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Research areaThe study was conducted in the Yunyang Experimental Station (108° 54′ E, 30° 55′ N; altitude of 700 m), Southwest University, Chongqing, China. The study area has a subtropical monsoon humid climate with an average annual sunshine duration of 1500 h, average annual temperature of 18.4 °C average annual rainfall of 1100.1 mm, and the rain period predominantly prolongs from June to September. Local soil type is clay loam in texture and Dystric Purple-Udic Cambosols according to the Chinese Soil Taxonomy (CRGCST 2001). Basic properties of 0–20 cm soil layer were as follows: pH 7.29, total N 0.94 g kg−1, total C 7.14 g kg−1, available N 37.45 mg kg−1, available P 2.36 mg kg−1, and available K 72.58 mg kg−1, respectively.The tested biochar was purchased from the Nanjing Qinfeng Straw Technology Co., Ltd. (Nanjing, China), which was made by pyrolysis of the rice (Oryza sativa L.) straw with limited oxygen supply at 500 °C for 2 h. Its properties were as follows: total N 0.61 g kg−1, total P 1.99 g kg−1, total K 27.15 g kg−1, total C 537.97 g kg−1 and pH 8.70.Experimental designA two-year filed experiment (2017–2019) was performed in a completely randomized design with twelve treatments in triplicates including two factors. The first factor was the application of biochar including B0 (0 t ha−1), B10 (10 t ha−1), B20 (20 t ha−1) and B40 (40 t ha−1); and the second factor is the application level N fertilizer including conventional rate (application amount by local farmers)-180 kg N ha−1 (N100), 80% of conventional rate-144 kg N ha−1 (N80) and 60% of conventional rate-108 kg N ha−1 (N60). The plot size was 3 m × 6 m with a border (0.5 m wide) between plots. Biochar was applied to soil only in the first year before the sowing of rapeseed. Each treatment plot received the same amount of potassium (90 kg K2O ha−1) and phosphorus (90 kg P2O5 ha−1). Further details of fertilizer application have been reported by Tian et al.24, being the same for the two-year experiment. Weed, pesticide, and pest management kept the same with the local farmers’ rapeseed management practices. Winter rapeseed (Sanxiayou No.5) was used in the experiment, which was sowed on 21 October 2017 and on 16 October 2018, respectively, and was harvested on 1 May in both years (2018 and 2019).Sampling and analysis of soil and cropCrop yieldRapeseed was hand-harvested when 70–80% of total seeds changed their color from green to black on 1 May 2019, and each plot was separately harvested for seed yield. Seed yield was calculated using 6% as standard seed moisture content.Soil indicesAfter the rapeseed harvest, soil samples were collected from all plots. Five sampling points were randomly selected within each plot. At each point, twenty soil cores of 2.5 cm diameter and 20.0 cm depth were taken in a 1 m radius of the point. All soil cores from each point were put in a plastic bag and thoroughly bulked, crumbled and mixed for physical, chemical and biological analyses. By dividing each soil sample into two subsamples, one subsample was ground, passed through a 2-mm sieve and was air-dried for the analyses of soil organic matter (SOM), total nitrogen (TN), total phosphorus (TP), total potassium (TK), alkali-hydrolyzale nitrogen (AN), available phosphorus (AP), available potassium (AK)25, particulate organic carbon (POC), water-soluble organic carbon (DOC), easily oxidized organic carbon (AOC)26, sucrase (SUC) and urease (URE)27, and another one was ground, passed through a 2-mm sieve and was stored in a refrigerator at − 20 °C for the analyses of structural and functional characteristics of soil microbial community28. At the same time, mixed soil samples (0–20 cm) from five points in each plot were taken using a shovel for soil aggregates analyses24.Drying method was used to determine soil water content (SWC); soil temperature (ST) was measured by temperature probe on the LI6400–09 (LI-COR Inc., Lincoln, NE); potassium dichromate oxidation method was used to determine SOM and DOC content; TN was measured by the Kjeldahl method; TP