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DataWe tested our models on ten publicly available datasets. In Fig. 4 we show examples of images from each of the datasets. When applicable, the training and test splits were kept the same as in the original dataset. For example, the ZooScan, Kaggle, EILAT, and RSMAS datasets lack a specific training and test set; in these cases, benchmarks come from k-fold cross-validation51,52, and we followed the exact same procedures in order to allow for a fair comparison.Figure 4Examples of images from each of the datasets.(a) RSMAS (b) EILAT (c) ZooLake (d) WHOI (e) Kaggle (f) ZooScan (g) NA-Birds (h) Stanford dogs (i) SriLankan Beetles (j) Florida Wildtrap.Full size imageRSMAS This is a small coral dataset of 766 RGB image patches with a size of (256times 256) pixels each53. The patches were cropped out of bigger images obtained by the University of Miami’s Rosenstiel School of Marine and Atmospheric Sciences. These images were captured using various cameras in various locations. The data is separated into 14 unbalanced groups and whose labels correspond to the names of the coral species in Latin. The current SOTA for the classification of this dataset is by52. They use the ensemble of best performing 11 CNN models. The best models were chosen based on sequential forward feature selection (SFFS) approach. Since an independent test is not available, they make use of 5-fold cross-validation for benchmarking the performances.EILAT This is a coral dataset of 1123 64-pixel RGB image patches53 that were created from larger images that were taken from coral reefs near Eilat in the Red sea. The image dataset is partitioned into eight classes, with an unequal distribution of data. The names of the classes correspond to the shorter version of the scientific names of the coral species. The current SOTA52 for the classification of this dataset uses the ensemble of best performing 11 CNN models similar to RSMAS dataset and 5-fold cross-validation for benchmarking the performances.ZooLake This dataset consists of 17943 images of lake plankton from 35 classes, acquired using a Dual-magnification Scripps Plankton Camera (DSPC) in Lake Greifensee (Switzerland) between 2018 and 2020 14,54. The images are colored, with a black background and an uneven class distribution. The current SOTA22 on this dataset is based on a stacking ensemble of 6 CNN models on an independent test set.WHOI This dataset 55 contains images of marine plankton acquired by Image FlowCytobot56, from Woods Hole Harbor water. The sampling was done between late fall and early spring in 2004 and 2005. It contains 6600 greyscale images of different sizes, from 22 manually categorized plankton classes with an equal number of samples for each class. The majority of the classes belonging to phytoplankton at genus level. This dataset was later extended to include 3.4M images and 103 classes. The WHOI subset that we use was previously used for benchmarking plankton classification models51,52. The current SOTA22 on this dataset is based on average ensemble of 6 CNN models on an independent test set.Kaggle-plankton The original Kaggle-plankton dataset consists of plankton images that were acquired by In-situ Ichthyoplankton Imaging System (ISIIS) technology from May to June 2014 in the Straits of Florida. The dataset was published on Kaggle (https://www.kaggle.com/c/datasciencebowl) with images originating from the Hatfield Marine Science Center at Oregon State University. A subset of the original Kaggle-plankton dataset was published by51 to benchmark the plankton classification tasks. This subset comprises of 14,374 greyscale images from 38 classes, and the distribution among classes is not uniform, but each class has at least 100 samples. The current SOTA22 uses average ensemble of 6 CNN models and benchmarks the performance using 5-fold cross-validation.ZooScan The ZooScan dataset consists of 3771 greyscale plankton images acquired using the Zooscan technology from the Bay of Villefranche-sur-mer57. This dataset was used for benchmarking the classification models in previous plankton recognition papers51,52. The dataset consists of 20 classes with a variable number of samples for each class ranging from 28 to 427. The current SOTA22 uses average ensemble of 6 CNN models and benchmarks the performance using 2-fold cross-validation.NA-Birds NA-Birds58 is a collection of 48,000 captioned pictures of North America’s 400 most often seen bird species. For each species, there are over 100 images accessible, with distinct annotations for males, females, and juveniles, totaling 555 visual categories. The current SOTA59 called TransFG modifies the pure ViT model by adding contrastive feature learning and part selection module that replaces the original input sequence to the transformer layer with tokens corresponding to informative regions such that the distance of representations between confusing subcategories can be enlarged. They make use of an independent test set for benchmarking the model