Alvaro R Gutierrez Falcón

Bruno S Zossi

Hagay Amit

Ana G Elias

Geomagnetic secular variation consequences on the trajectories of radiation belt trapped particles

Keywords: Magnetic fields, Trapped particles, Radiation belts, Geomagnetic field, Trapped particle trajectory, Mirror point

The trajectories of energetic particles trapped by the geomagnetic field, as those composing the Earth's Van Allen radiation belts, are usually defined by three cyclic motions: gyration, bounce along field lines and drift around the Earth, which are all controlled by this field. The geomagnetic dipole, in turn, has been declining at a rate of ∼5% every hundred years since at least ∼1840. Even with the possibility of a recovery without an extreme event, the global field intensity will very probably continue to decrease in the near future with a consequent weakening of our planet's magnetic shield capacity. The expected variations in trapped particle trajectories are analyzed in the present work through an analytical approach considering the observed axial dipolar geomagnetic field component and its secular variation. The variations expected on the mirror point altitude and on the boundary of Störmer forbidden zone are assessed along the period 1900-2020. The structures here analyzed could approximate plausible radiation belt changes for a continuously weakening geomagnetic dipole which might have numerous consequences for technologies that operate in space.

Introduction

The Earth's magnetosphere varies in timescales from minutes, or less, to thousands of years in response to changes in solar conditions and the geomagnetic field [START_REF] Siscoe | Long-term aspects of magnetospheric variability[END_REF]. Regarding long-term timescales over a century, averaging all variability sources, the geomagnetic field secular variation becomes the main forcing of trends that may remain. Over such long timescales, even the Gleissberg cycle which may be present in forcings of solar origin (e.g. [START_REF] Feynman | The Centennial Gleissberg Cycle and its association with extended minima[END_REF]) is expected to be averaged out. The main Earth's magnetic field has been decaying at a rate of ∼5% per century since at least 1840 [START_REF] Olson | Changes in earth's dipole[END_REF][START_REF] Finlay | Historical variation of the geomagnetic axial dipole[END_REF][START_REF] Finlay | Gyre-driven decay of the Earth's magnetic dipole[END_REF] or even earlier [START_REF] Gubbins | Fall in Earth's Magnetic Field Is Erratic[END_REF][START_REF] Poletti | Continuous millennial decrease of the Earth's magnetic axial dipole[END_REF]. The intensity of the global field will most likely continue to decrease in the near future [START_REF] Aubert | Geomagnetic forecasts driven by thermal wind dynamics in Earth's core[END_REF][START_REF] Sanchez | Predictions of the geomagnetic secular variation based on the ensemble sequential assimilation of geomagnetic field models by dynamo simulations[END_REF]) with a consequent weakening of our planet's magnetic shield [START_REF] Heirtzler | The Future of the South Atlantic Anomaly and implications for radiation damage in space[END_REF][START_REF] Tarduno | Subterranean clues to the future of our planetary magnetic shield[END_REF].

Changes in the geomagnetic field have a major impact on trapped particles in Earth's Van Allen radiation belts. In radiation belt models, trapped particle fluxes are mapped in the (B, L) magnetic coordinate system [START_REF] Mcilwain | Coordinates for mapping the distribution of magnetically trapped particles[END_REF]. The shell parameter L denotes the dipole-like shell where the particle is trapped and corresponds to the equatorial distance to the field line in Earth radii units (R E = 6371 km), while B is the magnetic intensity at which the particle is locally mirroring. [START_REF] Lindstrom | B-L Space and Geomagnetic Field Models[END_REF] were among the first to study the consequences of the Earth's magnetic field changes on trapped particles composing the radiation belts. Their study was mainly focused on comparing the B-L space for 0.20 < B < 0.24 (in Gauss units) and 1.2 < L < 1.8 estimated with different geomagnetic field models. When particle losses or redistributions during a drift period are negligible, the flux and spectra are the same for all points having the same B and L. [START_REF] Lindstrom | B-L Space and Geomagnetic Field Models[END_REF] analyzed the time variation of B-L trace in altitude for different longitudes. They obtained an altitude decrease from 1965 to 1975 for particles at the same B and L values due to the variation of the non-dipolar components of the field. If the field would have been purely dipolar, with a decreasing dipole moment, the altitude of particles at the same B and L would remain unchanged with time. In fact, they analyzed the South Atlantic Anomaly region, SAA, where the differences between the actual Earth's magnetic field and a dipolar field are strongest [START_REF] Bloxham | Geomagnetic secular variation[END_REF][START_REF] Terra-Nova | Relating the South Atlantic Anomaly and geomagnetic flux patches[END_REF]. This height decrease is then obtained as a result of using a time-dependent field to assess the B-L space.

In a later work, [START_REF] Heckman | Response of trapped particles to a collapsing dipole moment[END_REF] analyzed the secular decrease of the axial dipolar moment of the Earth's magnetic field, M z , in connection again to B-L space but this time through the theoretical analysis of a particle's motion in an ideal dipolar field considering the adiabatic invariants. They obtained an inward drift of particles accompanied with an increase in energy and particle flux, a decrease of the mirror point height, and the invariance of the pitch angle α (angle between the particle's velocity direction and the magnetic field line) and of the mirror point latitude, λ m . [START_REF] Farley | Effects of the Secular Magnetic Variation on the Distribution Function of Inner-Zone Protons[END_REF] considered the effects of the secular decrease of M z on the energetic protons of the Van Allen inner radiation belt in two ways: over the diffusion coefficient in the equation of the distribution function, and through the induced electric field that drives these protons radially inward and accelerates them to higher energies. Solving the case of equatorially trapped particles, they obtained an increase in the distribution function at the geomagnetic equator for energetic inner radiation belt protons. [START_REF] Schulz | Secular Magnetic Variation and the Inner Proton Belt[END_REF], in a similar analysis to Heckman and [START_REF] Heckman | Response of trapped particles to a collapsing dipole moment[END_REF], also demonstrated that the secular decrease of M leads to a contraction of the adiabatic drift shells and to an acceleration, or energization, of the trapped particles. They used the field intensity at the magnetic equator at the Earth's surface instead of M z , which was 0.31 Gauss at the time, and obtained a 0.05% decrease per year in L that means a drift shell contraction. They also demonstrated that a drift shell secular contraction occurs at constant α and λ m , and an increase of the kinetic energy.

In a different approach, [START_REF] Pu | Drift Shell Tracing and Secular Variation of Inner Radiation Environment in the SAA Region[END_REF] designed a drift shell tracing method (DSTM) and obtained results consistent with the previous mentioned works. They showed that the inductive electric field associated with the long-term variation of the magnetic field of the Earth drives the trapped particles inward increasing the fluxes at low L values. The DSTM, in addition to the dipolar term, also takes into account the secular changes of multipole terms which reinforce the deformation of the L-shells and provide additional contributions to the long term variation of the radiation environment, especially in the SAA.

Another approach to predict the variability of some magnetosphere characteristics in terms of the geomagnetic field is through scaling relations in terms of M z and the dynamic pressure of the solar wind P dyn , which lead to first-order approximations of the average geomagnetic field effects. A strong assumption for scaling relations validity is that the system reacts in a self-similar way to changes in the dipole strength. This can only be valid as long as the internal field is dominated by its dipolar part and if the dipole axis remains more or less aligned with the rotation axis, as emphasized by [START_REF] Vogt | Modelling the paleomagnetosphere: Strategy and first results[END_REF].

The magnetopause position, R mp , even though not directly linked to radiation belts, is a key magnetosphere parameter which serves as a characteristic measure of its size and an external limit for charged particle populations, such as radiation belts, the ring current and the plasmasphere. It scales with M z and P dyn according to [START_REF] Siscoe | Two magnetic tail models for 'Uranus[END_REF][START_REF] Siscoe | The paleomagnetosphere[END_REF][START_REF] Saito | The Earth's palaeomagnetosphere as the third type of planetary magnetosphere[END_REF])

R mp ∝ M z 1/3 P -1/6 dyn (1)
This equation can be derived analytically based on the equilibrium condition between the dynamic pressure of the solar wind and the magnetic pressure of the magnetosphere at the subsolar point on the magnetopause, assuming that the thermal solar wind and magnetosphere pressures are negligible, and the magnetic pressure of the solar wind is also negligible with respect to the two pressures considered. [START_REF] Siscoe | The paleomagnetosphere[END_REF] also deduced that the plasmapause position, R pp , decreases with M z slower than R mp , thus the fraction of the magnetosphere occupied by the plasmasphere increases with decreasing M z . [START_REF] Schulz | Paleomagnetospheric radial diffusion[END_REF] analyzed scaling laws for the radialdiffusion coefficients, D , that affect the Earth's radiation belts, in particular inner radiation belt protons, for a time varying axial dipolar field configuration. Two mechanisms were considered for magnetospheric radial diffusion, both conserving adiabatic invariants. One involves magnetic impulses caused by fluctuations in the solar wind pressure, denoted by the diffusivity D (m) . The other involves electrostatic impulses caused by fluctuations in the rate of energy dissipation at the magnetopause and neutral sheet, denoted by the diffusivity D (e) . Neglecting the solar modulation, that is considering constant average solar conditions, and considering particles' mirror point at the equator, the scaling relationships for a given (the third adiabatic invariant) were derived analytically [START_REF] Schulz | Paleomagnetospheric radial diffusion[END_REF]:

D (m) αM z
16/3 -6 and D (e) αM z 17/3 -6

(2)

Consequently, both diffusivities (D (m) and D (e) ) undergo roughly the same modulation with M z : for a 50% M z increase D is expected to increase by a factor of ∼10 and for a 50% M z decrease it is expected to decrease by a factor of ∼40. [START_REF] Fei | Radial diffusion and MHD particle simulations of relativistic electron transport by ULF waves in the September 1998 storm[END_REF] studied the radial diffusion mechanism during magnetic storms. They solved a radial diffusion equation to obtain scaling relations for the diffusion coefficients focusing on short-term variations. Their diffusion coefficients are not in complete agreement with those given in Eq. (2) but, nevertheless, they also increase with increasing M z and with decreasing . [START_REF] Vogt | On the location of trapped particle populations in quadrupole magnetospheres[END_REF] analyzed a more extreme scenario, considering a quadrupolar geomagnetic field as a feasible high-altitude morphology during a polarity transition to understand possible features of the paleomagnetosphere and of its trapped particle populations. They focused on a field capability of capturing particles based on geometrical criteria for a pure axial quadrupole configuration. They obtained two separate regions of trapped particles, one on each hemisphere, which led to two radiation belts. [START_REF] Vogt | Modelling the paleomagnetosphere: Strategy and first results[END_REF] highlighted that the validity of these scaling relations depends essentially on whether the magnetopause scaling given by Eq. ( 1) describes correctly R mp variations, and on the whole system actually behaving self-similarly with respect to changes of the dipole strength. This last condition may be violated for strong southward interplanetary magnetic field, B z , and high geomagnetic activity levels. However, solar wind average steady behavior on the timescale of interest in secular or paleo studies may guarantee self-similarity. Under a stronger additional assumption, a scaling equation for the ring current extent, R R , is deduced which can be roughly written as

