Wei Ding

Tianning Chen

Chen Chen

Dimitrios Chronopoulos

Badreddine Assouar

Yongzheng Wen

Jian Zhu

Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering

Chiral phononic crystals provide unique properties not offered by conventional metamaterial based on classic Bragg scattering and local resonance. However, it is insufficient to only consider the inertial amplification effect to describe its bandgap mechanism due to the absence of the bandgap caused by the chirality in some specific chiral structures. Here, we theoretically and experimentally introduce an analogy with Thomson scattering in electromagnetic waves to characterize the bandgap phenomena in chiral phononic crystals with translation-rotation coupling. Another phononic structures with translation-translation coupling are proposed to illustrate the universality of the analogy. We evidence that the coupling motion in chiral unit cells is similar to the result of Thomson scattering, which quantitatively formulizing as inertial amplification in theory and, twice elastic Thomson scattering allows the waves in the same polarization mode to superpose in antiphase, which is essence of the bandgap formation. This finding sheds a new light on the physics of the elastodynamic wave manipulation in chiral phononic crystals, thus opening a definite route for the pragmatic exploitation of chiral phononic crystals as well as other structures with motion coupling in achieving low-frequency and broad bandgaps.

Introduction

Phononic crystals (PnCs), one artificial periodical structure with the capability of flexible manipulation of acoustic and elastic waves, have received much attention [START_REF] Wu | A brief review of dynamic mechanical metamaterials for mechanical energy manipulation[END_REF][START_REF] Jia | Engineering lattice metamaterials for extreme property, programmability, and multifunctionality[END_REF][START_REF] Hussein | Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook[END_REF]. The remarkable feature, i.e. the bandgap, provides a powerful ability to control the wave propagation, such as the design of the non-reciprocal device [START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF][START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF][START_REF] Haghpanah | Programmable Elastic Metamaterials[END_REF] and the recent valley state formation with Dirac points degeneracy operation in its bandgap [START_REF] Foehr | Spiral-Based Phononic Plates: From Wave Beaming to Topological Insulators[END_REF][START_REF] Yang | Optically Reconfigurable Spin-Valley Hall Effect of Light in Coupled Nonlinear Ring Resonator Lattice[END_REF].

Local resonance [START_REF] Liu | Locally Resonant Sonic Materials[END_REF] and Bragg scattering [START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF] are regarded as the mainstream bandgap mechanisms.

In essence, both mechanisms mean two methods of realizing destructive interference. For the distinction, the destructive interference of the former features by the opposite movements between the main structure and resonant subsystem [START_REF] Meng | Optimal design of rainbow elastic metamaterials[END_REF][START_REF] Meng | Rainbow metamaterials for broadband multi-frequency vibration attenuation: Numerical analysis and experimental validation[END_REF][START_REF] Barnhart | Experimental demonstration of a dissipative multi-resonator metamaterial for broadband elastic wave attenuation[END_REF][START_REF] Liu | Analytic model of phononic crystals with local resonances[END_REF]. Therefore, we can observe the negative mass or the negative stiffness from a macro perspective. For the latter, in classical Bragg scattering, the destructive interferences are caused by the distance difference that satisfies 𝜆/2 (𝜆 is one wavelength) [START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF][START_REF] Liu | Wave Motion in Periodic Flexural Beams and Characterization of the Transition Between Bragg Scattering and Local Resonance[END_REF].

Distinguished from electromagnetic Bragg scattering, elastic waves are matter waves and their propagation relies on the vibration of the matter. Therefore, for elastic Bragg scattering, the tailored matrix can constrain the wave-propagation path. As a result, the bandgaps don't meet the principle of the classical Bragg scattering as shown in Eq. ( 1) [START_REF] Fei | Three-dimensional anti-chiral auxetic metamaterial with tunable phononic bandgap[END_REF][START_REF] Muhammad | Design and manufacturing of monolithic mechanical metastructures governing ultrawide low frequency three-dimensional bandgaps[END_REF][START_REF] Park | Continuum flexural metamaterial for broadband low-frequency band gap[END_REF], called deep sub-wavelength bandgaps. In other words, as long as realizing the destructive interferences in the propagation path, a bandgap would be available.

2 𝑎sin 𝜃 = 𝜆 (1)
Independent of local resonance and Bragg scattering gap, there is a design method called inertial amplification [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF]. Ideally, its dynamic inertia can be amplified to infinity [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF]. Therefore, the disadvantages of local resonance and Bragg scattering, i.e., low frequency and broad band are incompatible with high stiffness and lightweight, can be overcome by this bandgap [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Delpero | Inertia Amplification in Phononic Crystals for Low Frequency Band Gaps[END_REF][START_REF] Yilmaz | Inertial Amplification Induced Phononic Band Gaps in a Chiral Elastic Metamaterial[END_REF][START_REF] Orta | Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism[END_REF][START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF].

Although some remarkable advances have been achieved [START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF][START_REF] Banerjee | Inertial amplification band-gap generation by coupling a levered mass with a locally resonant mass[END_REF][START_REF] Yuksel | Realization of an ultrawide stop band in a 2-D elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF][START_REF] Taniker | Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges[END_REF][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF], the structure configurations with inertial amplification effect always originate from the classic geometric model to extend their versatility and applicability [START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF][START_REF] Banerjee | Inertial amplification band-gap generation by coupling a levered mass with a locally resonant mass[END_REF][START_REF] Yuksel | Realization of an ultrawide stop band in a 2-D elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF][START_REF] Li | Multiple band gaps for efficient wave attenuation by inertial amplification in periodic functionally graded beams[END_REF], rather than for further progress in amplifying dynamic inertia.

