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The overall magnitude of changes in SOC, TN, and C:N in response to chemical nitrogen fertilizers reductionThe results showed that chemical nitrogen fertilizers reduction significantly decreased SOC and TN by 2.76% and 4.19% respectively, while increased C:N by 6.11% across all database (Fig. 1). SOC mainly derives from crop residues and secretions which closely related to crops growths, and crops growths were affected by fertilization, especially nitrogen fertilization20,21. The reduction of chemical nitrogen fertilizer led to poor crop growth, which reduced the amount of crop residues return, and then decreased SOC. Similarly, TN from crops was reduced due to poor crop growth. In addition, the reduction of chemical nitrogen fertilizers directly reduced the input of soil nitrogen. The increase of C:N was the result of the decrease of TN being greater than that of SOC. The responses of C:N to chemical nitrogen fertilizers reduction enhanced the comprehension of the couple relationship between SOC and TN, which was beneficial to the evolution of the C-N coupling models. Moreover, the accuracy of C-N coupling models depends on the precise quantification of the responses of SOC and TN to nitrogen fertilization. And our results accurately quantified the difference responses of SOC and TN to different nitrogen fertilization regimes, thus optimizing the C-N coupling models.Figure 1The weighted response ratio (RR++) for the responses to chemical nitrogen fertilizers of soil organic carbon (SOC, a), total nitrogen (TN, b), and their ratios (C:N, c). Bars denote the overall mean response ratio RR++ and 95% confidence intervals (CI). The star (*) indicates significance when the 95% CI that do not go across the zero line. The vertical lines are drawn at lnRR = 0. The value represents independent sample size.Full size imageResponses of SOC, TN and C:N to chemical nitrogen fertilizers reduction magnitudeWhen grouped by chemical nitrogen fertilizers reduction magnitude, SOC showed a significant increase by 6.9% in medium magnitude, while SOC was significantly decreased by 3.10% and 7.26% in high and total magnitude respectively (Fig. 1a). Liu and Greaver22 also stated the reduction of medium nitrogen fertilizer increased the average microbial biomass from 15 to 20%, thereby increasing the SOC content. Previous studies had reported that there were strong positive correlations between soil organic matter and soil microbial biomass in both the agricultural ecosystem and natural ecosystem23,24. Numerous researchers have demonstrated the significance of nitrogen availability in soil for the plant biomass across most ecosystems25,26. Moreover, nitrogen deficient would inhibit the activity of extracellular enzymes and root activities27. Generally, soil degradation caused by continuous rising chemical nitrogen fertilizers application may inhibit the growth of crops and ultimately reduce the SOC28.TN showed no significant difference in low and medium chemical nitrogen fertilizers reduction magnitude (p  > 0.05), while TN in high magnitude and total chemical nitrogen fertilizers reduction magnitude exhibited a decrease with 3.10% and 9.37% respectively (Fig. 1b). Numerous studies described that the amount of nitrogen fertilizers used in China was higher than the demand of N for crop, which caused serious N leaching and runoff29,30. Chemical nitrogen fertilizers in low and medium magnitude would not decrease the TN of soil by reducing N leaching and runoff. However, the residual nitrogen in soil cannot meet the requirement for the sustainable growth of plant with litter or without exogenous nitrogen supplement, which resulted in the decrease of TN in high and total chemical nitrogen fertilizers magnitude. Consequently, optimal nitrogen fertilizers application rates will take into account crops yield and environment friendliness.Additionally, C:N had a significant increase with ranging from 1.82% to 8.98% under the four chemical nitrogen fertilizers reduction magnitude (Fig. 1c), suggesting the relative increase of SOC compared to TN. Previous studies have revealed that C:N had significantly influence on the soil bacterial community structures31. And there were also considerable studies indicated that chemical nitrogen fertilizers have impact on the soil bacterial communities32,33. We may speculate that the change of C:N would bring about the variations of soil bacteria communities under the chemical nitrogen fertilizers regimes.Responses of SOC, TN, and C:N to chemical nitrogen fertilizers reduction durationNegative response of SOC to short-term chemical nitrogen fertilizers reduction was observed in our study, which was consistent with the study of Gong, et al.34 that chemical nitrogen fertilizers reduction decreased SOC by reducing crop-derived carbon by one year. However, SOC was significantly increased by 1.06% and 4.65% at mid-term and long-term chemical nitrogen fertilizers reduction respectively, which was similar with the findings of Ning, et al.11 that SOC was significantly increased under more than 5 years of chemical nitrogen fertilizers reduction treatment. TN was significantly decreased by 1.96% at short-term chemical nitrogen fertilizers reduction duration, while the results converted at mid-term chemical nitrogen fertilizers reduction duration. The effect of long-term chemical nitrogen fertilizers reduction on TN was not significant (p  > 0.05). The divergent response of TN to different chemical nitrogen fertilizers duration was mainly caused by the various treatments. In terms of C:N, a greater positive response was observed at short-term chemical nitrogen fertilizers duration (9.06%) than mid-term and long-term duration (1.99%). Moreover, with the prolongation of the chemical reduction time of nitrogen, the response ratio tends to zero, suggesting that the effect of chemical fertilizers gradually decrease. This may be ascribed to the buffer capacity of soil to resist the changes from external environment, including nutrients, pollutants, and redox substances35.Responses of SOC, TN, and C:N to different chemical nitrogen fertilizers reduction