Boris Gralak
email: boris.gralak@fresnel.fr

Spatial dispersion and the sign of the imaginary part of the permeability

In this paper, arguments are provided to support that, in passive media, the imaginary part of the permeability can take positive and negative values. These arguments are based on the analysis of two different situations: i) the hydrodynamical model and ii) the spatially dispersive effective parameters of a multilayered structure. In addition, the electromagnetic energy is considered in a medium described by a spatially dispersive permittivity tensor.

I. INTRODUCTION

Magnetodielectric media can be described using two different models. The magnetodielectric properties can be defined by a dielectric permittivity ε(ω) and a magnetic permeability µ(ω) depending on the frequency ω, or by a permittivity tensor ε(k, ω) depending on the frequency ω and the wave vector k [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Agranovich | Spatial dispersion and negative refraction of light[END_REF][START_REF] Veselago | Negative refractive index materials[END_REF]. These two descriptions are related. For example, in the case of an isotropic medium, the permittivity tensor ε(k, ω) can be decomposed onto its longitudinal and transverse components, ε(k, ω) = ε lg (k, ω)P lg + ε tr (k, ω)P tr , where

P lg = kk k 2 , P tr = 1 - kk k 2 , (1)
and the permittivity and permeability can be retrieved (for bulk infinite media) according to the well-known relations [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Agranovich | Spatial dispersion and negative refraction of light[END_REF]

ε(ω) = lim k→0 ε lg (k, ω) , 1 ωµ(ω) = 1 ωµ 0 + lim k→0 ωε lg (k, ω) -ωε tr (k, ω) k 2 . ( 2
)
Although these two descriptions of magnetodielectric media seem to be interchangeable, the introduction of spatial dispersion and the gathering, in the permittivity tensor, of all the magnetic and dielectric properties of the materials, break the symmetry between the fields E and H in the Maxwell's equations. However, the symmetry between these fields and between the permittivity and permeability is widely used in textbooks of classical electrodynamics [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Van Bladel | IEEE Press Series on Electromagnetic Wave Theory[END_REF]. In particular, the Kramers-Kronig relations and the passivity requirement for the permittivity are generally also considered as valid for the permeability: for instance, it is generally considered that the imaginary part of the permittivity is positive [1, section 80],

Im µ(ω) ≥ 0 for ω ≥ 0 .

(

Nevertheless, it turns out that the behaviors of the permittivity and the permeability are different in the static regime [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Jackson | Classical Electrodynamics[END_REF]: unlike the static permittivity ε(0) [i.e. ε(ω) at the limit ω → 0], which always takes real value greater than ε 0 , the static permeability µ(0) [i.e. µ(ω) at the limit ω → 0] can take real values either greater (paramagnetic media) or less (diamagnetic media) than µ 0 . This alternative for the static permeability seems in contradiction with the causality principle and the passivity. Indeed, the Kramers-Kronig relations applied to the permeability imply

µ(0) = µ 0 + 2 π ∞ 0 dν Im µ(ν) ν , (4)
which requires for µ(0) to be greater than µ 0 if Im µ(ω) ≥ 0 for all positive real frequency ω [START_REF] Markel | Can the imaginary part of permeability be negative?[END_REF]. In the textbook [START_REF] Landau | Electrodynamics of Continuous Media[END_REF], this contradiction is explained by the frequency range where the macroscopic magnetic permeability makes sense, which is limited to the relatively low frequencies. Consequently, it is specified in [1, section 82] that the Kramers-Kronig relations and derived expressions like (4) must be modified for the magnetic permeability. Hence the requirement on the imaginary part of the permeability, Im µ(ω) ≥ 0, has been confirmed in reference [START_REF] Landau | Electrodynamics of Continuous Media[END_REF].

The introduction of metamaterials and negative index materials with effective permeability led to revisit these statements. Indeed, it has been found that the effective parameters of metamaterials present anomalous dispersion as in [START_REF] Koschny | Resonant and antiresonant frequency dependence of the effective parameters of metamaterials[END_REF]: "the resonant behavior of the effective magnetic permeability is accompanied by an antiresonant behavior of the effective permittivity" and "the imaginary parts of the effective permittivity and permeability are opposite in sign". These numerical results first generated some controversy [START_REF] Depine | Comment i on "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials[END_REF][START_REF] Efros | Comment ii on "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials[END_REF], but next the anomalous dispersion and the possibility for negative imaginary part of the effective permeability have been confirmed by several studies, which generated a series of investigations. For instance, fundamental questions on the Poynting vector and the energy have been adressed [START_REF] Markel | Can the imaginary part of permeability be negative?[END_REF][START_REF] Silveirinha | Poynting vector, heating rate, and stored energy in structured materials: A first-principles derivation[END_REF], the validity of the Kramers-Kronig relations for the magnetic permeability has been examined [START_REF] Silveirinha | Examining the validity of kramers-kronig relations for the magnetic permeability[END_REF], the validity of the causality principle and the physical meanining of the metamaterials constitutive parameters have been analyzed [START_REF] Alù | Restoring the physical meaning of metamaterial constitutive parameters[END_REF][START_REF] Alù | Causality relations in the homogenization of metamaterials[END_REF].

In this paper, arguments are provided to support that, in passive media, the imaginary part of the magnetic permeability µ(ω) can take both positive and negative values, i.e. that the relation ( 3) is not valid [START_REF] Gralak | Negative index materials: at the frontier of macroscopic electromagnetism[END_REF]. These arguments are based on the analysis of two different situations: i) the hydrodynamical model and ii) the spatially dispersive effective anisotropic permeability of a multilayered structure.

II. THE HYDRODYNAMICAL MODEL

The simple hydrodynamical model [START_REF] Boardman | Electromagnetic Surface Modes[END_REF] is considered with the permittivity tensor expression

ε(k, ω) = ε 0 -ε 0 Ω 2 ω 2 + iωγ -ω 2 0 -v 2 k 2 P lg -ε 0 Ω 2 ω 2 + iωγ -ω 2 0 P tr . (5)
Using the relation (2), it is possible to define from this model the following magnetic permeability

1 ωµ(ω) = 1 ωµ 0 -ωε 0 Ω 2 v 2 (ω 2 + iωγ -ω 2 0 ) 2 , (6)
and to obtain, for its imaginary part at real frequency ω,

Im 1 ωµ(ω) = ε 0 2γΩ 2 v 2 ω 2 (ω 2 -ω 2 0 ) [(ω 2 -ω 2 0 ) 2 + ω 2 γ 2 ] 2 . ( 7
)
This expression clearly shows that the imaginary part of the obtained permeability can take both positive and negative values, depending on ω 2 is smaller or larger than ω 2 0 . In addition, from the expression (6) of the permeability, the following identity, similar to the relation (4), can be derived

∞ 0 dν Im 1 νµ(ν) = π 2 1 µ(0) - 1 µ 0 = 0 , (8)
where it has been used that µ(0) = µ 0 to obtain that the integral vanishes. This last identity confirms that the imaginary part of the permeability (6) must take positive and negative values.

III. THE EFFECTIVE PERMITTIVITY TENSOR OF A MULTILAYERED STRUCTURE

The possibility for the imaginary part of the permeability to take positive and negative values is also investigated through the effective permeability of a stack of non-magnetic homogeneous layers (see Fig. 1, left part). Indeed, the simplicity of such a structure makes it possible to define exactely, using the retrieval method [START_REF] Gralak | Negative index materials: at the frontier of macroscopic electromagnetism[END_REF], effective anisotropic permittivity ε eff (k , ω) and permeability µ eff (k , ω) for all frequencies ω and wave vectors k parallel to the interfaces (see Fig. 1, right part).