R R α M z 1/4
(3) [START_REF] Glassmeier | Concerning long-term geomagnetic variations and space climatology[END_REF], even though not treating directly trapped energetic particles in the magnetosphere in their derivation of Dst index scaling, explicitly showed that the volume of the magnetospheric region where particle trapping is possible, V RC , scales with M z as

V RC α R mp 3 α M z (4)
and derived R pp scaling, neglecting solar wind effect variations, as

R pp α M z -5/6
(5)

which is opposite to the trend suggested by [START_REF] Siscoe | The paleomagnetosphere[END_REF]. However, [START_REF] Glassmeier | Concerning long-term geomagnetic variations and space climatology[END_REF] also noticed that, since R mp and R pp have opposite variation with M z , for extremely low M z values the plasmapause position may be larger than the magnetopause distance, which is "unreasonable". Therefore, the scaling derived is only applicable for a certain range of M z values.

The scaling relation for R mp was also analyzed by [START_REF] Zieger | Scaling relations in the paleomagnetosphere derived from MHD simulations[END_REF] through MHD simulations of the paleomagnetosphere varying M z at different values of B z . They found a power law with scaling exponents different than 1/3 (the theoretically expected one), where the extent of the deviation was found to be controlled by the magnitude of B z , being greater for strongest values and almost zero when it vanishes. [START_REF] Vogt | Energetic particles in the paleomagnetosphere: Reduced dipole configurations and quadrupolar contributions[END_REF] analyzed the variation of differential particle fluxes during periods of reduced dipole moment together with the energy and rigidity cutoff changes through scaling relations. In the case of particle dynamics constrained by adiabatic invariants, as it is the case of radiation belt trapped particles, they studied their ability to reach the upper atmosphere through the open field lines, but they did not consider the possibility of an inward drift of trajectory shells, together with the consequent energization of trapped particles at that shell, due to M decrease. [START_REF] Stadelmann | Cosmic ray and solar energetic particle flux in paleomagnetospheres[END_REF], in a study of energetic particles in the magnetosphere, presented results of a simulated magnetosphere with a decreasing dipole moment together with the case of an increasing quadrupolar component. They analyzed the trajectories of energetic particles but considered those which finally reach Earth's surface rather than trapped ones. They found that the impact area increases with increasing particle energy or with decreasing magnetic field. [START_REF] Cnossen | The dependence of the coupled magnetosphere-ionosphere-thermosphere system on the Earth's magnetic dipole moment[END_REF] used simulations with the Coupled Magnetosphere-Ionosphere-Thermosphere model. In the case of the magnetopause stand-off distance they found that the theoretical scaling of R mp is too weak (that is, the scaling exponent in Eq. ( 1) should be greater than 1/3). The reason is a strong increase of the ionospheric conductance with decreasing dipole moment, which leads to a strengthening of the field-aligned currents. These currents generate magnetic fields that are opposite to the main field on the nose of the magnetosphere, while they add to it on the flanks. This results in a stronger compression of the nose of the magnetosphere with diminishing dipole strength larger than expected from the theoretical scaling, while the flanks are less compressed than expected. [START_REF] Tsareva | Radiation Belts during a Magnetic Field Reversal[END_REF] studied changes in the shape of the Earth's radiation belts and energy spectra of charged particles in them considering the superposition of axial dipolar and quadrupolar components. A reversal scenario was proposed consisting in a decrease of the dipole component while keeping constant the quadrupole. The structure and dynamics of the radiation belts were determined by the ratio between the sources and losses of charged particles. A gradual reduction of radiation belts was found during the reversal, where the regions of trapped charged particles became asymmetrical with respect to the equatorial plane. Finally, two symmetric trapped regions, one on each hemisphere were obtained for a pure axial quadrupole magnetosphere as in [START_REF] Vogt | On the location of trapped particle populations in quadrupole magnetospheres[END_REF]. [START_REF] Bourdarie | Impact of the Earth's magnetic field secular drift on the low altitude proton radiation belt from years 1900 to 2050[END_REF] studied the slow variation of highenergy (E > 82 MeV) proton flux at 800 km height due to the secular drift of the Earth's main magnetic field and to solar cycle variations for the period 1900-2050. They considered IGRF-12 during 1900-2015 and a field prediction for the period 2015-2050 based on data assimilation using the coupled Earth numerical dynamo model of [START_REF] Aubert | Bottom-up control of geomagnetic secular variation by the Earth's inner core[END_REF]. This proton flux, mostly concentrated over the SAA region, is assessed from the OPAL (Onera Proton Altitude Low) model. Based on the results of [START_REF] Boscher | A new proton model for low altitude high energy specification[END_REF] which showed that considering changes in the maximum equatorial pitch-angle, MePA, is equivalent to taking into account the effect of the SAA drift over time on the trapped particle distribution at L shells with L < 2.5, Bourdarie et al. ( 2019 2019) obtained a modulation of the energetic proton flux amplitude clearly linked to the solar activity cycle and a global decrease along the period 1900-2050 linked to the secular drift of the Earth's core magnetic field. Note that, under a larger dipole tilt, with the axial dipolar component still the strongest, the MePA variation is less pronounced, and that for a perfect axial centered dipole with only its dipolar moment intensity change, there is no variation in MePA at all.

An alternative estimation of the geomagnetic field secular variation effect on trapped particles can be done considering the analysis of charged particles trajectories in a dipolar field [START_REF] Störmer | The Polar Aurora[END_REF]. He showed that there are regions that a charged particle cannot reach, whose size depends on Earth's magnetic field characteristics and the particle properties. These regions' boundary shrinkage or expansion in response to changes in the Earth's field may also serve as an indication of changes in trapped particles trajectories.

In the present work we analyze the consequences of the secular decrease of the axial dipole component of the geomagnetic field over aspects of trapped energetic particles in the Van Allen inner radiation belt that have not been analyzed so far: the mirror point altitude and the inner limit of the shielded region in Störmer theory. For this purpose, we take advantage of the theoretical analysis made by Heckman and [START_REF] Heckman | Response of trapped particles to a collapsing dipole moment[END_REF] and [START_REF] Schulz | Secular Magnetic Variation and the Inner Proton Belt[END_REF] together with the previously derived scaling relations. The geomagnetic field secular variation considered is described in Sect. 2, followed by the theory and assumptions in Sect. 3. Results are presented in Sect. 4 and discussed in Sect. 5.

Geomagnetic field secular variation

The motion of a charged particle, even in a purely dipolar magnetic field, is complex and cannot be solved analytically.

However, there are some key aspects that can be analyzed using fully analytical expressions under this simplified field. In order to be able to do this, we consider a centered dipolar geomagnetic field, aligned with the Earth's rotation axis, i.e. an axial dipole field.

Indeed, the current geomagnetic field is dominantly dipolar. At present, the dipolar component accounts for ∼93% of the mean square total field at the Earth's surface, and the axial dipole for ∼90%. These contributions increase further above the surface, reaching for example, at three R E distance from Earth's center, to ∼99% and ∼97% respectively. In fact, the dipolar geometry has dominated the geomagnetic field through the historical era (e.g. [START_REF] Jackson | Four centuries of geomagnetic secular variation from historical records[END_REF]. In addition, indirect magnetic measurements suggest that the field was dipole dominated during the Holocene [START_REF] Gubbins | Fall in Earth's Magnetic Field Is Erratic[END_REF][START_REF] Poletti | Continuous millennial decrease of the Earth's magnetic axial dipole[END_REF]) and during much of the geological past as well [START_REF] Siscoe | Long-term aspects of magnetospheric variability[END_REF][START_REF] Panovska | One hundred thousand years of geomagnetic field evolution[END_REF][START_REF] Biggin | Quantitative estimates of average geomagnetic axial dipole dominance in deep geological time[END_REF], though archeomagnetic and paleomagnetic field models are far less certain than the historical models which are based on direct measurements. While the axial dipole dominance may decrease and even collapse during geomagnetic reversal transitions, the field recovers to spend most of its time in an axial dipole dominated configuration [START_REF] Siscoe | Long-term aspects of magnetospheric variability[END_REF][START_REF] Biggin | Quantitative estimates of average geomagnetic axial dipole dominance in deep geological time[END_REF]. Considering thus the secular variation of a pure axial dipolar field, which means taking into account only time changes in the axial dipolar moment, M z , is a sensible approximation, especially far from Earth's core.

For this purpose, we considered M z estimated from g 1 0 Gauss coefficient in IGRF-13 [START_REF] Alken | International Geomagnetic Reference Field: the thirteenth generation[END_REF], that is

M z = g 1 0 R E 3 .
Its time evolution is shown in Fig. 1. The full centered dipole moment intensity M is also shown for comparison where M = [(g 1 0 ) 2 + (g 1 1 ) 2 + (h 1 1 ) 2 ] 1/2 R E 3 . Both present almost the same decreasing rate of ∼0.06% per year. The small mean difference between M and M z throughout the period 1900-2020, which does not exceed 2%, reflects the relatively small dipole tilt of ∼10-11.5°with respect to the rotation rate [START_REF] Amit | Geomagnetic dipole tilt changes induced by core flow[END_REF].