performances.Stanford Dogs The Stanford Dogs dataset comprises 20,580 color images of 120 different dog breeds from all around the globe, separated into 12,000 training images and 8,580 testing images60. The current SOTA59 makes use of modified ViT model called TransFG as explained above in NA-Birds dataset. They make use of an independent test set for benchmarking the model performances.Sri Lankan Beetles The arboreal tiger beetle data61 consists of 380 images that were taken between August 2017 and September 2020 from 22 places in Sri Lanka, including all climatic zones and provinces, as well as 14 districts. Tricondyla (3 species), Derocrania (5 species), and Neocollyris (1 species) were among the nine species discovered, with six of them being endemic . The current SOTA61 makes use of CNN-based SqueezeNet architecture and was trained using pre-trained weights of ImageNet. The benchmarking of the model performances was done on an independent test set.Florida Wild Traps The wildlife camera trap62 classification dataset comprises 104,495 images with visually similar species, varied lighting conditions, skewed class distribution, and samples of endangered species, such as Florida panthers. These were collected from two locations in Southwestern Florida. These images are categorized in to 22 classes. The current SOTA62 makes use of CNN-based ResNet-50 architecture and the performance of the model was benchmarked on an independent test set.ModelsVision transformers (ViTs)31 are an adaptation to computer vision of the Transformers, which were originally developed for natural language processing30. Their distinguishing feature is that, instead of exploiting translational symmetry, as CNNs do, they have an attention mechanism which identifies the most relevant part of an image. ViTs have recently outperformed CNNs in image classification tasks where vast amounts of training data and processing resources are available30,63. However, for the vast majority of use cases and consumers, where data and/or computational resources are limiting, ViTs are essentially untrainable, even when the network architecture is defined and no architectural optimization is required. To settle this issue, Data-efficient Image Transformers (DeiTs) were proposed32. These are transformer models that are designed to be trained with much less data and with far less computing resources32. In DeiTs, the transformer architecture has been modified to allow native distillation64, in which a student neural network learns from the results of a teacher model. Here, a CNN is used as the teacher model, and the pure vision transformer is used as the student network. All the DeiT models we report on here are DeiT-Base models32. The ViTs are ViT-B16, ViT-B32, and ViT-L32 models31.ImplementationTo train our models, we used transfer learning65: we took a model that was already pre-trained on the ImageNet43 dataset, changed the last layers depending on the number of classes, and then fine-tuned the whole network with a very low learning rate. All the models were trained with two Nvidia GTX 2080Ti GPUs.DeiTs We used DeiT-Base32 architecture, using the Python package TIMM66, which includes many of the well-known deep learning architectures, along with their pre-trained weights computed from the ImageNet dataset43. We resized the input images to 224 x 224 pixels and then, to prevent the model from overfitting at the pixel level and help it generalize better, we employed typical image augmentations during training such as horizontal and vertical flips, rotations up to 180 degrees, small zoom up’s to 20%, a small Gaussian blur, and shearing up to 10%. To handle class imbalance, we used class reweighting, which reweights errors on each example by how present that class is in the dataset67. We used sklearn utilities68 to calculate the class weights which we employed during the training phase.The training phase started with a default pytorch69 initial conditions (Kaiming uniform initializer), an AdamW optimizer with cosine annealing70, with a base learning rate of (10^{-4}), and a weight decay value of 0.03, batch size of 32 and was supervised using cross-entropy loss. We trained with early stopping, interrupting training if the validation F1-score did not improve for 5 epochs. The learning rate was then dropped by a factor of 10. We iterated until the learning rate reached its final value of (10^{-6}). This procedure amounted to around 100 epochs in total, independent of the dataset. The training time varied depending on the size of the datasets. It ranged between 20min (SriLankan Beetles) to 9h (Florida Wildtrap). We used the same procedure for all the datasets: no extra time was needed for hyperparameter tuning.ViTs We implemented the ViT-B16, ViT-B32 and ViT-L32 models using the Python package vit-keras (https://github.com/faustomorales/vit-keras), which includes pre-trained weights computed from the ImageNet43 dataset and the Tensorflow library71.First, we resized input images to 128 × 128 and employed typical image augmentations during