Philippa Kaur

EDITORIAL
14 July 2020

The pandemic has set back efforts to achieve the original 2015 targets. The need for change to make them more attainable is stronger than ever.

Lockdown home schooling in Mendoza province, Argentina. These children’s teachers drive 400 kilometres every month to deliver books and food to their students.Credit: Andres Larrovere/AFP/Getty

The United Nations has confirmed an unwelcome suspicion: the coronavirus pandemic has put the Sustainable Development Goals (SDGs) out of reach. Most of the goals to end poverty, protect the environment and support well-being by 2030 were already off course. Now, what little progress had been made has been stopped in its tracks.
This week, as government representatives join a virtual UN meeting to decide how best to achieve the goals, it cannot be business as usual for the 2030 agenda. Researchers both outside and inside the UN are questioning whether the goals are fit for the post-pandemic age. The goals’ ambition is as important as ever, but fresh thinking is needed on the best ways to achieve them.
Of the 17 SDGs, just 2 — eliminating preventable deaths among newborns and under-fives, and getting children into primary schools — were close to being achieved pre-pandemic. But COVID-19 has turned back the clock. The UN’s 2020 report on the SDGs reveals that childhood vaccination programmes have stalled in 70 countries, and that school closures have kept 90% of the world’s students — some 1.57 billion children — out of school.
The rise in domestic abuse brought about by lockdown measures has put paid to progress in the goal for gender equality and women’s empowerment. Many women have been unable to access sexual- and reproductive-health services, which could result in as many as 2.7 million extra unsafe abortions being carried out, according to Clare Wenham, a health-policy researcher at the London School of Economics, and her colleagues (C. Wenham et al. Nature 583, 194–198; 2020).

At the same time, at least 270 million people face hunger, and the World Food Programme is preparing its biggest humanitarian response in history. More than 70 million people will be forced into extreme poverty this year — potentially wiping out recent gains. That’s in addition to the more than 750 million who were already living on less than US$1.90 a day.
All in all, the goals to eliminate poverty, hunger and inequality, and to promote health, well-being and economic growth are headed for extinction. In many instances, countries will be unable to even record what is happening: according to a survey of 122 national statistics offices by the UN and the World Bank, 96% of such offices have fully or partially stopped face-to-face data collection.
What, then, needs to be done? Even before the pandemic, ideas were being floated to find ways to make the goals more achievable. Under one proposal from a group of UN science advisers, the 17 SDGs and 169 associated targets would be redistributed into 6 “entry points”. These would be human well-being (which would include eliminating poverty and improving health and education); sustainable economies; access to food and nutrition; access to, and decarbonization of, energy; urban development; and the global environmental commons (combining biodiversity and climate change).
A related proposal, but from a different group of advisers, the Sustainable Development Solutions Network (SDSN), also redistributes the 17 goals into 6, which it calls “transformations”. These are: education, gender and inequality; health, well-being and demography; energy decarbonization and sustainable industry; sustainable food, land, water and oceans; sustainable cities and communities; and digital revolution for sustainable development.

In both cases, however, countries would still be required to meet the actual SDGs — and their targets. Guido Schmidt-Traub, the SDSN’s executive director, told Nature the SDGs should still guide post-COVID-19 recovery. “There is nothing else to replace the SDGs right now.”
But such a measure is no longer realistic, according to Robin Naidoo, a lead scientist at the conservation group WWF-US in Washington DC, and Brendan Fisher, an environmental scientist at the University of Vermont in Burlington. Last week, they described how COVID-19 has irreparably altered at least some of the SDGs’ underpinning assumptions (R. Naidoo and B. Fisher Nature 583, 198–201; 2020).
When the goals were set, in 2015, the picture was one of rising economic growth and positive international cooperation — which led to the Paris climate agreement — both essential to meeting many of the SDGs’ targets. Now that the world is reeling from coronavirus and is on the brink of a once-in-a-century depression, governments are cooperating much less; crucial international meetings on protecting the climate, biodiversity and wetlands have been postponed; and aid to help the poorest countries meet their goals is set to fall.
Separate goals from growth
A more radical overhaul of the SDGs is also being advocated from inside the UN. In an excoriating report, Philip Alston, until recently the organization’s special rapporteur on extreme poverty and human rights, urged the SDGs’ supporters to acknowledge that things have changed. “The official response to date has been that ‘the 2030 Agenda must be preserved, and the SDGs must be reached’,” he wrote. “But doubling down on an inadequate and increasingly out-of-date approach is especially problematic.” He’s right.
One priority — as Alston, Naidoo and Fisher, among many others, advocate — is to decouple the SDGs from economic-growth targets. It is not just that growth is unachievable — at least for the foreseeable future — but that there is evidence to show that its benefits have not been equitably shared, and that it assigns value to undesirable things — what Naidoo and Fisher call “dangerous jobs, traffic jams and pollution”. Because of the way growth is measured, an increase in any of these three translates to positive growth figures. This creates perverse incentives for policymakers to put cars on the roads or invest further in fossil fuels.
Yet without growth, where will funding be found to achieve the many transformations needed? Naidoo and Fisher respond by pointing to one eye-watering figure: in 2015, government subsidies to the fossil-fuel industry came to $4.7 trillion, a figure that probably now exceeds $5 trillion. Each year, citizens are paying the equivalent of the gross domestic product of Japan to prop up an industry that is among the principal causes of climate change and unsustainable development. This money should be spent on achieving the goals, not undermining them.
Recalibrating the SDGs — especially in the current climate — won’t be easy. But the evidence that there is a need for a changed approach is accumulating. If the pandemic has shown us anything, it’s that countries can drastically change the way they think and act. The pandemic is radically altering economic and social realities. It shows that radical action can be taken to tackle poverty and inequality, health, education, biodiversity and climate.
When country representatives and the UN’s science-advice teams wrap up their meeting this week, they must heed their own poverty adviser and “avoid sleepwalking towards assured failure, while pumping out endless bland reports”.

Nature 583, 331-332 (2020)
doi: 10.1038/d41586-020-02002-3

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Notation
The notations described in ANCOM-BC methodology are summarized in Table 1.
Table 1 Summary of notations.
Full size table

Data preprocessing
We adopted the methodology of ANCOM-II22 as the preprocessing step to deal with different types of zeros before performing DA analysis.
There are instances where some taxa are systematically absent in an ecosystem. For example, there may be taxa present in a soil sample from a desert that might absent in a soil sample from a rain forest. In such cases, the observed zeros are called structural zeros. Let pij denote proportion non-zero samples of the ith taxon in the jth group, and let ({hat{p}}_{ij}=frac{1}{{n}_{j}}mathop{sum }nolimits_{k = 1}^{{n}_{j}}I({O}_{ijk}ne 0)) denote the estimate of pij. In practice, we declare the ith taxon to have structural zeros in the jth group if either of the following is true:
(a)
({hat{p}}_{ij}=0.)

(b)
({hat{p}}_{ij}-1.96scriptstyle{sqrt{frac{{hat{p}}_{ij}(1-{hat{p}}_{ij})}{{n}_{j}}}}le 0).

If a taxon is considered to be a structural zero in an experimental group then, for that specific ecosystem, the taxon is not used in further analysis. Thus, suppose there are three ecosystems A, B, and C and suppose taxon X is a structural zero in ecosystems A and B but not in C, then taxon X is declared to be differentially abundant in C relative to A and B and not analyzed further. If taxon Y is a structurally zero in ecosystem A but not in B and C, in that case we declare that taxon Y is differentially abundant in B relative to A as well as differentially abundant in C relative to A. We then compare the absolute abundance of taxon Y between B and C using the methodology described in this section. Taxa identified to be structural zeros among all experimental groups are ignored from the following analyses.
In a similar fashion, we address the outlier zeros as well as sampling zeros using the methodology developed in ANCOM-II22.
Model assumptions
Assumption 0.1.

$$E({O}_{ijk}| {A}_{ijk})={c}_{jk}{A}_{ijk}\ Var({O}_{ijk}| {A}_{ijk})={sigma }_{w,ijk}^{2},$$
(1)

where ({sigma }_{w,ijk}^{2}) = variability between specimens within the kth sample from the jth group. Therefore, ({sigma }_{w,ijk}^{2}) characterizes the within-sample variability. Typically, researchers do not obtain more than one specimen at a given time in most microbiome studies. Consequently, variability between specimens within sample is usually not estimated.
According to Assumption 0.1, in expectation the absolute abundance of a taxon in a random sample is in constant proportion to the absolute abundance in the ecosystem of the sample. In other words, the expected relative abundance of each taxon in a random sample is equal to the relative abundance of the taxon in the ecosystem of the sample.
Assumption 0.2. For each taxon i, Aijk, j = 1, …, g, k = 1, …, nj, are independently distributed with

$$E({A}_{ijk}| {theta }_{ij}) ={theta }_{ij}\ Var({A}_{ijk}| {theta }_{ij}) ={sigma }_{b,ij}^{2},$$
(2)

where ({sigma }_{b,ij}^{2}) = between-sample variation within group j for the ith taxon.
The Assumption 0.2 states that for a given taxon, all subjects within and between groups are independent, where θij is a fixed parameter rather than a random variable.
Regression framework
From Assumptions 0.1 and 0.2, we have:

$$E({O}_{ijk}) ={c}_{jk}{theta }_{ij}\ Var({O}_{ijk}) =f({sigma }_{w,ijk}^{2},{sigma }_{b,ij}^{2}):= {sigma }_{t,ijk}^{2}.$$
(3)

Motivated by the above set-up, we introduce the following linear model framework for log-transformed OTU counts data:

$${y}_{ijk}={d}_{jk}+{mu }_{ij}+{epsilon }_{ijk},$$
(4)

with

$$E({epsilon }_{ijk}) =0,\ E({y}_{ijk}) ={d}_{jk}+{mu }_{ij},\ Var({y}_{ijk}) =Var({epsilon }_{ijk}):= {sigma }_{ijk}^{2}.$$
(5)

