(iφ). This leads to the following expression for amplitude and phase:hilbert.m in MATLAB on the fitted model obtained from analysing the detrended oscillatory data with KOUosc. Amplitude is detected in the detrended data and translated back to real intensity units (or relative units where appropriate) by summation with the long-term trend  denotes the number of mRNA molecules in one cell at time t, P(t) denotes the number of HES5 protein molecules, and τ represents transcriptional delay, i.e. the average time that is required for individual mRNA molecules to be transcribed and transported to the ribosome. The parameters μm, μp,αm and αp are rate constants denoting the rates of mRNA degradation, protein degradation, basal transcription in the absence of protein, and protein translation, respectively. The rate of transcription in the model is modulated in dependence of HES5 protein P at time t−τ by the Hill function63We modelled protein expression dynamics emerging from transcriptional autorepression and delay using an established mathematical modelRNA Fig. . The modξm and ξp denote Gaussian white noise, which is characterised by delta-distributed autocorrelations50The variables Note that Chemical Langevin equations allow non-integer values for the molecule numbers of mRNA and Protein.Numerical implementation of the model: We calculate summary statistics from the long-term trend of the model to enable model-data comparison. To investigate model behaviour at a unique combination of model parameters (in the following referred to as parameter point), we generate Np = 200 model traces at this parameter point and calculate averaged summary statistics from these traces. For each trace, the first teq = 1000 min are discarded and the remaining tobs = 7500 min are used for the evaluation of summary statistics. The equilibration time teq is chosen such that that summary statistics do not depend on the initial condition. Initial conditions to evaluate the model at a given parameter point are Min = 10 mRNA molecules, and Pin = P0 protein molecules. We impose that no mRNA transcription events were initiated at negative times by inhibiting transcription in the model for t < τ. Specifying model behaviour for negative times is necessary when evaluating delay differential equations. Chemical Langevin equations were implemented numerically using an Euler–Maruyama scheme and a time step of Δt = 1 min All numerical simulation parameters are listed in Supplementary Table Coherence and other summary statistics: To characterise the model behaviour at individual parameter points we collect the following summary statistics: (i) the mean protein expression level relative SD of protein expression σP denotes the absolute SD of protein expression, (iii) the coherence and (iv) the mean observed period. Means and SDs of expression (i) and (ii) are calculated across all traces at one parameter point and across all discretized timesteps in the observation window.In order to calculate the oscillation coherence in Fig.  * denotes complex conjugation and the average is taken over all Np simulated traces at one parameter point. The discrete Fourier transform implemented in the Python numpy.fft package is used, introduced by Cooley and Tukey65. In order to minimise the influence of finite-size effects on the power spectrum we apply smoothing via a Savitzky–Golay filter66 with polynomial order three and a filtering window of 0.001/min, which corresponds to seven discrete frequency values. The coherence is then defined as C = Amax/Atot67, where Amax is the area under the power spectrum in a 20% band centred at the frequency corresponding to the maximum of the power spectrum, and Atot is the total area of the power spectrum. Typically, coherence is low for non-oscillatory cells and high for oscillatory cells. In Fig. 44 was used to calculate the power spectrum analytically before extracting coherence values, which reduces computational complexity. To calculate coherence values for the deterministic system63  to simulated traces. At each parameter point, we generate one equilibrated trace of 12-h duration, a similar observation window as the experimental data. Period values are identified as time differences between consecutive descending zero-crossings of the instantaneous phase, and the mean period across the measurement interval is recorded. In Fig. Parameter inference: In order to parameterise the model we use a combined approach of experimental parameter measurements and approximate Bayesian computation (ABC)70. Specifically, we use experimentally measured values for the mRNA and protein degradation rates, corresponding to half-life values of 30 and 90 min, respectively  predicted coherence values above 0.1, and (iv) period values below 5 h. The likelihood of an individual parameter change to induce oscillations in agreement with (i)–(iv) is then defined Losc = Nosc/Nsteady. We normalise the values of Losc to sum to a total probability of one in Fig. Ntot to 2,000,000 in Fig. Nsteady, with Nsteady = 20872.When investigating the impact of parameter changes on model predictions in Fig. Background on Bayesian inference: In Bayesian statistics, the joint probability distribution p of a parameter vector (parameter point) Θ and observed data vector D is used to calculate the posterior distribution p(Θ|D), the probability distribution of the parameters given the data. The calculation