training such as horizontal and vertical flips, rotations up to 180 degrees, small zooms up to 20%, small Gaussian blur, and shearing up to 10%. To handle class imbalance, we calculated the class weights and use them during the training phase.Using transfer learning, we imported the pre-trained model and froze all of the layers to train the model. We removed the last layer, and in its place we added a dense layer with (n_c) outputs (being (n_c) the number of classes), was preceded and followed by a dropout layer. We used the Keras-tuner72 with Bayesian optimization search73 to determine the best set of hyperparameters, which included the dropout rate, learning-rate, and dense layer parameters (10 trials and 100 epochs). After that, the model with the best hyperparameters was trained with a default tensorflow71 initial condition (Glorot uniform initializer) for 150 epochs using early stopping, which involved halting the training if the validation loss did not decrease after 50 epochs and retaining the model parameters that had the lowest validation loss.CNNs CNNs included DenseNet38, MobileNet39, EfficientNet-B240, EfficientNet-B540, EfficientNet-B640, and EfficientNet-B740 architectures. We followed the training procedure described in Ref.22, and carried out the training in tensorflow.Ensemble learningWe adopted average ensembling, which takes the confidence vectors of different learners, and produces a prediction based on the average among the confidence vectors. With this procedure, all the individual models contribute equally to the final prediction, irrespective of their validation performance. Ensembling usually results in superior overall classification metrics and model robustness74,75.Given a set of n models, with prediction vectors (vec c_i~(i=1,ldots ,n)), these are typically aggregated through an arithmetic average. The components of the ensembled confidence vector (vec c_{AA}), related to each class (alpha ) are then$$begin{aligned} c_{AA,alpha } = frac{1}{n}sum _{i=1}^n c_{i,alpha },. end{aligned}$$
(2)
Another option is to use a geometric average,$$begin{aligned} c_{GA,alpha } = root n of {prod _{i=1}^n c_{i,alpha }},. end{aligned}$$
(3)
We can normalize the vector (vec c_g), but this is not relevant, since we are interested in its largest component, (displaystyle max _alpha (c_{GA,alpha })), and normalization affects all the components in the same way. As a matter of fact, also the nth root does not change the relative magnitude of the components, so instead of (vec c_{GA}) we can use a product rule: (displaystyle max _alpha (c_{GA,alpha })=max _alpha (c_{PROD,alpha })), with (displaystyle c_{PROD,alpha } = prod _{i=1}^n c_{i,alpha }).While these two kinds of averaging are equivalent in the case of two models and two classes, they are generally different in any other case33. For example, it can easily be seen that the geometric average penalizes more strongly the classes for which at least one learner has a very low confidence value, a property that was termed veto mechanism36 (note that, while in Ref.36 the term veto is used when the confidence value is exactly zero, here we use this term in a slightly looser way). More

Our results show that GoM BFT larval survey samples can provide the crucial mark events for eventual CKMR estimates of adult abundance. The adult parents marked by larval samples can be directly recaptured in the fishery many years later as POPs, and indirectly through their progeny in future samples of larvae, as evidenced by the two cross-cohort HSPs (XHSPs) recovered in this study, which imply that a parent survived and spawned in the GoM in consecutive years. As more cohorts are sampled in future, the growing number of XHSPs could be used to estimate average adult survival rates, in addition to helping with the estimation of adult abundance31, as is now done for southern blue tuna40.There is a modest level of sibship within our 2016 samples, and a high level (involving over half the samples) in 2017, but it turns out not to be high enough to cause serious problems for POP-based CKMR. High sibship per se does not lead to bias in CKMR by virtue of the statistical construction of the estimate, but it does increase variance, which can be summarized through a reduction in effective sample size. In a POP-based CKMR model, our effective sample size would be about 75% of nominal for the two years combined, or 66% of nominal for the targeted sampling of 2017. Since it is actually the product of adult and juvenile sample sizes which drives precision in CKMR14, one way to think about the 75% is that we will need about 33% more adult samples to achieve a given precision on abundance estimates than if we had somehow been able to collect the same number of “independent” juvenile samples (i.e. without oversampling siblings). That increase is appreciable but entirely achievable; for WBFT, it is logistically much easier to collect more feeding-ground adult samples than to collect more larvae, and at present there is no known practical way to collect large