patternsUnder the pattern of chemical nitrogen fertilizers reduction without organic fertilizers supplement, SOC and TN significantly decreased by 3.83% and 11.46% respectively, however, chemical nitrogen fertilizers reduction with organic fertilizers supplement significantly increased SOC and TN by 4.92% and 8.33% respectively. Moreover, C:N significantly increased under the two chemical nitrogen fertilizers patterns (p  0.05), but there was a negative effect on SOC in high and total magnitude (p  0.05). The no significant decrease at mid-term duration might result from the limited information reported in original studies of this meta-analysis36. TN showed no significant response to chemical nitrogen fertilizers without organic fertilizers supplement in the low and medium magnitude (p  > 0.05). However, TN was significantly decreased by 8.62% and 16.7% respectively in the high and total magnitude. When regarding to chemical nitrogen fertilizers reduction duration, TN was significantly reduced at all of the categories, ranging from 3.13% to 13.4% (Fig. 2c). In the pattern of chemical nitrogen fertilizers reduction with organic fertilizers supplement, chemical nitrogen fertilizers reduction at medium, high, and total magnitudes significantly increased SOC by 13.85%, 13.03%, and 5.46%respectively, however, the response of SOC in the low chemical nitrogen fertilizers magnitude was not significant. Chemical nitrogen fertilizers reduction duration significantly increased SOC by 7.01%, 1.71%, and 22.02% in the short-term, mid-term, and long-term respectively. Comparatively, TN showed a significantly increase in most chemical nitrogen fertilizers categories expect for the long-term chemical nitrogen fertilizers duration, with an increasing from 4.90% to 14.69% (Fig. 2d).Figure 2The weighted response ratio (RR++) for the responses to chemical nitrogen fertilizers of soil organic carbon (SOC, a), total nitrogen (TN, b), and their ratios (C:N, c) under the two patterns (with organic fertilizers ; without organic fertilizers). Bars denote the overall mean response ratio RR++ and 95% confidence intervals (CI). The star (*) indicates significance when the 95% CI that do not go across the zero line. The vertical lines are drawn at lnRR = 0. The values represent independent sample size.Full size imageOrganic fertilizers were mainly derived from animal manure or crops straws, which contained large amount of organic matter and nitrogen elements37,38. The application of organic fertilizers increased the input of SOC and TN directly. Moreover, organic fertilizer could promote the growth of crops by releasing phenols, vitamins, enzymes, auxins and other substances during the decomposition process, thus the SOC derived from crops would be increased37,39. In addition, organic fertilizers provide various nutrients for microbial reproduction, which increase the microbial population and organic carbon and total nitrogen content37. More importantly, the application of organic fertilizers could improve organic carbon sequestration and maintain its stability in aggregates, thereby reducing losses of SOC and TN40.C:N showed an increase under all of the chemical nitrogen fertilizers reduction with organic fertilizer supplement. The positive response of C:N to organic fertilizer supplement may be related to the higher C:N of organic fertilizer than soil. The average values of C:N of the commonly used organic fertilizers including animal manure, crop straws and biochar were 14, 60 and 30 respectively, while the C:N of soil was lower than 10 in average according to extensive literature researches41. Therefore, organic fertilizers would be a favorable alternative of chemical fertilizers for the sustainable development of agriculture.The correlation between the response of SOC, TN, and C:N and environmental variablesThe analysis of linear regression was conducted to analyze the environmental variables including mean annual temperature (MAT), mean annual precipitation (MAP), accumulated temperature above 10 °C (MATA), which may exert influence on SOC, TN and C:N. No significant correlation among the lnRR of SOC, TN, C:N and environmental variables were observed among the whole database (p  > 0.05; Fig. S1). Rule out the interference of organic fertilizers supplement, we analyzed the relationship between lnRR of SOC, TN, C:N and environmental variables as the Figures showed in Figs. 3 and 4 respectively. Under chemical nitrogen fertilizers without organic fertilizers supplement, there was a significant negative correlation between lnRR of SOC and MAT (p  More

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ChemicalsErbium (III) oxide nanoparticles (Er2O3-NPs) were purchased from Sigma-Aldrich Chemical Company (Saint Louis, USA) with pink appearance and product number (203,238). Powders of Er2O3-NPs with 99.9 trace metals basis were suspended in deionized distilled water to prepare the required concentrations and ultra-sonicated prior use.Cell lineHuman hepatocellular carcinoma (Hep-G2) cells were obtained from Nawah Scientific Inc., (Mokatam, Cairo Egypt). Cells were maintained in Dulbecco’s Modified Eagle Medium (DMEM) media supplemented with streptomycin (100 mg/mL), penicillin (100 units/mL) and heat-inactivated fetal bovine serum (10) in humidified, 5% (v/v) CO2 atmosphere at 37 °C.Characterization of Er2O3-NPsThe purchased powders of Er2O3-NPs were characterized using a charge coupled device diffractometer (XPERT-PRO, PANalytical, Netherlands) to determine its X-ray diffraction (XRD) pattern. Zeta potential and particles’ size distribution of Er2O3-NPs were also detected using Malvern Instrument Zeta sizer Nano Series (Malvern Instruments, Westborough, MA) equipped with a He–Ne laser (λ = 633 nm, max 5mW). Moreover, transmission electron microscopy (TEM) imaging was done to detect the shape and average particles’ size of Er2O3-NPs suspension.Sulforhodamine B (SRB) cytotoxicity assaySulforhodamine B (SRB) assay was conducted to assess the influence of Er2O3-NPs on the proliferation of cancerous Hep-G2 cells12. Aliquots of 100 µl of Hep-G2 cells suspension containing 5 × 103 cells were separately cultured in 