was determined by Mo-Sb colorimetric method; TK was determined by NaOH melting and analyzed using an atomic spectrophotometry; AN was determined by diffusion-absorption method; AP was quantified by colorimetric analysis following extraction of soil with 0.5 mol L−1 NaHCO3; AK was measured using 1.0 mol L−1 CH3COONH4 extraction; POC was determined by sodium hexametaphosphate dispersion method; AOC was measured by potassium permanganate oxidation method; SUC was measured by 3,5-dinitrosalicylic acid colorimetric determination method; URE was measured by phenol-sodium hypochlorite indophenol colorimetry method; amount of bacteria (B), fungi (F), actinomycetes (A), gram-positive bacteria (GP), gram-negative bacteria (GN) was measured by the Bligh–Dyer method; utilization of sugars (S), amino acids (AA), phenolic acids (PA), carboxylic acids (CA), amines (AM) and polymers (P) by microorganism was measured using commercial Biolog EcoPlate (Biolog Inc., CA, USA).Shannon index (H), Simpson index (D), and evenness index (E) were calculated by the following equations:$$ {text{AWCD}} = sum {(C_{i} – R_{i} )} /n $$$$ {text{H}} = – sum {P_{i} } (ln P_{i} )quad P_{i} = (C_{i} – R_{i} )/sum {(C_{i} – R_{i} } ) $$$$ {text{D}} = 1 – sum P _{i}^{2} $$$$ {text{E}} = {text{H}}/ln {text{S}} $$where n is the 31 carbon sources on the ECO board; Ci and Ri and are the optical density values of the microwell and the control well respectively; Pi is the ratio of the absorbance of a particular well i to the sums of absorbance of all 31well at 120 h; S is the number of color change holes, which represents the number of carbon source used by the microbial community; Average well color development (AWCD), representing the overall carbon substrate utilization potential of cultural microbial communities across all wells per plate.In order to investigate the aggregate structure, all bulk clod samples from each plot were carefully mixed and then gently sieved to pass through a 10-mm sieve. According to the wet-sieving and dry-sieving protocol, the tested soil was fractionated into  > 5, 2 ~ 5, 1 ~ 2, 0.25 ~ 1 and  0.25} right)} }}{{sumnolimits_{{i = 1}}^{n} {(w_{i} )} }} times 100% $$$$ {text{D – MWD}}left( {{text{W – MWD}}} right) = sumlimits_{{i = 1}}^{n} {(bar{d}_{i} w_{i} )} $$$$ {text{D – GMD}}left( {{text{W – GMD}}} right) = exp left[ {frac{{sumlimits_{{i = 1}}^{n} {m_{i} ln bar{d}_{i} } }}{{sumlimits_{{i = 1}}^{n} {m_{i} } }}} right] $$where DR0.25 and WR0.25 are the proportion of  > 0.25 mm soil mechanical-stable aggregates and water-stable aggregates, respectively; D-MWD and W-MWD are the mean weight diameter of mechanical-stable aggregates and water-stable aggregates (mm), respectively; D-GMD and W-GMD are the mean geometric diameter of mechanical-stable aggregates and water-stable aggregates (mm), respectively; mi is mass in size fraction i; and wi is the proportion (%) of the total sample mass in size fraction i and di is mean diameter of size fraction i.Evaluation of soil fertilityGrey correlation analysisGrey correlation analysis refers to a method of quantitative description and comparison of a system’s development and change. The basic idea is to determine whether they are closely connected by determining the geometric similarity of the reference data column and several comparison data columns, which reflects the degree of correlation between the curves29. The grey relational coefficient ξi (k) can be expressed as follows:$$ xi (k) = frac{{mathop {min }limits_{i} mathop {min }limits_{k} left| {x_{0} (k) – x_{i} (k)} right| + rho mathop {max }limits_{i} mathop {max }limits_{k} left| {x_{0} (k) – x_{i} (k)} right|}}{{left| {x_{0} (k) – x_{i} (k)} right| + rho mathop {max }limits_{i} max left| {mathop {x_{0} (k)}limits_{k} – x_{i} (k)} right|}} $$$$ x_{i}^{k} = frac{{x_{i}^{k} }}{{mathop {max }limits_{i} x_{i}^{k} }} $$$$ gamma _{i} = frac{1}{n}sumlimits_{{k = i}}^{n} {xi _{i} } (k) $$$$ omega _{{i(gamma )}} = frac{1}{n}sumlimits_{{i = 1}}^{n} {gamma _{i} } $$$$ G_{i}^{k} = sumlimits_{{i = 1}}^{n} {left( {xi _{i} times omega _{{i(gamma )}} } right),quad k = 