numbers of older, more dispersed, and thus more independent, juvenile western origin bluefin tuna (WBFT).This study was motivated by the concern that sibship might be a serious impediment to use of WBFT larvae for CKMR. High levels of sibship have been found in larval collections for other taxa despite a pelagic larval phase, suggesting that abiotic factors can impede random mixing of larvae after a spawning event41. Our larval samples were only a few days old (4–11) and thus had little time to disperse since fertilization; our concern beforehand was that each tow might sample the offspring of a very small number of adults (one spawning group in one night), and in 2017 that repeatedly towing the same water mass might simply be resampling the same “family”. In practice, though, the cumulative effect was limited. Samples were not dominated by progeny from just a few adults; the maximum DPG size (i.e., number of offspring from any one adult) was 5, which is under 2% of the larval sample size. There are several possible reasons for this finding. First, plankton sample tows are typically standardized to a ten-minute duration, covering on average about 0.3 nautical miles. Based on continuous plankton cameras42, each tow is likely to tow through multiple patches of zooplankton, and therefore potentially multiple patches of BFT larvae. Second, spawning aggregations of BFT may contain many adults. For example, on the spawning grounds near the Balearic Islands in the Mediterranean, purse seine fisheries target spawning fish and individual net sets routinely capture upwards of 500 mature individuals43. These numbers suggest that BFT spawner aggregations can be quite large, although the number of individuals that contribute gametes to a single spawning event may be lower. The results of this study pose intriguing scenarios for understanding BFT larval ecology and spawning behavior, which could be explored with larger sample sizes paired with data on oceanographic conditions, direct observation of spawning aggregations, and modeling to compare observed and predicted dispersal. The results of this study are based on just two years of sampling, and numerous practical and theoretical challenges remain to fully understand BFT reproduction in the GoM.Our sibship impact calculations assume use of an unmodified adult-size-based CKMR POP model, where each juvenile is compared to each adult taking into account the latter’s size (e.g.,14). That will give unbiased estimates, which we regard as essential in a CKMR model. However, for WBFT the estimates are not fully statistically efficient, in that some adults receive more statistical weight than others because they are marked more often (by having a large DPG), and thus variance might not be the lowest achievable. Modifying the model to fix that would be simple in a “cartoon” CKMR setting where all adults are identical (e.g., Fig. 1 of14), simply by first condensing each DPG to a single representative, then only using those representatives (rather than all the larvae) in POP comparisons. Each marked parent then receives the same weight, giving maximum efficiency. For the cartoon, this condensed-DPG model still gives an unbiased estimate of abundance, because each DPG has one parent of given sex, and the chance of any sampled cartoon adult of that sex being that parent is 1/N. The DPG-condensed effective sample size is simply half the number of distinct parents, which would be a little larger than the effective sample sizes for the unmodified model shown in Table 3; e.g., in 2017, 504/2 = 252 versus 209. However, no such straightforward improvement is available for an adult-size-based CKMR model such as is needed for WABFT. Using condensed DPGs directly would bias the juvenile sampling against larger more-fecund adults, whose DPGs will tend on average to be larger and thus to experience disproportionate condensation. Those adults would be marked less often by the DPG-condensed juveniles than the model assumes, violating the basic requirements for unbiased CKMR in14. A more sophisticated model might be able to combine unbiasedness with higher efficiency but, since the unmodified adult-size-based POP model that we expect to use is unbiased and only mildly inefficient (at worst 209/252 = 83% efficient, in 2017) there seems no particular need for extra complications at present. However, that may not hold true if we eventually move to a POP + XHSP model, where the impact on unmodified CKMR variance is worse (though there is still no bias, for the same reason as with POPs). Intuitively, the biggest impact that a DPG of size 5 can have in a POP model is to suddenly raise the number of POPs by 5 if its parent happens to be sampled; within a useful total of, say, 75 POPs, the influence is not that large. But if two DPGs both of size 5 in different cohorts happen to share a parent, then the total of XHSPs suddenly jumps by 25— likely a substantial proportion of total XHSPs. Supplementary Material B also includes effective sample size formulae for a simplified XHSP-only