96-well plates and incubated for 24 h in complete media. Hep-G2 Cells were then treated with five different concentrations of Er2O3-NPs (0.01, 0.1, 1, 10 and 100 µg/ml) incubated for 24 h or (0.1, 1, 10, 100 and 1000 µg/ml) incubated for 72 h. After 24 or 72 h of Er2O3-NPs exposure, cultured cells were fixed by replacing media with 10% trichloroacetic acid (TCA) and incubated for one hour at 4 °C. Cells were then washed five times with distilled water, SRB solution (0.4% w/v) was added and incubated cells in a dark place at room temperature for 10 min. All plates were washed three times with 1% acetic acid and allowed to air-dry overnight. Then, protein-bound SRB stain was dissolved by adding TRIS (10 mM) and the absorbance was measured at 540 nm using a BMG LABTECH-FLUO star Omega microplate reader (Ortenberg, Germany).Cells treatmentCancerous Hep-G2 cells were cultured at the appropriate conditions and dived into control and treated cells. The control cells were treated with an equal volume of the vehicle (DMSO; final concentration, ≤ 0.1%), while the treated cells were treated with the IC50 of Er2O3-NPs. All cells were left for 72 h after nanoparticles treatment and were harvested by brief trypsinization and centrifugation. Each treatment was conducted in triplicate. Cells were washed twice with ice-cold PBS and used for different molecular assays.Estimation of genomic DNA integrityThe impact of Er2O3-NPs exposure on the integrity of genomic DNA in cancerous Hep-G2 cells was estimated using alkaline Comet assay13,14. Treated and control cells were mixed with low melting agarose and spread on clean slides pre-coated with normal melting agarose. After drying, slides were incubated in cold lysis buffer for 24 h in dark and then electrophoresed in alkaline electrophoresis buffer. Electrophoresed DNA was neutralized in Tris buffer and fixed in cold absolute ethanol. For analysis slides were stained with ethidium bromide, examined using epi-fluorescent microscope at magnification 200× and fifty comet nuclei were analyzed per sample using Comet Score software.Estimation of intracellular ROS generationThe effect of Er2O3-NPs exposure on intracellular ROS production in cancer Hep-G2 cells was studied using 2,7-dichlorofluorescein diacetate dye15. Cultured cells were washed with phosphate buffered saline (PBS) and then 2,7-dichlorofluorescein diacetate dye was added. Mixed cells and dye were left for 30 min in dark and spread on clean slides. The resultant fluorescent dichlorofluorescein complex from interaction of intracellular ROS with dichlorofluorescein diacetate dye was examined under epi-fluorescent at 20× magnification.Measuring the expression levels of apoptotic and anti-apoptotic genesQuantitative real time Polymerase chain reaction (RT-PCR) was conducted to measure the mRNA expression levels of apoptotic (p53 and Bax) and anti-apoptotic (Bcl2) genes in control and treated Hep-G2 cells. Whole cellular RNA was extracted according to the instructions listed by the GeneJET RNA Purification Kit (Thermo scientific, USA) (Thermo scientific, USA) and using Nanodrop device purity and concentration of the extracted RNAs were determined. These RNAs were then reverse transcribed into complementary DNA (cDNA) using the instructions of the Revert Aid First Strand cDNA Synthesis Kit (Thermo scientific, USA). For amplification, RT-PCR was performed using the previously designed primers shown in Table 116,17 by the 7500 Fast system (Applied Biosystem 7500, Clinilab, Egypt). A comparative Ct (DDCt) method was conducted to measure the expression levels of amplified genes and GAPDH gene was used as a housekeeping gene. Results were expressed as mean ± S.D.Table 1 Sequences of the used primers in qRT-PCR.Full size tableAnalysis of cell cycle distributionDistribution of cell cycle was analyzed using flow cytometry. Control and treated cancer Hep-G2 cells with IC50 of Er2O3-NPs for 72 h were harvested, washed with PBS and re-suspended in 1 mL of PBS containing RNAase A (50 µg/mL) and propidium iodide (10 µg/mL) (PI). Cells were incubated for 20 min in dark at 37 C and analyzed for DNA contents using FL2 (λex/em 535/617 nm) signal detector (ACEA Novocyte flow cytometer, ACEA Biosciences Inc., San Diego, CA, USA). For each sample, 12,000 events are acquired and cell cycle distribution is calculated using ACEA NovoExpress software (ACEA Biosciences Inc., San Diego, CA, USA).Estimation of apoptosis inductionApoptotic and necrotic cell populations were determined using Annexin V- Fluorescein isothiocyanate (FITC) apoptosis detection kit (Abcam Inc., Cambridge Science Park Cambridge, UK) coupled with two fluorescent channels flow cytometry. After treatment with Er2O3-NPs for 72 h and doxorubicin as a positive control, Hep-G2 cells were collected by trypsinization and washed twice with ice-cold PBS (pH 7.4). Harvested cells are incubated in dark with Annexin V-FITC/ propidium iodide (PI) solution for 30 min at room temperature, then injected via ACEA Novocyte flowcytometer (ACEA Biosciences Inc., San Diego, CA, USA) and analyzed for FITC and PI fluorescent signals using FL1 and FL2 signal detector, respectively (λex/em 488/530 nm for FITC and λex/em 535/617 nm for PI). For each sample, 12,000 events were acquired and positive FITC and/or PI cells are quantified by quadrant analysis and calculated using ACEA NovoExpress software (ACEA Biosciences Inc., San Diego, CA, USA).Statistical analysisResults of the current study are expressed as mean ± Standard Deviation (S.D) and were analyzed using the Statistical Package for the Social Sciences (SPSS) (version 20) at the significance level p  More