In this paper, the exact expressions of these effective anisotropic parameters are derived and their properties are analyzed [START_REF] Gralak | Negative index materials: at the frontier of macroscopic electromagnetism[END_REF]. In particular, it is shown that the parameters ε eff (k , ω) and µ eff (k , ω) are analytic functions in the half plane of complex frequencies ω with positive imaginary part [START_REF] Gralak | Negative index materials: at the frontier of macroscopic electromagnetism[END_REF]. Then, a relation similar to the equations ( 4) and ( 8) is derived for the effective permeability µ eff (k , ω) at k = 0:

1 µ eff (0, 0) = 1 µ 0 + 2 π ∞ 0 dν Im 1 νµ eff (0, ν) . ( 9
)
multilayered stack with symmetry plane effective medium with spatial dispersion

h x k ε eff (k , ω) , µ eff (k , ω) ε(x, ω)
Fig. 1: Left: the considered multilayered structure with a plane of symmetry described by the frequency dependent and isotropic permittivity ε(x, ω). Right: the equivalent effective medium described by the effective permittivity ε eff (k , ω) and permeability µ eff (k , ω).

Next, using that the starting multilayered structure is non-magnetic, the quasistatic limit provides µ eff (0, 0) = µ 0 and the relation ( 9) becomes [START_REF] Gralak | Negative index materials: at the frontier of macroscopic electromagnetism[END_REF]

∞ 0 dν Im 1 νµ eff (0, ν) = 0 . (10)
This last equation leads to the conclusion that the imaginary part of the effective permeability µ eff (0, ν) must take positive and negative values. Finally, the electromagnetic energy is considered in a medium described by a spatially dispersive permittivity tensor ε(k, ω) and the passivity requirement on this permittivity tensor is derived.

IV. CONCLUSION

In this paper, it has been shown in two situations that the passivity requirement on imaginary part of the magnetic permeability, Im µ(ω) ≥ 0, is not valid. In contrast, these arguments support that the Kramers-Kronig relations make sense for the permeability µ(ω). These results appear in contradiction with the electrodynamics based on the symmetry between, in one hand, the electric field E and the permittivity ε(ω) and, in the other hand, the magnetic induction field H and the permeability µ(ω). Hence the description of magnetodielectric media by a permittivity tensor ε(k, ω) depending on the frequency ω and the wave vector k appears as the most appropriate.

th International Congress on Artificial Materials for Novel Wave Phenomena -Metamaterials 2021 New York, USA, Aug. 2 nd -7 th 2021


Louise Mair
email: louise.mair@newcastle.ac.uk

Leon A Bennun

Thomas M Brooks

Stuart H M Butchart

Friederike C Bolam

Neil D Burgess

Jonathan M M Ekstrom

E J Milner-Gulland

Michael Hoffmann

Keping Ma

Nicholas B W Macfarlane

Domitilla C Raimondo

Ana S L Rodrigues

Xiaoli Shen

Bernardo B N Strassburg

Craig R Beatty

Carla Gómez-Creutzberg

Alvaro Iribarrem

Meizani Irmadhiany

Eduardo Lacerda

Bianca C Mattos

Karmila Parakkasi

Marcelo F Tognelli

Elizabeth L Bennett

Catherine Bryan

Giulia Carbone

Abhishek Chaudhary

Maxime Eiselin

Gustavo A B Da Fonseca

Russell Galt

Arne Geschke

Louise Glew

Romie Goedicke

Jonathan M H Green

Richard D Gregory

Samantha L L Hill

David G Hole

Jonathan Hughes

Jonathan Hutton

Marco P W Keijzer

Laetitia M Navarro

Eimear Nic Lughadha

Andrew J Plumptre

Philippe Puydarrieux

Hugh P Possingham

Aleksandar Rankovic

Eugenie C Regan

Carlo Rondinini

Joshua D Schneck

Juha Siikamäki

Cyriaque Sendashonga

Gilles Seutin

Sam Sinclair

Andrew L Skowno

Carolina A Soto- Navarro

Simon N Stuart

Helen J Temple

Antoine Vallier

Francesca Verones

Leonardo R Viana

James Watson

Simeon Bezeng

Monika Böhm

Ian J Burfield

Viola Clausnitzer

Colin Clubbe

Neil A Cox

Jörg Freyhof

Leah R Gerber

Craig Hilton-Taylor

Richard Jenkins

Ackbar Joolia

Lucas N Joppa

Lian Pin Koh

Thomas E Lacher

Penny F Langhammer

Barney Long

David Mallon

Michela Pacifici

Beth A Polidoro

Caroline M Pollock

Malin C Rivers

Nicolette S Roach

Jon Paul Rodríguez

Jane Smart

Bruce E Young

Frank Hawkins

Philip J K Mcgowan

A metric for spatially explicit contributions to science-based species targets

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The Convention on Biological Diversity's post-2020 Global Biodiversity Framework will likely include a goal to stabilise and restore the status of species. Its delivery would be facilitated by making the actions required to halt and reverse species loss spatially explicit. We develop a "Species Threat Abatement and Restoration" (STAR) metric, which is scalable across species, threats and geographies. STAR quantifies the contributions that abating threats and restoring habitats in specific places offer towards reducing extinction risk. While every nation can contribute towards halting biodiversity loss, Indonesia, Colombia, Mexico, Madagascar and Brazil combined have stewardship over 31% of total STAR values for terrestrial amphibians, birds and mammals. Among actions, sustainable crop production and forestry dominate, contributing 41% of total STAR values for these taxonomic groups. Key Biodiversity Areas cover 9% of the terrestrial surface but capture 47% of STAR values. STAR could support governmental and non-state actors to quantify their contributions to meeting science-based species targets within the framework.

The Convention on Biological Diversity (CBD) sets the policy framework for biodiversity conservation and sustainable use through the commitments of 195 countries and the European Union. The Strategic Plan for Biodiversity 2011-2020 included Aichi Biodiversity Target 12, which set the goal for 2020 of preventing the extinction of known threatened species and improving and sustaining their conservation status. Despite government commitments and successful efforts for certain species [START_REF] Hoffmann | The Impact of Conservation on the Status of the World's Vertebrates[END_REF] , overall extinction risk continues to increase, and widespread implementation shortfalls will prevent Target 12 from being met [START_REF] Butchart | Assessing progress towards meeting major international objectives related to nature and nature's contributions to people[END_REF] . A new global framework with revised goals and targets is currently being negotiated, which places the stabilisation and restoration of species' populations as an outcome goal for 2030, as a stepping-stone towards the CBD's 2050 Vision [START_REF] Cbd | Zero draft of the Post-2020 Global Biodiversity Framework[END_REF][START_REF] Rounsevell | A biodiversity target based on species extinctions[END_REF] .