model, which demonstrate the increased impact of within-cohort sibship; for our WBFT samples, it turns out that the XHSP-effective size is slightly lower for the targeted 2017 samples (110) than for the 2016 samples (130), unlike the POP-only effective size. Dropping from a maximum theoretical effective sample size of 252 (half the number of DPGs) down to 110 would be rather inefficient and would increase the number of years of sampling required to yield a useful XHSP dataset. This motivates developing a modified POP + XHSP model that retains unbiasedness without sacrificing too much efficiency. In principle, that can be done by condensing each DPG but then conditioning its comparison probabilities on the DPG’s original size, in accordance with the framework in14. This is a topic for subsequent research, and the results will inform future sampling strategy decisions for WBFT.One potential difficulty for western BFT CKMR might occur if a substantial proportion of animals reaching maturity are the offspring of “Western” (in genetic terms) adults who persistently spawn in the western North Atlantic but outside the GoM. However, as long as the adults marked by GoM larvae are well mixed at the time of sampling with any western adults that do spawn outside of the GoM, the total POP-based population estimate of genetically-western BFT from CKMR will remain unbiased. Given evidence from tagging of widespread adult movements within the western North Atlantic2, good mixing in the sampled feeding grounds seems likely; so, even if successful non-GoM western BFT spawning really is commonplace, there should not be a problem with relying on GoM larvae for at least the POP component of CKMR14.Studies of fish early life history have long been considered to have great potential to provide novel insight into the unique population dynamics of fishes44,45,46. Sampling efforts aimed at estimating fish recruitment dynamics have spawned a diversity of larval survey programs. Examples of these long-term programs include the California Cooperative Oceanic Fisheries Investigations, International Council for the Exploration of the Sea (ICES) surveys in the North Atlantic and adjacent areas, Southeast Monitoring and Assessment Program (SEAMAP) in the GoM, Ecosystem Monitoring (EcoMon) in the Northeast U.S., and numerous others, many of which provide indices of larval abundance widely used in fisheries and ecosystem assessments. Yet, as a result of the inherent patchiness of larvae42, sampling variability, and highly variable density dependent mortality45, fisheries scientists have often struggled to determine how larval surveys relate to the adult fish populations. Inclusion of estimates of sibship among larvae collected in surveys could refine estimates of adult spawning stock biomass estimated from these surveys.The results of this study also represent products of decades of work and coordination in obtaining high-quality DNA from larval specimens. Key steps to successful genotyping of larvae include ensuring that larvae are preserved, sorted, and handled in 95% non-denatured ethanol. In addition, strict instrument cleaning protocols must be followed, and stomachs should be removed or avoided (this study used larval tails and, when possible, eyes to avoid cross contamination of prey contents, including possible congeners and other BFT individuals). Exposure to hot lamps during the sorting and dissection processes should also be minimized to ensure that DNA quality is sufficiently high for genotyping-by-sequencing. Although the tissues available for genetic analysis were limited by the needs of other experiments that required BFT tissues, otoliths, gut contents, and other information from the same larvae, we were able to successfully genotype most larvae greater than 6 mm SL and identify thousands of informative SNPs. The lower size limit of larvae could likely be decreased if whole specimens were available for genotyping, although the use of younger larvae could increase the incidence of sibship.In summary, while we observed both FSPs and HSPs in larval collections, with elevated sibship overall and with siblings being more prevalent within tows and in nearby tows, the level of sibship was sufficiently low that collections of GoM BFT larvae can still provide the critical genetic mark of parental genotypes required for CKMR. Our results demonstrate a crucial proof of concept and are the first step towards an operational CKMR modelling estimate of spawning stock abundance for western BFT. More