General description of the experimental areaThe experiment was performed for two years at the National Key Irrigation Experimental Station located on the Songnen Plain in Heping town, Qing’an County, Suihua, Heilongjiang, China, with a geographical location of 45° 63′ N and 125° 44′ E at an elevation of 450 m above sea level (Fig. 1). This region consists of plain topography and has a semiarid cold temperate continental monsoon climate, i.e., a typical cold region with a black soil distribution area. The average annual temperature is 2.5 °C, the average annual precipitation is 550 mm, the precipitation is concentrated from June to September of each year, and the average annual surface evaporation is 750 mm. The growth period of crops is 156–171 days, and there is a frost-free period of approximately 128 days year−122. The soil at the study site is albic paddy soil with a mean bulk density of 1.01 g/cm3 and a porosity of 61.8% prevails. The basic physicochemical properties of the soil were as follows: the mass ratio of organic matter was 41.8 g/kg, pH value was 6.45, total nitrogen mass ratio was 15.06 g/kg, total phosphorus mass ratio was 15.23 g/kg, total potassium mass ratio was 20.11 g/kg, mass ratio of alkaline hydrolysis nitrogen was 198.29 mg/kg, available phosphorus mass ratio was 36.22 mg/kg and available potassium mass ratio was 112.06 mg/kg.Figure 1Location of the study area. The map and inset map in this image were drawn by the authors using ArcGIS software. The software version used was ArcGIS software v.10.2, and its URL is http://www.esri.com/.Full size imageHumic acid fertilizerHumic acid fertilizer was produced by Yunnan Kunming Grey Environmental Protection Engineering Co., Ltd., China (Fig. 2). The organic matter was ≥ 61.4%, and the total nutrients (nitrogen, phosphorus and potassium) were ≥ 18.23%, of which N ≥ 3.63%, P2O5 ≥ 2.03%, and K2O ≥ 12.57%. The moisture content was ≤ 2.51%, the pH value was 5.7, the worm egg mortality rate was ≥ 95%, and the amount of faecal colibacillosis was ≤ 3%. The fertilizer contained numerous elements necessary for plants. The contents of harmful elements, including arsenic, mercury, lead, cadmium and chromium, were ≤ 2.8%, 0.01%, 7.6%, 0.1% and 4.7%, respectively; these were lower than the test standard.Figure 2Humic acid fertilizer in powder form.Full size imageExperimental design and observation methodsIrrigationIn this experiment, three irrigation practices, namely, control irrigation (C), wet irrigation (W) and flood irrigation (F), were designed (Table 1).Table 1 Different irrigation methods.Full size tableControl irrigation (C) of rice had no water layer in the rest of the growing stages, except for the shallow water layer at the regreen stage of rice, which was maintained at 0–30 mm, and the natural dryness in the yellow stage. The irrigation time and irrigation quota were determined by the root soil moisture content as the control index. The upper limit of irrigation was the saturated moisture content of the soil, the lower limit of soil moisture at each growth stage was the percentage of saturated moisture content, and the TPIME-PICO64/32 soil moisture analyser was used to determine the soil moisture content at 7:00 a.m. and 18:00 p.m., respectively. When the soil moisture content was close to or lower than the lower limit of irrigation, artificial irrigation occurred until the upper irrigation limit was reached. The soil moisture content was maintained between the upper irrigation limit and the lower irrigation limit of the corresponding fertility stage. Under the wet irrigation (W) and flood irrigation (F) conditions, it was necessary to read the depth of the water layer through bricks and a vertical ruler embedded in the field before and after 8:00 am every day to determine if irrigation was needed. If irrigation was needed, then the water metre was recorded before and after each irrigation. The difference between before and after was the amount of irrigation23.FertilizationIn our research, five fertilization methods were applied, as shown in Table 2. In this experiment, the rice cultivar “Suijing No. 18” was selected. Urea and humic acid fertilizer were applied according to the proportion of base fertilizer:tillering fertilizer:heading fertilizer (5:3:2). The amounts of phosphorus and potassium fertilizers were the same for all treatments, and P2O5 (45 kg ha−1) and K2O (80 kg ha−1) were used. Phosphorus was applied once as a basal application. Potassium fertilizer was applied twice: once as a basal fertilizer and at 8.5 leaf age (panicle primordium differentiation stage) at a 1:1 ratio22.Table 2 The fertilizer methods.Full size tableThis study was performed with a randomized complete block design with three replications. Three irrigation practices and five fertilizer methods were applied, for a total of 15 treatments as follows: CT1, CT2, CT3, CT4, CT5; WT1, WT2, WT3, WT4, WT5; FT1, FT2, FT3, FT4, and FT5 (C, W, and F represent control irrigation, wet irrigation, and flood irrigation; T represents fertilizer treatment).Measurements of the samplesA soil temperature sensor (HZTJ1-1) was buried in each experimental plot to monitor the temperature of each soil layer (5 cm, 10 cm, 15 cm, 20 cm and 25 cm depth). The transmission of photosynthetically active radiation was measured from 11:00 to 13:00 by using a SunScan Canopy Analysis System (Delta T Devices, Ltd., Cambridge, UK), and data during the crop-growing season were recorded every day24.Plant measurements were taken during the periods of tillering to ripening on days with no wind and good light. The fluorescence parameters were measured by a portable fluorescence measurement system (Li-6400XT, America). The detection light intensity was 1500 μmol m−2 s−1, and the saturated pulsed light intensity was 7200 μmolm−2 s−1. The functional leaves were dark adapted for 30 min, and then the maximum photosynthetic efficiency of PSII (Fv/Fm) was measured. Photochemical quenching (QP) and nonphotochemical quenching (NPQ) were measured with natural light. Simultaneously, the leaf chlorophyll relative content (SPAD) was monitored using SPAD 502 (Konica Minolta, Inc., Tokyo, Japan). For plant agronomic characteristics, the distance from the stem base to the stem tip was measured with a straight ruler to quantify plant height24.Statistical analysisExperimental data obtained for different parameters were analysed statistically using the analysis of variance technique as applicable to randomized complete block design. Duncan’s multiple range test was employed to assess differences between the treatment means at a 5% probability level. All statistical analyses were performed using SPSS 22.0 for Windows24.