Theory and method

General effects of axial dipolar moment variation on trapped particles in radiation belts

The theory used to assess the consequences of geomagnetic field secular variation over the mirror point height and the Störmer forbidden region boundary is mainly described in [START_REF] Heckman | Response of trapped particles to a collapsing dipole moment[END_REF] and [START_REF] Schulz | Secular Magnetic Variation and the Inner Proton Belt[END_REF]. We start from their theoretical analysis of a particle's motion in an ideal axial dipolar field, with moment M z , considering the three adiabatic invariants: μ associated with the motion around field lines (gyromotion) and expressing the conserved magnetic flux enclosed by the particle's gyromotion, J associated with the bouncing motion along the magnetic field between mirror points and implying the conserved total length of the particle trajectory along this line, and associated with the azimuthal drift around the Earth and representing the conserved magnetic flux encompassed by the guiding drift shell. The conservation of the third invariant for an axial dipolar field is given by

= 2πM z L (6)
that is, the ratio L/M z is also conserved. In terms of the rates of temporal change this implies that

1 L dL dt = 1 M z dM z dt (7)
Another way of deducing the L decrease induced by a weakening field is through Faraday's law. A time varying magnetic field B will induce an electric field E ind given by

E ind • dl = ∂B ∂t • dS = ∂A ∂t • dl (8)
where A is the magnetic vector potential, which for an axial dipolar field in polar or cylindrical coordinates has only an azimuthal component, A ϕ , given by

A ϕ = μ o 4π M z r 2 cos λ (9)
where μ o is the vacuum permeability, r is the radial distance and λ the latitude angle. Equation ( 8) results then

E ind = - μ o 4π cos λ r 2 dM z dt φ (10)
This induced field has an azimuthal direction and produces a drift which results from the cross product of E ind and B. For a decreasing M z the drift direction will be inward for positive and negative charges, producing a shell contraction and particles' energization, while it will be outward for an M z increase producing a shell expansion (see Chap. 3 in Roederer and Zhang 2014). Given that it is not easy to estimate the shell shrinking following this reasoning, an alternative approach relies on the third invariant, given by Equation ( 6) which allows an easy estimation of the drift shell contraction for a decreasing M z [START_REF] Heckman | Response of trapped particles to a collapsing dipole moment[END_REF][START_REF] Schulz | Secular Magnetic Variation and the Inner Proton Belt[END_REF].

The variation in M z implies also changes in the magnetic field intensity B along the particle's trajectory. For an axial dipolar field BαM z /L 3 and accounting for (7) results

1 B dB dt = - 2 M z dM z dt (11)
From the conservation of the first and second adiabatic invariants, the invariance of α and of λ m are also deduced. In addition, the following relation holds for the particle's moment, p,

1 p dp dt = - 1 M z dM z dt (12)
In terms of the nonrelativistic kinetic energy, E, this condition can be written as

1 E dE dt = - 2 M z dM z dt (13)

Assumptions

One of the key aspects of a trapped particle's motion in Earth's magnetic field is its bounce motion between mirror points along a field line, and in particular the mirror point height, r m (measured from the Earth's center) or h m = r m -R E (measured from the Earth's surface). In order to obtain an analytical expression for the mirror point altitude in terms of M z (Sect. 3.3) and its time variation considering the theory described in Sect. 3.1, and in order to deduce the changes in the Störmer forbidden region boundary through the time derivative of this boundary equation (Sect. 3.4), we apply the following assumptions:

(1) Conservation of particle's magnetic moment, μ (that is the first adiabatic invariant), in its gyro-motion around the magnetic field line, which is given by

μ = mv 2 ⊥ 2B ( 14
)
where v ⊥ is the particle's velocity component perpendicular to the magnetic field line, m is its mass and B is the magnetic field intensity at the point under consideration. This means that the magnetic field variation in space is very small within a gyro-radius, and that the timescale of variation is much longer than the gyro-period.

(2) Conservation of the kinetic energy. This implies that

v 2 + v 2 ⊥ = constant (15)
where v || is the particle's velocity component parallel to the magnetic field line.

(3) Drift motion around the Earth is negligible with respect to bouncing motion along the field line, which also guarantees the validity of the adiabatic invariants.

Mirror point altitude

Considering the field equation for an axial dipole, the radial distance of the mirror point, r m , is

r m = LR E (cos λ m ) 2 (16)
where λ m is the latitude of the mirror point. This latitude position can be estimated from the pitch angle when the particle is at the magnetic equator, α eq , through [START_REF] Soni | L-shell and energy dependence of magnetic mirror point of charged particles trapped in Earth's magnetosphere[END_REF])

sin α eq 2 = (cos λ m ) 6 1 + 3 (sin λ m ) 2 (17)
From Eqs. ( 16) and ( 17) it can be deduced that r m (or h m ) does not depend neither on the particle's energy nor on the magnetic field intensity, just on the pitch angle and the line upon which it is bouncing given by L.

The temporal variation of r m as a consequence of the geomagnetic field secular variation will be determined by that of L which can be estimated from Eq. ( 7). Differentiating Eq. ( 16) by time and substituting Eq. ( 7) gives

dr m dt = R E (cos λ m ) 2 L M z dM z dt (18)
Assuming that the height, h, at which trapped energetic particles are lost due to their interaction with a dense enough atmosphere is h ≤ 500 km, then all charged particles trapped in shells where r m is such as to lower h m to 500 km or less will be lost or absorbed by the atmosphere. Considering r m at 1900 as an initial value, its time evolution is forward iterated using Eqs. ( 16) and ( 18) in steps of 5 years (that is the time resolution of IGRF g 0 1 coefficient to estimate M z ). Because the re-organization time of the geomagnetic axial dipole is on the order of ∼1000 years [START_REF] Amit | On equatorially symmetric and antisymmetric geomagnetic secular variation timescales[END_REF]), the 5-year time step is clearly small enough to guarantee proper numerical resolution. [START_REF] Störmer | The Polar Aurora[END_REF] showed that there are regions that a charged particle cannot access. By analyzing the Hamiltonian of the motion of a charged particle in an axial dipolar magnetic field (that is with azimuthal symmetry), he reduced the problem to the motion in a potential field where the potential function provides qualitative information about the resulting motion [START_REF] Stern | Charged particle motions in a magnetic field that reduce to motions in a potential[END_REF]. Although there are two constants of motion, the energy and the azimuthal component of the canonical momentum, they are not enough to provide an analytical solution for the motion. In a breakthrough contribution, [START_REF] Störmer | The Polar Aurora[END_REF] noticed that the constancy of azimuthal canonical moment reduced the Hamiltonian to that of a motion in a two-dimensional potential. By qualitative arguments based on this potential, he deduced the existence of trapped orbits in the dipole field and of forbidden regions whose boundaries can be deduced analytically from the Hamiltonian or Lagrangian equation, as is the case of Eq. ( 19).

Störmer forbidden region boundary

The size of these regions depends on Earth's magnetic field characteristics, that is M z for the field here considered, and some particle properties, which are its charge, q, mass, m, and energy, E. The boundary of this region is defined by the equation for its distance from the Earth's center, r, in terms of the latitude angle, λ [START_REF] Störmer | The Polar Aurora[END_REF][START_REF] Shepherd | Störmer theory applied to magnetic spacecraft shielding[END_REF])

r = μ o 4π M z q √ 2Em (cos λ) 2 1 + 1 + (cos λ) 3 (19)
The root factor corresponds to the Störmer length, C st .

A decrease in M z clearly induces an approach of this boundary towards the Earth's surface. Considering also that E increases with M z decrease according to Eq. ( 13), it would enhance the boundary approach towards the Earth. Taking the temporal derivative of r, and considering the temporal variations of M z and E, the following equation is obtained

dr dt = 1 2 r 1 M z dM z dt - 1 4 r 1 E dE dt = r 1 M z dM z dt (20)
Equation ( 19) cannot be readily solved for the time evolution of r because E is unknown. However, Eq. ( 20) is independent of E hence can be used to forward iterate r in time. Finally, we normalize the solution by the initial value of r at 1900 to obtain a non-dimensional Störmer forbidden region boundary which is independent of any choice of an initial E.

Results

Mirror point altitude

Figure 2 shows the time variation of λ m and h m , for three different initial (i.e. at 1900) altitudes corresponding to three different initial values of L: 1.16 (∼1000 km above Earth's surface), 1.5, and 2. The 500 km mirror point height level is shown as a dashed black line, to indicate that at lower heights charged particles are lost in the atmosphere. It can be clearly seen that as M z decreases with time h m approaches the atmosphere, and that this approach is slower for higher λ m . It is fastest for particles bouncing close to the equator, that is at low λ m , but the trapped particles perform their trajectories in this case, for a given L value, much higher. This can be deduced from Eq. ( 6), since for a fixed L value, a lower λ m results in a cos(λ m ) factor closer to 1 and thus a higher r m .

For larger L values, the particle can mirror at higher latitudes. This is logical considering that the Earth has a finite volume, and for a dipolar field assuming there is no atmosphere surrounding the Earth, the maximum attainable λ m would correspond to the invariant latitude, , of a given L at the Earth surface, that is

cos = 1 L (21)
In other words the maximum attainable λ m is the latitude where a magnetic field line touches the Earth. For L = 1.16, 1.5 and 2, = 22.4, 35.3 and 45 degrees respectively, which corresponds to the maximum λ m for 1900 in each panel of Fig. 2. As M z decreases, L decreases also and consequently , or the maximum attainable λ m (for a dipolar field assuming that a particle can reach the Earth's surface as the closer mirror distance) is also observed to decrease. That is the downward slope with time noticed for each height level in Fig. 2.

Inner limit of the shielded region in Störmer theory

Figure 3 shows the boundary of the forbidden region, in a meridional plane, considering M z in 1900 (solid), for a proton of 2 GeV energy as an example. Its numerically iterated 20), is also shown (dashed). Taking into account the axial symmetry and also the symmetry with respect to 19) and its temporal variation forward iterated from Eq. ( 20) the equatorial plane, this figure illustrates a quadrant of the meridian plane where the abscissa corresponds to r cos(λ) and the ordinate to r sin(λ).

Figure 4 presents the time variation of the distance of the Störmer forbidden region from the Earth's center at the equator, which is ( √ 2 -1)C st . Even though C st , used here as a proxy for the trapped particles drift shell contraction, is too distant for some regular particles trapped in the Earth's radiation belts [START_REF] Lemaire | The effect of a southward interplanetary magnetic field on Störmer's allowed regions[END_REF]) as compared to their theoretical drift shells, it clearly shows how the shield provided by the geomagnetic field approaches the Earth's surface as M z decreases. In the 120 years studied period, the distance of this shield to Earth's surface decreased by ∼7% (Fig. 4), at the same rate as the decrease of the geomagnetic axial dipole (Fig. 1). In addition, the well-known non-linear undulations with typical periodicity of ∼60 years observed in the geomagnetic axial dipole secular variation [START_REF] Olson | Changes in earth's dipole[END_REF][START_REF] Buffett | Geomagnetic fluctuations reveal stable stratification at the top of the earth's core[END_REF][START_REF] Buffett | Evidence for MAC waves at the top of Earth's core and implications for variations in length of day[END_REF][START_REF] Finlay | Gyre-driven decay of the Earth's magnetic dipole[END_REF][START_REF] Huguet | Geomagnetic Dipole Changes and Upwelling/Downwelling at the Top of the Earth's Core[END_REF]) correlate well with the undulations in the inner limit of the shielded region (Fig. 4).