1,2,3, ldots ,n;quad i = 1,2,3, ldots ,n} $$where (x_{i}^{k}) The i trait observation value of treatment k; (mathop {max }limits_{i} x_{i}^{k}) The maximum value of the i trait in all treatments; (mathop {min }limits_{i} x_{i}^{k}) The minimum value of the i trait in all treatments; (mathop {min }limits_{i} mathop {min }limits_{k} left| {x_{0} (k) – x_{i} (k)} right|) Second level minimum difference; (mathop {max }limits_{i} mathop {max }limits_{k} left| {x_{0} (k) – x_{i} (k)} right|) Second level maximum difference; (rho) Resolution coefficient (0.5).Principal component analysisPrincipal component analysis refers to a multivariate statistical method that converts multiple indicators into several comprehensive indicators by the idea of dimensionality under the premise of losing little information. It simplifies the complexity in high-dimensional data while retaining trends and patterns30.Cluster analysisCluster analysis comprises a range of methods for classifying multivariate data into subgroups. Using the euclidean distance as a measure of the difference in the fertility of each treatment, the shortest distance method was used to systematically cluster according to the degree of intimacy and similarity of soil fertility levels. By organizing multivariate data into such subgroups, clustering can help reveal the characteristics of any structure or patterns present31.Statistical analysisCorrelation analysis was performed to assess the relationships between rapeseed yield and soil attributes. Grey correlation analysis and principal component analysis were performed to establish comprehensive score for soil fertility and the main soil factors affecting rapeseed yield. Cluster analysis was used to cluster the soil fertility of each treatment. All the statistical analyses were performed using Excel 2018 (Office Software, Inc., Beijing, China) and SPSS 17.0 (SPSS Inc., Chicago, Illinois, USA). The comparisons of treatment means were based on LSD test at the P  More

The cell modelGrowth rate of a cellThe growth rate of a bacteria cell depends on the acquisition of C (from the particle) and N (from the particle and through ({{rm{N}}}_{2}) fixation), as well as on metabolic expenses in terms of C.Uptake of C and NBacteria get C from glucose and both C and N from amino acids. The total amount of C available for the cell from monomers is (units of C per time)$${J}_{{rm{DOC}}}={f}_{{rm{G,C}}}J_{G}+{f}_{{rm{A}},{rm{C}}}{J}_{{rm{A}}},$$
(8)
and the amount of N available from monomer is (N per time)$${J}_{{rm{DON}}}={f}_{{rm{A}},{rm{N}}}{J}_{{rm{A}}},$$
(9)
where ({J}_{rm{G}}) and ({J}_{rm{A}}) are uptake rates of glucose and amino acids, ({f}_{rm{G,C}}) is the fraction of C in glucose, and ({f}_{rm{A,C}}) and ({f}_{rm{A,N}}) are fractions of C and N in amino acids.The rate of obtaining N through ({{rm{N}}}_{2}) fixation is:$${J}_{{{rm{N}}}_{2}}({{psi }})={{psi }}{M}_{{{rm{N}}}_{2}},$$
(10)
where ({psi },(0 < {{psi }} < 1)) regulates ({{rm{N}}}_{2}) fixation rate and fixation can happen at a maximum rate ({M}_{{{rm{N}}}_{2}}). ({{rm{N}}}_{2}) fixation is only limited by the maximum ({{rm{N}}}_{2}) fixation rate as dissolved dinitrogen (({{rm{N}}}_{2})) gas in seawater is assumed to be unlimited70.The total uptake of C and N from different sources becomes$${J}_{{rm{C}}}={J}_{{rm{DOC}}}$$ (11) $${J}_{{rm{N}}}({{psi }})={J}_{{rm{DON}}}+{J}_{{{rm{N}}}_{2}}({{psi }})$$ (12) CostsRespiratory costs of cellular processes together with ({{rm{N}}}_{2}) fixation and its associated ({{rm{O}}}_{2}) removal cost depend on the cellular ({{rm{O}}}_{2}) concentration. Two possible scenarios can be observed: Case 1: When ({O}_{2}) concentration is sufficient to maintain aerobic respiration Respiratory costs for bacterial cellular maintenance can be divided into two parts: one dependent on limiting substrates and the other one is independent of substrate concentration71. Here we consider only the basal respiratory cost ({R}_{rm{B}}{x}_{rm{B}}), which is independent of the limiting substrates and is assumed as proportional to the mass of the cell ({x}_{B}) (μg C). In order to