Ethics approvalExperimental research and field studies on plants, including the collection of plant material, comply with relevant institutional, national, and international guidelines and legislation. We had appropriate permissions/licences to perform the experiment in the study area. More

The bee health modelThe conceptual model of the interactions of various stressors with honey bee health is described by the following system of ordinary differential equations (ODEs)$${{tau }_{{HB}}dot{x}}_{{HB}}= {-{delta }_{{HB}}x}_{{HB}}+{g}_{{TC}}left({x}_{{TC}}right)+{g}_{{VA}}left({x}_{{VA}}right)+{g}_{{VI}}left({x}_{{VI}}right) \ +{bar{f}}_{S,C}left({u}_{S},{u}_{C},{x}_{{TC}},{x}_{{VA}}right)+{bar{f}}_{P}left({u}_{P},{x}_{{TC}}right)+{underline{f}}_{{HB}}left({u}_{T}right)$$
(1)
$${{tau }_{{TC}}dot{x}}_{{TC}}={-{delta }_{{TC}}x}_{{TC}}+{g}_{{HB}}left({x}_{{HB}}right)$$
(2)
$${{tau }_{{VA}}dot{x}}_{{VA}}={-{delta }_{{VA}}x}_{{VA}}+{h}_{{VA}}left({{x}_{{HB}},x}_{{TC}},varepsilon {x}_{{VI}}right)+{underline{f}}_{{VA}}left({u}_{T}right)$$
(3)
$${{tau }_{{VI}}dot{x}}_{{VI}}={-{delta }_{{VI}}x}_{{VI}}+{h}_{{VI}}left({{x}_{{HB}},x}_{{TC}},{varepsilon x}_{{VI}}right)$$
(4)
for the state variables ({x}_{{HB}}) representing honey bee health, ({x}_{{TC}}) the stress due to toxic compounds (e.g., neonicotinoid insecticides), ({x}_{{VA}}) the stress due to parasites (e.g., V. destructor) and ({x}_{{VI}}) the stress due to pathogens (e.g., DWV). The system includes the effects of external inputs as sugar ({u}_{S}), pollen ({u}_{P}), absolute deviation from desired temperature ({u}_{T}) and sub-optimal temperature ({u}_{C}). All the inputs and possible parameters are non-negative; the coefficients (tau) denote the time constants; the coefficients (delta) denote the self-regulation parameters; (varepsilon) in the last two equations allows to account for pathogens that can ((varepsilon , > , 0)) or cannot ((varepsilon=0)) impair the immune system (through link m in Fig. 1). We assume that the functions (g) are smooth, bounded, positive, convex and decreasing to 0; the functions (bar{f}) are smooth, bounded, non-negative, concave and increasing with respect to (w.r.t.) (u) arguments (vanishing only when the first (u) argument vanishes) while convex and decreasing to 0 w.r.t. (x) arguments; the functions ({underline{f}}) are smooth, bounded, non-positive and decreasing (vanishing only when (u=0)); the functions (h) are smooth, bounded, positive, convex and decreasing to 0 w.r.t. the first argument while concave and increasing w.r.t. all the other arguments. For a detailed description of the various functions, together with a summary of the biological effects they account for and a reference to the conceptual model in Fig. 1, see Supplementary Table 3.Structural analysis of the bee health modelWe describe here the structural considerations and computations that yield the structural influence matrix for the honey bee health system.The structural influence matrix (M) is defined as follows. (M) is a symbolic matrix with entries ({M}_{{ij}}) chosen among: +,−,0,?, according to the criteria described below. Consider an equilibrium point (bar{x}) and a constant perturbation (u) applied on the (j)-th system variable (small enough not to compromise the stability of the equilibrium). The equilibrium value will be modified as (bar{x}+delta bar{x}). Consider the sign of the perturbation of the (i)-th variable, (delta bar{{x}_{i}}). Then ({M}_{{ij}}) = + if (delta bar{{x}_{i}}) always has the same sign as (u); ({M}_{{ij}}=) − if (delta bar{{x}_{i}}) always has the opposite sign as (u); ({M}_{{ij}}) = 0 if always (delta bar{{x}_{i}}=0); regardless of the system parameters. Conversely, if the sign does depend on the system parameters, we set ({M}_{{ij}}) = ?.In this section we prove that the influence matrix of the honey bee health system is structurally determined, i.e., there are no “?”‘ entries in (M).We start with the following proposition.