Note that with a slight abuse of notation for simplicity of exposition, the above log-transformation of data is inspired by the Box–Cox family of transformations23 which are routinely used in data analysis. Note that d in the above equation is not exactly log(c) due to Jensenʼs inequality, it simply reflects the effect of c
An important distinction from standard ANOVA: Before we describe the details of the proposed methodology, we like to draw attention to a fundamental difference between the current formulation of the problem and the standard one-way ANOVA model. For simplicity, let us suppose we have two groups, say a treatment and a control group. Let us also suppose that there is only one taxon in our microbiome study and n subjects are assigned to the treatment group and n are assigned to the control group. Suppose the researcher is interested in comparing the mean absolute abundance of the taxon in the ecosystems of the two groups. Then under the above assumptions, the model describing the study is given by:

$${y}_{jk}={d}_{jk}+{mu }_{j}+{epsilon }_{jk},j=1,2,k=1,2,ldots ,n.$$

Then trivially the sample mean absolute abundance of jth group is given by ({bar{y}}_{j.}=frac{1}{n}mathop{sum }nolimits_{k = 1}^{n}{y}_{jk}) and (E({bar{y}}_{j.})=frac{1}{n}mathop{sum }nolimits_{k = 1}^{n}{d}_{jk}+{mu }_{j}={bar{d}}_{j.}+{mu }_{j}). The difference in the sample means between the two groups is ({bar{y}}_{1.}-{bar{y}}_{2.}) and its expectation is (E({bar{y}}_{1.}-{bar{y}}_{2.})=({bar{d}}_{1.}-{bar{d}}_{2.})+({mu }_{1}-{mu }_{2})). Under the null hypothesis μ1 = μ2, (E({bar{y}}_{1.}-{bar{y}}_{2.})={bar{d}}_{1.}-{bar{d}}_{2.}ne ,0), unless ({bar{d}}_{1.}={bar{d}}_{2.}). Thus because of the differential sampling fractions, which are sample specific, the numerator of the standard t-test under the null hypothesis for these microbiome data is non-zero. This introduces bias and hence inflates the Type I error. On the other hand, the standard one-way ANOVA model for two groups, which is not applicable for the microbiome data described in this paper, is of the form:

$${y}_{jk}=d+{mu }_{j}+{epsilon }_{jk},j=1,2,k=1,2,ldots ,n.$$

Hence under the null hypothesis μ1 = μ2, (E({bar{y}}_{1.}-{bar{y}}_{2.})=0). Thus, in this case the standard t-test is appropriate. Hence in this paper we develop methodology to eliminate the bias introduced by the differential sampling fraction by each sample. To do so, we exploit the fact that we have a large number of taxa on each subject and we borrow information across taxa to estimate this bias, which is the essence of the following methodology.
Bias and variance of bias estimation under the null hypothesis: From the above model (equation (4)), for each j, note that (E({bar{y}}_{ijcdot })={bar{d}}_{jcdot }+{mu }_{ij}) and (E({bar{y}}_{cdot jk})={d}_{jk}+{bar{mu }}_{cdot j}), where (bar{w}) represents the arithmetic mean over the suitable index. Using the least squares framework, we therefore estimate μij and djk as follows:

$${hat{d}}_{jk} ={overline{y}}_{cdot jk}-{overline{y}}_{cdot jcdot },,k=1,,ldots ,,{n}_{j},j=1,2,ldots g,\ {hat{mu }}_{ij} ={overline{y}}_{ijcdot }-{overline{hat{d}}}_{jcdot }={overline{y}}_{ijcdot },,i=1,,ldots ,,m.$$
(6)

Note that (E({hat{mu }}_{ij})=E({overline{y}}_{ijcdot })={mu }_{ij}+{overline{d}}_{jcdot }). Thus, for each j = 1, 2, …g, ({hat{mu }}_{ij}) is a biased estimator and (E({hat{mu }}_{i1}-{hat{mu }}_{i2})=({mu }_{i1}-{mu }_{i2})+{overline{d}}_{1cdot }-{overline{d}}_{2cdot }). Denote (delta ={overline{d}}_{1cdot }-{overline{d}}_{2cdot }.) To begin with, in the following we shall assume there are two experimental groups with balanced design, i.e., g = 2 and n1 = n2 = n. Later the methodology is easily extended to unbalanced design and multigroup settings. Suppose we have two ecosystems and for each taxon i, i = 1, 2, …m, we wish to test the hypothesis

$${H}_{0} :{mu }_{i1}={mu }_{i2}\ {H}_{1} :{mu }_{i1}ne {mu }_{i2}.$$
(7)

Under the null hypothesis, (E({hat{mu }}_{i1}-{hat{mu }}_{i2})=delta , ne , 0), and hence biased. The goal of ANCOM-BC is to estimate this bias and accordingly modify the estimator ({hat{mu }}_{i1}-{hat{mu }}_{i2}) so that the resulting estimator is asymptotically centered at zero under the null hypothesis and hence the test statistic is asymptotically centered at zero. First, we make the following observations. Since (E({overline{y}}_{ijcdot })={overline{d}}_{jcdot }+{mu }_{ij}) and ({hat{mu }}_{ij}={overline{y}}_{ijcdot }), therefore ({hat{mu }}_{ij}) is an unbiased estimator of ({overline{d}}_{jcdot }+{mu }_{ij}). From (5) and Lyapunov central limit theorem, we have:

$$frac{{hat{mu }}_{ij}-({mu }_{ij}+{overline{d}}_{jcdot })}{{sigma }_{i}}{to }_{d}N(0,1),,,{rm{as}} nto infty ,\ {rm{where}} {sigma }_{ij}^{2}=Var({hat{mu }}_{ij})=Var({overline{y}}_{ij.})=frac{1}{{n}^{2}}mathop{sum }limits_{k=1}^{n}{sigma }_{ijk}^{2}.$$
(8)

Let Σjk denote an m × m covariance matrix of ({epsilon }_{jk}={({epsilon }_{1jk},{epsilon }_{2jk},ldots ,{epsilon }_{mjk})}^{T}), where ({sigma }_{ii^{prime} jk}) is the ({(i,i^{prime} )}^{th}) element of Σjk and ({sigma }_{ijk}^{2}) is the ith diagonal element of Σjk. Furthermore, suppose
Assumption 0.3.

$${sigma }_{ijk}^{2} νi0. Thus, without loss of generality for κ1, κ2  > 0, let νi1 = νi0 + κ1 and νi2 = νi0 + κ2. While this assumption is not a requirement for our method, it is reasonable to assume that variability among differentially abundant taxa is larger than that among the null taxa. By making this assumption, we speed-up the computation time.
Assuming samples are independent between experimental groups, we begin by first estimating ({nu }_{i0}^{2}={rm{V}}{rm{ar}}({hat{mu }}_{i1}-{hat{mu }}_{i2})={rm{V}}{rm{ar}}({hat{mu }}_{i1})+{rm{V}}{rm{ar}}({hat{mu }}_{i2})). Using the estimator stated in (14), we estimate ({nu }_{i0}^{2}) consistently as follows:

$${hat{nu }}_{i0}^{2}=mathop{sum }limits_{j=1}^{2}{hat{sigma }}_{ij}^{2}=mathop{sum }limits_{j=1}^{2}frac{1}{{n}^{2}}mathop{sum }limits_{k=1}^{n}{({y}_{ijk}-{hat{d}}_{jk}-{hat{mu }}_{ij})}^{2}.$$
(20)

In all future calculations, we plug in ({hat{nu }}_{i0}^{2}) for ({nu }_{i0}^{2}). This is similar in spirit to many statistical procedures involving nuisance parameters. The following lemma is useful in the sequel.
Lemma 0.1.
(frac{partial }{partial theta }{mathrm{log}},f(x)={E}_{f(z| x)}[frac{partial }{partial theta }{mathrm{log}},f(z)+frac{partial }{partial theta }{mathrm{log}},f(x| z)]).25
Let (Theta ={(delta ,{pi }_{1},{pi }_{2},{pi }_{3},{l}_{1},{l}_{2},{kappa }_{1},{kappa }_{2})}^{T}) denote the set of unknown parameters, then for each taxon the log-likelihood can be reformulated using Lemma 0.1, as follows:

$$Theta leftarrow arg {max }_{Theta }mathop{sum }limits_{i=1}^{m}mathop{sum }limits_{r=0}^{2}{p}_{r,i}[{mathrm{log}},{rm{P}}{rm{r}}(iin {C}_{r})+{mathrm{log}},f({Delta }_{i}| iin {C}_{r})].$$
(21)

Then the E–M algorithm is described as follows:

E-step: Compute conditional probabilities of the latent variable. Define ({p}_{r,i}={rm{P}}{rm{r}}(iin {C}_{r}| {Delta }_{i})=frac{{pi }_{r}phi (frac{{Delta }_{i}, -, (delta , +, {l}_{r})}{{nu }_{ir}})}{{sum }_{r}{pi }_{r}phi (frac{{Delta }_{i}, -, (delta , + , {l}_{r})}{{nu }_{ir}})},r=0,1,2;i=1,ldots ,m), which are conditional probabilities representing the probability that an observed value follows each distribution. Note that l0 = 0.

M-step: Maximize the likelihood function with respect to the parameters, given the conditional probabilities.

We shall denote the resulting estimator of δ by ({hat{delta }}_{{rm{EM}}}.)
Next we estimate ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{EM}}})). Since the likelihood function is not a regular likelihood and hence it is not feasible to derive the Fisher information. Consequently, we take a simpler and a pragmatic approach to derive an approximate estimator of ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{EM}}})) using ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})), which is defined below. Extensive simulation studies suggest that ({hat{delta }}_{{rm{EM}}}) and ({hat{delta }}_{{rm{WLS}}}) are highly correlated (Supplementary Fig. 9) and it appears to be reasonable to approximate ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{EM}}})) by ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})).
Let {Cr} = mr, r = 0, 1, 2, then

$${hat{delta }}_{{rm{WLS}}} =frac{{sum }_{iin {C}_{0}}frac{{Delta }_{i}}{{hat{nu }}_{i0}^{2}}+{sum }_{iin {C}_{1}}frac{{Delta }_{i}-{hat{l}}_{1}}{{hat{nu }}_{i1}^{2}}+{sum }_{iin {C}_{2}}frac{{Delta }_{i}-{hat{l}}_{2}}{{hat{nu }}_{i2}^{2}}}{{sum }_{iin {C}_{0}}frac{1}{{hat{nu }}_{i0}^{2}}+{sum }_{iin {C}_{1}}frac{1}{{hat{nu }}_{i1}^{2}}+{sum }_{iin {C}_{2}}frac{1}{{hat{nu }}_{i2}^{2}}}\ =frac{{sum }_{iin {C}_{0}}frac{{Delta }_{i}}{{nu }_{i0}^{2}}+{sum }_{iin {C}_{1}}frac{{Delta }_{i}-{l}_{1}}{{nu }_{i1}^{2}}+{sum }_{iin {C}_{2}}frac{{Delta }_{i}-{l}_{2}}{{nu }_{i2}^{2}}}{{sum }_{iin {C}_{0}}frac{1}{{nu }_{i0}^{2}}+{sum }_{iin {C}_{1}}frac{1}{{nu }_{i1}^{2}}+{sum }_{iin {C}_{2}}frac{1}{{nu }_{i2}^{2}}}+{o}_{p}(1).$$
(22)