solubilize particles, particle-attached bacteria produce ectoenzymes that cleave bonds to make molecules small enough to be transported across the bacterial cell membrane. Cleavage is represented by a biomass-specific ectoenzyme production cost ({R}_{rm{E}})72. The metabolic costs associated with the uptake of hydrolysis products and intracellular processing are assumed to be proportional to the uptake (({J}_{i})): ({R}_{{rm{G}}}{J}_{{rm{G}}}) and ({R}_{{rm{A}}}{J}_{{rm{A}}}) where the ({R}_{i})’s are costs per unit of resource uptake. In a similar way, the metabolic cost of ({{rm{N}}}_{2}) fixation is assumed as proportional to the ({{rm{N}}}_{2}) fixation rate: ({R}_{{{rm{N}}}_{2}}{rho }_{{rm{CN}},{rm{B}}}{J}_{{{rm{N}}}_{2}}), where ({rho }_{{rm{CN}},{rm{B}}}) is the bacterial C:N ratio. If we define all the above costs as direct costs, then the total direct respiratory cost becomes$${R}_{{rm{D}}}({{psi }})={R}_{{rm{B}}}{x}_{{rm{B}}}+{R}_{{rm{E}}}{x}_{{rm{B}}}+{R}_{{rm{G}}}{J}_{{rm{G}}}+{R}_{{rm{A}}}{J}_{{rm{A}}}+{R}_{{{rm{N}}}_{2}}{rho }_{{rm{CN}},{rm{B}}}{J}_{{{rm{N}}}_{2}}({{psi }}).$$ (13) Indirect costs related to ({{rm{N}}}_{2}) fixation arises from the removal of ({{rm{O}}}_{2}) from the cell and the production/replenishment of nitrogenase as the enzyme is damaged by ({{rm{O}}}_{2}). The cell can remove ({{rm{O}}}_{2}) either by increasing respiration73 or by increasing the production of nitrogenase enzyme itself74. Here we consider only the process of ({{rm{O}}}_{2}) removal by increasing respiration. To calculate this indirect cost, the concentration of ({{rm{O}}}_{2}) present in the cell needs to be estimated.Since the time scale of ({{rm{O}}}_{2}) concentration inside a cell is short, we have assumed a pseudo steady state inside the cell; the ({{rm{O}}}_{2}) diffusion rate inside a cell is always balanced by the respiration rate14, which can be expressed as$${rho }_{{rm{CO}}}{F}_{{{rm{O}}}_{2}}={R}_{{rm{D}}}left({{psi }}right).$$ (14) Here ({rho }_{{rm{CO}}}) is the conversion factor of respiratory ({{rm{O}}}_{2}) to C equivalents and ({F}_{{{rm{O}}}_{2}}) is the actual ({{rm{O}}}_{2}) diffusion rate into a cell from the particle and can be calculated as$${F}_{{{rm{O}}}_{2}}=4{rm{pi }}{r}_{{rm{B}}}{K}_{{{rm{O}}}_{2}}left({X}_{{{rm{O}}}_{2}}-{X}_{{{rm{O}}}_{2},{rm{C}}}right),$$ (15) where ({r}_{{rm{B}}}) is the cell radius, ({X}_{{{rm{O}}}_{2}}) is the local ({{rm{O}}}_{2}) concentration inside the particle, ({X}_{rm{{O}}_{2},{rm{C}}}) is the cellular ({{rm{O}}}_{2}) concentration, and ({K}_{{{rm{O}}}_{2}}) is the effective diffusion coefficient of ({{rm{O}}}_{2}) over cell membrane layers. The effective diffusion coefficient can be calculated according to Inomura et al.14 in terms of diffusion coefficient inside particles (({bar{D}}_{{{rm{O}}}_{2}})), the diffusivity of cell membrane layers relative to water (({varepsilon }_{{rm{m}}})), the radius of cellular cytoplasm (({r}_{{rm{C}}})), and the thickness of cell membrane layers (({L}_{{rm{m}}})) as$${K}_{{{rm{O}}}_{2}}={bar{D}}_{{{rm{O}}}_{2}}frac{{varepsilon }_{{rm{m}}}({r}_{{rm{C}}}+{L}_{{rm{m}}})}{{varepsilon }_{{rm{m}}}{r}_{{rm{C}}}+{L}_{{rm{m}}}}.$$ (16) The apparent diffusivity inside particles (({bar{D}}_{{{rm{O}}}_{2}})) is considered as a fraction ({f}_{{{rm{O}}}_{2}}) of the diffusion coefficient in seawater (({D}_{{{rm{O}}}_{2}}))$${bar{D}}_{{{rm{O}}}_{2}}={f}_{{{rm{O}}}_{2}}{D}_{{{rm{O}}}_{2}}.$$ (17) Combining (14) and (15) gives the cellular ({{rm{O}}}_{2}) concentration ({X}_{{rm{O}}_{2},{rm{C}}}) as$${X}_{{{rm{O}}}_{2},{rm{C}}}={{max }}left[0,{X}_{{{rm{O}}}_{2}}-frac{{R}_{{rm{D}}}left({{psi }}right)}{4{rm{pi }}{r}_{{rm{B}}}{K}_{{{rm{O}}}_{2}}{rho }_{{rm{CO}}}}right].$$ (18) If there is excess ({{rm{O}}}_{2}) present in the cell after respiration (({X}_{{rm{O}}_{2},{rm{C}}} , > , 0)), then the indirect cost of removing the excess ({{rm{O}}}_{2}) to be able to perform ({{rm{N}}}_{2}) fixation can be written as$${R}_{{{rm{O}}}_{2}}left({{psi }}right)=Hleft({{psi }}right){rho }_{{rm{CO}}}4{rm{pi }}{r}_{{rm{B}}}{K}_{{{rm{O}}}_{2}}{X}_{rm{{O}}_{2},{rm{C}}},$$
(19)