Proposition 1
Assume that a matrix
(J)
is Hurwitz stable (i.e., all its eigenvalues have negative real part) and has the sign pattern
$${sign}left(Jright)=left[begin{array}{cccc}- & – & – & -\ – & – & 0 & 0\ – &+& – &+\ – &+& 0 & -end{array}right]$$
Then, the sign pattern of
({adj}left(-Jright))
, the adjoint of
(-J)
, is
$${sign}left({adj}left(-Jright)right)=left[begin{array}{cccc}+& – & – & -\ – &+&+&+\ – &+&+&+\ – &+&+&+end{array}right]$$
Proof To prove the statement, we just change the sign of the first variable, hence we change sign to the first row and column of matrix (J). The resulting matrix (M) is such that$${sign}left(Mright)=left[begin{array}{cccc}- &+&+&+\+& – & 0 & 0\+&+& – &+\+&+& 0 & -end{array}right]$$We observe that (M) is a Metzler matrix, namely, all its off-diagonal entries are non-negative. Moreover, the matrix is Hurwitz stable. Then, we can proceed as in the proof of Proposition 4 in a previous report16. Given a Metzler matrix that is Hurwitz stable, its inverse has non-positive entries; hence, the inverse of (-M) has non-negative entries: ({left(-Mright)}^{-1}ge 0) elementwise. Moreover, we observe that(,M) is an irreducible matrix, i.e., there is no variable permutation that brings the matrix in a block (either upper or lower) triangular form. This implies that the inverse of (-M) has strictly positive entries: ({left(-Mright)}^{-1} , > , 0) elementwise. Also, stability implies that the determinant of (-M) is positive: ({det }left(-Mright) , > , 0). Then, ({adj}left(-Mright)={left(-Mright)}^{-1}{det }left(-Mright) > 0), hence the adjoint of (-M) is also positive elementwise. To consider again the original sign of the variables, we change sign to the first row and column of ({adj}left(-Mright)), and we get the signature above for ({adj}left(-Jright)).The next step is the characterization of the structural influence matrix, which corresponds to the sign pattern of the adjoint of the negative Jacobian matrix in Proposition 1.To this aim, we first consider the linearized system and write it in a matrix-vector form$$dot{x}left(tright)={Jx}left(tright)+{e}_{j}u$$where (dot{x}left(tright)) is the time derivative of the four-dimensional vector (xleft(tright)) and ({e}_{k}), (k={{{{mathrm{1,2,3,4}}}}}), is an input vector, constant in time, with a single non-zero component, the (k)-th, equal to 1, while the scalar (u , > , 0) is the magnitude of the input. We wish to assess the (i)-th component of (xleft(tright)), ({x}_{i}left(tright)={e}_{i}^{T}xleft(tright)). If (J) is Hurwitz, as assumed, the steady-state value of variable ({x}_{i}left(tright)) due to the input perturbation ({e}_{k}) applied to the equation of variable ({x}_{k}left(tright)) is achieved for$$0=Jbar{x}+{e}_{k}u,$$namely$${x}_{i}=-{e}_{i}^{T}{J}^{-1}{e}_{k}u,$$which implies that the sign of the steady-state value ({bar{x}}_{i}) of variable ({x}_{i}) due to a persistent positive input acting on the (k)-th equation has the same sign as ({(-{J}^{-1})}_{{ik}}), the (left(i,kright)) entry of matrix ({left(-Jright)}^{-1}). Since we assume Hurwitz stability, we have that ({det }left(-Jright)) is positive, hence the sign pattern of the inverse ({left(-Jright)}^{-1}) corresponds to the sign pattern of the adjoint, ({adj}left(-Jright)). In fact, ({adj}left(-Jright)={left(-Jright)}^{-1}{det }left(-Jright)).We next consider the nonlinear system under investigation, which we write in the form$$dot{x}left(tright)=fleft(xleft(tright)right)$$and without restriction we assume that the zero vector is an equilibrium point: (0=fleft(0right)). This condition can be always achieved, without loss of generality, by a translation of coordinates. We also consider a stable equilibrium: we assume that the linearized system at the equilibrium is asymptotically stable, namely its Jacobian (J), which has the sign pattern considered in Proposition 1 above, is Hurwitz. We also assume that a constant input perturbation of magnitude (u) is applied to the system, affecting the (k)-th equation, i.e.,$$dot{x}left(tright)=fleft(xleft(tright)right)+{e}_{k}u,$$and that the perturbation is small enough to keep the state in the domain of attraction of the considered equilibrium. Due to this perturbation, a new steady state (bar{x}left(uright)) is reached that satisfies the condition$$0=fleft(bar{x}left(uright)right)+{e}_{k}u$$To determine the sign of the new equilibrium components (bar{x}left(uright)), we consider this new equilibrium vector as a function of (u) in a small interval (left[0,{x}_{{MAX}}right]). Adopting the implicit function theorem yields$$frac{d}{{dx}}bar{x}left(uright)=-J{left(uright)}^{-1}{e}_{k}u,$$where we have denoted by (Jleft(uright)) the Jacobian matrix computed at the perturbed equilibrium (bar{x}left(uright)). Hence, for (u) small enough, the sign of the derivatives of the entries of the new, perturbed equilibrium are, structurally, the same as those in the (k)-th column of matrix (-{J}^{-1}). Since, by construction, (xleft(0right)=0), this is also the sign of the elements of vector (bar{x}left(uright)), for (u) in the interval (left[0,{x}_{{MAX}}right]).We have therefore proved that the original nonlinear system describing honey bee health admits the following structural influence matrix:$$left[begin{array}{cccc}+& – & – & -\ – &+&+&+\ – &+&+&+\ – &+&+&+end{array}right]$$System equilibriaThe results concerning the system equilibria were obtained through a standard analytical treatment of the nonlinear equations describing the equilibrium conditions of the system of differential Eqs. (1), (2), (3), (4). A detailed description of methods is reported in Supplementary Methods.Laboratory experiments using honey beesTo confirm the bistability of the system representing honey bee health as affected by multiple stressors, we used data from several survival experiments, carried out in a laboratory environment according to the same standardized method, over a 6-year period (Source data file).All experiments involved Apis mellifera worker bees, sampled at the larval stage or before eclosion, from the hives of the experimental apiary