The above expression is of the form

$$frac{{a}_{1}^{T}{x}_{1}+{a}_{2}^{T}{x}_{2}+{a}_{3}^{T}{x}_{3}}{{a}_{1}^{T}{bf{1}}+{a}_{2}^{T}{bf{1}}+{a}_{3}^{T}{bf{1}}}equiv frac{{alpha }^{T}u}{{alpha }^{T}{bf{1}}},$$
(23)

where
(1)
1 = (1, …, 1)T,

(2)
({a}_{r}={({a}_{r1},{a}_{r2},ldots ,{a}_{r{m}_{r}})}^{T}:= {(frac{1}{{nu }_{ir}^{2}})}^{T},,iin {C}_{r},,r=0,1,2),

(3)
({x}_{r}={({x}_{r1},{x}_{r2},ldots ,{x}_{r{m}_{r}})}^{T}:= {({Delta }_{i}-{l}_{i})}^{T},,iin {C}_{r},,r=0,1,2). Note that l0 = 0,

(4)
(alpha ={({alpha }_{1},{alpha }_{2},ldots ,{alpha }_{m})}^{T}equiv {({a}_{1}^{T},{a}_{2}^{T},{a}_{3}^{T})}^{T}),

(5)
(u={({u}_{1},{u}_{2},ldots ,{u}_{m})}^{T}equiv {({x}_{1}^{T},{x}_{2}^{T},{x}_{3}^{T})}^{T}).

For the simplicity of notation, we relabel a and x by α and u, respectively. Denote Cov(x) = Cov(u) by Ω, and let ({omega }_{ii^{prime} }) denotes the ((i,i^{prime} )) element of Ω. As in Assumption 0.3, we make the following assumption
Assumption 0.4.

$$frac{mathop{sum }nolimits_{ine i^{prime} }^{m}{omega }_{ii^{prime} }}{{m}^{2}}=o(1).$$
(24)

Using the above expressions, we compute the variance as follows:

$${rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})={rm{V}}{rm{ar}}(frac{{alpha }^{T}u}{{alpha }^{T}{bf{1}}})=frac{mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i}^{2}{omega }_{ii}}{{(mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i})}^{2}}+frac{mathop{sum }nolimits_{ine i^{prime} }^{m}{alpha }_{i}{alpha }_{i^{prime} }{omega }_{ii^{prime} }}{{(mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i})}^{2}}.$$
(25)

Recall that (a) for i ∈ C0, ({omega }_{ii}={rm{V}}{rm{ar}}({Delta }_{i})={nu }_{i0}^{2}=O({n}^{-1})), (b) for i ∈ C1, ({omega }_{ii}={rm{V}}{rm{ar}}({Delta }_{i})={nu }_{i1}^{2}={nu }_{i0}^{2}+{kappa }_{1}=O(1)), and (c) for i ∈ C2, ({omega }_{ii}={rm{V}}{rm{ar}}({Delta }_{i})={nu }_{i2}^{2}={nu }_{i0}^{2}+{kappa }_{2}=O(1)). Note that ({alpha }_{i}=frac{1}{{rm{V}}{rm{ar}}({Delta }_{i})}=frac{1}{{omega }_{ii}}), thus we have:

$${rm{V}}{rm{ar}}(frac{{alpha }^{T}u}{{alpha }^{T}{bf{1}}})=frac{mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i}^{2}{omega }_{ii}}{{(mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i})}^{2}}+frac{mathop{sum }nolimits_{ine i^{prime} }^{m}{alpha }_{i}{alpha }_{i^{prime} }{omega }_{ii^{prime} }}{{(mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i})}^{2}}=frac{1}{mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i}}+frac{mathop{sum }nolimits_{ine i^{prime} }^{m}{alpha }_{i}{alpha }_{i^{prime} }{omega }_{ii^{prime} }}{{(mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i})}^{2}}.$$
(26)

Since ({nu }_{i0}^{2}=O({n}^{-1})), ({nu }_{i1}^{2}=O(1)), and ({nu }_{i2}^{2}=O(1)), consequently, a1i = O(n), a2i = a3i = O(1), and

$$mathop{sum }limits_{i=1}^{m}{alpha }_{i} ={{bf{1}}}^{T}{a}_{1}+{{bf{1}}}^{T}{a}_{2}+{{bf{1}}}^{T}{a}_{3} = sum _{iin {C}_{0}}O(n)+sum _{iin {C}_{1}}O(1)+sum _{iin {C}_{2}}O(1)\ = O({m}_{0}n)+O({m}_{1})+O({m}_{2})\ =O({m}_{0}n),,,{rm{if}}{m}_{0}nge max {{m}_{1},{m}_{2}}.$$
(27)

Using these facts and Assumption 0.4 in (26), we get

$${rm{V}}{rm{ar}}(frac{{alpha }^{T}u}{{alpha }^{T}{bf{1}}}) = O({m}_{0}^{-1}{n}^{-1})+frac{mathop{sum }nolimits_{ine i^{prime} }^{m}{{n}^{-1}{m}^{-1}{alpha }_{i}}{{n}^{-1}{m}^{-1}{alpha }_{i^{prime} }}{omega }_{ii^{prime} }}{{n}^{-2}{m}^{-2}{(mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i})}^{2}}\ = O({m}_{0}^{-1}{n}^{-1})+frac{1}{{m}^{2}}frac{mathop{sum }nolimits_{ine i^{prime} }^{m}{{n}^{-1}{alpha }_{i}}{{n}^{-1}{alpha }_{i^{prime} }}{omega }_{ii^{prime} }}{{(mathop{sum }nolimits_{i = 1}^{m}{n}^{-1}{m}^{-1}{alpha }_{i})}^{2}}\ = O({m}_{0}^{-1}{n}^{-1})+frac{1}{{m}^{2}}frac{O(1)o({m}^{2})}{O(1)}\ = O({m}_{0}^{-1}{n}^{-1}).$$
(28)

Thus, under Assumption 0.4 regarding ({omega }_{ii^{prime} }), the contribution of the covariance terms in the above variance expression is negligible as long as m is very large compared with n, which is usually the case. Hence

$${rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})={rm{V}}{rm{ar}}(frac{{alpha }^{T}u}{{alpha }^{T}{bf{1}}})=O({m}_{0}^{-1}{n}^{-1}).$$
(29)

Furthermore, appealing to Cauchy–Schwartz inequality we get

$${rm{Cov}}({hat{mu }}_{i1}-{hat{mu }}_{i2},{hat{delta }}_{{rm{WLS}}}) le sqrt{{rm{V}}{rm{ar}}({hat{mu }}_{i1}-{hat{mu }}_{i2}){rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})}\ le O({n}^{-1/2})O({m}_{0}^{-1/2}{n}^{-1/2})=O({n}^{-1}{m}_{0}^{-1/2}).$$
(30)

Hence, as long as m0 is large, the contribution made by ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})) and ({rm{Cov}}({hat{mu }}_{i1}-{hat{mu }}_{i2},{hat{delta }}_{{rm{WLS}}})) relative to ({rm{V}}{rm{ar}}({hat{mu }}_{i1}-{hat{mu }}_{i2})) is negligible.
Neglect the covariance term in (26), let ({hat{C}}_{r}) denote the estimator of Cr, r = 0, 1, 2 from the E–M algorithm, define

$$widehat{{rm{V}}{rm{ar}}}({hat{delta }}_{{rm{WLS}}})=frac{1}{{sum }_{iin {hat{C}}_{0}}frac{1}{{hat{nu }}_{i0}^{2}}+{sum }_{iin {hat{C}}_{1}}frac{1}{{hat{nu }}_{i1}^{2}}+{sum }_{iin {hat{C}}_{2}}frac{1}{{hat{nu }}_{i2}^{2}}},$$
(31)

an estimator of ({rm{V}}{rm{ar}}({hat{delta }}_{{rm{WLS}}})) under the Assumption 0.4. Then

$$widehat{{rm{V}}{rm{ar}}}({hat{delta }}_{{rm{WLS}}}){to }_{p}frac{1}{mathop{sum }nolimits_{i = 1}^{m}{alpha }_{i}}=frac{1}{{sum }_{iin {C}_{0}}frac{1}{{nu }_{i0}^{2}}+{sum }_{iin {C}_{1}}frac{1}{{nu }_{i1}^{2}}+{sum }_{iin {C}_{2}}frac{1}{{nu }_{i2}^{2}}},{rm{as}},, m,nto infty .$$
(32)

We performed extensive simulations to evaluate the bias and variance of ({hat{delta }}_{{rm{EM}}}) and that of ({hat{delta }}_{{rm{WLS}}}). The scatter plot (Supplementary Fig. 9) of ({hat{delta }}_{{rm{EM}}}) and ({hat{delta }}_{{rm{WLS}}}) are almost perfectly linear with correlation coefficient nearly 1 in all cases. This suggests that ({hat{delta }}_{{rm{WLS}}}) is a very good approximation for ({hat{delta }}_{{rm{EM}}}). The variance of ({hat{delta }}_{{rm{EM}}}) as well as that of ({hat{delta }}_{{rm{WLS}}}) are roughly of the order ({n}^{-1}{m}_{0}^{-1}), as we expected. In addition, they are approximately unbiased (Supplementary Table 6).
Hypothesis testing for two-group comparison
For taxon i, we test the following hypothesis

$${H}_{0} :{mu }_{i1}={mu }_{i2}\ {H}_{1} :{mu }_{i1},,ne ,,{mu }_{i2}$$

using the following test statistic which is approximately centered at zero under the null hypothesis:

$${W}_{i}=frac{{hat{mu }}_{i1}-{hat{mu }}_{i2}-{hat{delta }}_{{rm{EM}}}}{sqrt{{hat{sigma }}_{i1}^{2}+{hat{sigma }}_{i2}^{2}}}.$$
(33)

From Slutsky’s theorem, we have:

$${W}_{i}{to }_{d}N(0,1),{rm{as}},, m,,n,to infty .$$
(34)