where (H({{psi }})) is the Heaviside function:$$Hleft({{psi }}right)=left{begin{array}{cc}0,&{rm{if}}{,}{{psi }}=0\ 1, &{rm{if}}{,}{{psi }} , > , 0end{array}right..$$
(20)
Therefore, the total aerobic respiratory cost becomes:$${R}_{{rm{tot}},{rm{A}}}left({{psi }}right)={R}_{{rm{D}}}left({{psi }}right)+{R}_{{{rm{O}}}_{2}}left({{psi }}right).$$
(21)

Case 2: Anaerobic respiration
When available ({{rm{O}}}_{2}) is insufficient to maintain aerobic respiration (({R}_{{rm{tot}}}left({{psi }}right) , > , {rho }_{{rm{CO}}}{F}_{{{rm{O}}}_{2},{{max }}})), cells use ({{{rm{NO}}}_{3}}^{-}) and ({{{rm{SO}}}_{4}}^{2-}) for respiration. The potential ({{{rm{NO}}}_{3}}^{-}) uptake, ({J}_{{{rm{NO}}}_{3},{rm{pot}}}), is$${J}_{{{rm{NO}}}_{3},{rm{pot}}}={M}_{{{rm{NO}}}_{3}}frac{{A}_{{{rm{NO}}}_{3}}{X}_{{{rm{NO}}}_{3}}}{{A}_{{{rm{NO}}}_{3}}{X}_{{{rm{NO}}}_{3}}+{M}_{{{rm{NO}}}_{3}}},$$
(22)
where ({M}_{{{rm{NO}}}_{3}}) and ({A}_{{{rm{NO}}}_{3}}) are maximum uptake rate and affinity for ({{{rm{NO}}}_{3}}^{-}) uptake, respectively. However, the actual rate of ({{{rm{NO}}}_{3}}^{-}) uptake, ({J}_{{{rm{NO}}}_{3}}), is determined by cellular respiration and can be written as$${J}_{{{rm{NO}}}_{3}}={{min }}left({J}_{{{rm{NO}}}_{3},{rm{pot}}},{{max }}left(0,frac{{R}_{{rm{tot}},{rm{A}}}left({{psi }}right)-{rho }_{{rm{CO}}}{F}_{{{rm{O}}}_{2},{{max }}}}{{rho }_{{rm{C}}{{rm{NO}}}_{3}}}right)right),$$
(23)
where ({rho }_{{rm{C}}{{rm{NO}}}_{3}}) is the conversion factor of respiratory ({{{rm{NO}}}_{3}}^{-}) to C equivalents and the maximum ({{rm{O}}}_{2}) diffusion rate into a cell ({F}_{{{rm{O}}}_{2},{{max }}}) can be obtained by making cellular ({{rm{O}}}_{2}) concentration ({X}_{{{rm{O}}}_{2},{rm{c}}}) zero in (15) as$${F}_{{{rm{O}}}_{2},{{max }}}=4{rm{pi }}{r}_{{rm{B}}}{K}_{{{rm{O}}}_{2}}{X}_{{{rm{O}}}_{2}},$$
(24)
Further, in the absence of sufficient ({{{rm{NO}}}_{3}}^{-}), the cell uses ({{{rm{SO}}}_{4}}^{2-}) as an electron acceptor for respiration. Since the average concentration of ({{{rm{SO}}}_{4}}^{2-}) in seawater is 29 mmol L−1 75, ({{{rm{SO}}}_{4}}^{2-}) is a nonlimiting nutrient for cell growth and the potential uptake rate of ({{{rm{SO}}}_{4}}^{2-}) is mainly governed by the maximum uptake rate as$${J}_{{{rm{SO}}}_{4},{rm{pot}}}={M}_{{{rm{SO}}}_{4}},$$
(25)
where ({M}_{{{rm{SO}}}_{4}}) is the maximum uptake rate for ({{{rm{SO}}}_{4}}^{2-}) uptake. The actual rate of ({{{rm{SO}}}_{4}}^{2-}) uptake, ({J}_{{{rm{SO}}}_{4}}), can be written as$${J}_{{{rm{SO}}}_{4}}={{min }}left({J}_{{{rm{SO}}}_{4},{rm{pot}}},{{max }}left(0,frac{{R}_{{rm{tot}},{rm{A}}}left({{psi }}right)-{rho }_{{rm{CO}}}{F}_{{{rm{O}}}_{2},{{max }}}-{rho }_{{rm{CN}}{{rm{O}}}_{3}}{F}_{{{rm{NO}}}_{3},{rm{pot}}}}{{rho }_{{rm{C}}{{rm{SO}}}_{4}}}right)right),$$
(26)
where ({rho }_{{rm{C}}{{rm{SO}}}_{4}}) is the conversion factor of respiratory ({{{rm{SO}}}_{4}}^{2-}) to C equivalents.According to formulations (23) and (26), ({{{rm{NO}}}_{3}}^{-}) and ({{{rm{SO}}}_{4}}^{2-}) uptake occurs only when the diffusive flux of ({{rm{O}}}_{2}), and both ({{rm{O}}}_{2}) and ({{{rm{NO}}}_{3}}^{-}) are insufficient to maintain respiration(.) Moreover, the uptake rates of ({{{rm{NO}}}_{3}}^{-}) and ({{{rm{SO}}}_{4}}^{2-}) are regulated according to the cells’ requirements.Uptakes of ({{{rm{NO}}}_{3}}^{-}) and ({{{rm{SO}}}_{4}}^{2-}) incur extra metabolic costs ({R}_{{{rm{NO}}}_{3}}{rho }_{{rm{C}}{{rm{NO}}}_{3}}{J}_{{{rm{NO}}}_{3}}) and ({R}_{{{rm{SO}}}_{4}}{rho }_{{rm{C}}{{rm{SO}}}_{4}}{J}_{{{rm{SO}}}_{4}}), where ({R}_{{{rm{NO}}}_{3}}) and ({R}_{{{rm{SO}}}_{4}}) are costs per unit of ({{{rm{NO}}}_{3}}^{-}) and ({{{rm{SO}}}_{4}}^{2-}) uptake. The total respiratory cost can be written as$${R}_{{rm{tot}}}left({{psi }}right)={R}_{{rm{tot}},{rm{A}}}left({{psi }}right)+{R}_{{{rm{NO}}}_{3}}{rho }_{{rm{C}}{{rm{NO}}}_{3}}{J}_{{{rm{NO}}}_{3}}+{R}_{{{rm{SO}}}_{4}}{rho }_{{rm{C}}{{rm{SO}}}_{4}}{J}_{{{rm{SO}}}_{4}}.$$
(27)