of the University of Udine (46°04′54.2″N, 13°12′34.2″E). Previous studies indicated that the local bee population consists of hybrids between A. mellifera ligustica and A.m. carnica62,63. Ethical approval was not required for this study.We considered experiments on the effect of the following stressors: infection with 1000 DWV genome copies administered through the diet before pupation, feeding with a 50 ppm nicotine in a sugar solution at the adult stage, exposition to a sub-optimal temperature of 32 °C at the adult stage. All experiments were replicated 3 to 13 times, using, in total, the number of bees reported in Table 1.For the artificial infection with DWV, we collected with soft forceps individual L4 larvae from the brood cells of several combs. Groups of 20–30 of such larvae were placed in Petri dishes with an artificial diet made of 50% royal jelly, 37% distilled water, 6% glucose, 6% fructose, and 1% yeast. 25 DWV copies per mg of diet were added or not to the diet according to the experimental group (note that a bee larva at this stage consumes about 40 mg of larval food per day, thus the viral infection per bee was 1000 viral copies). After 24 h larvae were transferred onto a piece of filter paper to remove the residues of the diet and then into a clean Petri dish, where they were maintained until eclosion. At the day of emergence, bees were transferred to plastic cages in a thermostatic cabinet, where they were kept until death. The DWV extract was prepared according to previously described protocols64 and quantified according to standard methods.For the treatment with nicotine, 10 µL of pure nicotine were added to 200 g of the sugar solution used for the feeding of the caged bees, to reach the concentration of 50 ppm.Finally, to expose bees to a 32 °C temperature, the plastic cages with the adult bees were kept in a thermostatic cabinet whose temperature was set accordingly.To monitor the survival of the adult bees treated as above, they were maintained from eclosion until death in plastic cages in a dark incubator at 34.5 °C (or 32 °C, according to the experiment), 75% R.H.; two syringes were used to supply a sugar solution made of 2.4 mol/L of glucose and fructose (61% and 31%, respectively) and water, respectively; dead bees were counted daily.All the results of these experiments are reported in Source data file.All experiments were carried out during the summer months, from June to September for 6 consecutive years. Previous data indicated that, in this region, virus prevalence increases along the active season starting from very low levels in spring and reaching 100% of virus-infected honey bees by the end of the summer; virus abundance in infected honey bees follows a similar trend28. For this reason, it can be assumed that bees sampled early in the season are either uninfected or they bear only a very low viral infection level, whereas bees sampled later in the season are likely to be virus-infected, bearing moderate to high viral infections. To confirm this assumption and identify a method for filtering our data according to viral infection, we assessed viral infection in a sample of bees from the untreated control group of each experiment, by means of qRT-PCR. According to standard practice, we assumed that Ct values below 30 are indicative of an effective viral infection, whereas Ct above that threshold are more likely in virus negative bees. As expected, we found that virus prevalence increases from June to September (Supplementary Figure 1a), in such a way that up to mid July only the minority of bees can be considered as viral infected (Supplementary Figure 1b). Therefore, we classified as “early” all the samples collected up to mid July and assumed that viral infection in those samples was low; on the other hand, samples collected from mid July till September were classified as “late” and we assumed that viral infection in those samples was high.qRT-PCR analysis of viral infection was carried out as follows. At the beginning of every experiment (i.e., at day 0), two to five bees for each replication were sampled in liquid nitrogen and transferred in a −80 °C refrigerator. After defrosting of samples in RNA later, the gut of each honey bee was eliminated to avoid the clogging of the mini spin column used after. The whole body of sampled bees was homogenized using a TissueLyser (Qiagen®, Germany). Total RNA was extracted from each bee according to the procedure provided with the RNeasy Plus mini kit (Qiagen®, Germany). The amount of RNA in each sample was quantified with a NanoDrop® spectrophotomer (ThermoFisher™, USA). cDNA was synthetized starting from 500 ng of RNA following the manufacturer specifications (PROMEGA, Italy). Additional negative control samples containing no RT enzyme were included. DWV presence was verified by qRT-PCR considering as positive all samples with a Ct value lower than 30. The following primers were adopted: DWV (F: GGTAAGCGATGGTTGTTTG, R: CCGTGAATATAGTGTGAGG65). 10 ng of cDNA from each sample were analyzed using SYBR®green dye (Ambion®) according to the manufacturer specifications, on a BioRad CFX96 Touch™ Real time PCR Detector. Primer efficiency was calculated according to the formula (E={10}^{left(-1/{{{{{{rm{slope}}}}}}}-1right)*100}). The following thermal cycling profiles were adopted: one cycle at 95 °C for 10 min, 40 cycles at 95 °C for 15 s and 60 °C for 1 min, and one cycle at 68 °C for 7 min.Individual survival and colony stabilityTo investigate how the death rate of forager bees affects colony growth, a compartment model of honey bee colony population dynamics was proposed50. This model showed that death rates over a critical threshold led to colony failure. Here we modified this model to include premature death of bees at younger age, as predicted by our model of individual bee health in the presence of an immuno-suppressive virus. We show that the critical threshold found in the previously published model50 becomes a decreasing function of the death rate of the younger individuals, so that premature death (and, in turn, immune-suppression) favors colony collapse.In more details, we first summarize the results of the previously published model50 where two populations (F) (forager) and (H) (hive) of bees are considered and where conditions are provided on the mortality (m) of (F) under which the whole population collapses: namely, mathematically stated, the system admits the zero equilibrium only. Here we extend the model partitioning (H) in two categories, (Y) (younger hive bees) and (O) (older hive bees), asintroducing an early mortality factor (n) for the young population, showing how such a factor worsens the collapsing condition.The previously published model50 concerns the interaction between hive bees (H) and forager bees (F) and is described by the ODEs$$dot{H}=Lfrac{H+F}{w+H+F}-Hleft(alpha -sigma frac{F}{H+F}right)$$$$dot{F}=Hleft(alpha -sigma frac{F}{H+F}right)-{mF}.