If the sample size is not very large and/or the number of non-null taxa are very large, then we modify the above test statistic as follows:

$${W}_{i}^{* }=frac{{hat{mu }}_{i1}-{hat{mu }}_{i2}-{hat{delta }}_{{rm{WLS}}}}{sqrt{{hat{sigma }}_{i1}^{2}+{hat{sigma }}_{i2}^{2}+widehat{{rm{V}}{rm{ar}}}({hat{delta }}_{{rm{WLS}}})+2sqrt{({hat{sigma }}_{i1}^{2}+{hat{sigma }}_{i2}^{2})widehat{{rm{V}}{rm{ar}}}({hat{delta }}_{{rm{WLS}}})}}}.$$
(35)

To control the FDR due to multiple comparisons, we recommend applying the Holm–Bonferroni method26 or Bonferroni27,28 correction rather than the Benjamini–Hochberg (BH) procedure29 to adjust the raw p values as research has showed that it is more appropriate to control the FDR when p values were not accurate30, and the BH procedure controls the FDR provided you have either independence or some special correlation structures such as perhaps positive regression dependence among taxa29,31. In our simulation studies, since the absolute abundances for each taxon are generated independently, we compared the ANCOM-BC results adjusted either by Bonferroni correction (Fig. 4) or BH procedure (Supplementary Fig. 10), it is clearly that the FDR control by Bonferroni correction is more conservative while implementing BH procedure results in FDR around the nominal level (5%). Obviously, ANCOM-BC has larger power when using BH procedure.
Hypothesis testing for multigroup comparison
In some applications, for a given taxon, researchers are interested in drawing inferences regarding DA in more than two ecosystems. For example, for a given taxon, researchers may want to test whether there exists at least one experimental group that is significantly different from others, i.e., to test

$${H}_{0,i} :{cap }_{jne j^{prime} in {1,ldots ,g}}{mu }_{ij}={mu }_{ij^{prime} }\ {H}_{1,i} :{cup }_{jne j^{prime} in {1,ldots ,g}}{mu }_{ij},, ne ,, {mu }_{ij^{prime} }.$$

Similar to the two-group comparison, after getting the initial estimates of ({hat{mu }}_{ij}) and ({hat{d}}_{jk}), setting the reference group r (e.g., r = 1), and obtaining the estimator of the bias term ({hat{delta }}_{rj}) through E–M algorithm, the final estimator of mean absolute abundance of the ecosystem (in log scale) are obtained by transforming ({hat{mu }}_{ij}) of (6) into:

$${hat{mu }}_{ij}^{* }:= left{begin{array}{l}{}hskip -42pt {hat{mu }}_{ir},,,, qquad j=r\ {hat{mu }}_{ij}+{hat{delta }}_{rj},,,,j ,, ne ,, rin 1,ldots ,gend{array}right..$$
(36)

Thus, based on (18) and the E–M estimator of δrj, as (m,min ({n}_{j},{n}_{j^{prime} })to infty)

$${{hat{mu }}_{ij}^{* }-{hat{mu }}_{ij^{prime} }^{* }to }_{p}left{begin{array}{ll}hskip -30pt 0&{rm{if}} {rm{taxon}} i {rm{is}} {rm{not}} {rm{differentially}} {rm{abundant}} {rm{between}} {rm{group}} j {rm{and}} j^{prime} ,\ {mu }_{ij}-{mu }_{ij^{prime} }&hskip -219pt {rm{otherwise}}.end{array}right.$$
(37)

Similarly, the estimator of the sampling fraction is obtained by transforming ({hat{d}}_{jk}) of (6) into

$${hat{d}}_{jk}^{* }:= left{begin{array}{l}hskip -45pt {hat{d}}_{rk},,,,, qquad j=r\ {hat{d}}_{jk}-{hat{delta }}_{rj},,,,j ,, ne ,, rin 1,ldots ,gend{array}right..$$
(38)

As by (13) and the E–M estimator of δrj

$${hat{d}}_{jk}^{* }{to }_{p}{d}_{jk}-{overline{d}}_{rcdot }{rm{as}} m,,min ({n}_{j},{n}_{j^{prime} })to infty,$$
(39)

which indicates that we are only able to estimate sampling fractions up to an additive constant (({overline{d}}_{rcdot })).
Define the test statistic for pairwise comparison as:

$${W}_{i,jj^{prime} }=frac{{hat{mu }}_{ij}^{* }-{hat{mu }}_{ij^{prime} }^{* }}{sqrt{{hat{sigma }}_{ij}^{2}+{hat{sigma }}_{ij^{prime} }^{2}}},,,,i=1,,ldots ,,m,j ,, ne ,, j^{prime} in {1,,ldots ,,g}.$$
(40)

For computational simplicity, inspired by the William’s type of test32,33,34,35, we reformulate the global test with the following test statistic:

$${W}_{i}=mathop{max }limits_{jne j^{prime} in {1,ldots ,g}}| {W}_{i,jj^{prime} }| ,,,,i=1,,ldots ,,m.$$
(41)

Under null, ({W}_{i,jj^{prime} }{to }_{d}N(0,1)), thus we can construct the null distribution of Wi by simulations, i.e., for each specific taxon i,
(a)
Generate ({W}_{i,jj^{prime} }^{(b)} sim N(0,1),j, ne , j^{prime} in {1,,ldots ,,g},b=1,ldots ,B.)

(b)
Compute ({W}_{i}^{(b)}=mathop{max }limits_{jne j^{prime} in {1,ldots ,g}}{W}_{i,jj^{prime} }^{(b)}.)

(c)
Repeat above steps B times (e.g., B = 1000), we then get the null distribution of Wi by ({({W}_{i}^{(1)},ldots ,{W}_{i}^{(B)})}^{T}.)

Finally, p value is calculated as

$${p}_{i}=frac{1}{B}mathop{sum }limits_{b=1}^{B}I({W}_{i}^{(b)} > {W}_{i}),,,,i=1,,ldots ,,m$$
(42)

and the Bonferroni correction is applied to control the FDR.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. More

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Ethics statement
The work was carried under the ethical Permit Number VARELY/711/2016 issued by the regional environmental centre of Finland. Every possible effort was made to minimise disturbance.
Field system set-up
The study was performed in a farmland area in Southern Finland, near Lammi Biological Station (61° N, 25° E). We selected the lapwing as the study species, a common breeder in the study area. This ground nesting medium-sized farmland bird is listed as IUCN Near Threatened globally but Endangered within Europe17. The main threat to the species is agricultural intensification, and particularly the destruction of nests in arable land due to large scale mechanical operations, such as sowing17. In Finland, as in many other countries, a fraction of lapwing nests on arable land are located by volunteers, and their protection secured. However, locating these nests is highly time consuming and challenging, and most nests are left unprotected, and likely destroyed, every spring18.
We selected nine field parcels with a total of 22 lapwing nests in 2016, and three field parcels with nine nests in 2017. Study field parcels were intensively searched for nests by experienced observers during the week preceding the start of the recording work (see below), and individual nest locations were recorded with a hand-held GPS (model Garmin GPSmap 62 s, position accuracy ~ 3 m). Overall five of the field parcels were represented by ploughed fields (with irregular substrate owing to the upper soil being turned over) and seven by unploughed fields (direct sowing or winter cereals). Flight routes were constructed and saved in MapPilot 1.5.1 (Drones Made Easy, San Diego, USA) for areas around nests that covered approximately 0.5–1 ha (see Fig. 4). A drone cruised along the pre-programmed route, each flight lasting approximately 6–10 min (the time was constrained by the drone battery). Flights were done between 5-25.5.2016 and 16-18.5.2017 at altitudes of 15 or 25 m above ground level. Distance between transects was approximately 7 m for flights at 15 m of altitude and 12 m for flights at 25 m of altitude. We aimed to achieve a minimum of at least one flight over each field at each altitude (15 and 25 m) and each period of the day (07—11 a.m. and 14—20 p.m.). Overall, 54 flights were completed in 2016 and 19 in 2017. Thermal images were acquired by using a DJI Phantom 3 Advanced quadcopter with a FLIR Tau 2, 336 * 256 pixel, 9 mm lens thermal camera mounted on it. To ensure quality, thermal images were continuously saved during flights directly to a USB stick using a ThermalCapture unit (TeAx Technology GmbH, Wilnsdorf, Germany). Image acquisition occurred with the camera pointed in nadir position. Temperature, cloud cover, air humidity and wind speed data for each flight were accessed afterwards through Airdata.com service. In all circumstances, the incubating adult left the nest as we approached the edge of the field before the start of the drone flight.
Figure 4

Schematic representation of the key steps of this study, from extensive nest search at selected fields (A), to flying the drone carrying the thermal sensor along a pre-programmed route over the field (B), preparing the thermal images by extracting the coordinates of the box drawn around the nest (C), and finally applying a neural network deep learning algorithm to classify images as having or not having a nest (D). The satellite image in (A) and (B) is taken from Google maps (www.google.it/maps, map data: Google). Figure created in Microsoft Office PowerPoint 2016 (www.microsoft.com).