Synthesis and growth rateThe assimilated C and N are combined to synthesize new structure. The synthesis rate is constrained by the limiting resource (Liebig’s law of the minimum) and by available electron acceptors such that the total flux of C available for growth ({J}_{{rm{tot}}}) (μg C d−1) is:$${J}_{{rm{tot}}}left({{psi }}right)={{min }}left[{J}_{{rm{C}}}-{R}_{{rm{tot}}}left({{psi }}right),{rho }_{{rm{CN,B}}}{J}_{{rm{N}}}left({{psi }}right),{rho }_{{rm{CO}}}{F}_{{{rm{O}}}_{2}}+{rho }_{{rm{C}}{{rm{NO}}}_{3}}{J}_{{{rm{NO}}}_{3}}+{rho }_{{rm{C}}{{rm{SO}}}_{4}}{J}_{{{rm{SO}}}_{4}}right].$$
(28)
Here, the total available C for growth is ({J}_{{rm{C}}}-{R}_{{rm{tot}}}({{psi }})), the C required to synthesize biomass from N source is ({rho }_{{rm{CN}},B}{J}_{rm{N}}), and the C equivalent inflow rate of electron acceptors to the cell is ({rho }_{{rm{CO}}}{F}_{{rm{O}}_{2}}+{rho }_{{rm{C}}{{rm{NO}}}_{3}}{J}_{{{rm{NO}}}_{3}}+{rho }_{{rm{C}}{{rm{SO}}}_{4}}{J}_{{{rm{SO}}}_{4}}). We assume that excess C or N is released from the cell instantaneously.Synthesis is not explicitly limited by a maximum synthesis capacity; synthesis is constrained by the C and N uptake in the functional responses (Eqs. 34 and 35). The division rate (mu) of the cell (d−1) is the total flux of C available for growth divided by the C mass of the cell (({x}_{rm{B}})):$$mu ({{psi }})={J}_{{rm{tot}}}({{psi }})/{x}_{rm{B}}.$$
(29)
The resulting division rate, (mu), is a measure of the bacterial fitness and we assume that the cell regulates its ({{rm{N}}}_{2}) fixation rate depending on the environmental conditions to gain additional N while maximizing its growth rate. The optimal value of the parameter regulating ({{rm{N}}}_{2}) fixation ({{psi }}) ((0le {{psi }}le 1)) then becomes:$${{{psi }}}^{ast }={{arg }}mathop{{{max }}}limits_{{{psi }}}{mu ({{psi }})},$$
(30)
and the corresponding optimal division rate becomes$${mu }^{ast }=mu left({{{psi }}}^{ast }right).$$
(31)
The particle modelWe consider a sinking particle of radius ({r}_{{rm{P}}}) (cm) and volume ({V}_{{rm{P}}}) (cm3) (Supplementary Fig. S1). The particle contains facultative nitrogen-fixing bacterial population (B(r)) (cells L−1), polysaccharides ({C}_{{rm{P}}}(r)) (μg G L−1), and polypeptides ({P}_{{rm{P}}}(r)) (μg A L−1) at a radial distance (r) (cm) from the center of the particle, where G and A stand for glucose and amino acids. We assume that only fractions ({f}_{{rm{C}}}) and ({f}_{{rm{P}}}) of these polymers are labile (({C}_{{rm{L}}}(r)={f}_{{rm{C}}}{C}_{{rm{P}}}(r),) ({P}_{{rm{L}}}(r)={f}_{{rm{P}}}{P}_{{rm{P}}}(r))), i.e., accessible by bacteria. Bacterial enzymatic hydrolysis converts the labile polysaccharides and polypeptides into monosaccharides (glucose) ((G) μg G L−1) and amino acids ((A) μg A L−1) that are efficiently taken up by bacteria. Moreover, the particle contains ({{rm{O}}}_{2}), ({{{rm{NO}}}_{3}}^{-}), and ({{{rm{SO}}}_{4}}^{2-}) with concentrations ({X}_{{{rm{O}}}_{2}}(r)) (μmol O2 L−1), ({X}_{{{rm{NO}}}_{3}}(r)) (μmol NO3 L−1), and ({X}_{{{rm{SO}}}_{4}}(r)) (μmol SO4 L−1). Glucose and amino acids diffuse out of the particle whereas ({{rm{O}}}_{2}) and ({{{rm{NO}}}_{3}}^{-}) diffuse into the particle from the surrounding environment. Due to the high concentration of ({{{rm{SO}}}_{4}}^{2-}) in ocean waters, we assume that ({{{rm{SO}}}_{4}}^{2-}) is not diffusion limited inside particles, its uptake is limited by the maximum uptake capacity due to physical constraint. The interactions between particle, cells, and the surrounding environment are explained in Supplementary Fig. S1 and equations are provided in Table 1 of the main text.We assume that labile polysaccharide (({C}_{{rm{L}}})) and polypeptide (({P}_{{rm{L}}})) are hydrolyzed into glucose and amino acids at rates ({J}_{{rm{C}}}) and ({J}_{{rm{P}}}) with the following functional form$${J}_{{rm{C}}}={h}_{{rm{C}}}frac{{A}_{{rm{C}}}{C}_{{rm{L}}}}{{h}_{{rm{C}}}+{A}_{{rm{C}}}{C}_{{rm{L}}}}$$
(32)
$${J}_{{rm{P}}}={h}_{{rm{P}}}frac{{A}_{{rm{P}}}{P}_{{rm{L}}}}{{h}_{{rm{P}}}+{A}_{{rm{P}}}{P}_{{rm{L}}}}$$
(33)
where ({h}_{{rm{C}}}) and ({h}_{{rm{P}}}) are maximum hydrolysis rates of the carbohydrate and peptide pool, and ({A}_{{rm{C}}}) and ({A}_{{rm{P}}}) are respective affinities. ({J}_{{rm{G}}}) and ({J}_{{rm{A}}}) represent uptake of glucose and amino acids:$${J}_{{rm{G}}}={M}_{{rm{G}}}frac{{A}_{{rm{G}}}G}{{A}_{{rm{G}}}G+{M}_{{rm{G}}}}$$