$$Above, (L) is the queen’s eggs laying rate, (w) is the rate at which (L) is reached as the total population (H+F) gets large, (alpha) is the maximum rate at which hive bees become forager bees in the absence of the latter, (sigma) measures the reduction of recruitment of hive bees in the presence of forager bees and, finally, (m) is the death rate of forager bees (while the death rate of hive bees is assumed to be negligible).We first summarize the main results in terms of a threshold value for (m) in view of colony collapse, as our further analysis will follow a similar approach. All the parameters are assumed to be positive.The search for the equilibria of the above ODEs leads to the unique nontrivial equilibrium (beyond the trivial one)$$bar{H}=frac{L}{{mJ}}-frac{w}{1+J}$$$$bar{F}=Jbar{H}$$for$$J=Jleft(mright):=frac{alpha -sigma -m+sqrt{{left(alpha -sigma -mright)}^{2}+4malpha }}{2m}.$$Note that (J) is alway positive (and, moreover, it is independent of (L) and (w)). It follows that (bar{F}) and (bar{H}) have the same sign, so that the existence of the nontrivial equilibrium is equivalent to (bar{F}+bar{H} , > , 0). It is not difficult to recover that$$bar{F}+bar{H}=frac{w}{m}left(lfrac{1+J}{J}-mright)$$where (l:=L/w) is introduced for brevity. Then if (alpha le l) we get$$bar{F}+bar{H}=frac{w}{m}left(lfrac{1+J}{J}-mright)ge frac{w}{m}left(alpha frac{1+J}{J}-mright)=frac{w}{m}left(sigma+{mJ}right) , > , 0,$$with the last equality following from$$alpha -sigma frac{J}{1+J}-{mJ}=0,$$which in turn comes from annihilating the right-hand side of the second ODE and from using (J=bar{F}/bar{H}) while searching for equilibria. We conclude that, independently of (m), the colony never collapses if the recruitment rate (alpha) of forager bees is sufficiently low.Hence, we assume (alpha , > , l). Observe that$$bar{F}+bar{H}iff l , > , Jleft(m-lright)$$guarantees existence whenever (m) is sufficiently small, viz. (mle l). Assume then (m , > , l), so that the above condition reads$$J , < , frac{l}{m-l}$$leading to the threshold condition$$m , < , bar{m}:=frac{l}{2}frac{alpha+sigma+sqrt{{left(alpha -sigma right)}^{2}+4sigma l}}{alpha -l}$$by using the definition of (J), see Eq. (2) the previously published model50.A standard stability analysis shows that, assuming (alpha,m , > , l), the nontrivial equilibrium is (globally) asymptotically stable whenever it exists (positive), i.e., whenever (m , < , bar{m}). Otherwise, the only (globally) attracting equilibrium is the trivial one, corresponding to colony collapse (see Fig. 5 for the previously published model50 or Fig. 4 for (n=0)). In the mathematical jargon, the disappearance of the positive equilibrium, for (m) exceeding (bar{m}), is referred to as a transcritical bifurcation43.Now, in view of the outcome of the analysis of our model of individual bee health, we introduce a mortality term for the younger bees. As forager bees are recruited from adult hive bees, we divide the class of hive bees (H) in younger (Y) and older (O), assuming that the former die at a rate (n), while the death rate of the latter remains negligible according to the previously published model50. Obviously, (H=Y+O). The original ODEs are consequently modified as$$dot{Y}=Lfrac{H+F}{w+H+F}-Y$$$$dot{O}=left(1-nright)Y-Hleft(alpha -sigma frac{F}{H+F}right)$$$$dot{F}=Hleft(alpha -sigma frac{F}{H+F}right)-{mF}.$$Note that the sum of the first two equations above gives$$dot{H}=Lfrac{H+F}{w+H+F}-Hleft(alpha -sigma frac{F}{H+F}right)-{nY}.$$The new negative mortality term for younger hive bees, (-{nY}), models the fact that only the younger hive bees die prematurely while the rest of the dynamics is unchanged with respect to the original model.The search for equilibria soon gives$$bar{Y}=Lfrac{bar{H}+bar{F}}{w+bar{H}+bar{F}}$$from the first ODE above, so that the remaining two equilibrium conditions lead to$$bar{H}=frac{{L}_{n}}{{mJ}}-frac{w}{1+J}$$$$bar{F}=Jbar{H}$$for the same (J) originally defined and ({L}_{n}:=Lleft(1-nright)) (note that (nin left({{{{mathrm{0,1}}}}}right)), and the case (n=0) brings us back to the original model). From this point on the analysis is the same as that previously summarized for the original model, but for replacing (L) with ({L}_{n}) and (l) with (l:=lleft(1-nright)). Consequently, by assuming (alpha,m , > , {l}_{n}) (which is less restrictive when (n , > , 0)), the threshold condition (m < bar{m}) becomes$$m , < , bar{m}left(nright):=frac{{l}_{n}}{2}frac{alpha+sigma+sqrt{{left(alpha -sigma right)}^{2}+4sigma {l}_{n}}}{alpha -{l}_{n}},$$which clearly returns the original threshold condition when (n=0). Since$$frac{dbar{m}}{{dn}}left(nright) , < , 0$$as it can be immediately verified, it follows that the critical value for (m), (bar{m}left(nright)), beyond which the colony system admits only the zero equilibrium, i.e., the transcritical bifurcation value, decreases with (n) (Fig. 4). We thus conclude that colony collapse is favored by the premature death of younger hive bees, possibly caused by a virus impairing the immune system as shown by the analysis of our model of individual bee health.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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