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Isotope labelling and phylogenetic analysis of a natural marine bacterioplankton population at seaMediterranean seawater was collected during August 2017 (station N1200, 32.45° N, 34.37 °E) from 11 depths by Niskin bottles and divided into triplicate 250 ml polycarbonate bottles. Two bottles from each depth were labelled with 1 mM sodium bicarbonate-13C and 1 mM ammonium-15N chloride (Sigma-Aldrich), and all three bottles (two labelled and one control) were incubated at the original depth and station at sea for 3.5 h around mid-day. The stable isotopes were chosen to enable direct comparison of C and N uptake in single cells, and the short incubation time was chosen to minimize isotope dilution and potential recycling and transfer of 13C and 15N between community members25. After incubation, bottles were brought back on board and the incubations were stopped by fixing with 2× electron-microscopy-grade glutaraldehyde (2.5% final concentration) and stored at 4 °C until sorting analysis. Cell sorting, NanoSIMS analyses and the calculation of uptake rates were performed as described in Roth-Rosenberg et al.26.DNA collection and extraction from seawaterSamples for DNA were collected on 0.22 µm Sterivex filters (Millipore). Excess water was removed using a syringe, 1 ml lysis buffer (40 mM EDTA, 50 mM Tris pH 8.3, and 0.75 M sucrose) was added and both ends of the filter were closed with parafilm. Samples were kept at −80 °C until extraction. DNA was extracted by using a semi-automated protocol including manual chemical cell lysis before automated steps using the QIAamp DNA Mini Protocol: DNA Purification from Blood or Body Fluids (Spin Protocol, starting from step 6, at the BioRap unit, Faculty of Medicine, Technion). The manual protocol began with thawing the samples, then the storage buffer was removed using a syringe and 170 µl lysis buffer added to the filters. Thirty microlitres of Lysozyme (20 mg ml−1) were added to the filters and incubated at 37 °C for 30 min. After incubation, 20 µl proteinase K and 200 µl buffer AL (from the Qiagen kit) were added to the tube for 1 h at 56 °C (with agitation). The supernatant was transferred to a new tube, and DNA was extracted using the QIAcube automated system. All DNA samples were eluted in 100 μl DNA-free distilled water.ITS PCR amplificationPCR amplification of the ITS was carried out with specific primers for Prochlorococcus CS1_16S_1247F (5′-ACACTGACGACATGGTTCTACACGTACTACAATGCTACGG) and Cs2_ITS_Ar (5′-TACGGTAGCAGAGACTTGGTCTGGACCTCACCCTTATCAGGG)21,22. The first PCR was performed in triplicate in a total volume of 25 μl containing 0.5 ng of template, 12.5 μl of MyTaq Red Mix (Bioline) and 0.5 μl of 10 μM of each primer. The amplification conditions comprised steps at 95 °C for 5 min, 28/25 (16 S/ITS) cycles at 95 °C for 30 s, 50 °C for 30 s and 72 °C for 1 min followed by one step of 5 min at 72 °C. All PCR products were validated on a 1% agarose gel, and triplicates were pooled. Subsequently, a second PCR amplification was performed to prepare libraries. These were pooled and after a quality control sequenced (2 × 250 paired-end reads) using an Illumina MiSeq sequencer. Library preparation and pooling were performed at the DNA Services facility, Research Resources Center, University of Illinois at Chicago. MiSeq sequencing was performed at the W.M. Keck Center for Comparative and Functional Genomics at the University of Illinois at Urbana-Champaign.ITS sequence processingPaired-end reads were analysed using the Dada2 pipeline46. The quality of the sequences per sample was examined using the Dada2 ‘plotQualityProfile’ command. Quality filtering was performed using the Dada2 ‘filterAndTrim’ command with parameters for quality filtering truncLen=c(290,260), maxN=0, maxEE=c(2,2), truncQ=2, rm.phix=TRUE, trimLeft=c(20,20). Following error estimation and dereplication, the Dada2 algorithm was used to correct sequences. Merging of the forward and reverse reads was done with minimum overlap of 4 bp. Detection and removal of suspected chimaeras was done with command ‘removeBimeraDenovo’. In total, 388,417 sequences in 484 amplicon sequence variants were counted. The amplicon sequence variants were aligned in MEGA6 (ref. 47), and the first ~295 nucleotides, corresponding to the 16S gene, were trimmed. The ITS sequences were then classified using BLASTn against a custom database of ITS sequences from cultured Prochlorococcus and Synechococcus strains as well as from uncultured HL and LL clades.Individual-based modelPlanktonIndividuals.jl (v0.1.9) was used to run the individual-based simulations48. Briefly, the cells fix inorganic carbon through photosynthesis and nitrogen, phosphorus and DOC from the water column into intracellular quotas and grow until division or grazing. Cell division is modelled as a probabilistic function of cell size. Grazing is represented by a quadratic probabilistic function of cell population. Cells consume nutrient resources, which are represented as Eulerian, density-based tracers. A full documentation of state variables and model equations are available online at https://juliaocean.github.io/PlanktonIndividuals.jl/dev/. Equations related to mixotrophy are shown below as an addition to the online documentation.$$V_{{mathrm{DOC}}} = V_{{mathrm{DOC}}}^{{mathrm{max}}} cdot {{mathrm{max}}}left( {0.0,{{mathrm{min}}}left( {1.0,,frac{{q_{mathrm{C}}^{{mathrm{max}}} – q_{mathrm{C}}}}{{q_{mathrm{C}}^{{mathrm{max}}} – q_{mathrm{C}}^{{mathrm{min}}}}}} right)} right) cdot frac{{{mathrm{DOC}}}}{{{mathrm{DOC}} + K_{{mathrm{DOC}}}^{{mathrm{sat}}}}}$$