Full size image

Deep learning algorithm for nest identification
To automatically classify images as having or not having a nest, we trained a deep learning model (more details in extended methods in Supplementary information) to detect nests while minimising the occurrence of false positives (hereafter false presences) and false negatives (hereafter false absences). In doing so, we choose a convolutional neural network (CNN) approach19 and YOLOv3 training program20, as done in a similar study3.
We selected all images with nests in them (positive images, n = 1969) from each flight, and a larger number of randomly selected images without a nest (negative images, n = 3,469). We manually labelled all positive images using software ImageJ 1.8.021 with bounding boxes, which coordinates were then indexed by corresponding image names to be used by the YOLOv3. Next, we split the set of positive images into a training set and a test set by randomly selecting 70% of the flights (and all associated images) for training, and using the remaining 30% of flight images for testing. Selection was done at the flight rather than the image level to avoid that similar images from the same flight would be used in both training and testing, causing testing results to be positively biased.
Training was executed for 800 epochs, whereby an epoch is defined as the entire dataset passing forward and backward through the neural network (training program available at http://github.com/ultralytics/yolov3). After training was completed, a weights file was generated, which included the final value stored by each neuron, represented by floating point numbers in series. We then developed a program which read the configuration file of the YOLOv3 architecture and constructed the network according to the architecture specified in PyTorch implementation. Developing our own program was done to increase flexibility of expanding functionalities without interrupting unfamiliar code flow. The program read the trained weights file in series and stored the values to proper neurons, resulting in the construction of a deep learning model specifically for nest detection. This was then used to validate and test all the trained weight files to select the best model.
Through the testing, an output value was produced representing the confidence that an image included a nest. Values close to zero indicate that an image contained no nests with a high confidence, while values close to one that an image included a nest with a high confidence. In the cases where multiple spots appeared as possible nests, the overall confidence for that image was taken from the spot with the highest confidence values.
Performance of the neural network in discriminating images as having a nest or not was assessed by means of calculating four standard evaluation metrics for these types of systems: Precision, recall (propensity to avoid false positives and false negatives, respectively), accuracy (overall discrimination accuracy) and F1 score (the harmonic mean of precision and recall; more details in extended methods in Supplementary information). Performance was evaluated separately for the training and for the testing parts of the system.
Statistical analyses
We quantified the effect of environmental factors on the performance of the deep learning algorithm. We ran two separate analyses, one focused on factors affecting occurrence of false presences (i.e. the algorithm erroneously identifies an image without a nest as having a nest), and one on factors affecting occurrence of false absences (algorithm erroneously identifies an image with a nest as not having a nest). As the response of the former model we used the confidence as given by the algorithm for all the test images known to have no nest in them (values close to one indicate high probability of false presence). This model used data from images without a nest from all available flights (n = 3,469 images from 73 flights, as these were not used for training). The response for the false absence model was the confidence of the test images (n = 497 images with a nest from the 30% flights, n = 14 flights, not used for training) to having no nest in them (values close to one indicate high probability of false absence). Because the response variable is a proportion, we used beta regression with logit link. In each model we also included flight identity as a random factor to account for pseudo-replication stemming from multiple images per flight.
Models were run using the glmmTMB package22 in R version 3.6.123. Covariates considered are: air temperature (hereafter temperature), cloud cover, wind speed, air humidity (hereafter humidity), drone height (either 15 or 25 m above ground level) and substrate type (two classes: ploughed or un-ploughed). These variables are deemed potentially relevant in affecting the nest visibility in thermal images, and consequent performance of the deep learning algorithm. Air temperature, humidity, cloud cover and wind speed may reduce the thermal contrast between a nest (warm) and the remaining landscape, and create heterogeneous thermal landscape that hampers nest detection24. Moreover, ploughing alters the physical structure of the substrate, making it rougher and more heterogeneous, with many physical objects that may resemble a nest shape, thereby affecting nest detection accuracy. Drone height may affect detection accuracy due to the decreasing size and sharpness of a nest at higher altitudes.
Prior to analyses, we checked for outliers and ran variance inflation (VIF) analyses to assess the collinearity level between covariates. For the false presence model, humidity had a high VIF value of 3.9 being strongly correlated to temperature (R = − 0.7) and was excluded from following analyses. For the false absence model, humidity and wind speed had high VIF (28.8 and 2.3, both correlated with temperature, R = − 0.9 and 0.7, respectively), and were excluded from analyses. The remaining covariates had a VIF  More

Test-case species
Perca fluviatilis (Actinopterygii, Percidae) is a common widely spread Eurasian species living in freshwater and brackish habitats49. Its economic (high market value) and recreational (i.e. fishing) interests have led to the development of its RAS (i.e. recirculating aquaculture systems) farming since the 1990’s49. Perca fluviatilis is among the most interesting species for the development of inland aquaculture in Europe49. However, despite its economic potential, the production is still limited. This is mainly due to major bottlenecks, especially in first-life stages, such as low growth rate, low survival rate, and high cannibalism rate49. This highlights the potential interest of re-starting new domestication programmes. An intraspecific differentiation was already showed for this species in standardised conditions (e.g. growth19,50, behaviour16, development51).
Biological material collection and pre-experiment rearing conditions
Population sampling and rearing conditions were adapted from Toomey et al.16. Egg ribbons (one ribbon corresponding to one female) were obtained during the May 2018 and May 2019 spawning seasons from four lakes (Appendix S1): Valkea-Müstajärvi (VAL; 2018; Finland; 61° 13′ 08″ N, 25° 07′ 05″ E), Iso-Valkjärvi (ISO; 2018; Finland; 60° 57′ 21″ N, 26° 13′ 3″ E), Geneva (GEN; 2019; France; 46° 22′ 7.20″ N, 6° 27′ 14.73″ E), and Balaton (BAL ; 2019 ; Hungary ; 46° 54′ 23.375″ N, 18° 2′ 43.119″ E). We chose these populations since a phenotypic differentiation is known between the Finnish and Geneva populations16 while genetic specificities of central Europe populations have been already observed52,53. After transportation, 13–19 egg ribbons per lake were incubated at the experimental platform of aquaculture (Unit of Animal Research and Functionality of Animal Products, University of Lorraine, Vandœuvre-lès-Nancy, France) at 13 °C in incubators (110 × 64 × 186 cm; one population per incubator to avoid potential disease transmission between populations, one to two incubators per population) containing nine racks each (45 × 7 × 12 cm). Each incubator had its own temperature control and recirculated water system (flow rate of 4 m3 h−1). Water was UV-sterilized. Temperature (13.0 ± 0.4 °C) and oxygen rate (10.0 ± 0.5 mg L−1) were checked daily while pH (8.0 ± 0.2) was monitored three times a week (± standard error). Ammonium and nitrite concentrations (lower than 0.05 mg L−1) were measured three times a week until hatching. Light intensity was 400 lx at the water surface. Photoperiod was 12L:12D (12 h of light and 12 h of darkness).
Experimental rearing protocol
Two independent experimental phases were performed: phase I from hatching until the end of weaning (i.e. transition from live feed to inert pellets; 26 days post-hatching, dph) and phase II from 27 dph until the end of nursery, at 60 dph. The larval period was split in two phases in order to ensure availability of larvae across the whole larval period since there is a very high mortality rate during first-life stages. Because wild egg ribbons are not available the same time for all populations (i.e. asynchronous spawning seasons) and in order to prevent potential pathogen transmission, all populations were reared in independent structures. Since there are increasing concerns about the epizootic disease Perhabdovirus in Europe49, all populations were tested for the occurrence of this virus (Laboratoire Département d’Analyses du Jura, Poligny, France). All results were negative to the presence of the Perhabdovirus.
Regarding phase I, larvae from the different egg ribbons of each population were mixed after hatching and transferred to three green (RGB: 137, 172, 118) internal-wall 71 L cylindro-conical tanks (three replicates per population; RAS) at a density of 50 larvae L−1. Photoperiod was 12L:12D (simulation of dawn and dusk for 30 min) and light intensity was 400 lx at the water surface. Temperature was gradually increased during the first 2 weeks to 20 °C (1 °C day−1). Larvae were hand-fed with newly hatched Artemia nauplii (Sep-Art, INVE; seven times a day, every 1 h 30 from 8.30 am to 5.30 pm) from 3 days post-hatching until at 16 dph, which corresponds to the beginning of the weaning (i.e. transition from live feed to inert dry artificial diet) period. At 16 dph, Artemia ration was decreased by 25% every 3 days and dry feed ration (BioMar, 300 µm until 21 dph, then 500 µm) was increased by the same ratio. After 25 dph, larvae were fed with dry feed ad libitum (BioMar 500 µm, then 700 µm at 44 dph until 60 dph). At 26 dph, the larvae left in the cylindro-conical tanks were removed in order to start phase II.
Regarding phase II, after hatching, larvae left (i.e. not sampled for phase I) were held in 2 m3 tanks (RAS). The same conditions as phase I were used (temperature, light intensity, feeding, and photoperiod regimes). At 27 dph, these larvae were transferred to the three cylindro-conical tanks in order to start phase II at a density of 19 larvae L−1. Light intensity was 80 lx at water surface, all else remaining the same as phase I (except for density).
During the two phases, oxygen concentration (8.5 ± 2.3 mg L−1) and temperature (20.0 ± 0.6 °C) were checked daily in all tanks (± standard error). Ammonium and nitrite concentrations (means inferior to 0.05 mg L−1) and pH (7.7 ± 0.6 mg L−1) were monitored three times a week (± standard error). Tanks were cleaned daily after first feeding and dead individuals were removed every morning.
Larviculture performance assessment
A trait is considered in this study at the replicate level. Each trait value is obtained from the mean of individual values.
Survival and development traits
Survival rate is one of the key traits contributing to the success of larviculture production49,54. Because of fast decomposition of dead larvae, it was not possible to count dead larvae during the first 5 days post-hatching. Consequently, the daily count of dead larvae was only performed in phase II. Therefore, survival rate in phase I was calculated for each cylindro-conical tank thanks to the final count of larvae using the following formula: Nf × 100/(Ni − Ns), where Nf is the final number of fish counted at the end of phase I, Ni the initial number of fish, and Ns the number of fish sampled along the phase (i.e. for behaviour experiments, see below). For phase II, the Bergot survival rate55 was used since it takes into account the number of fish removed for sampling and the daily mortality. Two traits related to the development of individuals and essential for larviculture were considered49: swim bladder inflation rate and deformity rate. Swim bladder inflation rate was estimated at the end of each experimental phase (following the protocol used in Jacquemond56; 20 g L−1 of sea salt and 70 mg L−1 of MS-222): 100 × (SB + /Nf) with SB + the number of larvae with swim bladder and Nf the final number of larvae. Deformity rate was estimated in the final counting of each experimental phase using the following formula: 100 × (Nm/Nf) with Nm the number of deformed larvae (only visible column deformities) and Nf the final number of larvae counted. Finally, the volume of the yolk sac was also evaluated at 1 day post-hatching since it reflects the quantity of nutritional reserves available before exogenous feeding49. It is calculated using the following formula: π/6 × YSL×YSH2, where YSL is the length of the yolk sac and YSH the height of the yolk sac57.
Behavioural traits
The ability of a biological unit to be efficiently produced in intensive conditions (i.e. intensive farming is an increasing trend in the aquaculture development) also depends on (1) inter-individual relationships, (2) inter-individual distances, and (3) activity16. Indeed, tolerance to conspecifics in a restricted area is essential for production since it can impact individual welfare58. Highlighting populations which present a cohesive group structure would be advantageous. Nevertheless, living in group is not costless because it can trigger aggressive behaviours which can lead to uneven competition for food, mortalities, stress, or immune suppression16. Therefore, both inter-individual distances and inter-individual interactions need to be considered. Finally, activity is also important since it contributes to the total energetic budget16. Furthermore, less active individuals could contribute to decrease the occurrence of inter-individual contacts and subsequent potential aggressive interactions.
Regarding aggressive interaction quantification, aggressiveness was calculated for phase II using the formula : 100 × (Ni − (Nf + Nd + Ns) + Nc)/(Nf − Ns) with Ni the initial number of larvae, Nf the final number of larvae, Nd the sum of dead larvae counted daily, Ns the number of sampled larvae, and Nc the number of truncated or enucleated (enucleation being a specific indicator of aggressiveness in perch16) larvae among the dead ones (not possible in phase I since it was not possible to count dead larvae during the first 5 days post-hatching due to fast decomposition; in addition, no truncated or enucleated individuals were recorded for phase I).
Regarding the evaluation of inter-individual distances and activity, the detailed protocol is available in Toomey et al.16. Briefly, for each population, three replicates for each cylindro-conical tank were performed (nine replicates per population) over 2 days for phase I (25 and 26 dph) and phase II (44 and 45 dph). For each population, 90 individuals (n = 30 per cylindro-conical tank, 10 individuals per replicate) were sampled and transferred to three aquaria (58 L; light intensity of 80 lx light intensity, 20 °C). After one night of acclimatisation, populations were tested per groups of ten individuals in circular arenas (30 cm diameter, 1.5 cm of water depth, 10 lx) to study inter-individual distances and activity16. After 30 min acclimatisation, individuals were filmed for 30 min. After the experiment, individuals were euthanized (overdose of MS-222) and kept in formalin 4% for later length measurements. Larvae tested in the circular arenas from ISO, VAL, BAL, and GEN were respectively 14.05 ± 0.55 mm, 12.90 ± 0.62 mm, 10.62 ± 0.47 mm, and 11.81 ± 1.01 mm during phase I and 26.74 ± 1.67 mm, 26.28 ± 1.99 mm, 19.24 ± 1.22 mm, and 12.26 ± 0.45 mm during phase II (± standard error). Analyses were performed using the ImageJ software59. Images were extracted from videos at 5-min interval (six images per video). For each image, coordinates of individuals were noted using the middle point between the eyes. The mean of inter-individual distances was evaluated per replicate. It corresponds to the mean of distances between a focal fish and all the other fish of the group and it is an indicator of the group cohesion. Detailed calculation is available in Colchen et al.60. Activity was analysed in ImageJ. Every 5 min, one image per second was extracted for six consecutive seconds. For each image, coordinates of each individual were noted. This allowed calculating distance swam every second during the 5 s. The mean allowed obtaining for each individual the mean distance swam per second. From these values, we were able to calculate an average activity per replicate.
Growth traits
Growth traits are important in larviculture production, more particularly the length at hatching, specific growth rate, and growth heterogeneity49. To evaluate these traits, 30 larvae per population (i.e. ten larvae per cylindro-conical tank) were sampled the first and last days of each experimental phase, euthanized with an overdose of MS-222, and kept in formalin 4%. For phases I and II, larvae were measured using ImageJ (± 0.01 mm). For phase II, larvae were also individually weighted (± 0.1 mg; not possible in phase I due to the imprecision of measure at 1 day post-hatching). Since specific growth rate (SGR) is a trait of interest at the end of larviculture, it was calculated only in phase II using the following formula: SGR = 100 × (ln(Xf) − ln(Xi)) × ∆T−1 where Xi and Xf are respectively the average initial and final weight/length and ∆T the duration of phase II. Final growth heterogeneity was calculated for both phases in the following way: CVXf/CVXi in which CV is the coefficient of variation (100 × standard deviation/mean) and Xi and Xf the initial and final weight/length, respectively.
Statistical analyses
All statistical analyses were performed in R 3.0.361 to assess if there were statistical differences (p value  More