The Aichi Biodiversity Targets were largely approached as a list of twenty discrete targets, not making explicit how progress towards pressure-reduction targets would support progress towards biodiversity-outcome targets [START_REF] Marques | A framework to identify enabling and urgent actions for the 2020 Aichi Targets[END_REF] . In contrast, the proposed post-2020 Global Biodiversity Framework explicitly states the need to reduce threats to halt the loss of biodiversity, and proposes specific sub-targets for threat reduction [START_REF] Cbd | Zero draft of the Post-2020 Global Biodiversity Framework[END_REF] . While the major direct threats to species are well-documented [START_REF] Butchart | Assessing progress towards meeting major international objectives related to nature and nature's contributions to people[END_REF] , establishing specific targets for threat reduction is complex, because there are large numbers of threatened species (>30,000 species assessed as threatened on the IUCN Red List 6 ), rapid deteriorations (revealed by the Red List Index [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Measuring global trends in the status of biodiversity: Red list indices for birds[END_REF] ), and large spatial variation in species' distributions, extinction risk trends and the threats impacting them [START_REF] Rodrigues | Spatially Explicit Trends in the Global Conservation Status of Vertebrates[END_REF] . Tools that support actions to address these threats include the documentation of species recovery [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF] , identification of important sites [START_REF] Iucn | A Global Standard for the Identification of Key Biodiversity Areas, Version 1.0[END_REF] , and systematic conservation planning [START_REF] Moilanen | Spatial Conservation Prioritization: Quantitative Methods and Computational Tools[END_REF] . However, no mechanisms yet exist to quantify the contributions that particular actions in particular places could make towards abating threats to and restoring habitat for threatened species worldwide, to support achievement of the goals of the post-2020 biodiversity framework.

Results and Discussion

The Species Threat Abatement and Restoration metric

We develop and analyse a "Species Threat Abatement and Restoration" (STAR) metric, which evaluates the potential benefit for threatened species of actions to reduce threats and restore habitat. Like the Red List Index [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Measuring global trends in the status of biodiversity: Red list indices for birds[END_REF] , STAR is derived from existing data in the IUCN Red List and is intended to help address an urgent need. STAR is spatially explicit, enabling identification of specific threat abatement and habitat restoration opportunities in particular places, which if implemented, could reduce species extinction risk to levels that would exist without ongoing human impact.

Abatement of threats to species encompasses reduction in threat intensity and/or action to mitigate the impacts of threats. Positive population and/or distribution changes, and resulting reduction of species extinction risk, have been documented in response to threat abatement [START_REF] Sodhi | Conservation successes at micro-, mesoand macroscales[END_REF] . STAR assumes that for the great majority of species (see Supplementary Discussion) complete alleviation of threats would reduce extinction risk through halting decline and/or permitting sufficient recovery in population and distribution, such that the species could be downlisted to the IUCN Red List category of Least Concern. We recognize that complete threat reduction is difficult, incremental conservation gains will need to be tracked at the species level [START_REF] Brown | Spatial Variation in Abundance[END_REF] and species recovery will vary across a species' range [START_REF] Brown | Spatial Variation in Abundance[END_REF] .

For each species, a global STAR threat-abatement (START) score is defined. This varies from zero for Least Concern species to 100 for Near Threatened, 200 for Vulnerable, 300 for Endangered and 400 for Critically Endangered (using established weighting ratios [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Measuring global trends in the status of biodiversity: Red list indices for birds[END_REF] ). The sum of START values across all species represents the global threat abatement effort needed for all species to become Least Concern. START scores can be disaggregated spatially, based on the area of habitat currently available for each species in a particular location (as a proxy for population proportion). This shows the potential contribution of conservation actions in that location to reducing the extinction risk for all species globally. The local START score can be further disaggregated by threat, based on the known contribution of each threat to the species' risk of extinction (see Methods). This quantifies how actions that abate a specific threat at a particular location contribute to the global abatement of extinction risk for all species.

The STAR metric also includes a complementary habitat restoration component to reflect the potential benefits to species of restoring lost habitat. During the UN Decade on Ecosystem Restoration (2021-2030), restoration efforts are likely to expand. The STAR restoration component applies a similar logic to the STAR threat-abatement component, but for habitat that has been lost and is potentially restorable ('restorable area of habitat'). The STAR restoration component does not make assumptions about the extent of habitat restoration required for individual species, but instead quantifies the potential contribution that habitat restoration activities could make to reducing species' extinction risk. For a particular species at a particular location, the STAR restoration (STARR) score reflects the proportion that restorable habitat at the location represents of the global area of remaining habitat for that species. Importantly, a multiplier is applied to STARR scores to reflect the slower and lower success rate in delivering benefits to species from restored habitat compared with conserved existing habitat [START_REF] Jones | Restoration and repair of Earth's damaged ecosystems[END_REF] . Again, STARR scores can be disaggregated by threat, and summed across species within the location.

STAR is intended to provide a metric to underpin the establishment of science-based targets as explicit contributions from individual actors towards the post-2020 biodiversity framework, by allowing assessment of actions and locations according to their potential ability to deliver towards international conservation targets. Individual spatially-based START and STARR scores, for all species present in a particular location or country, represent a proportion of the global opportunity to reduce species' extinction risk through threat abatement and restoration respectively. For each species, the total START score could be achieved by the complete abatement of all threats in remaining habitat, or an equivalent value of the STAR metric can be achieved by a combination of threat abatement in remaining habitat and restoration of lost habitat (with concomitant threat abatement therein). The metric can support establishment of science-based targets by a range of actors across spatial scales. By enabling governments and non-state actors to quantify their potential contributions, STAR, along with other tools, could facilitate achievement of global policy goals, notably the species component of the Sustainable Development Goals and the expected post-2020 Global Biodiversity Framework. STAR uses existing publicly available datasets: species' extinction risk categories and threats available from the IUCN Red List 6 (or, for country endemics not yet assessed globally, from national red lists), and species' area of habitat estimated using species' ranges, habitat associations, and elevation limits, along with digital elevation models and current and historical land cover maps (here, we used back-cast land cover maps of the distribution of habitat pre-human impact, as in [START_REF] Strassburg | Global priority areas for ecosystem restoration[END_REF] ).

To demonstrate the utility of STAR, we calculated global STAR scores for those groups of terrestrial vertebrate species that are comprehensively assessed on the IUCN Red List, i.e. threatened and Near Threatened species of amphibians, birds and mammals globally (n=5,359).

Potential to reduce species extinction risk

Globally, the greatest contribution that could be made to reduce the extinction risk of these groups is tackling threats from annual and perennial non-timber crop production, which account for 24.5% of the global START score (Figure 1). A further 16.4% is contributed by logging and wood harvesting.

There are likely to be specific targets for reducing agriculture and forestry threats in the post-2020 framework [START_REF] Cbd | Zero draft of the Post-2020 Global Biodiversity Framework[END_REF] and applying STAR quantifies the large potential contribution that mitigating these threats could make to the goal for species conservation. Appropriate activities to deliver on such targets range along a continuum from land sharing through to land sparing [START_REF] Phalan | Reconciling Food Production and Biodiversity Conservation: Land Sharing and Land Sparing Compared[END_REF] .

STAR can be used in combination with existing policy and planning tools to quantify the potential contribution of action targets towards species conservation outcomes. The proposed post-2020 framework includes an action target for the protection of sites of particular importance to biodiversity [START_REF] Cbd | Zero draft of the Post-2020 Global Biodiversity Framework[END_REF] . Key Biodiversity Areas [START_REF] Iucn | A Global Standard for the Identification of Key Biodiversity Areas, Version 1.0[END_REF] , which include Important Bird & Biodiversity Areas [START_REF] Donald | Important Bird and Biodiversity Areas (IBAs): the development and characteristics of a global inventory of key sites for biodiversity[END_REF] and Alliance for Zero Extinction sites [START_REF] Ricketts | Pinpointing and preventing imminent extinctions[END_REF] , correspond to such sites. Key Biodiversity Areas so far cover 8.8% of the terrestrial surface (www.keybiodiversityareas.org; identification is ongoing), but already capture 47% of the global START score for the vertebrate groups analysed. They represent large proportions of some national START scores: >70% in Mexico and Venezuela, and >50% in Madagascar, Ecuador, the Philippines and Tanzania.