(34)
$${J}_{{rm{A}}}={M}_{{rm{A}}}frac{{A}_{{rm{A}}}A}{{A}_{{rm{A}}}A+{M}_{{rm{A}}}}$$
(35)
where ({M}_{{rm{G}}}) and ({M}_{{rm{A}}}) are maximum uptake rates of glucose and amino acids, whereas ({A}_{{rm{G}}}) and ({A}_{{rm{A}}}) are corresponding affinities. Hydrolyzed monomers diffuse out of the particle at a rate ({D}_{{rm{M}}}).({mu }^{ast }) is the optimal division rate of cells (Eq. 31) and ({m}_{rm{B}}) represents the mortality rate (including predation) of bacteria. ({F}_{{{rm{O}}}_{2}}) and ({J}_{{{rm{NO}}}_{3}}) represent the diffusive flux of ({{rm{O}}}_{2}) and the consumption rate of ({{{rm{NO}}}_{3}}^{-}), respectively, through the bacterial cell membrane. ({bar{D}}_{{{rm{O}}}_{2}}) and ({bar{D}}_{{{rm{NO}}}_{3}}) are diffusion coefficients of ({{rm{O}}}_{2}) and ({{{rm{NO}}}_{3}}^{-}) inside the particle.At the center of the particle ((r=0)) the gradient of all quantities vanishes:$${left.frac{partial G}{partial r}right|}_{r=0}={left.frac{partial A}{partial r}right|}_{r=0}={left.frac{partial {X}_{{{rm{O}}}_{2}}}{partial r}right|}_{r=0}={left.frac{partial {X}_{{rm{N}}{{rm{O}}}_{3}}}{partial r}right|}_{r=0}=0$$
(36)
At the surface of the particle ((r={r}_{{rm{P}}})) concentrations are determined by the surrounding environment:$${left.Gright|}_{r={r}_{{rm{P}}}}={G}_{infty },{left.Aright|}_{r={r}_{{rm{P}}}}={A}_{infty },{left.{X}_{{{rm{O}}}_{2}}right|}_{r={r}_{{rm{P}}}}={X}_{{{rm{O}}}_{2},infty },{left.{X}_{{{rm{NO}}}_{3}}right|}_{r={r}_{{rm{P}}}}={X}_{{{rm{NO}}}_{3},infty }$$
(37)
where ({G}_{infty },) ({A}_{infty ,}) ({X}_{{{rm{O}}}_{2},infty }) and ({X}_{{{rm{NO}}}_{3},infty }) are concentrations of glucose, amino acids, ({{rm{O}}}_{2}), and ({{{rm{NO}}}_{3}}^{-}) in the environment.Calculation of total N2 fixation rateThe total amount of fixed ({{rm{N}}}_{2}) in a specific size class of particle, ({{rm{N}}}_{{rm{fix}},{rm{P}}}) (({rm{mu }})g N particle−1), is calculated as$${{rm{N}}}_{{rm{fix}},{rm{P}}}=int int 4pi {{r}_{{rm{B}}}}^{2}B{J}_{{{rm{N}}}_{2}}{rm{d}}{r}_{{rm{P}}}{rm{dz}},$$
(38)
where ({r}_{{rm{P}}}) (cm) is the particle radius and z (m) represents the water column depth.({{rm{N}}}_{2}) fixation rate per unit volume of water, ({{rm{N}}}_{{rm{fix}},{rm{V}}}left(tright)) (({rm{mu }}{rm{mol}}) N m−3 d−1), is calculated as$${{rm{N}}}_{{rm{fix}},{rm{V}}}=int int 4pi {{r}_{{rm{B}}}}^{2}rho B{J}_{{{rm{N}}}_{2}}n(x){rm{d}}{r}_{{rm{P}}}{rm{d}}x,$$
(39)
Here (x) (cm) represents the size range (radius) of particles, (rho) is the fraction of diazotrophs of the total heterotrophic bacteria, and (n(x)) (number of particles per unit volume of water per size increment) is the size spectrum of particles that is most commonly approximated by a power law distribution of the form$$n(x)={n}_{0}{(2x)}^{xi }$$
(40)
where ({n}_{0}) is a constant that controls total particle abundance and the slope (xi) represents the relative concentration of small to large particles: the steeper the slope, the greater the proportion of smaller particles and the flatter the slope, and the greater the proportion of larger particles34.Depth-integrated ({{rm{N}}}_{2}) fixation rate, ({{rm{N}}}_{{rm{fix}},{rm{D}}}) (({rm{mu }}{rm{mol}}) N m−2 d−1), can be obtained by$${{rm{N}}}_{{rm{fix}},{rm{D}}}left(tright)=int {{rm{N}}}_{{rm{fix}},{rm{V}}}{rm{d}}z.$$
(41)
Assumptions and simplification in the modeling approachAccording to our current model formulation, the particle size remains constant while sinking. However, in nature, particle size is dynamic due to processes like bacterial remineralization, aggregation, and disaggregation. We neglect these complications to keep the model simple and to focus on revealing the coupling between particle-associated environmental conditions and ({{rm{N}}}_{2}) fixation by heterotrophic bacteria. These factors can, however, possibly be incorporated by using in situ data or by using the relationship between carbon content and the diameter of particles48 and including terms for aggregation and disaggregation55.Our model represents a population of