Discussion

The motion of charged particles trapped in radiation belts by the Earth's magnetic field is complicated even for the axial dipolar field approximation. It can be simplified for a slowly varying purely axial dipole field by decomposing the total motion into three components: gyration around a guiding field line, bounce along this line between mirror points, and drift in the azimuthal direction around the Earth. For a static dipolar magnetic field, and in the absence of any loss mechanism, these particles should remain stably trapped indefinitely following their trajectories in the magnetosphere. The time variation of the Earth's field is expected to cause changes in the particle's trajectory characteristics not only due to the dependence on the field, but also due to the force related to the induced electric field associated with the timedependent magnetic field [START_REF] Roederer | Dynamics of Magnetically Trapped Particles[END_REF]. This can be easily taken into account if the field time variation is much slower than the corresponding timescale of each type of motion, as is the case for the secular variation here considered. This can be done by considering the third adiabatic invariant which leads to Eq. (2). In the axial dipolar field assumed in the present work, a decrease of the dipole moment causes the guiding drift shells to contract, and this change in drift shell is accompanied by a change in particle energy given by Eq. ( 13).

Considering the actual secular change in M z associated to the axial dipolar field, we calculated for the first time the temporal evolution of the mirror point height and the boundary of the Störmer forbidden region. Both quantities decrease, as qualitatively expected from a drift shell contraction, at the same percentage rate as M z does. Their absolute changes, however, depend not only on M z but on L and α eq (or λ m ) in the case of the mirror point height, and on energy and particle type as well in the case of the boundary of the Störmer forbidden region.

If we compare the variations in h m and the boundary of the Störmer forbidden region to the magnetopause approach which can be estimated from the scaling relation in Equation (1), a much slower approach is obtained for the latter. In fact, R mp rate of change with M z , considering steady solar wind conditions, would be 1 R mp

dR mp dt = 1 3 1 M z dM z dt (22)
that is one third of h m and C st rate of change. This means that, if we were able to extrapolate our hypothesis without limits, there would be a period of time with a magnetosphere cavity surrounding the Earth but not acting as a shield.

An interesting aspect to consider in future works would be the comparison between h m decrease induced by the secular geomagnetic field dipolar component weakening, and the shrinking of the upper atmosphere due to the cooling induced by greenhouse gases increasing concentration during the last decades [START_REF] Lastovicka | Long-Term Trends in the Upper Atmosphere[END_REF]). There will be particles whose mirror points altitudes' lowering would result in their absorption and loss, unless it is smaller than the atmosphere contraction due to its long-term cooling.

  ) introduced MePA predicted variation in the OPAL model. In fact, Boscher et al. (2014) observed a clear, almost linear, pitch angle increase with the SAA drift along the period 1978-2008. Bourdarie et al. (

Fig. 1

 1 Fig. 1 Time evolution of the dipolar moment of Earth's magnetic field in Am 2 obtained from degree 1 Gauss coefficients in IGRF-13 for an axial dipole (M z ; solid line) and a full centered dipole (M; dashed line)

Fig. 2

 2 Fig. 2 Variation of the mirror point height, h m [km], corresponding to a given mirror point latitude, λ m [°], with the Earth's axial dipole moment variation during 1900-2020, for initial L(t = 1900) = 1.16 (a), 1.5 (b) and 2 (c). The axial dipole moment is estimated from the time-dependent g 0 1 Gauss coefficient in IGRF-13. Dashed black lines indicate 500 km height

Fig. 3 Fig. 4

 34 Fig. 3 Boundaries of Störmer forbidden region estimated with Eq. (19) considering M z in 1900 (solid line) and with Eq. (20) to estimate the boundary in 2020 (dashed line). (Note: origin is the Earth's center; black surface is the Earth)

Acknowledgements Argentina's Consejo Interuniversitario Nacional provided financial support to A.R. Gutierrez Falcon through the research fellowship Beca-EVC-CIN. A.G. Elias and B.S. Zossi also acknowledge PIUNT E642 and PIP 2957. IGRF is freely available at https://www.ngdc.noaa.gov/IAGA/vmod/igrf.html.

Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Author Contribution All authors contributed to the study and design of this work. The first draft of the manuscript was written by A.R. Gutierrez Falcon and A.G. Elias and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declarations

Ethical statement All authors comply with ethical standards.

Conflict of Interest

The authors declare they have no financial interests.

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Yves Bourne
email: yves.bourne@afmb.univ-mrs.fr

Gerlind Sulzenbacher

Zoran Radić

Rómulo Aráoz

Morgane Reynaud

Evelyne Benoit

Armen Zakarian

Denis Servent

Jordi Molgó

Palmer Taylor

Zoran Radiae

Pascale Marchot
email: pascale.marchot@univ-amu.fr.

Marine macrocyclic imines, pinnatoxins A and G: structural determinants and functional properties to distinguish neuronal α7 from muscle α1 2 βγδ nAChRs

Keywords: acetylcholine binding protein, competitive antagonism, crystal structure, macrocyclic imine, dinoflagellate toxin, marine phycotoxin, nicotinic acetylcholine receptor, pinnatoxin, nAChR subtype, seafood poisoning, subunit interface

published or not. The documents may come

Introduction

The "rapid-acting" macrocyclic imines toxins consist of a recently identified and widely distributed group of phycotoxins with increasing prevalence in warming oceanic environments [START_REF] Stivala | Synthesis and biology of cyclic imine toxins, an emerging class of potent, globally distributed marine toxins[END_REF]. Among them, pinnatoxins (Fig. 1) and pteriatoxins represent one of the largest subgroups of 27-membered macrocyclic imine phycotoxins first detected in coastal Japan and isolated from the digestive glands of the bivalve mollusks Pinna attenuata and Pinna muritica (pinnatoxins A-D) [START_REF] Zheng | The isolation and bioactivities of pinnatoxin[END_REF][START_REF] Uemura | Pinnatoxin A: a toxic amphoteric macrocycle from the Okinawan bivalve Pinna muricata[END_REF] and Pteria penguin (pteriatoxins A-C) [START_REF] Takada | Structural determination of pteriatoxins A, B and C, extremely potent toxins from the bivalve Pteria penguin[END_REF]. Novel congeners (pinnatoxins E-G) were rapidly isolated from a range of shellfish samples from Australia and New Zealand, and pinnatoxin-G (PnTx-G) was proposed to represent a biosynthetic progenitor for all pinnatoxins, as exemplified by metabolic oxidation of PnTx-G into pinnatoxin-A (PnTx-A) after ingestion by shellfish [START_REF] Selwood | Isolation, structural determination and acute toxicity of pinnatoxins E, F and G[END_REF]. PnTx-A and PnTx-G, along with iso-spirolide C and 20-methyl spirolide G, are widespread in shellfish samples from Norwegian waters [START_REF] Rundberget | Pinnatoxins and spirolides in Norwegian blue mussels and seawater[END_REF]. Pinnatoxins and pteriatoxins contain the 7-membered spirocyclic imine structure common to most spirolides, but an unusual 6,5,6-bis-spiroketal moiety replaces the spirolide 5,5,6 ring system (reviewed in [START_REF] Molgó | Cyclic imine toxins: chemistry, origin, metabolism[END_REF][START_REF] Otero | Cyclic imines: chemistry and mechanism of action: a review[END_REF][START_REF] Stivala | Synthesis and biology of cyclic imine toxins, an emerging class of potent, globally distributed marine toxins[END_REF]). They also contain a bulky bridged 5,6-bicycloketal substructure instead of a smaller allylic alcohol linker and a quaternary stereocenter at the core of the spiroimine ring, along with varying functional group substitutions. Fatty acid ester metabolites of PnTx-G were recently identified in mussels from Eastern Canada and these esterified forms may display higher toxicity [START_REF] Mccarron | Identification of pinnatoxins and discovery of their fatty acid ester metabolites in mussels (Mytilus edulis) from Eastern Canada[END_REF].

Numerous shellfish-poisoning instances have been attributed to these bioactive alkaloids, and PnTx-A has been proposed as one of the major compounds responsible for a poisoning outbreak in China and Japan [START_REF] Zheng | The isolation and bioactivities of pinnatoxin[END_REF]. When administered intraperitoneally or orally to mice, all (+)-pinnatoxin congeners so far identified induce rapid lethality associated with central neurological symptoms [START_REF] Selwood | Isolation, structural determination and acute toxicity of pinnatoxins E, F and G[END_REF] and flaccid paralysis [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF] suggesting action at Ca 2+ -channels and/or acetylcholine receptors. They also display the highest oral toxicity among cyclic imine toxins [START_REF] Munday | Acute toxicity of pinnatoxins E, F and G to mice[END_REF]. In fact, the high oral toxicity of pinnatoxin F compared to the E and G congeners (Fig. 1) has raised recent concerns about possible adverse effects for shellfish consumers [START_REF] Munday | Acute toxicity of pinnatoxins E, F and G to mice[END_REF]. In contrast, the non-natural (-)-PnTx-A is inactive in mouse assays [START_REF] Munday | Toxins: Pharmacology, Physiology and Detection[END_REF]. Identification of successful strategies for the synthesis of (+)-PnTx-A and PnTx-G, and subsequent electrophysiological, competition binding and computational analyses established PnTx-A as a potent antagonist of nicotinic acetylcholine receptors (nAChR), with high selectivity for the human neuronal α7 subtype [START_REF] Stivala | Total synthesis of (+)-pinnatoxin A[END_REF][START_REF] Nakamura | Total synthesis of pinnatoxin A[END_REF][START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF]. In contrast, a synthetic aminoketone derivative of PnTx-A (PnTx-A AK) with an open imine ring shows no action on various nAChR subtypes, a feature supporting the central role of the imine ring for potent nAChR antagonism previously observed for the related toxins, 13-desmethyl spirolide C (SPX) and gymnodimine A (GYM) [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF][START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF]. Hence, the unusual stability of the imine ring in pinnatoxins has implications for their potent toxicity when administered orally [START_REF] Jackson | Stability of cyclic imine toxins: interconversion of pinnatoxin amino ketone and pinnatoxin A in aqueous media[END_REF].