START scores can also support target-setting at national and sub-national scales to help meet international policy goals. The control and eradication of invasive species forms one of the CBD's proposed post-2020 action targets [START_REF] Cbd | Zero draft of the Post-2020 Global Biodiversity Framework[END_REF] . New Zealand has already set a "Predator Free 2050" goal that aims to eradicate three invasive mammal species by 2050. New Zealand contributes 0.8% to the global START score for the three vertebrate groups included in this study. Achieving the Predator Free 2050 goal would contribute 30% of the total START score for New Zealand, amounting to 0.2% of the global START score.

All countries contribute towards the global START score, but scores are highly skewed, with a few countries having high START scores and most having low scores for the vertebrate groups analysed (Figure 2a; Extended Data 1). The highest scoring countries are located in biodiverse regions with many threatened endemic species [START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF] : Indonesia contributes 7.1% of the global START score, Colombia 7.0%, Mexico 6.1% Madagascar 6.0% and Brazil 5.2%. These top five countries contribute 31.3% of the global START score. In contrast, the lowest scoring 88 countries together contribute only 1% of the global START score. This does not imply that these low-scoring countries have negligible species conservation responsibilities: the global decline in even common species indicates that all countries must act to reverse the degradation of nature and restore the diversity and abundance of species and integrity of ecosystems [START_REF] Ceballos | Biological annihilation via the ongoing sixth mass extinction signaled by vertebrate population losses and declines[END_REF] as well as preventing extinctions at a national scale. Moreover, most countries have a Red List Index [START_REF]SDG Indicators[END_REF] , or an equivalent, quantifying their progress or failure in reducing global extinction risk of assessed species relative to their national responsibility for global species conservation; STAR provides a means to guide the reduction of extinction risk and so assist all countries in meeting national species conservation targets.

At the global level, we estimated that an equivalent to 55.9% of the global START score for vertebrates could, theoretically, be achieved by restoring lost habitat within current range (Figure 1). Ecosystem restoration objectives have been identified in many national biodiversity strategies for the CBD, as well as in many countries' commitments under the Bonn Challenge, and as part of Nationally Determined Contributions under the United Nations Framework Convention on Climate Change (UNFCCC). The STAR metric has the potential to support restoration initiatives alongside species conservation targets by quantifying the potential benefit to particular species of restoring habitat in specific places [START_REF] Strassburg | Strategic approaches to restoring ecosystems can triple conservation gains and halve costs[END_REF] (Figure 2b). Restoration may be particularly important for some species, including those assessed under Red List sub-criteria D/D1 (with a very small population), or Bac (with a small range with severe fragmentation, plus extreme fluctuations). For species uniquely assessed under these criteria (2.8% of those included in this study), threat abatement alone is unlikely to eliminate extinction risk, and so might need to be complemented by restoration in order to achieve Least Concern status (see Supplementary Discussion). Moreover, depending on habitat loss and threat type, restoration of habitat may be beneficial for a larger proportion of threatened species.

Application of STAR at the landscape scale

We tested the landscape-scale application of the STAR metric in the southern part of Bukit Tigapuluh landscape, in central Sumatra, Indonesia (Figure 3a). The Bukit Tigapuluh Sustainable Landscape and Livelihoods Project is a sustainable commercial rubber initiative. The study area (approximately 88,000 ha) includes a 5 km buffer, which is set aside to support local livelihoods, wildlife conservation areas and forest protection and restoration, and two ecosystem restoration areas, which form a conservation management zone that protects the Bukit Tigapuluh National Park from encroachment.

The total START score for the study area represents 0.2% of the START score for Sumatra, 0.04% of that for Indonesia and 0.003% of the global START. The major threats are from annual and perennial non-timber crops, logging and wood harvesting, and collecting terrestrial animals (Figure 3b). The proximate causes of these pressures in the project area are rubber cultivation, oil palm cultivation, industrial logging, subsistence wood cutting, and hunting. STAR analysis shows that areas with the greatest potential to contribute to species conservation through threat mitigation are in remaining natural habitat close to the National Park, with a small area of high potential also to the west, where the relatively small distribution of the Orbiculus leaf-nosed bat (Hipposideros orbiculus) overlaps the site (Figure 3A). Additionally, due to recent forest loss, 47% of the START score for the study area could be achieved through habitat restoration (i.e. STARR). Investment in these management actions has the potential to deliver these quantified contributions to national and global biodiversity targets.

Operationalisation and future development

The STAR metric makes use of the best available data, producing results that are relevant to policy and practice. However, there is scope for future refinement as the underlying data improves. Here, the STAR metric covers amphibians, birds and mammals globally, constituting a well-studied but small proportion of taxonomic diversity (see Extended Data 2 and 3 for variation among taxa). STAR can be expanded to other taxonomic groups, including freshwater and marine species, as data become available (reptiles, cacti, cycads, conifers, freshwater fish and reef-building corals are among the groups imminently available for incorporation). Global application of STAR will require comprehensive assessment of taxonomic groups, testing of the transferability of the STAR metric assumptions among taxa as Red List coverage expands, and further development of methods to calculate area of habitat. Area of habitat calculation does not currently capture spatial variation in species' population density, which will be important for many species [START_REF] Brown | Spatial Variation in Abundance[END_REF] ; such data have not been gathered on a global scale yet and could be incorporated as available.