(1)
$$f_{{mathrm{PS}}} = frac{{P_{mathrm{S}}}}{{P_{mathrm{S}} + V_{{mathrm{DOC}}}}}$$
(2)
$$V_{{mathrm{DOC}}} = 0,,{mathrm{if}},f_{{mathrm{PS}}} < f_{{mathrm{PS}}}^{{mathrm{min}}}$$ (3) where VDOC is the cell-specific DOC uptake rate (mol C cell−1 s−1), (V_{{mathrm{DOC}}}^{{mathrm{max}}}) is the maximum cell-specific DOC uptake rate (mol C cell−1 s−1), (q_{mathrm{C}}^{{mathrm{max}}}) is the maximum cell carbon quota (mol C cell−1), (q_{mathrm{C}}^{{mathrm{min}}}) is the minimum cell carbon quota (mol C cell−1). The maximum and minimum functions here is used to keep qC between (q_{mathrm{C}}^{{mathrm{min}}}) and (q_{mathrm{C}}^{{mathrm{max}}}). (K_{{mathrm{DOC}}}^{{mathrm{sat}}}) is the half-saturation constant for DOC uptake (mol C m−3). fPS is the fraction of fixed C originating from photosynthesis (PS, mol C cell−1 s−1). DOC uptake stops when fPS is smaller than (f_{{mathrm{PS}}}^{{mathrm{min}}})(minimum fraction of fixed C originating form photosynthesis, 1% by default) according to laboratory studies of Prochlorococcus that showed that they cannot survive long exposure to darkness (beyond several days) even when supplied with organic carbon sources13. (1 − fPS) is also shown in Fig. 3 as the contribution of DOC uptake.We set up two separate simulations; each of them has a population of either an obligate photo-autotroph or a mixotroph that also consumes DOC. The initial conditions and parameters (Supplementary Table 3) are the same for the two simulations except the ability of mixotrophy. The simulations were run with a timestep of 1 min for 360 simulated days to achieve a steady state. We run the two simulations for multiple times in order to get the range of the stochastic processes.Evaluation of autotrophic growth ratesWe evaluated the carbon-specific, daily-averaged carbon fixation rate, ℙ as a function of light intensity (I, µE), following Platt et al.33:$${Bbb P} = frac{1}{{Delta t}}{int}_0^{Delta t} {frac{{q_{{mathrm{Chl}}}}}{{q_{mathrm{C}}}}} P_{mathrm{S}}^{{mathrm{Chl}}}left( {1 - e^{ - alpha _{{mathrm{Chl}}}I/P_{mathrm{S}}^{{mathrm{Chl}}}}} right)e^{ - beta _{{mathrm{Chl}}}I/P_{mathrm{S}}^{{mathrm{Chl}}}}Delta t$$ (4) Here, (P_{mathrm{S}}^{{mathrm{Chl}}}), αChl and βChl are empirically determined coefficients representing the chlorophyll-a-specific carbon fixation rate (mol C (mol Chl)−1 s−1), the initial slope of the photosynthesis–light relationship and photo-inhibition effects at high photon fluxes, respectively. We impose empirically determined values for (P_{mathrm{S}}^{{mathrm{Chl}}}), αChl and βChl from the published study of Moore and Chisholm24. The natural Prochlorococcus community comprises HL and LL ecotypes, which have different values of (P_{mathrm{S}}^{{mathrm{Chl}}}), αChl and βChl, and the community growth rate is expected to be between that of HL extremes and LL extremes. Therefore, we use photo-physiological parameters for an HL-adapted ecotype (MIT9215), acclimated at 70 µmol photons m−2 s−1 and an LL-adapted ecotype (MIT9211), acclimated 9 µmol photons m−2 s−1. The models with these values are shown as the different lines in Fig. 2b,d. I is the hourly PAR, estimated by scaling the observed noon value at each depth with a diurnal variation evaluated from astronomical formulae based on geographic location and time of year37,38.(frac{{q_{{mathrm{Chl}}}}}{{q_{mathrm{C}}}}) is the molar chlorophyll-a to carbon ratio, which is modelled as a function of growth rate and light intensity using the Inomura34 model (equation 17 therein) where parameters were calibrated with laboratory data from Healey49. In addition, the maximum growth rate ((mu _{{mathrm{max}}}^I)) based on macromolecular allocation is also estimated using the Inomura model (equation 30 therein). An initial guess of the growth rate and the empirically informed light intensity are used to estimate (frac{{q_{{mathrm{Chl}}}}}{{q_{mathrm{C}}}}), which is then used to evaluate the light-limited, photoautotrophic growth rate$${Bbb V}_{mathrm{C}}^{{mathrm{auto}}} = min left( {{Bbb P} - K_{mathrm{R}},mu _{{mathrm{max}}}^I} right)$$ (5) from which the (frac{{q_{{mathrm{Chl}}}}}{{q_{mathrm{C}}}}) is again updated. The light-limited growth rate is used to re-evaluate the (frac{{q_{{mathrm{Chl}}}}}{{q_{mathrm{C}}}}). Repeating this sequence until the values converge, ({Bbb V}_{mathrm{C}}^{{mathrm{auto}}}) and (frac{{q_{{mathrm{Chl}}}}}{{q_{mathrm{C}}}}) are solved iteratively.The nitrogen-specific uptake rate of fixed nitrogen (day−1) is modelled as$${Bbb V}_{{{mathrm{N}}}} = {Bbb V}_{mathrm{N}}^{{mathrm{max}}}frac{1}{{Q_{mathrm{N}}}}frac{N}{{N + K_{{{mathrm{N}}}}}}$$ (6) where values of the maximum uptake rate, ({Bbb V}_{mathrm{N}}^{{mathrm{max}}}), and half-saturation, KN, are determined from empirical allometric scalings35, along with a nitrogen cell quota QN from Bertilsson et al.39.The P-limited growth rate, or the phosphorus-specific uptake rate of phosphate (day−1), is modelled as$${Bbb V}_{mathrm{P}} = {Bbb V}_{mathrm{P}}^{{mathrm{max}}}frac{1}{{Q_{mathrm{P}}}}frac{{{mathrm{PO}_{4}}^{3 - }}}{{{mathrm{PO}_{4}}^{3 - } + K_{mathrm{P}}}}$$ (7) where values of the maximum uptake rate, ({Bbb V}_{mathrm{P}}^{{mathrm{max}}}). and half-saturation, KP, are determined from empirical allometric scalings35, along with a nitrogen cell quota QP from Bertilsson et al.39.Iron uptake is modelled as a linear function of cell surface area (SA), with rate constant ((k_{{mathrm{Fe}}}^{{mathrm{SA}}})) following Lis et al.36.