The nAChRs are prototypical cation-selective, ligand-gated ion channels that mediate fast neurotransmission in the central and peripheral nervous systems [START_REF] Changeux | The nicotinic acetylcholine receptor: the founding father of the pentameric ligand-gated ion channel superfamily[END_REF]. They belong to the Cys-loop superfamily of the ligand-gated ion channels and are formed by distinct combinations of five subunits that confer selectivity in pharmacological properties and regional tissue locations. The diversity in the nAChR subunit subtypes and assemblies is most evident in the central nervous system where up to nine α and three β subunits have been described [START_REF] Novère | The diversity of subunit composition in nAChRs: evolutionary origins, physiologic and pharmacologic consequences[END_REF]. The various receptors used in this study are briefly described in the Supplementary Information (SI) and their subunit sequences are displayed in Suppl. Fig. S1.

The soluble acetylcholine (ACh) binding proteins (AChBP) from mollusks form homopentameric assemblies of subunits homologous to the N-terminal, extracellular ligandbinding domain of the nAChR [START_REF] Brejc | Crystal structure of an ACh-binding protein reveals the ligand-binding domain of nicotinic receptors[END_REF][START_REF] Hansen | Tryptophan fluorescence reveals conformational changes in the acetylcholine binding protein[END_REF][START_REF] Celie | Nicotine and carbamylcholine binding to nicotinic acetylcholine receptors as studied in AChBP crystal structures[END_REF] (Suppl. Fig. S1). In addition to the overall structural features of the subunits, the aromatic side chains that form the ligand binding pocket at the subunit interfaces are well conserved in the nAChR family, with greater variation of residues at the complementary (or (-)) face than at the principal (or (+)) face of each interface. The binding pocket of AChBP possesses all the functional residues identified in the nAChR ligand-binding domain, and its extension towards apical and "membrane" directions along the interface provides multiple means for selective accommodation of nicotinic full and partial agonists and competitive antagonists [START_REF] Celie | Nicotine and carbamylcholine binding to nicotinic acetylcholine receptors as studied in AChBP crystal structures[END_REF][START_REF] Hansen | Structures of Aplysia AChBP complexes with nicotinic agonists and antagonists reveal distinctive binding interfaces and conformations[END_REF][START_REF] Hibbs | Structural determinants for interaction of partial agonists with acetylcholine binding protein and neuronal α7 nicotinic acetylcholine receptor[END_REF]. Overall, agonists recognize a "core agonist signature motif" central to the binding pocket, whereas the larger antagonists extend towards the apical or the membrane directions, resulting in opening of loop C on the (+) face, and often in greater subtype selectivity than seen for agonists. Novel α7/AChBP chimeras harboring a fully functional α7-like ligand-binding site show a marked enhancement in ligand affinity correlated with conservation of the overall architecture, thus justifying their use for investigating ligands selective for the neuronal α7 subtype [START_REF] Li | Ligand-binding domain of an α7-nicotinic receptor chimera and its complex with agonist[END_REF][START_REF] Nemecz | Creating an α7 nicotinic acetylcholine recognition domain from the acetylcholinebinding protein: crystallographic and ligand selectivity analyses[END_REF].

Previously we documented the mode of binding of two closely related macrocyclic imine toxins, SPX and GYM, to Aplysia californica and Lymnaea stagnalis AChBPs (A-and L-AChBP) and showed why they are extremely potent nicotinic antagonists with broad specificity toward various nAChR subtypes [START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF]. In particular, crystal structures of the SPX-and GYM-A-AChBP complexes revealed neatly imbedded phycotoxins within the AChBP nest of aromatic side chains that accommodates all nicotinic agonists and competitive antagonists. Predominant interactions were observed with residues in loop C from the (+) face of the subunit interface but only sparing interactions with residues in loop F on the (-) face, consistent with loop F dictating nAChR subtype specificity. Moreover, the protonated imine nitrogen in the toxins exhibits tight hydrogen bonding with the backbone carbonyl oxygen of loop C residue Trp147 on the (+) face in AChBP. The requirements for hydrogen bonding orientation and distance (~3.0 Å) constrain the orientation of the toxin core centered within the pocket. More recent investigations involving several nAChR subtypes and muscarinic receptors along with complementary binding and molecular docking studies confirmed the nAChRs as the primary targets of SPX and GYM with no significant binding to a wide range of other receptors [START_REF] Hauser | Comparison of acetylcholine receptor interactions of the marine toxins, 13desmethylspirolide C and gymnodimine[END_REF].

To delineate experimentally the molecular determinants that govern the unique selectivity of PnTx-A for the neuronal α7 nAChR and to explore the capacity of PnTx-G for subtype selectivity, we conducted a comprehensive pharmacological and structural study of chemically synthesized PnTx-A, PnTx-G and the PnTx-A AK derivative, relative to the neuronal α7, α4β2 and α3β2 and muscle-type α1 2 βγδ nAChRs and to A-and L-AChBP and the previously developed α7/AChBP chimera III with enhanced α7 binding properties [START_REF] Nemecz | Creating an α7 nicotinic acetylcholine recognition domain from the acetylcholinebinding protein: crystallographic and ligand selectivity analyses[END_REF]. Our data, when compared with those reported for the SPX and GYM toxins, point to the bicyclic EF-ketal ring found in the pinnatoxins (Fig. 1) as a novel binding determinant for mediating polar versus non-polar interactions with loop F that confers nAChR subtype selectivity, and extend toxicological insights for these prevalent marine biotoxins.

Results and Discussion

Functional and binding characteristics and nAChR subtype selectivity of PnTx-A and PnTx-G Functional analyses of PnTx-A and PnTx-G were performed by voltage-clamp recording of ACh-evoked currents in oocytes expressing the neuronal α7 or α4β2 nAChRs or in oocytes microtransplanted with the muscle-type nAChR. PnTx-A blocked the current in both neuronal nAChRs without affecting the desensitization rates, although with a ~300-fold greater potency (denoted by the IC 50 values) and apparent irreversibility toward α7 [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF] (Fig. 2, Suppl. Fig. S2A, Table 1). PnTx-G also blocked these currents but, compared with PnTx-A, it displayed ~50-fold lower potency towards α7 and no selectivity between the two neuronal subtypes. Both PnTx-A and PnTx-G also blocked the AChevoked current in the membrane-bound muscle-type α1 2 βγδ nAChR, with nanomolar potencies equivalent to that of PnTx-G for the two neuronal nAChRs. Hence, the inhibitory potency of PnTx-A depends on the nAChR subtype, with IC 50 values in the 0.1-10 nM range and a α7 > α1 2 βγδ > α4β2 ranking in selectivity, while those values of PnTx-G lie in the same nanomolar range for all three nAChR subtypes. Finally, analysis of the open-ring PnTx-A AK derivative led to IC 50 values in the sub-millimolar range for the three nAChR subtypes, consistent with the initial report of their absence of activity [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF].

Binding analyses were performed in the membrane-bound muscle-type α1 2 βγδ nAChR and in mammalian cells transiently expressing the neuronal α7 (as a α7/5-HT3 chimera), α4β2 and α3β2 nAChRs, by competition against radiolabeled derivatives of the competitive antagonist α-bungarotoxin (BgTx) and the full agonist (+/-)-epibatidine (EPI), respectively [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF] (Suppl. Fig. S2B, Table 2). Similar sub-nanomolar binding potencies (denoted by the K i values) of PnTx-A and PnTx-G were obtained for the α7 nAChR, whereas potencies ranging from 10 to100 nM where recorded for the α4β2 and α3β2 subtypes, pointing to selectivity of PnTx-A and PnTx-G for the homopentameric α7 versus heteropentameric neuronal nAChRs. By contrast, on the muscle-type nAChR, PnTx-G interacted with 25-fold higher affinity than PnTx-A, leading to a rank ordering of α7 > α1 2 βγδ > α3β2 ~ α4β2 and α1 2 βγδ > α7 > α3β2 ~ α4β2 in the PnTx-A and PnTx-G specificities, respectively. Analysis of the PnTx-A AK derivative led to K i values greater than 1-10 μM, consistent with the electrophysiological analyses.

Finally, complementary binding analyses were performed using mammalian cells stably expressing the neuronal α7 or α4β2 nAChRs along with a Ca 2+ sensing fluorescent tag [START_REF] Yamauchi | Characterizing ligand-gated ion channel receptors with genetically encoded Ca 2++ sensors[END_REF] (Table 3). PnTxA binding to the α7 nAChR led to a K d value similar to the IC 50 and K i values obtained upon voltage-clamp and competitive binding experiments (Table 1 andTable 2, respectively), whereas PnTx-A binding to the α4β2 nAChR and PnTx-G binding to either receptor led to more divergent values, presumably because of the distinct methodological approaches and time required to reach equilibrium conditions. Yet, for both toxins the same α7 > α4 ranking in selectivity reported above was retained.

Hence, the higher blocking potencies and higher binding affinities of PnTx-A and PnTx-G for the neuronal α7 and muscle-type α1 2 βγδ nAChRs, respectively, compared with the two neuronal heteropentameric nAChRs, should arise from distinguishing structural determinants in pinnatoxins.

Binding characteristics of PnTx-A and PnTx-G with A-and L-AChBPs and a α7/A-AChBP chimera

The sequence differences and distinctive binding affinities of the two AChBPs for the nicotinic ligands render them useful templates for studying ligand binding to various nAChR subtypes (Table 4). Binding of nicotinic agonists and competitive antagonists to A-and L-AChBPs, in affecting the connectivity of the aromatic side chains in the binding pocket, is associated with quenching of the intrinsic Trp fluorescence [START_REF] Hansen | Tryptophan fluorescence reveals conformational changes in the acetylcholine binding protein[END_REF]. Quenching is sufficient to monitor rates of complex formation and dissociation by stopped-flow kinetic instrumentation and to correlate the ratio of dissociation and association rates with the dissociation constants measured by equilibrium titrations with ligand. To ascertain kinetic constants for slow dissociating complexes and confirm the equilibrium constants for the lower affinity complexes, a complementary scintillation proximity assay was also used.