The completeness of threat data in the IUCN Red List is uneven but is continually improving. The STAR metric does not currently reflect spatial variation in threat magnitude within species' ranges; more broadly there is a lack of information on the spatial distribution of threats [START_REF] Joppa | Filling in biodiversity threat gaps[END_REF] . Most species included in this study have relatively small ranges; total current area of habitat is <5,000 km 2 for 55%, <1,000 km 2 for 33%, and within a single country for 66% (Extended Data 4). This prevalence of small ranges may reduce the significance of spatial variation in threats. Nevertheless, threats may vary spatially for any species not confined to a single location, and there is scope to use threat mapping to inform the likely spatial distribution of threats [START_REF] Joppa | Filling in biodiversity threat gaps[END_REF] . Application of STAR at the landscape or site level, for instance to set targets or identify management actions (e.g. Figure 3), will therefore require verification of the presence and distribution of threatened species (including restorable habitat), and assessment of the distribution and severity of threats. Such assessments should examine synergies among threats [START_REF] Brook | Synergies among extinction drivers under global change[END_REF] and potential leakage in response to threat mitigation [START_REF] Ewers | Estimates of reserve effectiveness are confounded by leakage[END_REF] ; context-specific processes that cannot be accounted for in the global metric. At the global level, periodic recalibration of STAR scores based on updated Red List assessments will be necessary to account for the emergence of new threats [START_REF] Sutherland | Future novel threats and opportunities facing UK biodiversity identified by horizon scanning[END_REF] and the changing extinction risk of species [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Measuring global trends in the status of biodiversity: Red list indices for birds[END_REF] as well as the inclusion of additional groups not previously assessed. Where uncertainty cannot be reduced in a given application of STAR, sensitivity analyses (for example see Methods section below) can be used to explore and quantify uncertainty. For a summary of sources of uncertainty and approaches to quantify and reduce uncertainty in STAR calculations, see Supplementary Table 1 and Extended Data 5. STAR alone does not identify conservation priorities, but could be harnessed alongside other data, for example on costs and benefits of conservation actions, to support conservation planning and prioritisation [START_REF] Moilanen | Spatial Conservation Prioritization: Quantitative Methods and Computational Tools[END_REF] . The STAR metric identifies what, in principle, needs to be done for species to achieve Least Concern status; however, the feasibility of abating threats will depend on the specific threat and context. Threats such as climate change or infectious disease cannot be reduced significantly through local action only. However, they may be mitigated through measures such as (for climate change) conservation translocations or increasing habitat connectivity to support distribution shifts [START_REF] Mcguire | Achieving climate connectivity in a fragmented landscape[END_REF] . Habitat restoration is a particularly important strategy to mitigate climate change impacts, and STAR quantifies the contribution of habitat restoration in combination with threat abatement to reducing species' extinction risk. Appropriate prioritisation [START_REF] Strassburg | Strategic approaches to restoring ecosystems can triple conservation gains and halve costs[END_REF] and local planning are needed to identify the spatial urgency, feasibility and expected benefit from restoration. Furthermore, while in principle complete delivery of START would achieve downlisting to Least Concern for the great majority of species, the varying reasons for raised extinction risk reflected in different Red List criteria are -necessarily -not conveyed when creating a standardised index (see Supplementary Discussion). Moreover, delivery of START does not equate to long-term species recovery. Other tools exist to support more ambitious goals, notably the IUCN Green Status of Species, which is complementary to STAR in its data inputs and requirements, scope and audience, and in that it assesses progress towards species' full recovery and ecological functionality [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF] . Over time, the Green Status approach may also provide additional data that could enhance STAR, but the urgent need is to quantify how actions can contribute to achieving species goals using already available data.

Finally, countries with high START scores face intense pressures on biodiversity, but these pressures often originate from beyond their borders. This is owing both to global-scale threats, such as climate change and infectious disease, and to market forces operating beyond national boundaries. Global-scale and transboundary threats cannot necessarily be addressed within habitats, but require concerted actions within and among countries, for example through national commitments to reduce greenhouse gas emissions, implementation of biosecurity measures to prevent the spread of invasive alien species, and enforcement of restrictions imposed by the Convention on International Trade in Endangered Species of Wild Fauna and Flora (CITES). STAR scores can indicate the need for such actions, which then require implementation in a non-local context. International trade in commodities and services is an important and growing driver of biodiversity loss. Some countries with high consumption per capita (e.g. in Northern Europe) have relatively low in-country START scores, suggesting that it is important to consider embodied (i.e. full lifecycle) as well as direct impacts for products and processes. For example, Germany contributes only 0.01% of the global START score, but is the third biggest importer of biodiversity impacts through commodity supply chains [START_REF] Lenzen | International trade drives biodiversity threats in developing nations[END_REF] . There is therefore urgent need to advance supply chain analyses [START_REF] Lenzen | International trade drives biodiversity threats in developing nations[END_REF] in order to quantify and account for the biodiversity impacts driven by end-consumers.

Policy implications

STAR can be disaggregated to identify and quantify the opportunities for both countries and nonstate actors to contribute their shares of action towards a global species conservation goal. In doing so, STAR can support a framework analogous to the UNFCCC's 2015 Paris Agreement, which provided a new model for global environmental governance. Uptake of this model among non-state actors has been promising, with 476 companies [START_REF]Science Based Targets initiative[END_REF] and 98 cities 31 (as of 5 October 2020) establishing science-based targets for greenhouse gas emissions reduction at the level necessary to deliver the Paris Agreement. Moreover, the approach will doubtless be applied to analyse whether the sum of Nationally Determined Commitments, set by individual countries, is indeed sufficient to hold climate change to 1.5-2°C [START_REF] Fawcett | Can Paris pledges avert severe climate change?[END_REF] . STAR provides an equivalent metric to guide establishment of science-based targets for conserving species-level biodiversity. STAR will need to sit alongside equivalent metrics for ecosystems (e.g. [START_REF] Keith | Scientific Foundations for an IUCN Red List of Ecosystems[END_REF] ) and potentially also genetic diversity [START_REF] Laikre | Post-2020 goals overlook genetic diversity[END_REF] , consistent with the CBD's definition of biological diversity, in supporting the establishment of science-based targets in the post-2020 framework.

The application of STAR would have important implications for conservation and sustainable development. In terms of the post-2020 biodiversity framework, it could facilitate the establishment of targets to mitigate threats to the level necessary to halt and reverse biodiversity loss. Such an approach could be extended across the other biodiversity-related conventions, with, for example, the Ramsar Convention on Wetlands calibrating its global target as the START score for wetland biodiversity. It could similarly be extended to inform delivery of the biodiversity-related targets of Sustainable Development Goals 14 (Life below water) and 15 (Life on land); aligned with the role of the Red List Index [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Measuring global trends in the status of biodiversity: Red list indices for birds[END_REF][START_REF] Rodrigues | Spatially Explicit Trends in the Global Conservation Status of Vertebrates[END_REF] as an official indicator. Finally, and perhaps most fundamentally, the approach provides a common metric for the conservation of threatened species that stands to incentivise voluntary contributions from actors beyond national governments: cities, states and provinces, the private sector, and indigenous and local communities. The increasing recognition of the importance of polycentric governance in addressing global environmental challenges [START_REF] Ostrom | Polycentric systems for coping with collective action and global environmental change[END_REF] suggests that such broadening of contributions is not only desirable, but essential and urgent.

Methods

Data inputs

Calculation of the STAR metric requires information on species' extinction risk, threats, and current and restorable Area of Habitat 36 (AOH). Species' extinction risk categories and threat classification data were obtained for amphibians, birds and mammals from the IUCN Red List version 2019-3 [START_REF]The IUCN Red List of Threatened Species[END_REF] .

These taxonomic groups are comprehensively assessed on the IUCN Red List (meaning >80% of the taxonomic group assessed; recent taxonomic splits mean that 16% of amphibian species have been recently recognised and not yet assessed for the IUCN Red List) and range maps are available for nearly all species. Species assessed as Near Threatened and threatened (Vulnerable, Endangered and Critically Endangered) were included in the analysis. Least Concern species were not included, as threats are not coded for the majority of Least Concern species on the IUCN Red List. Data Deficient species were also excluded, as these are too poorly known to classify their extinction risk, and they often lack data on threats, habitats, elevation and/or distribution [START_REF]The IUCN Red List of Threatened Species[END_REF] .

The IUCN/Conservation Measures Partnership Threat Classification Scheme is hierarchical [START_REF]Threats Classification Scheme[END_REF][START_REF] Salafsky | A Standard Lexicon for Biodiversity Conservation: Unified Classifications of Threats and Actions[END_REF] , and threats to species are classified at the most detailed level possible. For each threat to each species, the scope (proportion of the global population impacted), severity (rate of decline driven by the threat within its scope) and timing (past, ongoing, or future) of the threat are coded as part of the Red List assessments. Threats that were recorded as "past and unlikely to return" were excluded from the analysis. Threats that were not expected to cause a population decline were also excluded; these were threats with a severity scored as "no decline", and threats with a combination of severity scored as "negligible decline" and scope scored as either the minority or majority of the species' distribution (see explanation in STAR calculation below and Supplementary Table 2). Consequently, any species recorded as suffering only from threats that were not expected to cause a population decline were excluded from the analysis.