Monthly climatic variables
Our dataset is based on simulations of monthly mean temperature (°C), precipitation (mm month−1), cloudiness (%), relative humidity (%) and wind speed (m s−1) of the HadCM3 general circulation model6,12,13. At a spatial grid resolution of 3.75° × 2.5°, these data cover the last 120,000 years in 72 snapshots (2,000 year time steps between 120,000 BP and 22,000 BP; 1,000 year time steps between 22,000 BP and the pre-industrial modern era), each representing climatic conditions averaged across a 30-year post-spin-up period. We denote these data by

$$begin{array}{c}{T}_{{rm{HadCM3}}}(m,t),;{P}_{{rm{HadCM3}}}(m,t),;{C}_{{rm{HadCM3}}}(m,t),\ {H}_{{rm{HadCM3}}}(m,t),;{W}_{{rm{HadCM3}}}(m,t),end{array}$$
(1)

where (m=1,ldots ,12) represents a given month, and (tin {T}_{{rm{120}}{rm{k}}}) represents a given one of the 72 points in time for which simulations are available, denoted ({T}_{{rm{120}}{rm{k}}}).
We downscaled and bias-corrected these data in two stages (Fig. 1). Both are based on variations of the Delta Method14, under which a high-resolution, bias-corrected reconstruction of climate at some time t in the past is obtained by applying the difference between modern-era low-resolution simulated and high-resolution observed climate – the correction term – to the simulated climate at time t. The Delta Method has previously been used to downscale and bias-correct palaeoclimate simulations, e.g. for the widely used WorldClim database9. A recent evaluation of three methods commonly used for bias-correction and downscaling15 showed that the Delta Method reduces the difference between climate simulation data and empirical palaeoclimatic reconstructions overall more effectively than two alternative methods (statistical downscaling using Generalised Additive Models, and Quantile Mapping). We therefore used this approach for generating our dataset.
Fig. 1

Method of reconstructing high-resolution climate. Yellow boxes represent raw simulated and observed data, the dark blue box represents the final data. Maps, showing modern-era climate, correspond to the datasets represented by the bottom three boxes.

Full size image

Downscaling to ~1° resolution
A key limitation of the Delta Method is that it assumes the modern-era correction term to be representative of past correction terms15. This assumption is substantially relaxed in the Dynamic Delta Method used in the first stage of our approach to downscale the data in Eq. (1) to a ~1° resolution. This involves the use of a set of high-resolution climate simulations that were run for a smaller but climatically diverse subset of ({T}_{120k}). Simulations at this resolution are very computationally expensive, and therefore running substantially larger sets of simulations is not feasible; however, these selected data can be very effectively used to generate a suitable time-dependent correction term for each (tin {T}_{120k}). In this way, we are able to increase the resolution of the original climate simulations by a factor of ~9, while simultaneously allowing for temporal variability in the correction term. In the following, we detail the approach.
We used high-resolution simulations of the same variables as in Eq. (1) from the HadAM3H model13,16,17, available at a 1.25° × 0.83° resolution for the last 21,000 years in 9 snapshots (2,000 year time steps between 12,000 BP and 6,000 BP; 3,000 year time steps otherwise). We denote these by

$$begin{array}{c}{T}_{{rm{HadAM3H}}}(m,t),;{P}_{{rm{HadAM3H}}}(m,t),;{C}_{{rm{HadAM3H}}}(m,t),\ {H}_{{rm{HadAM3H}}}(m,t),;{W}_{{rm{HadAM3H}}}(m,t),end{array}$$

respectively, where (tin {T}_{21k}), represents a given one of the 9 points in time for which simulations are available, denoted ({T}_{{rm{21}}{rm{k}}}).
For each variable (Xin {T,P,C,H,W}), we considered the differences between the medium- and the high-resolution data at times (tin {T}_{21k}) for which both are available,

$$Delta {X}_{{rm{HadAM3H}}}^{{rm{HadCM3}}}(m,t),:={X}_{{rm{HadAM3H}}}(m,t)-{X}_{{rm{HadCM3}}}^{ boxplus }(m,t),$$

where the ( boxplus )-notation indicates that the coarser-resolution data was interpolated to the grid of the higher-resolution data. For this, we used an Akima cubic Hermite interpolant18, which (unlike a bilinear interpolant) is smooth but (unlike a bicubic interpolant) avoids potential overshoots. For each (tin {T}_{120k}) and each (tau in {T}_{21k}),

$${X}_{{rm{HadCM3}}}^{ boxplus }(m,t)+Delta {X}_{{rm{HadAM3H}}}^{{rm{HadCM3}}}(m,tau )$$
(2)

provides a 1.25° × 0.83° resolution downscaled version of the data ({X}_{{rm{HadCM3}}}(m,t)) in Eq. (1). The same is true, more generally, for any weighted linear combination of the Δ({X}_{{rm{HadAM3H}}}^{{rm{HadCM3}}}(m,tau )), which is the approach taken here, yielding

$$X{{prime} }_{ sim {1}^{circ }}(m,t),:={X}_{{rm{HadCM3}}}^{ boxplus }(m,t)+mathop{underbrace{sum _{tau in {T}_{21k}}w(t,tau )cdot Delta {X}_{{rm{HadAM3H}}}^{{rm{HadCM3}}}(m,tau )}}limits_{{rm{time-variable}},{rm{correction}},{rm{term}}},$$
(3)

where ({sum }_{tau in {T}_{21k}}mathop{overbrace{w(t,tau )}}limits^{ge 0}=1) for any given (tin {T}_{125k}). We discuss the choice of an additive approach for all climatic variables later on. Crucially, in contrast to the classical Delta Method – which, for all (tin {T}_{125k}), would correspond to (w(t,{rm{present}},{rm{day}})=1) and (w(t,tau )=0) otherwise (cf. Eq. (5)) –, the resolution correction term that is added to ({X}_{{rm{HadCM3}}}^{ boxplus }(m,t)) in Eq. (3) need not be constant over time. Instead, the high-resolution heterogeneities that are applied to the medium-resolution HadCM3 data are chosen from the broad range of patterns simulated for ({T}_{21k}). The strength of this approach lies in the fact that the last 21,000 years account for a substantial portion of the range of climatic conditions present during the whole Late Quaternary. Thus, by choosing the weights (w(t,tau )) for a given time (tin {T}_{125k}) appropriately, we can construct a ({T}_{21k})-data-based correction term corresponding to a climatic state that is, in a sense yet to be specified, close to the climatic state at time t. Here, we used atmospheric CO2 concentration, a key determinant of the global climatic state19, as the metric according to which the (w(t,tau )) are chosen; i.e. we assigned a higher weight to Δ({X}_{{rm{HadAM3H}}}^{{rm{HadCM3}}}(m,tau )) the closer the CO2 level at time (tau ) was to that at time t. Specifically, we used

$$w{prime} (t,tau )=frac{1}{{({{rm{CO}}}_{2}(t)-{{rm{CO}}}_{2}(tau ))}^{2}},$$

and rescaled these to (w(t,tau )=frac{w{prime} (t,tau )}{{sum }_{tau in {T}_{21k}}w{prime} (t,tau )}) (Supplementary Fig. 1). In the special case of (tin {T}_{21k}), we have (w(t,t)=1) and (w(t,tau )=0) for (tau ne t), for which Eq. (3) simplifies to

$$X{{prime} }_{ sim {1}^{circ }}(m,t)={X}_{{rm{HadAM3H}}}(m,t)quad {rm{for}},{rm{all}},tin {T}_{21k}.$$