of the posterior is achieved by applying Bayes’ rulep(D|Θ) represents the probability of D given Θ, and is usually referred to as the likelihood, and p(D), the probability of observing the data, is the marginal likelihood. The likelihood of the parameter p(Θ) is referred to as the prior.where 47, which approximates the posterior through random sampling of parameters from the prior distribution and evaluating the model for each sample. Summary statistics are used in order to compare the model with data. This is necessary since the modelled data is high-dimensional: in our case it comprises the protein and mRNA numbers at each simulated timepoint in Np = 200 simulated cells per parameter point. As detailed above, we use the mean protein expression and the SD of protein expression as summary statistics for fitting. In rejection-based ABC, samples for which the chosen summary statistics fall within an experimentally observed range are accepted as samples of the posterior distribution, otherwise they are rejected. Here, we generate Ntot = 200,000 samples from the prior distribution, providing us with 4901 samples of the posterior distribution. The choice of prior distribution is discussed in the following section.ABC can be used to estimate parameters of complex models for which the likelihood is not analytically tractable. Here, we apply rejection-based ABCPrior information on parameters: When applying ABC it is necessary to define a prior probability p(Θ)70. This prior is typically used to restrict inferred parameter combinations to biophysically realistic values, by imposing uniform and uncorrelated distributions on each parameter. We follow this approach by considering biophysical principles and literature values to define marginal prior distributions on each parameter as detailed below and summarised in Supplementary Table basal transcription rate αm, we assume a logarithmically uniform prior, representing that we do not initially know the order of magnitude of this parameter. We consider a maximal, biophysically possible transcription rate of 60 transcripts per minute, which has previously been estimated72. We choose a lower bound of 0.1 transcripts per minute. This lower bound is chosen manually such that the prior bounds lie outside the support of the posterior. An example experimental estimate of a gene transcription rate is two transcripts per minute73. A genome-wide quantification of transcription rate estimates in mouse fibroblast cells revealed a distribution of transcription rates between 10−3 and eight transcripts per minute74.For the translation rate αp, which represents the number of protein molecules generated per mRNA molecule per time interval, we use a logarithmically uniform prior, similar to the prior in the basal transcription rate. We consider biophysical principles to impose an upper boundary on the translation rate as follows: ribosomes translate individual mRNA molecules with a speed of approximately six codons per second77. The footprint of an individual ribosome is approximately ten codons78. Thus individual ribosomes require ~1.6 s in order to free the translation start site, which limits the maximal possible translation rate to ~40 translation events per mRNA molecule per minute. We hence use 40/min as upper value for our prior on the translation rate, and the lower bound is chosen to be 0.5/min, which is chosen sufficiently small to lie on the outside of the support of the posterior. Genome-wide quantification of translation rate estimates included values of 20 translation events per mRNA per minute and above74.For the repression threshold P0 represents the amount of protein required to reduce HES5 transcription by half. For the repression threshold we use a uniform prior covering the range 0–120,000. The upper bound is chosen as twice the value of the experimentally observed number of protein molecules per cell , which is sufficiently large to include parameter regimes corresponding to no genetic auto-repression.The τ corresponds to the time required to transcribe RNA and move it out of the nucleus for translation. Previous estimates of this parameter for a variety of genes varied between 5 and 40 min79, and typically assumed values are around 20–30 min63. Based on these values we used a uniformly distributed prior of 5–40 min.The transcriptional delay n describes the steepness of the auto-repression response. Previously used values range between two and five80. Here, we use a uniform prior in the range of 2–6. Importantly, we do not consider values above n = 6, since (i) the change in slope of G decreases for increasing n, and (ii) high values of n correspond to steep, step-like, response curves, which are unrealistic.The Hill coefficient Prediction of mean and variance correlation: In order to estimate how mean and variance inter-depend in our model we obtained Bayesian posterior predictions for mean HES5 expression and for its variance after changing the repression threshold (P0), which we varied from 10% to 200% of its original value in 10% intervals. For each relative change in the repression threshold, these Bayesian posterior predictions are probability distributions analogous to those in Fig. Further information on research design is available in the Supplementary information_newPeer ReviewSupplementary Movie 1Supplementary Movie 2Reporting SummaryDescription of Additional Supplementary FilesSource Data