$${Bbb V}_{{mathrm{Fe}}} = k_{{mathrm{Fe}}}^{{mathrm{SA}}} cdot {mathrm{SA}}frac{1}{{Q_{{mathrm{Fe}}}}}{mathrm{Fe}}$$ (8) The potential light-, nitrogen-, phosphorus- and iron-limited growth rates (({Bbb V}_{mathrm{C}},{Bbb V}_{mathrm{N}},{Bbb V}_{mathrm{P}},{Bbb V}_{{mathrm{Fe}}})) were evaluated at each depth in the water column and the minimum is the local modelled photo-autotrophic growth rate estimate, assuming no mixotrophy (Fig. 2b,d, blue lines). Parameters used in this evaluation are listed in Supplementary Table 2.An important premise of this study is that heterotrophy is providing for the shortfall in carbon under very low light conditions, but not nitrogen. It is known that Prochlorococcus can assimilate amino acids9 and therefore the stoichiometry of the heterotrophic contribution might alter the interpretations. However, it is also known that Prochlorococcus can exude amino acids40, which might cancel out the effects on the stoichiometry of Prochlorococcus.For the estimates of phototrophic growth rate from local environmental conditions (Fig. 2) we employed photo-physiological parameters from laboratory cultures of Prochlorococcus24. For the purposes of this study, we have assumed that the photosynthetic rates predicted are net primary production, which means that autotrophic respiration has been accounted for in the measurement. However, the incubations in that study were of relatively short timescale (45 min), which might suggest they are perhaps more representative of gross primary production. If this is the case, our estimates of photo-autotrophic would be even lower after accounting for autotrophic respiration, and thus would demand a higher contribution from heterotrophic carbon uptake. In this regard, our estimates might be considered a lower bound for organic carbon assimilation.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

All the results were reported relative to the control, unless specifically stated to the contrary or for clarity.Growth of cucumber plants in response to different biofertilizersSoilThere was no significant difference in cucumber growth before microbial fertilizer was applied. However, some microbial fertilizers significantly increased cucumber height and stem diameter when they were applied within 4 weeks from when the seedlings were planted (Fig. 1a,b,e,f). In the second week, SHZ and SMF increased plant height by 11.2 and 9.5%, respectively. In the third week, S267, SBS, SBH, SM and SHZ increased plant height by 12.0, 13.8, 15.0, 20.5 and 26.9%, respectively (Fig. 1a). In the fourth and fifth weeks, some treatments significantly increased cucumber height. In the second and third weeks, S267 significantly increased stem diameter by 21.2 and 16.8% (Fig. 1b).Figure 1Effect of different biofertilizer treatments on the growth of cucumber seedlings produced in soil or substrate in a greenhouse. S267 = Trichoderma Strain 267 added to soil; SBH = Bacillus subtilis and T. harzianum biofertilizers added to soil; SBS = B. subtilis biofertilizer added to the soil; SM = Compound biofertilizer added to soil; SHZ = T. harzianum biofertilizer added to soil; SCK = Untreated soil. US267 = T.267 biofertilizer added to substrate; USBH = B. subtilis and T. harzianum biofertilizers added to substrate; USBS = B. subtilis biofertilizer added to substrate; USM = Compound biofertilizer added to substrate; USHZ = T. harzianum biofertilizer added to substrate; USCK = Untreated substrate.Full size imageOver the subsequent 5 weeks, some microbial fertilizer treatments decreased cucumber height and stem diameter (Fig. 1g,h).SubstrateThere were no significant differences in cucumber growth before microbial fertilizer microbial fertilizer was applied (Fig. 1c,d,g,h). However, within 4 weeks of applying the microbial fertilizer, each biofertilizer treatment applied significantly increased cucumber height (Fig. 1c). US267 and USHZ significantly increased cucumber height by 39.8–75.4% and 56.1–86.1%, respectively. US267, USM and USHZ significantly increased the stem diameter by 76.8–108.9%, 71.1–97.6% and 80.4–122.4%, respectively (Fig. 1d).Over the subsequent 5 weeks, US267, USM and USHZ treatments continued to significantly increase cucumber height and stem diameter (Fig. 1g,h).Changes in the taxonomic composition of soil-borne fungal pathogensSoilBiofertilizers application significantly reduced the taxonomic composition of soil-borne fungal pathogens at different times during the cucumber growth period (Tables 1 and 2). Fusarium spp. were significantly reduced (T, 63.8% reduction, P  More