Formally, the correction term in Eq. (3) corresponds to an inverse square distance interpolation of the Δ({X}_{{rm{HadAM3H}}}^{{rm{HadCM3}}}) with respect to CO220. We also note that, for our choice of (w(t,tau )), the correction term is a smooth function of t, as would be desired. In particular, this would not the case for the approach in Eq. (2) (unless (tau ) is the same for all (tin {T}_{125k})).
The additive approach in Eq. (3) does not by itself ensure that the derived precipitation, relative humidity, cloudiness and wind speed are non-negative and that relative humidity and cloudiness do not exceed 100% across all points in time and space. We therefore capped values at the appropriate bounds, and obtain

$$begin{array}{lll}{T}_{ sim {1}^{circ }}(m,t) & := & T{{prime} }_{ sim {1}^{circ }}(m,t),\ {P}_{ sim {1}^{circ }}(m,t) & := & {rm{max }}left(0,P{{prime} }_{ sim {1}^{circ }}(m,t)right),\ {C}_{ sim {1}^{circ }}(m,t) & := & {rm{min }}left(100 % ,{rm{max }}(0 % ,C{{prime} }_{ sim {1}^{circ }}(m,t))right),\ {H}_{ sim {1}^{circ }}(m,t) & := & {rm{min }}left(100 % ,{rm{max }}(0 % ,H{{prime} }_{ sim {1}^{circ }}(m,t))right),\ {W}_{ sim {1}^{circ }}(m,t) & := & {rm{max }}left(0,W{{prime} }_{ sim {1}^{circ }}(m,t)right).end{array}$$
(4)

Supplementary Fig. 2 shows that this step only affects a very small number of data points, whose values are otherwise very close to the relevant bound.
Bias-correction and downscaling to 0.5° resolution
In the second stage of our approach, we applied the classical Delta Method to the previously downscaled simulation data. Similar to the approach in Eq. (3), this is achieved by applying a correction term, which is now given by the difference between present-era high-resolution observational climate and coarser-resolution simulated climate, to past simulated climate. This further increases the resolution and removes remaining biases in the data in Eq. (4).
Since our present-era simulation data correspond to pre-industrial conditions (280 ppm atmospheric CO2 concentration)6,12,13, it would be desirable for the observational dataset used in this step to be approximately representative of these conditions as well, so that the correction term can be computed based on the simulated and observed climate of a similar underlying scenario. There is generally a trade-off between the quality of observation-based global climate datasets of recent decades, and the extent to which they reflect anthropogenic climate change (which, by design, is not captured in our simulated data) – both of which increase towards the present. Fortunately, however, significant advances in interpolation methods21,22,23 have produced high-quality gridded datasets of global climatic conditions reaching as far back as the early 20th century23. Thus, here we used 0.5° resolution observational data representing 1901–1930 averages (~300 ppm atmospheric CO2) of terrestrial monthly temperature, precipitation and cloudiness23. For relative humidity and wind speed, we used a global data representing 1961–1990 average (~330 ppm atmospheric CO2) monthly values24 due to a lack of earlier datasets. We denote the data by

$${T}_{{rm{obs}}}(m,0),;{P}_{{rm{obs}}}(m,0),;{C}_{{rm{obs}}}(m,0),;{H}_{{rm{obs}}}(m,0),;{W}_{{rm{obs}}}(m,0).$$

We extrapolated these maps to current non-land grid cells using an inverse distance weighting approach so as to be able to use the Delta Method at times of lower sea level. The resulting data were used to bias-correct and further downscale the ~1° data in Eq. (3) to a 0.5° grid resolution via

$$X{{prime} }_{0.{5}^{circ }}(m,t),:={X}_{ sim {1}^{circ }}^{ boxplus }(m,t)+mathop{underbrace{{X}_{{rm{obs}}}(m,0)-{X}_{ sim {1}^{circ }}^{ boxplus }(m,0)}}limits_{{rm{correction}},{rm{term}}},$$
(5)

where (Xin {T,P,C,H,W}). In particular, the data for the present are identical to the empirically observed climate,

$$X{{prime} }_{0.{5}^{circ }}(m,0)={X}_{{rm{obs}}}(m,0).$$

Finally, we again capped values at the appropriate bounds, and obtained

$$begin{array}{lll}{T}_{0.{5}^{circ }}(m,t) & := & T{{prime} }_{0.{5}^{circ }}(m,t),\ {P}_{0.{5}^{circ }}(m,t) & := & {rm{max }}left(0,P{{prime} }_{0.{5}^{circ }}(m,t)right),\ {C}_{0.{5}^{circ }}(m,t) & := & {rm{min }}left(100 % ,{rm{max }}(0 % ,C{{prime} }_{0.{5}^{circ }}(m,t))right),\ {H}_{0.{5}^{circ }}(m,t) & := & {rm{min }}left(100 % ,{rm{max }}(0 % ,H{{prime} }_{0.{5}^{circ }}(m,t))right),\ {W}_{0.{5}^{circ }}(m,t) & := & {rm{max }}left(0,W{{prime} }_{0.{5}^{circ }}(m,t)right).end{array}$$
(6a)

Similar as in the analogous step in the first stage of our approach (Eq. (4)), only a relatively small number of data points is affected by the capping; their values are reasonably close to the relevant bounds, and their frequency decreases sharply with increasing distance to the bounds (Supplementary Fig. 2).
In principle, capping values, where necessary, can be circumvented by suitably transforming the relevant variable first, then applying the additive Delta Method, and back-transforming the result. In the case of precipitation, for example, a log-transformation is sometimes used, which is mathematically equivalent to a multiplicative Delta Method, in which low-resolution past simulated data is multiplied by the relative difference between high- and low-resolution modern-era data14; thus, instead of Eq. (5), we would have ({P}_{0.{5}^{circ }}(m,t),:={P}_{ sim {1}^{circ }}^{ boxplus }(m,t)cdot frac{{P}_{{rm{obs}}}(m,0)}{{P}_{ sim {1}^{circ }}^{ boxplus }(m,0)}). However, whilst this approach ensures non-negative values, it has three important drawbacks. First, if present-era observed precipitation in a certain month and grid cell is zero, i.e. ({P}_{{rm{obs}}}(m,0)=0), then ({P}_{0.{5}^{circ }}(m,t)=0) at all points in time, t, irrespectively of the simulated climate change signal. Specifically, this makes it impossible for current extreme desert areas to be wetter at any point in the past. Second, if present-era simulated precipitation in a grid cell is very low (or indeed identical to zero), i.e. ({P}_{ sim {1}^{circ }}^{ boxplus }(m,0)approx 0), then ({P}_{0.{5}^{circ }}(m,t)) can increase beyond all bounds. Very arid locations are particularly prone to this effect, which can generate highly improbable precipitation patterns for the past. In our scenario of generating global maps for a total of 864 individual months, this lack of robustness of the multiplicative Delta Method would be difficult to handle. Third, the multiplicative Delta Method is not self-consistent: applying it to the sum of simulated monthly precipitation does not produce the same result as applying it to simulated monthly precipitation first and then taking the sum of these values. The natural equivalent of the log-transformation for precipitation is the logit-transformation for cloudiness and relative humidity, however, this approach suffers from the same drawbacks.
Minimum and maximum annual temperature
Diurnal temperature data are not included in the available HadCM3 and HadAM3H simulation outputs. We therefore used the following approach to estimate minimum and maximum annual temperatures. Based on the monthly HadCM3 and HadAM3H temperature data, we created maps of the mean temperature of the coldest and the warmest month. In the same way as described above, we used these data to reconstruct the mean temperature of the coldest and warmest month at a 1.25° × 0.83° resolution by means of the Dynamic Delta Method, yielding

$${T}_{ sim {1}^{circ }}^{{rm{coldest}},{rm{month}}}(t),{rm{and}},{T}_{ sim {1}^{circ }}^{{rm{warmest}},{rm{month}}}(t),$$

for (tin {T}_{120k}). We then used observation-based 0.5° resolution global datasets of modern-era (1901–1930 average) minimum and maximum annual temperature23, denoted

$${T}_{{rm{obs}}}^{{rm{min }}}(0),{rm{and}},{T}_{{rm{obs}}}^{{rm{max }}}(0),$$

to estimate past minimum and maximum annual temperature as

$$begin{array}{lll}{T}_{0.{5}^{circ }}^{{rm{min }}}(t) & := & {T}_{ sim {1}^{circ }}^{{rm{coldest}},{rm{month}}, boxplus }(t)+{T}_{{rm{obs}}}^{{rm{min }}}(0)-{T}_{ sim {1}^{circ }}^{{rm{coldest}},{rm{month}}, boxplus }(0),\ {T}_{0.{5}^{circ }}^{{rm{max }}}(t) & := & {T}_{ sim {1}^{circ }}^{{rm{warmest}},{rm{month}}, boxplus }(t)+{T}_{{rm{obs}}}^{{rm{max }}}(0)-{T}_{ sim {1}^{circ }}^{{rm{warmest}},{rm{month}}, boxplus }(0),end{array}$$
(6b)

respectively. This approach assumes that the difference between past and present mean temperature of the coldest (warmest) month is similar to the difference between the past and present temperature of the coldest (warmest) day. Instrumental data of the recent past suggest that this assumption is well justified across space (Supplementary Fig. 3).
Land configuration
We used a reconstruction of mean global sea level25 and a global elevation and bathymetry map26, interpolated to a 0.5° resolution grid, to create land configuration maps for the last 120,000 years. Maps of terrestrial climate through time were obtained by cropping the global data in Eq. (6a and b) to the appropriate land masks. Values in non-land grid cells were set to missing values, except in the case of below-sea-level inland grid cells, such as the Aral, Caspian and Dead sea.
Bioclimatic data, net primary productivity, leaf area index, biome
Based on our reconstructions of minimum and maximum annual temperature, and monthly temperature and precipitation, we derived 17 bioclimatic variables27 listed in Table 1. In addition, we used the Biome4 global vegetation model28 to compute net primary productivity, leaf area index and biome type at a 0.5° resolution for all (tin {T}_{120k}), using reconstructed minimum annual temperature, and monthly temperature, precipitation and cloudiness. Similar to a previous approach21, we converted cloudiness to the percent of possible sunshine (required by Biome4) by using a standard conversion table and applying an additional latitude- and month-specific correction. Since Biome4 estimates ice biomes only based on climatic conditions and not ice sheet data, it can underestimate the spatial extent of ice. We therefore changed simulated non-ice biomes to ice, and set net primary production and leaf area index to 0, in grid cells covered by ice sheets according to the ICE-6g dataset29 at the relevant points in time. Whilst our data represent potential natural biomes, and as such do not account for local anthropogenic land use, maps of actual land cover can readily be generated by superimposing our data with available reconstructions of global land use during the Holocene30. More