facultative heterotrophic diazotrophs that grow at a rate similar to other heterotrophic bacteria but the whole community initiates ({{rm{N}}}_{2}) fixation when conditions become suitable. However, under natural conditions, diazotrophs may only constitute a fraction of the bacterial community, and their proliferation may be gradual21, presumably affected by multiple factors. In such case, our approach will overestimate diazotroph cell concentration and consequently the ({{rm{N}}}_{2}) fixation rate.For simplicity, our approach includes only aerobic respiration, ({{{rm{NO}}}_{3}}^{-}) and ({{{rm{SO}}}_{4}}^{2-}) respiration, although many additional aerobic and anaerobic processes likely occur on particles (e.g Klawonn et al.19). To our knowledge, a complete picture of such processes, their interactions and effects on particle biochemistry is unavailable. For example, we have assumed that when ({{rm{O}}}_{2}) and ({{{rm{NO}}}_{3}}^{-}) are insufficient to maintain respiration, heterotrophic bacteria start reducing ({{{rm{SO}}}_{4}}^{2-}). However, ({{{rm{SO}}}_{4}}^{2-}) reduction has been detected only with a significant lag after the occurrence of anaerobic conditions, suggesting it as a slow adapted process76, whereas we assume it to be instantaneous. On the other hand, the lag may not be real but due to a so called cryptic sulfur cycle, where ({{{rm{SO}}}_{4}}^{2-}) reduction is accompanied by concurrent sulfide oxidation effectively masking sulfide production77. Hopefully, future insights into interactions between diverse aerobic and anaerobic microbial processes can refine our modelling approach and fine-tune predictions of biochemistry in marine particles.Procedure of numerically obtaining optimal N2 fixation rateTo avoid making the optimization in Eq. (30) at every time step during the simulation, a lookup table of ({mu }^{ast }) (Eq. 31) over realistic ranges of the four resources (glucose, amino acids, ({{rm{O}}}_{2}), and ({{{rm{NO}}}_{3}}^{-})) and the parameter determining ({{rm{N}}}_{2}) fixation rate (({{psi }})) was created at the beginning of the simulation.The effects of temperature on N2 fixation rateTo examine the role of temperature variation on ({{rm{N}}}_{2}) fixation rate in sinking particles, we consider hydrolysis of polysaccharide and polypeptide, uptake of glucose and amino acids, uptake of ({{{rm{NO}}}_{3}}^{-}), respiration, and diffusion dependent on temperature. Apart from diffusion, all other processes are multiplied by a factor ({Q}_{10}) that represents the factorial increase in rates with ({10}^{0})C temperature increase. The rate (R) at a given temperature (T) is then$$R={R}_{{rm{ref}}}{{Q}_{10}}^{(T-{T}_{{rm{ref}}})/10}.$$
(42)
Here the reference rate ({R}_{{rm{ref}}}) is defined as the rate at the reference temperature ({T}_{{rm{ref}}}.) We set the reference temperature ({T}_{{rm{ref}}}) at room temperature of 20 °C. The effect of temperature on the diffusion coefficient D for glucose, amino acids, ({{rm{O}}}_{2}), and ({{{rm{NO}}}_{3}}^{-}) is described by Walden’s rule:$$D={D}_{{rm{ref}}}{eta }_{{rm{ref}}}T/(eta {T}_{{rm{ref}}})$$
(43)
where (eta) is the viscosity of water at the given temperature (T), and ({D}_{{rm{ref}}}) and ({eta }_{{rm{ref}}}) are diffusion coefficient and viscosity at ({T}_{{rm{ref}}}).({Q}_{10}) values for different enzyme classes responsible for hydrolysis (({Q}_{10,{rm{h}}})) lie within the range 1.1–2.978. Here, we have chosen ({Q}_{10,{rm{h}}}=2) for hydrolysis from the middle of the prescribed range. The ({Q}_{10}) values for uptake affinities (({Q}_{10,{rm{A}}})) are taken as 1.579. ({Q}_{10,{rm{R}}}=2) is chosen for all parameters related to respiration (({R}_{{rm{B}}}), ({R}_{{rm{E}}}), ({R}_{{rm{G}}}), ({R}_{{rm{A}}}), ({R}_{{{rm{N}}}_{2}}), ({R}_{{{rm{NO}}}_{3}}), ({R}_{{{rm{SO}}}_{4}}))80. ({R}_{{rm{ref}}}) and ({D}_{{rm{ref}}}) are the values of (R)’s and (D)’s provided in Table S1. The reference viscosity (({eta }_{{rm{ref}}})) and viscosities ((eta)) at different temperatures are taken from Jumars et al.80. More

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