The extent of current and restorable Area of Habitat 36 (AOH) for species was determined using 5 km resolution species' AOH rasters. We calculated species current AOH following [START_REF] Strassburg | Global priority areas for ecosystem restoration[END_REF] . We used the European Space Agency "Climate Change Initiative" (ESA CCI) land use and cover maps [START_REF]European Space Agency "Climate Change Initiative[END_REF] from 2015, with 300 x 300 m pixel size. The ESA CCI original 37 land cover classes were reclassified into ten major classes (forests, wetlands, arid ecosystems, natural grasslands, shrublands, croplands, cultivated grasslands, rock and ice, and urban areas), and then matched to the habitat classes from IUCN Red List assessments. Species' range maps [START_REF]The IUCN Red List of Threatened Species[END_REF][START_REF]BirdLife International and Handbook of the Birds of the World. Bird species distribution maps of the world[END_REF] were then overlaid with land cover and digital elevation maps to map the area of habitat within each species' range, constrained by the species' elevation range (from the IUCN Red List). Species' range map polygons are coded for presence and origin [START_REF]Mapping Standards and Data Quality for the IUCN Red List Categories and Criteria[END_REF] ; we excluded from current AOH parts of species' ranges where the species' presence was recorded as Extinct, Possibly Extant or Presence Uncertain, leaving only parts recorded as Extant, Probably Extant (a category that is being phased out) and Possibly Extinct. We also excluded parts of each species' range where the species' origin was recorded as Introduced, Vagrant or Origin Uncertain, thus leaving only parts recorded as Native, Reintroduced or present through Assisted Colonisation.

Original area of habitat represented the extent of original ecosystem types before human impact (i.e. the land cover before conversion to croplands, pasturelands or urban areas; following [START_REF] Strassburg | Global priority areas for ecosystem restoration[END_REF] ). ESA CCI land use and cover maps from 1992 were used to inform back-casting of the extent of original ecosystem types. Species range maps were then overlaid with this back-cast land cover and with digital elevation maps to map the original area of habitat within each species range. For the purposes of this analysis, the extent of species original AOH was constrained to within individual species' range maps according to the IUCN Red List; these range maps largely reflect current range limits due to a lack of consistent information across all species on their historical, recently extirpated range. As with current AOH, we included in original AOH only parts of each species' range where the species' origin was recorded as Native, Reintroduced or present through Assisted Colonisation according to the origin coding of the IUCN Red List assessments [START_REF]Mapping Standards and Data Quality for the IUCN Red List Categories and Criteria[END_REF] . We also excluded parts of each that species. Scores were calculated using the most detailed threat classification available and then aggregated to higher levels in the threat classification scheme by summing scores.

The STAR restoration score (R) for the potential contribution of habitat restoration (and threat abatement therein) at location i for threat t is calculated as:

𝑅 𝑡,𝑖 = ∑ 𝐻 𝑠,𝑖 𝑊 𝑠 𝐶 𝑠,𝑡 𝑀 𝑠,𝑖 𝑁 𝑠 𝑠
where Hs,i is the extent of restorable AOH for species s at location i, expressed as a percentage of the global species' current AOH, and Mi is a multiplier appropriate to the habitat at location i to discount restoration scores. Here, we use a global multiplier of 0.29 based on the median rate of recovery from a global meta-analysis [START_REF] Jones | Restoration and repair of Earth's damaged ecosystems[END_REF] assuming that restoration has been underway for ten years (the period of the post-2020 outcome goals).

The STAR metric assumes that abating all current and plausible future threats in species' current AOH would stabilise species populations and distributions, such that they would be downlisted to Least Concern (with few exceptions: see Supplementary Discussion).

START and STARR scores were mapped at the 5 km grid cell resolution. For each species, the START score per grid cell was calculated by multiplying each species' total START score by the proportion of the species' current AOH in the grid cell. The STARR score per grid cell was calculated by multiplying the species' total STARR score by the proportion of species' restorable AOH present in the grid cell.

Global maps of total START and STARR scores were produced by summing the respective score maps across all species. For presentation, maps were aggregated to the 50 km resolution by summing scores across cells.

We calculated START scores for 196 regions (195 recognised countries, including their dependencies, plus Antarctica). The proportion of species' current AOH within each country was estimated by overlaying species' current AOH with polygons of national boundaries. The STAR calculation was then applied at the country level.

START scores were calculated for Key Biodiversity Areas. The boundaries of Key Biodiversity Areas already formally identified were obtained from the World Database of Key Biodiversity Areas 43 on 13 January 2020. Polygon data were available for 15,782 sites. START scores for terrestrial sites were calculated by overlaying the Key Biodiversity Area polygons onto the global 5 km grid cell resolution rasters of START scores, which were generated as described above.

In order to relate START scores to conservation policy in the example of New Zealand, we calculated START scores per invasive species. Where species have been assessed as threatened by invasive nonnative/alien species or diseases, the invasive threat has been documented at genus or species level in 85% of cases. In the case of New Zealand, the invasive threat was documented in 97% of cases, allowing the START score for invasive species to be calculated at the level of the individual species.

Calculation of START and STARR scores for Bukit Tigapuluh landscape in Indonesia was carried out at a higher spatial resolution than for the global STAR analysis, in order to provide more detailed maps at the landscape scale. The Bukit Tigapuluh landscape is dominated by forest, and so only species associated with forest according to the IUCN Red List habitat classification scheme [START_REF][END_REF] were included.

We used species distribution polygons [START_REF]The IUCN Red List of Threatened Species[END_REF][START_REF]BirdLife International and Handbook of the Birds of the World. Bird species distribution maps of the world[END_REF] combined with Global Land Analysis and Discovery maps of forest cover change [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] at the 30 m resolution to calculate species' current AOH and restorable AOH at the location. Based on available forest change data, current AOH was calculated from forest cover in the year 2018, while restorable AOH was forest lost since 2000. Species AOH was clipped to species' elevation limitations using species' elevation data from the IUCN Red List combined with a digital elevation map [START_REF]USGS EROS Archive -Digital Elevation -Global 30 Arc-Second Elevation[END_REF] . Thus, species' current and restorable AOH were calculated at 30 m resolution for the extent of the Bukit Tigapuluh landscape. Species' global AOH (at 5 km resolution, as described above) was then used to calculate the proportion that species' current and restorable AOH at the location represented of the species global current AOH.

All data processing and analyses were carried out using the software R [START_REF]R: A language and environment for statistical computing[END_REF] .

Sensitivity analyses to inform STAR development

The sensitivity of START scores to variation in the metric's various components was explored in order to inform the development of the metric. All sensitivity analyses were carried out using data on birds, due to the completeness of their Red List assessment data (see Supplementary Methods for detailed methods).

Threat scope and severity data are largely complete for birds but missing for the majority of amphibian and mammal species; this information is recommended but not required documentation for Red List assessments, so is not consistently documented. Approaches to dealing with missing scope and severity data were explored (see Supplementary Methods and Extended Data 6) and it was concluded that using the median of possible values of scope and severity to replace missing data was a suitable approach (see also Supplementary Discussion).