Binding data for PnTx-A and PnTx-G reveal K d values in the 10 -7 M range for L-AChBP but in the 10 -11 -10 -9 M range for A-AChBP, with differences being primarily governed by the dissociation rates (k off ). The association rates approach the diffusion limitation for partly sequestered sites and show smaller differences between AChBP species (Table 4). In fact, PnTx-A binding to A-AChBP, with its dissociation constant below 50 pM and its >3400fold selectivity over L-AChBP, approaches the characteristics of SPX binding kinetics for A-AChBP [START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF], while PnTx-G binding appears only slightly less favorable (420-fold in selectivity). In contrast, binding of the PnTx-A AK derivative to A-and L-AChBP led to K d values in the mM range, confirming the critical role of the cyclic imine to confer high potency binding. However, PnTx-G binding to the α7/A-AChBP chimera mutant III, expected to reflect binding properties intermediate to the α7 nAChR and A-AChBP [START_REF] Nemecz | Creating an α7 nicotinic acetylcholine recognition domain from the acetylcholinebinding protein: crystallographic and ligand selectivity analyses[END_REF], involved 10-fold lower affinity, indicating that sequential mutagenesis cannot yield a full complement of residues to achieve the folding differences between the two proteins.

Hence, both PnTx-A and PnTx-G show higher affinity binding for A-AChBP over L-AChBP, with PnTx-A accounting for the greater selectivity and reflecting a similar preference for neuronal α7 nAChR. These features justify the use of A-AChBP for structural analyses.

Overall view of the crystalline PnTx-AChBP complexes

The crystal structures of PnTx-A and PnTx-G bound to A-AChBP, solved at 1.9 Å and 2.2 Å resolution, respectively, show the compact homopentameric ring assembly of subunits found in all AChBP structures (Fig. 3, Suppl. Table S1) (reviewed in [START_REF] Karlin | Emerging structure of the nicotinic acetylcholine receptors[END_REF][START_REF] Sixma | Acetylcholine binding protein (AChBP): a secreted glial protein that provides a high-resolution model for the extracellular domain of pentameric ligand-gated ion channels[END_REF]). The ligand binding pocket encompasses a nest of five electron-rich aromatic side chains provided by residues Tyr93 in loop A, Trp147 in loop B and Tyr188 and Tyr195 in loop C on the (+) face of the interface, and residue Tyr55 in loop D on the (-) face (Suppl. Fig. S1). The pocket is partially sheltered from the solvent by loop C, which is found at the outer perimeter of the pentamer and harbors at its tip the vicinal disulfide bridge linking the vicinal Cys190 and Cys191 residues, a signature determinant for nAChR α subunits. The well-defined electron densities reveal fully occupied subunit interfaces with identical orientations of all five PnTx-A or PnTx-G bound within a pentamer (Fig. 3), consistent with their sub-nanomolar affinities and low dissociation rates for A-AChBP (Table 1) (cf rmsd values in the Experimental Procedures). Each toxin molecule, shaped as a slightly curved crescent of overall dimensions 14 × 6 × 6 Å, tightly accommodates the curvature of the subunit interface, resulting in buried surface areas of ~380 Å 2 and ~370 Å 2 at the (+) and (-) faces of the interface, respectively (Fig. 3, Table 3).

Detailed description of the PnTx-AChBP complexes

PnTx-A, with its sub-nanomolar IC 50 , K i and K d values for the α7 nAChR and A-AChBP (Tables 123), shows higher specificity for these receptors over PnTx-G, which in turn appears to be more selective for the muscle-type nAChR. Despite the close resemblance of the two A-AChBP complexes, structural differences contributed by the distinctive R 1 substituent (carboxylate in PnTx-A versus vinyl in PnTx-G, Fig. 1) in the two toxins may well account for the functional differences.

At the subunit interfaces in both complexes, the bulky spirolide skeleton is oriented with its long axis aligned roughly parallel to the five-fold axis of the pentamer (Fig. 3). The 6,5,6 bis-spiroketal three-ring system (rings BCD on Fig. 1), being bulkier than the 6,5,5 bisspiroketal found in SPX, points in an apical direction along the interface while the R 1substitued cyclohexene ring (ring G), located at the opposite end of the molecule, is oriented in the membrane direction of the subunit interface. The bis-spiroketal framework bends toward the central protonated cyclic imine ring (ring A) and comfortably complements the shape of the binding pocket. This orientation of the carbon skeleton ideally positions the cyclic imine centrally to the subunit interface and within hydrogen bonding distance (~2.9 Å for PnTx-A and 2.8 Å for PnTx-G) to the carbonyl oxygen of Trp147 in A-AChBP loop B (Suppl. Fig. S1, Table S2), as found for the corresponding imines in SPX and GYM (~2.9 Å). The two vicinal methyl groups on the cyclic imine, also found in SPX, largely contribute to binding by abutting against the conserved phenol rings of Tyr188 and Tyr195 from loop C. Ideal positioning of the central cyclic imine in the binding pocket is consistent with absence of binding of the open-ring PnTx-A AK derivative to A-and L-AChBPs and the nAChR subtypes assayed (Tables 123).

On the apical side of the interface, the bis-spiroketal system emerges from the macrocyclic framework and abuts against the tip of loop C, where it buttresses the binding interface (Fig. 3). Here, this ring system perfectly complements the non-polar environment of the surrounding residues in A-AChBP, and retains the preferred cis stereochemistry elucidated by NMR analysis and confirmed by total synthesis [START_REF] Chou | Relative stereochemistry of pinnatoxin A, a potent shellfish poison from Pinna muricata[END_REF][START_REF] Mccauley | Total synthesis of pinnatoxin A[END_REF]. As a result, the proximal tetrahydropyran ring (ring D) perfectly abuts against Tyr188, Tyr195 and the vicinal Cys190 and Cys191 (loop C), while the central tetrahydrofuran ring (ring C) does not make direct contacts, and while the distal tetrahydropyran ring (ring B) interacts with Val148 (loop E) from the (-) face. Here, the hydroxyl and methyl substituents on the distal tetrahydropyran ring are respectively positioned to make a hydrogen bond with Tyr195 (loop C) and van der Waals contacts with Arg79 and Val108 (loop E). The additional methyl and hydroxyl groups that distinguish ring D of PnTx-D from the other congeners could alter the degree of closure of loop C.

In the membrane direction of the interface, the cyclohexene ring (ring G) in PnTx-A and PnTx-G makes nearly edge-to-face stacking interactions with Tyr93 (loop A) from the (+) face and Tyr55 (loop D) from the (-) face (Fig. 3). The carboxylate group (R 1 substituent) in PnTx-A, much smaller than the corresponding (γ)-butyrolactone moiety in SPX and GYM, is perfectly positioned to hydrogen bond with Ser167 (loop F) and make several watermediated contacts with Tyr188 (loop C). By contrast, the corresponding vinyl group in PnTx-G establishes weak interactions only with Ser167, since its more non-polar character does not permit formation of the solvation shell associated with bound PnTx-A. The mode of interaction of the PnTx R 1 substituent in AChBP is consistent with a reduced in vitro neuromuscular blocking activity and in vivo toxicity of a fluorophore-conjugated PnTxF derivative at this position [START_REF] Hellyer | In vitro labelling of muscle type nicotinic receptors using a fluorophore-conjugated pinnatoxin F derivative[END_REF].

In the solvent-exposed surface, the bulky bridged 5,6-bicycloketal substructure (ring EF), which is unique to the pinnatoxins and pteriatoxins, radially extends from the binding pocket to promote additional interactions with the complementary face (Fig. 3). Hence, the exocyclic hydroxyl (R 2 substituent) and methyl groups of the pinnatoxins mediate interactions with Thr36 in strand β1, Ile118 in loop E and Asp164 in loop F, along with water-mediated interactions between the hydroxyl group and the cluster of polar residues Asp164, Ser166 and Ser167 in loop F.

Comparison of the modes of binding of the pinnatoxins, spirolides and gymnodimines

Structural features of the A-AChBP complex with PnTx-A and PnTx-G closely resemble those observed for the SPX and GYM complexes (cf. rmsd values in Experimental Procedures). Indeed, deep anchoring of PnTx-A and PnTx-G at the subunit interface results in a buried surface increase of ~750 Å 2 (1.4 Å probe radius), close to the ~680 Å 2 and ~600 Å 2 surface areas buried by bound SPX and GYM, respectively [START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF] (Fig. 4). Also, comparative analysis of the buried surface at the binding pockets shows a significant contribution of the bridged 5,6-bicycloketal (ring EF) in PnTx-A and PnTx-G, which is replaced by a smaller allylic alcohol linker in SPX and GYM, along with subtle variations around the R 1 group at the base of the cyclohexene ring. The additional exocyclic hydroxyl (R 3 ) and methyl (R 4 ) groups in the tetrahydropyran ring (ring D) of the pinnatoxins D, E and F may lead to steric clashes with the facing Tyr188 and Tyr195 phenyl rings and require conformational adaptability of loop C associated with distinct binding properties. Concomitant variability in the R 1 group and absence of the hydroxyl group (R 2 ) in the bridged bicycloketal substructure that faces the complementary subunit surface may significantly affect the potency and subtype selectivity of the three pinnatoxin congeners. Most importantly, the bicycloketal substructure (ring EF), located in the solvent-exposed branch of PnTx-A and PnTx-G and harboring exocyclic hydroxyl (R 2 ) and methyl groups, distinguishes the pinnatoxins and pteriatoxins (cf. below) from the spirolides and gymnodimines. Compared to the methyl and hydroxyl groups found in SPX at positions 9 and 10, respectively, this substructure offers additional interactions and water-mediated contacts with Asp164 and the neighboring Ser166-Ser167 residue pair in the sequencevariable loop F (Figs. 3 and4). The acyl chain at position C28 in the bridged 5,6bicycloketal (EF) substructure of the esterified pinnatoxin conjugates may generate steric clashes with residues in loop F, with significant reductions in binding affinities for the AChBPs and nAChRs.

Other members of this emerging biotoxin family, such as the structurally related pteriatoxins, would be expected to bind in a similar way as PnTx-A and PnTx-G. Compared with the pinnatoxins, the pteriatoxins contain the same 6,5,6-bis-spiroketal and 5,6bicycloketal rings, but they differ in the chemical properties of the R 1 group that extends from the cyclohexene G ring at the bottom of the macrocyclic framework. This moiety, that replaces the (γ)-butyrolactone ring in SPX and GYM, extends toward the membrane side of the binding pocket where a highly ordered network of solvent molecules connects neighboring residues at the (-) face. In methyllycaconitine (MLA), the bulkier N-phenyl succinimide moiety, which represents a major determinant for α7 selectivity, overlaps only partially with the R 1 group of PnTx (Fig. 4) [START_REF] García-Guzmán | α-Bungarotoxin-sensitive nicotinic receptors on bovine chromaffin cells: molecular cloning, functional expression and alternative splicing of the α7 subunit[END_REF].