Distinctive temperature responses along a substrate gradient
Within the temperature range of ~14 to ~34 °C in our incubations, the observed AORs at the ambient substrate level (AORambient, see Methods) varied over 3 orders of magnitude, from 0.5 to ~4000 nM d−1, across a wide spectrum of ambient ammonium levels ranging from 14 nM to 96 μM (Fig. 1). Three different types of temperature response of AORambient patterns at estuarine, shelf, and sea basin stations were observed: (I) a positive response with a Topt of ≥34 °C (Fig. 1a, b); (II) a negative response, which has never been reported before, with a Topt of ≤14 °C (Fig. 1d–f); and (III) a dome-shaped response with a Topt of 20–29 °C (Fig. 1c, g–i).
The Type I pattern was observed at two of the three estuarine stations (JLR1 and JLR2, Fig. 1a, b) where ammonium concentrations were high (≥24 μM), and the AOR increased linearly as the temperature increased from 14 to 34 °C. In these cases, the Topt was equal to or higher than the maximum experimental temperature of 34 °C (Fig. 1a, b). The Type II pattern was observed at the shelf stations (N1, M1, and M2), where NH4+ concentrations ranged from 45 to 550 nM (Fig. 1d–f). In contrast to the Type I pattern, the Topt of the Type II pattern was equal to or lower than the minimum experimental temperature of 14 °C, showing a continuously decreasing AOR as temperature increased. The Type III pattern was observed at station JLR3 (outer estuary), N2 (shelf), N3 and J1 (basin), for which the Topt of the AOR varied from 20 to 29 °C, with rates decreasing toward both higher and lower temperatures (Fig. 1c, g–i). The NH4+ concentrations of the Type III stations ranged from 14 to 5000 nM. Nevertheless, the highest Topt values were observed at coastal sites with the highest ambient ammonium concentrations (Fig. 1).
Substrate regulates AOR and its thermal optimum temperature
For those stations with low ammonium concentrations, the AOR at in situ temperature increased when the substrate was enriched (AORenriched, additions of 2000 nM 15NH4+) (Fig. 1f, i). Meanwhile, the Topt of the AOR shifted significantly toward higher values (t test, p  Q10Vmax (Supplementary Fig. 2). This criterion was fulfilled in both J1 and JLR4 cases (Fig. 2c–f). We see the positive shift of Topt due to the ammonium addition (up to ~100 nM at J1 station and up to ~10 μM at JLR4) can be closely predicted (Fig. 3c, d; Supplementary Fig. 2). Overall, the DAMM model successfully predicts the entire thermal response curve, including rates and Topt, except when the manipulated temperatures exceed Topt-sat (Fig. 3; Supplementary Fig. 2). AOR drops significantly when temperature is greater than the Topt-sat, so heat-impaired biological enzyme activity27,28 might result in deviations from the relationship between Vmax (Km) values and temperature from the Arrhenius law.
Fig. 3: Validation plot for the rate predictions and observations.

a, b Scatter plot of the predicted rates via the Dual Arrhenius and Michaelis–Menten kinetics model (DAMM model) and the measured rates under different substrate concentrations and temperatures (below the optimum temperature in substrate-saturated conditions, Topt-sat). Linear regressions between the model predictions and the measurements are presented (two-sided t test was used to generate the p value (95% confidence) to measure the strength of correlation coefficient. p values are uncorrected). c, d The rate patterns (dots) against temperature under different substrate concentrations. Curves stand for the predicted rates derived from the DAMM model and the symbols represent the measured rates. The shades denote the uncertainty of model prediction. The dashed black vertical lines represent the Topt-sat. The measured rates in (a, c) are presented as mean values, instead of standard deviation the given bars indicate the variation range of two independent experiments. The measured rates in (b, d) and the predicted rates in (a–d) are expressed as the mean values ± SD (n = 10 in (a, c); n = 48 in (b, d); independent experiments).

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The substrate-dependent thermal optimum is attributable to the effect of temperature on biochemical kinetics and the structural stability of the enzymes. Increasing temperature promotes catalytic rate, thus, Vmax increases due to increasing kinetic energy of reactants and rates of collision, as well as higher structural flexibility of enzymes27,29. However, higher structural flexibility (lower stability) also results in active sites with a reduced ability for ligand recognition and binding, therefore, lower kinetic efficiency. Accordingly, one important physiological response of an organism to rising temperature will be a reduction in substrate affinity (Supplementary Table 2), and thus higher substrate demands (i.e. higher Km value)25,29,30,31. In other words, higher substrate levels help to compensate for enzyme structural stability losses and so promote growth rates at higher temperature. Note that some other microbes may respond differently to temperature, with Q10Km ≤ Q10Vmax for instance. This may lead to predictable yet unidirectional rate increases in response to warming (without substrate-regulated Topt) until the Topt-sat is reached, regardless of substrate changes.
Similarly, nutrient-dependent Topt has been reported for phytoplankton growth in pure cultures previously. For instance, Thomas et al.32 indicated that the Topt for growth of a marine diatom was a saturating function of major nutrient (nitrate and phosphate) concentration, and that the Topt could decrease by 3–6 °C at low concentrations relative to that at saturated nutrient levels. In addition to studies of pure cultures, field studies have also suggested that organisms may tolerate higher temperature stresses when nutrients are more abundant. For example, kelp (Laminaria saccarina) with high nitrogen reserves have more capacity for thermal adaptation33, while corals with symbionts limited by phosphate are more susceptible to heat-induced bleaching34. Although these examples are functionally and taxonomically distant from AOA and ammonia-oxidizing bacteria (AOB), strong similarities in substrate/nutrient regulation characteristics may imply a similar mechanism of enzymatic thermal responses between chemoautotrophs and photoautotrophs.
Nevertheless, the higher thermal optimum of AO in the estuarine system (e.g., JRL1, JLR2, and JLR3) than in the offshore environment (e.g., N3 and J1) can be explained by a substrate-regulated Topt. Note that field AOR represents explicitly the collective activity of the AO community composed of AOA and AOB, which may have distinctive thermal tolerances and affinities for substrate. Therefore, community structure very likely plays a role in modulating the thermal response patterns of community AOR in the field environments, in addition to substrate concentration.
Rate proportion and community thermal optimum
To further examine to what extent the community structure (proportions of AOA and AOB) might shape the thermal response patterns of community AOR observed in the field, we added allylthiourea (ATU) to inhibit the activity of AOB for rate discrimination (see Methods; Supplementary Discussions). Results showed that the inhibitory efficiency of AOR was gradually reduced with increasing offshore distance (Fig. 4). That is, from the estuary (JLR4, JLR1, JLR2, and JLR3) to the shelf (N1 and N2) and the sea basin (N3), the relative contribution of AOB to the community AOR dropped from as high as ~100% in the upper estuary down to ~70% in the shelf transition zone, and near 0% in the basin. Meanwhile, the AOA/AOB gene copies data (see Supplementary Methods) from estuary to sea basin (Fig. 4) also clearly show that the abundance of AOA relative to AOB increased exponentially with increasing offshore distance. A similar offshore pattern of community distribution was also observed in other regions, such as from the Pearl River estuary to the South China Sea35, and from the freshwater region of the Chesapeake Bay to the coastal and open ocean water column36. This pattern suggests that AOB strongly prefer substrate-replete niches, and vice versa for AOA20,37, agreeing well with our M–M experimental data that the substrate saturation condition for AOB-dominated water at JLR4 was several orders of magnitude higher than that for AOA-dominated water from J1 (Supplementary Table 2). The Km values of AOB in JLR4 varied from 7 to 55 μM in accordance with varying temperatures from 10 to 37 °C, while the Km values of AOA in J1 varied from 13 to 44 nM over a similar temperature range (Supplementary Table 2). Results were supportive of previous pure culture and field studies which showed the minimum ammonium demand for AOB is >1 μM and Km values range from 28 to 4000 μM38,39,40,41, while minimum ammonium demand and Km value for AOA are  Q10Vmax, which also results in a reduction in α during warming for both AOA and AOB. Yet, the relative reduction in AOA specific affinity as temperature increases is more significant (Fig. 5a), suggesting AOAs are more competitive in low temperature environments relative to AOBs, and so may not be favored in a warming ocean. On the other hand, the specific affinity of AOBs is insensitive to temperature change, suggesting their adaptation to nearshore environments with greater temperature fluctuation. The seaward gradient in temperature fluctuations and ammonium concentrations determine the nitrifier community, thus, thermal response pattern of community AOR observed in the field.
Fig. 5: Thermal response projections in near- and offshore regions.

a The thermal responses of specific affinity at the J1 and JLR4 stations. Data are expressed as the mean values ± SD (n = 10 in J1 station; n = 48 in JLR4 station; independent experiments). b Normalized warming-driven variations in ammonia oxidation rates. Rate changed (%) is relative to the ammonia oxidation rate (AOR) at in situ temperature. The mean increase (nearshore hollow dots) is denoted by the dashed line and the mean reduction (offshore, solid dots) is denoted by the solid black line. The shaded area represents the 4 °C increase in temperature mentioned in the IPCC study. Note that the stations where the surface salinities are lower than 32 are classified as nearshore station, and the others are classified as offshore stations. The data in (b) are presented as mean values, instead of standard deviation the given bars indicate the variation range of two independent experiments.

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To predict the trends of AOR in different geographic spaces in warming ocean, we compile the available marine AOR data to examine the AOR changes empirically. If we assume the biogeographic distribution of AO community remains unchanged and consider solely the warming effect on AOR relative to the onsite temperature, we found the thermal responses of AOR in nearshore and offshore are quite different (Fig. 5b). More specifically, the higher Topt of these AO communities in nearshore regimes allows ocean warming to promote coastal AOR when the temperature change increment is More

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