The effect of applying equal steps weighting, log steps weighting and no weighting to species Red List categories was investigated (Extended Data 7a-b). Equal steps weighting was selected, rather than relative extinction risk weights, for the same reasons as articulated for the Red List Index [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Using Red List Indices to measure progress towards the 2010 target and beyond[END_REF] , as relative extinction risk (log step) weights would make START values overwhelmingly dominated by threats to Critically Endangered species, whereas the 'equal steps' weights lead to START scores representing opportunities to improve the extinction risk of a much wider set of threatened and Near Threatened species. Importantly, equal steps align the weighting of species in STAR metric to the weighting of species in the well-established RLI.

The effect of giving greater weight to larger proportions of species' current AOH per location and lower weight to smaller proportions of species current AOH per location [START_REF] Saout | Protected Areas and Effective Biodiversity Conservation[END_REF] was explored (Extended Data 7c), with a view to reflecting the role of habitat configuration in species' persistence. However, this was not adopted, in order to maintain the scalability and additivity of the metric.

Figures

  Figures

Figure 1 .

 1 Figure 1. The contribution to the global STAR threat-abatement score of different threats and the

Figure 2 .

 2 Figure 2. Global STAR scores for amphibians, birds and mammals at a 50-km grid cell resolution for

Figure 3 .

 3 Figure 3. STAR results for the Bukit Tigapuluh Sustainable Landscape and Livelihoods Project. The

The Nature Conservancy, Arlington, Virginia 22203, USA.

Acknowledgements

We acknowledge the funding by the Luc Hoffmann Institute, Vulcan Inc, Synchronicity Earth, and Global Environment Facility, and the support from the Conservation International GEF Project Agency. We thank Lisa Genasci (ADM Foundation) for providing technical guidance for the field tests, James Deutsch for facilitating funding, and Resit Akçakaya, Julia P.G. Jones, Joseph R. Bennett and an anonymous reviewer for comments on the manuscript. LM is funded by Newcastle University and IUCN, LPK by the National Research Foundation, Singapore (NRF-RSS2019-007), MB by The Rufford Foundation, and AR by the "Investissements d'avenir" programme, managed by the French National Research Agency (ANR) (ANR-10-LABX-14-01).

The authors declare the following competing interests: E.C. Regan works for Springer Nature, the publishers of Nature Ecology & Evolution.

species' range where the species' presence was recorded as Possibly Extant or Presence Uncertain.

However, for original AOH, we additionally included parts of species' ranges where the species was recorded as Extinct, for all species for which this information was available. Species restorable AOH was then calculated as the difference between original and current AOH. A total of 5,359 species (2,055 amphibians, 1,957 birds and 1,347 mammals) were included in the analysis based on the availability of the necessary data.

STAR calculation

To calculate STAR values, we used data on the extent of species' current AOH and restorable AOH, extinction risk (IUCN Red List category) and the relative contribution of each threat to the species' extinction risk. The STAR metric is calculated for all Near Threatened and threatened species present at a location. 'Location' in this context represents any spatially defined area; the maximum size is the entire area of the globe, while the minimum practical size is determined by the spatial resolution of habitat maps available for species. The STAR threat-abatement score (T) for a location (i) and threat (t) is calculated among all species as:

where Ps,i is the current Area of Habitat 36 (AOH) of each species (s) within location (i), expressed as a percentage of the global species' current AOH; Ws is the IUCN Red List category weight of species s (Near Threatened = 1, Vulnerable = 2, Endangered = 3 and Critically Endangered = 4 [START_REF] Butchart | Improvements to the Red List Index[END_REF][START_REF] Butchart | Measuring global trends in the status of biodiversity: Red list indices for birds[END_REF] ); C is the relative contribution of threat [START_REF] Salafsky | A Standard Lexicon for Biodiversity Conservation: Unified Classifications of Threats and Actions[END_REF] t to the extinction risk of species s; and Ns is the total number of species at location (i). The relative contribution of each threat to the species' extinction risk was calculated as the percentage population decline from that threat (derived from the product of severity and scope for that threat in each species' IUCN Red List assessment as in [START_REF] Garnett | Metrics of progress in the understanding and management of threats to Australian birds[END_REF] ; see Supplementary Table 2) divided by the sum of percentage population declines from all threats to

The percentage population decline expected to be caused by a particular threat was the median value from within the range of expected percentage population declines for the particular combination of scope and severity scores (which represent bands of possible values). The effect of varying the expected percentage population decline within this range for each combination of scope and severity scores was explored, and the metric was found to be robust to this variation (Extended Data 8).

Data availability statement

Species' extinction risk category, threat data, elevation limitations, habitat associations and distribution polygons are publicly available under specified Terms and Conditions of Use from the IUCN Red List website [START_REF]The IUCN Red List of Threatened Species[END_REF] . KBA boundaries are available from the World Database of Key Biodiversity Areas [START_REF]Digital boundaries of Key Biodiversity Areas from the World Database of Key Biodiversity Areas[END_REF] , again under specified Terms and Conditions of Use. The European Space Agency "Climate Change Initiative" (ESA CCI) land use and cover maps are available at www.esa-landcover-cci.org [START_REF]European Space Agency "Climate Change Initiative[END_REF] .

Forest cover change maps are available from https://glad.umd.edu [START_REF] Hansen | High-Resolution Global Maps of 21st-Century Forest Cover Change[END_REF] . Digital elevation maps are available from https://earthexplorer.usgs.gov [START_REF]USGS EROS Archive -Digital Elevation -Global 30 Arc-Second Elevation[END_REF] . Global STAR threat-abatement and restoration scores for amphibians, birds and mammals at 50 km grid cell resolution are available in TIFF file format as Supplementary Data 1 and 2.

Author contributions

LM led on analysis, development and manuscript drafting. LAB, TMB, SHMB, FH and PJKM led on FH & PJKM contributed equally to conception and coordination.

Author information

Reprints and permissions information is available at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Louise Mair (louise.mair@newcastle.ac.uk).

Competing Interests Statement

Supplementary Information

Supplementary Information is available for this paper. availability of restorable habitat. Thus, the total STAR restoration score per grid cell (b) is the sum of 688 the individual species' STAR restoration scores per grid cell across all species included in this study. 689


Maya Topf

Edina Rosta

Gregory R Bowman

Massimiliano Bonomi
email: mbonomi@pasteur.fr

Editorial: Experiments and simulations: a pas de deux to unravel biological function

Understanding the molecular mechanisms used by biological systems to perform their functions is often essential to rationally target associated diseases. In many cases, the determination of the three-dimensional structure of these systems provides precious insights. However, it is more often the interplay between structural and dynamical properties that determines the behavior of complex systems [START_REF] Henzler-Wildman | Dynamic personalities of proteins[END_REF][START_REF] Orozco | A theoretical view of protein dynamics[END_REF]. While both experimental and computational methods are invaluable tools to study protein structure and dynamics, limitations in each individual technique can hamper their predictive capabilities [START_REF] Schneidman-Duhovny | Uncertainty in integrative structural modeling[END_REF]. On one hand, determining structural models solely from experimental data is challenging as these data often come from time and ensemble-averaged measurements over conformationally heterogeneous states, provide sparse and sometimes ambiguous information, and are always subject to random and systematic errors. On the other hand, structural models determined by computational approaches such as protein structure prediction methods and/or molecular dynamics (MD) are limited by the inaccuracies of the force fields used as well as by the challenge of exhaustively sampling the conformational landscape of complex systems [START_REF] Bonomi | Principles of protein structural ensemble determination[END_REF].