Structure-activity implications for PnTx binding to L-AChBP and the nAChRs

The presence of a Trp residue in loop D of L-AChBP and most nAChRs, in place of Tyr55 in A-AChBP, most likely contributes to the >3400-and ~400-fold lower affinities of PnTx-A and PnTx-G, as documented for other ligands [START_REF] Hansen | Structural and ligand recognition characteristics of an acetylcholine-binding protein from Aplysia californica[END_REF][START_REF] Hansen | Structures of Aplysia AChBP complexes with nicotinic agonists and antagonists reveal distinctive binding interfaces and conformations[END_REF][START_REF] Hibbs | Structural determinants for interaction of partial agonists with acetylcholine binding protein and neuronal α7 nicotinic acetylcholine receptor[END_REF][START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF][START_REF] Rucktooa | Structural characterization of the binding mode of smoking cessation drugs to nicotinic acetylcholine receptors through the study of ligand complexes with acetylcholine binding protein[END_REF][START_REF] Edink | Fragment growing induces conformational changes in acetylcholine-binding protein: a structural and thermodynamic analysis[END_REF]. As well, the side chains of Tyr164 in loop F and neighboring Lys36 in strand β1, being bulkier than the corresponding Ser167 and Thr36 side chains in A-AChBP (Fig. 4, Suppl. Fig. S1), are likely to encounter steric clashes with the exocyclic methyl and hydroxyl groups in the 5,6bicycloketal ring (ring EF) of PnTx-A and PnTx-G and contribute to the lower affinity for L-AChBP (Table 3).

Proximity of the sequence-variable loop F region, at the (-) face of the binding pocket, to the cyclohexene ring harboring the R 1 group with differences in charge (e.g. carboxylate versus vinyl), along with additional substructures in the various pinnatoxin congeners, clearly are key functional determinants in the pinnatoxins for dictating selectivity toward the α7 or α1 2 βγδ nAChRs compared to the α3β2 and α4β2 nAChRs (Tables 1 and2). Hence, replacement of A-AChBP loop F residue Ser167 (Gly167 in human and chicken α7, Ala in δ subunit of the Torpedo nAChR) to a charged Asp residue in the β2 subunits may contribute to the low affinity of PnTx-A and PnTx-G for the neuronal heteropentameric nAChRs. The recent crystal structure of the LBD of the neuronal α9 subunit bound to MLA in a monomeric state pointed to the lack of interaction of the N-phenyl succinimide moiety at the bottom of the binding pocket with shorter side chains Thr147 and I190 (corresponding to bulkier AChBP Q186 and K143) [START_REF] Zouridakis | Crystal structures of free and antagonist-bound states of human α9 nicotinic receptor extracellular domain[END_REF]. Molecular diversity amongst the PnTx congeners, as exemplified by a lactone ring attached to the cyclohexene ring in PnTx-F instead of a carboxylate in PnTx-A or vinyl group in PnTx-G, may contribute to the highest potency of PnTx-F in acute toxicity studies [START_REF] Munday | Acute toxicity of pinnatoxins E, F and G to mice[END_REF]. Yet, lack of a hydroxyl group in the bridged EF-ketal ring of PnTx-D, PnTx-E and PnTx-F could modulate interactions with residues from the complementary subunit at the base of the binding pocket and contribute to nAChR selectivity.

Hence, these structures confirm the importance of the cyclic imine and bis-spirocetal framework embedded within a macrocyclic skeleton as key determinants to confer high binding affinity and potent nAChR antagonism to the pinnatoxins, as previously documented for the spirolides and gymnodimines [START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF]. In contrast to the limited interactions of the latter two toxins with the sequence-variable loop F to preserve broad nAChR subtype specificity, the structures also point to the key role of the radially protruding bicycloketal ring (ring EF) unique to the pinnatoxins to promote an additional, either polar or non-polar, interaction network with loop F and govern subtype selectivity.

Concluding remarks

The pinnatoxins are members of the expanding superfamily of macrocyclic imine biotoxins that are structurally related to the previously characterized non-selective SPX and GYM toxins. Yet they contain unique substituents that target distinct structural determinants located on the (-) face of the subunit interface of the nAChRs and their AChBP surrogates and confer subtype selectivity. Indeed, all these phycotoxins embed a cyclic 6 or 7membered imine ring structure that tethers the toxin core at a pivotal place and displays exquisite shape and chemical complementarity with the ligand-binding pocket, while the unique substructures of the macrocyclic framework of the pinnatoxins radially extend from the binding pocket to engage in distinctive interactions with residues from the complementary face and the sequence-variable loop F of the nAChRs. This study unveils the specific positions of the functional variations in the substituents attached to the macrocyclic framework and offers new concepts for the design of new ligands with higher nAChR subtype selectivity. Since most other subtype-selective nAChR antagonists, such as the conotoxins, are polar molecules that do not cross the blood-brain barrier, the macrocyclic imine framework also offers new avenues for distinguishing nAChR subtype functions in the brain.

Experimental Procedures

Live animals and biological materials

Their sources are stated in the Supplementary Information.

Concentration-inhibition analysis of ACh currents

The procedures for nAChR microtransplantation and expression in oocytes and for voltageclamp recording are described in the SI. Dose-response curves for agonist activation were analyzed using equation: I = I max [L] n /(EC 50 + [L]) nH (equ. 2), where I is the measured agonist-evoked current, [L] is the agonist concentration, EC 50 is the agonist concentration that evoked half the maximal current (I max ) and nH is the Hill coefficient. Current (I) values were normalized to the I max value recorded from the same oocyte to yield fractional (%) response data. IC 50 values were determined from dose-response curves by fitting to equation: F = 1/[1+([X]/IC 50 ) nH ] (equ. 3), where F is the fractional response obtained in presence of the inhibitor at concentration [X] and IC 50 is the inhibitor concentration that reduced the ACh-evoked amplitude by half. Statistical significance of differences between controls and test values was assessed using the two-tailed Student's t-test or the Kolmogorov-Smirnov two-sample test and p < 0.05. Data are presented in Fig. 2, Suppl. Fig. S2A, andTable 1.

Binding assays on the nAChRs

The procedures for preparation of T. marmorata membranes enriched in the muscle-type α1 2 βγδ nAChR and for transient expression of the human neuronal α3β2 and α4β2 nAChRs and chicken α7/mouse 5-HT3 chimera from tsA-201 cells are described in the SI. Equilibrium competition on α1 2 βγδ used 0.05 μg of membranes (~100 pM in toxin binding sites), 0.5-1.5 nM 125 I-bungarotoxin (BgTx) (L*) and a range of PnTx concentrations (at least 4 h incubation, 20-25°C) [START_REF] Servent | Only snake curaremimetic toxins with a fifth disulfide bond have high affinity for the neuronal α7 nicotinic receptor[END_REF]. Equilibrium competition on the α7/5-HT3 chimera and α3β2 and α4β2 nAChRs used a cell density adjusted to specifically bind no more than 10% of the radioligand 125 I-BgTx (α7/5-HT3) or [ 3 H]-epibatidine (EPI) (α3β2 and α4β2), added at 0.9-0.5 nM (~25-30-fold the K d , see below) (L*), and a range of PnTx concentrations (4 h incubation, 20-25°C) [START_REF] Kharrat | The marine phycotoxin gymnodimine targets muscular and neuronal nicotinic acetylcholine receptor subtypes with high affinity[END_REF]. IC 50 values were determined by fitting the competition data to a binding isotherm and conversion to K i constants using equation: K i = IC 50 /(1+L*/K d ) (equ. 1) [START_REF] Cheng | Relationship between the inhibition constant (K1) and the concentration of inhibitor which causes 50 per cent inhibition (I50) of an enzymatic reaction[END_REF], and K d values of 50 pM for 125 I-BgTx binding on α1 2 βγδ and of 35 pM and 20 pM for [ 3 H]-EPI binding on α3β2 and α4β2, respectively. Data are presented in Suppl. Fig. S2B and Table 2.

Stable Ca 2+ sensing nAChR cell lines and binding assays

Stable Ca 2+ sensing tsA-201 cell lines (CNiFER) [START_REF] Yamauchi | Characterizing ligand-gated ion channel receptors with genetically encoded Ca 2++ sensors[END_REF] expressing human 7 or 42 nAChRs were harvested at 70-80% of confluence with a light trypsin treatment. They were diluted with Hank's Balanced Salt Solution (HBSS) buffer containing 1% BSA to a cell count of ~5-10,000 per ml and plated in 0.02ml into a 96-well low-profile plate.

[ 3 H]-EPI, wheat germ agglutinin, scintillation proximity assay beads (Amersham Biosciences) (~100 μg/well) and specified concentrations of competing toxin in HBSS buffer in 1% BSA were added to each well to a final volume of 80 μl. Plates were incubated at room temperature for 2 h before repetitive counting on a Wallac 1450 Microbeta Trilux counter. Data were analyzed by GraphPad Prism 4 and fitted to a sigmoidal concentrationresponse curve to determine IC 50 and K d values. They are presented in Table 3.

Ligand binding to AChBPs and the α7/A-AChBP chimera

The procedures for stable expression and purification of AChBPs and the α7/A-AChBP chimera are described in the SI. Rate constants for association (k on ) and dissociation (k off ) were determined by multiple kinetic means to cover the wide range of values observed, as previously described [START_REF] Hansen | Tryptophan fluorescence reveals conformational changes in the acetylcholine binding protein[END_REF][START_REF] Hansen | Structures of Aplysia AChBP complexes with nicotinic agonists and antagonists reveal distinctive binding interfaces and conformations[END_REF]. Measurement of k on 's (and k off 's in case of lower affinity ligands) entailed direct admixture of reactants and monitoring the quenching of the protein native Trp fluorescence. Additional measurements of k off employed occupation of free AChBP binding sites by a competing ligand (gallamine) in excess over its K d , to form a non-quenching complex. In brief, each equilibrated PnTx-AChBP complex at 250 pM in binding sites and slight molar excess of toxin was mixed with [ 3 H]-MLA for A-AChBP or [ 3 H]-EPI for L-AChBP at concentrations well above their respective K d value to titrate those binding sites made available upon toxin dissociation. The time course of [ 3 H]-ligand binding was monitored over several hours. First order dissociation rate constants were determined by nonlinear regression of data using simple mono-exponential relationship. K d/calc values were derived from the ratio of rate constants. Data are presented in Table 4.