Combining experiments and simulations is therefore a successful strategy to overcome the limitations of the individual approaches and to accurately characterize the behavior of biological systems [START_REF] Rout | Principles for Integrative Structural Biology Studies[END_REF][START_REF] Bottaro | Biophysical experiments and biomolecular simulations: A perfect match?[END_REF].

The goal of this Research Topic is to present some representative examples of synergistic use of experimental and computational techniques aimed at accurately characterizing the structure, dynamics, and ultimately function of biological systems. This collection of 14 articles explores different areas of experimental-computational integration: the use of computational approaches to assist the interpretation of existing experimental data or to predict the outcome of new measurements, the experimental validation of computational predictions, and the incorporation of experimental data to drive and/or refine molecular simulations. A wide spatial spectrum of systems will be covered, encompassing ordered and disordered peptides and proteins, small-molecules interacting with proteins, protein complexes, nucleic acids, and entire cells. The integration of molecular simulations with different types of experimental data will be illustrated, including cryo-electron microscopy (cryo-EM) and tomography, superresolution microscopy, Nuclear Magnetic Resonance (NMR) spectroscopy, biochemical, and Small Angle X-ray Scattering (SAXS) data.

One of the areas of research in which computationally approaches, and particularly MD simulations, have traditionally been used to complement experimental measurements is the prediction and/or rationalization of the effect of mutations on the structure and dynamics of biological systems. Cnrjar et al. use MD simulations to shed light into the effect of a singlepoint mutation in the outer lipid-facing helix (M4) of the 5-HT3A pentameric ligand-gated ion channels. The mutation of a tyrosine (Y441) in this area has been experimentally shown to inhibit the function of the receptor. The MD simulations reported in this paper reveal a network of interactions that connects Y441 to the ion channel hydrophobic gate on helix M2, thus rationalizing the effect of the mutation that has experimentally been observed. Ochoa et al.

build a set of scoring matrices using structural observables extracted from MD simulations to predict the effect of single-point mutations on peptide binders to the Major Histocompatibility Complex (MHC) class II receptors. The method developed integrates sequence, structural and dynamical information and can be used to guide the design of novel peptide binders to the MHC class II receptors.

Another area in which molecular simulations can complement experiments is the characterization of conformations that are often difficult to observe directly, such as short-lived conformational states and disordered motifs. Nierzwicki and Palermo illustrate the case of the CRISPR-Cas9 genome editing machinery, of which the catalytically active structure has been predicted by MD simulations and subsequently validated by high-resolution cryo-EM data. This paper also provides an overview of computational approaches that can be used to refine both single-structure models and conformational ensembles given a cryo-EM map. Lambrughi et al. use enhanced-sampling MD simulations to study the Ubiquitin Interacting Motif (UIM), a conserved, highly-dynamic segment used by several multi-domain proteins to interact with ubiquitin. While existing X-ray data could not capture the structural heterogeneity of UIM, the MD ensembles revealed an equilibrium between ordered and disordered states, in agreement with NMR chemical shifts data.

Computational techniques have become over the years an invaluable tool in the drug discovery field [START_REF] Brogi | Editorial: In silico Methods for Drug Design and Discovery[END_REF]. In this Research Topic, De Felice et al. present an in silico approach named "Computational Profiling for GPCRs" that repurposes a GPCR-binding ligand for a different GPCR. The method is tested on 3 different GPCR receptors and validated using docking calculations and pharmacological data. Sgrignani and Cavalli use computational docking and molecular simulations to investigate the mode of binding of bromhexine to the transmembrane serine protease TMPRSS2, an enzyme involved in the activation of several coronaviruses, including SARS-CoV-2. Their analysis reveals the existence of an allosteric pocket involved in the binding of bromhexine to TMPRSS2 and in its inhibition.

  Finally, Prerovská et al. perform MD simulations to predict the structure of the complex formed by ß-D-galactosyl Yariv reagent and oligo ß-D-(1→3)-galactan and ultimately to shed light into the structural basis of arabinogalactan protein precipitation by Yariv. Over the last decade, a novel class of methods that incorporate experimental information into molecular simulations has flourished. These so-called integrative or hybrid modeling approaches use experimental data to either guide or refine a posteriori [8] molecular simulations in order to determine individual structural models or conformational ensembles consistent with the available information. These techniques often based on i) Bayesian frameworks to properly balance the information provided by different types of experiments with prior, physico-chemical knowledge of the system; and ii) the Maximum Entropy Principle to resolve the ambiguity of determining conformational distributions based on the knowledge of ensemble-averaged experimental observations [4, 6, 9, 10]. In this Research Topics, 6 examples of this type of hybrid computational-experimental approaches are illustrated. The papers by Ahmed et al., Spill et al., and Paissoni et al. present different ways to characterize conformational ensembles of dynamic systems by integrating MD simulations with SAXS data. Ahmed et al. use the refinement Bayesian/Maximum Entropy (BME) technique to determine structural ensembles of α-synuclein starting from MD simulations performed with different force fields. Spill et al. propose a Bayesian weighting approach for SAXS data coupled with a selection of an ensemble of minimal size to characterize the conformational heterogeneity of a tandem of domains from the protein PTPN4. Paissoni et al. perform an exhaustive analysis of the statistical precision of the metainference technique coupled with metadynamics using as test system the chignolin peptide. Voelz et al. present a review of the features, advantages over other integrative approaches, and shortcomings of their Bayesian Inference of Conformational Populations (BICePs) method. Gaalswyk et al. illustrate how their Modeling Employing Limited Data (MELD) approach can be used to determine protein structural ensembles using sparse NMR data combined with physical modelling. Finally, Vakili and Habeck introduce a Bayesian technique to address the problem of reconstructing, in tomography, a 3D structure from 2D views along unknown random

directions.

The majority of applications of combined computational-experimental techniques presented in this Research Topic involve biological systems ranging in size, from small molecules to protein complexes. However, integrative approaches can in principle be applied to larger spatial scales, provided that appropriate experimental data are available. One illustrative example presented here is the work of Bianchi et al. In this paper, the authors combine single molecule super-resolution microscopy in cells with computational modeling to study how a small RNA (SgrS) regulates glucose-phosphate stress, or "sugar shock" in E. coli. Their stochastic simulations guided by the in cell experimental data enable the description of the cellular heterogeneity observed in the E. coli sugar shock response network.

To summarize, this Research Topic demonstrates how a synergistic use of experiments and simulations can be a powerful strategy to study structure, dynamics, and function of biological systems across a variety of spatial and temporal scales. These successes have been made possible by the continuous improvements in both experimental and computational techniques, the development of open-source software and web-servers for integrative studies [START_REF] Rout | Principles for Integrative Structural Biology Studies[END_REF], the dissemination of protocols in open-source notebooks and public repositories [START_REF]Promoting transparency and reproducibility in enhanced molecular simulations[END_REF], and the availability of raw data and structural models on public databases, such as the PDB-Dev 1 [START_REF] Burley | PDB-Dev: a Prototype System for Depositing Integrative/Hybrid Structural Models[END_REF].

There is a bright future ahead for integrative studies, especially these days, when a new class of methods based on artificial intelligence, such as DeepMind's AlphaFold [START_REF] Jumper | Highly accurate protein structure prediction with AlphaFold[END_REF] and RoseTTA fold [START_REF] Baek | Accurate prediction of protein structures and interactions using a three-track neural network[END_REF], has joined the game and already taken it by storm, reaching accuracy in structure prediction comparable to experimental techniques.