Structure determination and refinement

The procedures for complex formation and crystallization and for data collection are described in the SI. The structures were solved by molecular replacement with PHASER [START_REF] Mccoy | Structure and mechanism of an ADP-glucose phosphorylase from Arabidopsis thaliana[END_REF] using the apo A-AChBP structure (PDB code 2BYN [START_REF] Hansen | Structures of Aplysia AChBP complexes with nicotinic agonists and antagonists reveal distinctive binding interfaces and conformations[END_REF]) as a search model. Analysis of the binding site in a preliminary difference map of the PnTx-A AK complex revealed five empty interfaces in an otherwise well-ordered AChE pentamer, consistent with the extremely low binding affinity of the derivative (Table 4). The initial PnTx-AChBP models were improved by manual adjustment with COOT [START_REF] Emsley | Coot: model-building tools for molecular graphics[END_REF]) and were refined with REFMAC [START_REF] Murshudov | Refinement of Macromolecular Structures by the Maximum-Likelihood Method[END_REF]) including TLS refinement with each subunit defining a TLS group. In each case, a random set of reflections was set aside for cross-validation purposes. The molecular structures of PnTx-A and PnTx-G and the associated library files containing stereochemical and parametric data were generated with SKETCHER (CCP4, 1994). Automated solvent building used COOT. Data collection and refinement statistics are reported in Suppl. Table S1.

The final structures comprise A-AChBP residues His1-Arg208 for each of the five subunits in the two pentamers in the asymmetric unit and a fully resolved PnTx-A or PnTx-G molecule in each of the ten binding sites. The N-terminal FLAG sequence (cf. above) is fully resolved in one subunit of each pentamer, while high temperature factors and weak electron density are associated with the other FLAG sequences and with surface loop Asn15-Met19 in all subunits. The stereochemistry of the structures was analyzed with COOT and MOLPROBITY [START_REF] Davis | MolProbity: all-atom contacts and structure validation for proteins and nucleic acids[END_REF].

Structural analyses and comparisons

Comparison of the PnTx-A and PnTx-G AChBP complexes with other structures include those of A-AChBP in the apo form and as complexes with the phycotoxins SPX and GYM (PDB codes 2WZY and 2X00 [START_REF] Bourne | Structural determinants in phycotoxins and AChBP conferring high affinity binding and nicotinic AChR antagonism[END_REF]) and the diterpene alkaloid from plant, MLA (PDB code 2BYR [START_REF] Hansen | Structures of Aplysia AChBP complexes with nicotinic agonists and antagonists reveal distinctive binding interfaces and conformations[END_REF]). The average root mean square deviation between AChBP subunits bound to PnTx-A and PnTx-G is 0.3 Å for 208 Cα atoms; between AChBP bound to PnTx-A and the above-mentioned structures it is in the 0.36-0.67 Å range for 212-216 Cα atoms.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material. Compared with the accommodation modes of SPX and GYM within the ligand-binding pocket, the additional contact points with loop F residues mediated by the 5,6-bicycloketal ring (blue) found in PnTx-A are evident. In contrast, the N-phenyl succinimide moiety of MLA extends to the bottom of the binding pocket. Inhibition constants (IC 50 ) for pinnatoxin effect on ACh-evoked nicotinic currents, recorded from oocytes either expressing the human neuronal α7 or α4β2 nAChR subtypes or transplanted with the muscle-type α1 2 βγδ nAChR. b From [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF] c See Fig. 3 in [START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF] Structure. Author manuscript; available in PMC 2016 June 02.

Fig. 1 .

 1 Fig. 1. Chemical structures of the pinnatoxin variants and the open-ring PnTx-A AK derivative The 7-membered cyclic imines common to other spirolides are highlighted with a grey background while the substructures that distinguish the pinnatoxins from other spirolides are boxed. Locations of group substitutions (R 1 to R 4 ) that differ amongst the various pinnatoxins are indicated. PnTx-A: R 1 = CO 2 , R 2 = OH, R 3 = R 4 = H; PnTx-G: R 1 = CH=CH 2 /O-acyl esters (C14 to C24 acyl chain with variable numbers of double bonds), R 2 = OH, R 3 = R 4 = H; PnTx-B/C: R 1 = NH 2 CO 2 , R 2 = OH, R 3 = R 4 = H; PnTx-D: R 1 = CO(CH 2 ) 2 CO 2 , R 2 = H, R 3 = OH, R 4 = CH 3 ; PnTx-E: R 1 = COH(CH 2 ) 2 CO 2 , R 2 = H, R 3 = OH, R 4 = CH 3 ; PnTx-F: R 1 = C 4 H 6 O 2 (butyrolactone), R 2 = H, R 3 = OH, R 4 = CH 3 .

Fig. 2 .

 2 Fig. 2. Inhibition of ACh-evoked currents recorded from human α7 nAChR transiently expressed in oocytesThe amplitudes of the ACh-induced currents recorded in the presence of PnTx-A (filled circles) and of PnTx-G (empty circles) (mean ± SEM; 3-4 oocytes per concentration) were normalized to control currents and are expressed as relative current values. The holding membrane potential during recordings was -60 mV.

Fig. 3 .

 3 Fig. 3. Overall and close-up views of the pentameric PnTx-AChBP complexes, and structural comparisons (A) The pentamer in the PnTx-A complex is viewed from the "membrane" side (left), and in a radial perspective towards one subunit interface with the apical side at top and the membrane side at bottom (center). The main and side chains at the (+) and (-) faces of the exemplified interface are displayed in yellow and cyan, respectively. The position and orientation of bound PnTx-A and PnTx-G are identical. Bound PnTx-A (pink bonds and molecular surface, red oxygens, blue nitrogen) and PnTx-G (light blue bonds and molecular surface, red oxygens, blue nitrogen) are perfectly ordered, as assessed by the quality of the 1.9Å and 2.2 Å resolution 2Fo-Fc electron density maps contoured at 1.2σ (blue) (right). (B) Close-up views of bound PnTx-A (left) and PnTx-G (right) in the aromatic nest at the subunit interface, showing details of the cyclic imine environment (radial perspective). The Cys190-Cys191 disulfide bridge embedded at the tip of loop C is displayed in brown and loop F is in green. The 5,6-bicycloketal ring (i.e., ring EF in Fig. 1) is highlighted in blue. Side chains (in yellow and cyan) and solvent molecules (red dots) that interact specifically with bound PnTx-A/G are shown and labeled. The dashed lines denote hydrogen bonds. Note the distinctive interaction networks of the R 1 substituents at the membrane side of the interface. (C) Overlays of A-AChBP as bound to PnTx-A with L-AChBP (PDB code 1UW6) (left) and the α7/A-AChBP chimera mutant III (PDB code 3SH1) (right). Residues in the ligandbinding pockets of L-AChBP and α7 that differ from those in A-AChBP are colored in

Fig. 4 .

 4 Fig. 4. Surface complementarity of bound PnTx-A and PnTx-G with the A-AChBP subunit interface and comparison with the SPX, GYM and MLA complexes (A) The molecular surface buried by bound PnTx-A (pink toxin and surface) and PnTx-G (light blue toxin) at the (+) (yellow) and (-) (blue) faces of the A-AChBP subunit interface, is viewed radially from the pentamer outer periphery (transparent brown loop C, green loop F). (B-D) Superimposition of PnTx-A (pink toxin) with SPX (orange toxin and surface), GYM (green toxin and surface) and MLA (magenta toxin and surface) as bound to A-AChBP. Compared with the accommodation modes of SPX and GYM within the ligand-binding pocket, the additional contact points with loop F residues mediated by the 5,6-bicycloketal

a

  Values from individual experiments performed in triplicate or quadruplicate, with 95% confidence intervals provided in parentheses. The experimental curves associated with the data displayed on a grey background are reported as Suppl. Fig.S2A.

Table 1

 1

Table 2

 2 Binding constants (K i ) for pinnatoxin binding on cells expressing the neuronal and muscle-type nAChRs, determined by competition against binding of 125 I-BgTx (α7/5-HT3, α1 2 βγδ) or [ 3 H]-EPI (α4β2, α3β2) at equilibrium. Average ± variation or means ± SEM from two to five distinct experiments performed in duplicate. Experimental curves associated with the data displayed on a grey background are reported as Suppl. Fig.S2B.

			K i (nM) a
	nAChR subtype
		PnTx-A b	PnTx-G	PnTx-A AK b
	α7/5-HT3 chimera (chick) 0.35 ± 0.04 0.72 ± 0.03	>10 000
	α4β2 (human)	15.6 ± 5.2	101 ± 30	>2000
	α3β2 (human)	9.4 ± 1.9	63.9 ± 1.2	>2000
	α1 2 βγδ (Torpedo)	2.8 ± 0.03	0.11 ± 0.04	>1000
	a

b From

[START_REF] Aráoz | Total synthesis of pinnatoxins A and G and a revision of the mode of action of pinnatoxin A[END_REF]

Structure. Author manuscript; available in PMC 2016 June 02.

Structure.Author manuscript; available in PMC 2016 June 02.

Acknowledgments

We gratefully acknowledge the expert assistance of: Kwok-Yiu Ho, Kimberly Gomez and Dr. John Yamauchi (UCSD, La Jolla) for AChBP, α7 and α4β2 binding assays; Sandrine Conrod (CNRS/AMU, Marseille) for crystallization; the ID14-4 staff of the European Synchrotron Radiation Facility (ESRF, Grenoble) and the Proxima-1 staff of the SOLEIL synchrotron (Gif-sur-Yvette) during data collection. This work was supported by grant NIH/NIGMS R01 GM077379 (to AZ); grant AQUANEUROTOX-ANR-12-ASTR-0037 (to JM and DS); USPHS grants R37-GM18360 and P42-ES-010337 (to PT); the French Infrastructure for Integrated Structural Biology (FRISBI) ANR-10-INSB-05-01 (to the AFMB lab); and a CNRS-DRI PICS grant (to YB and PM).

Author Manuscript

Bourne et al.

Page 24

Table 3 Dissociation constants determined at equilibrium (K d/equ ) for pinnatoxin binding to Ca 2+ sensing cells stably expressing the full-length α7 and α4β2 nAChRs, determined by competition against [ 3 H]-EPI binding.