Eventually, the breakthrough in low-frequency bandgaps is stalled, which can be attributed to the lack of diverse structures superficially. Actually, the ultimate reason is the obscureness in the underlying physics of the bandgap. In detail, despite the existence of the inertial amplification (see Supplementary S1 and S2 or the reference [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF]) in the system, the bandgap formation cannot be ensured in different geometrical configurations [START_REF] Orta | Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism[END_REF][START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF][START_REF] Krushynska | Accordion-like metamaterials with tunable ultra-wide low-frequency band gaps[END_REF][START_REF] Zheng | Mechanical logic switches based on DNA-inspired acoustic metamaterials with ultrabroad low-frequency band gaps[END_REF]. Therefore, it deserves to study the essence of the inertial amplification and the underlying physics of this bandgap, to guide the advances of geometric diversification and thus provide more possible in lower-frequency bandgaps [START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF][START_REF] Taniker | Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges[END_REF][START_REF] Li | Multiple band gaps for efficient wave attenuation by inertial amplification in periodic functionally graded beams[END_REF][START_REF] Li | Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores[END_REF].

In this work, we theoretically and experimentally investigate the wave phenomena and bandgap generation in the compression-torsion coupled PnCs to concretize the bandgap mechanism. An incident wave polarizing in one mode passing the chiral subunit cells will be decomposed into the outgoing waves with two polarizations, the one in translation and the other in rotation. The decomposition of the polarization can be considered as a splitting process of the incident waves, which can be an analogy to the result of Thomson scattering in electromagnetic waves [START_REF]The Corpuscular Theory of Matter[END_REF] to be distinct from Bragg scattering [START_REF] Jackson | Classical electrodynamics[END_REF]. It demonstrates that a minimum of twice

Structure and results

Fig. 1 shows the schematics of the unit cells and calculated band structures. The lattices in Fig. 1 In fact, the study on such bandgaps in MTR can be traced back to Bergamini's work in 2019 [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF],

where the bandgap mechanism is attributed to inertial amplification and the nature of the coupling between the spins of the atoms which is provided by the relative orientation of adjacent chiral centers.

As revealed by Eq. (S58) and Eq. (S63) in Supplementary S2, the inertial matrixes of MTR and ATR have the inertial amplification effect, but the bandgap only exists in MTR. In other words, the current definition of inertial amplification lacks the universality principle of one mechanism like Bragg scattering and local resonance, even though inertial amplification has been studied for almost two decades. Therefore, this study aims to elucidate the underlying physics of inertial amplification and the coupling between the spins of the atoms. As illustrated in the analysis of the subunit cell Ⅰ (more details in Supplementary S1), the chiral effect essentially achieves the motion coupling and thus exhibits the function of the inertial amplification, but this is not sufficient to form a bandgap. Due to the ambiguous contribution of chirality to the bandgap generation, one may easily attribute the underlying physics of the bandgap in Fig. 1(b) to the chirality. For instance, in some specific functional structures, chirality can indeed bring novel phenomena [START_REF] Wu | Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review[END_REF], such as negative Poisson's ratio [START_REF] Lakes | Foam Structures with a Negative Poisson's Ratio[END_REF], high structural damping [START_REF] Baravelli | Internally resonating lattices for bandgap generation and low-frequency vibration control[END_REF], and spin mechanical metastructures [START_REF] Frenzel | Three-dimensional mechanical metamaterials with a twist[END_REF]. Some of these aforementioned properties have been realized in other non-chiral structures [START_REF] Fei | Three-dimensional anti-chiral auxetic metamaterial with tunable phononic bandgap[END_REF][START_REF] Taylor | Low Porosity Metallic Periodic Structures with Negative Poisson's Ratio[END_REF][START_REF] Yeh | Origins of broadband vibration attenuation empowered by optimized viscoelastic metamaterial inclusions[END_REF]. Therefore, chirality is one of the effective ways but not the exclusive way to achieve the desired function.

The designed non-chiral unit cells with translation-translation coupling consist of two subunit cells a are demonstrated in Fig. 1(c) and Fig. 1(d). The arrayed and mirrored translation-translation coupling unit cell with central and rotational symmetry in α is named ATT and MTT. The right panels in Fig. 1(c)-1(d) represent the calculated band structure. From the perspective of the bandgap, there is a phenomenon similar to that between MTR and ATR in Fig. 1(a)-1(b), i.e., MTT has a bandgap that is not in ATT, which is an encouraging result as it confirms that the bandgap possessed in MTR is not exclusive to chiral lattices. To elucidate the mechanism of the bandgap generation, Fig. 2 shows the initial polarization orientations of the oscillators when the wave propagates in four types of unit cells shown in Fig. 1. For the sake of simplicity, two assumptions are made during the analysis. The 𝑧-axis rotational freedom of 𝑚 " is restricted; the incident wave vibrates sinusoidally in translational form, and the initial direction of vibration is along +𝑧 axis with an initial phase of zero (refer to the coordinate system shown in Fig. 1). As shown in Fig. 2(a) or Fig. 2(b), when the incident wave passes through the first subunit cell, the translation 𝑢 " will split into the translation 𝑢 # and rotation 𝜑 # . Actually, the splitting process of the polarization is similar to the Thomson scattering. As illustrated in Fig. 3(a), in classical Thomson scattering, the plane electromagnetic wave forces the electron to vibrate; the vibration of electrons will create a harmonic electric field, thus resulting in two electromagnetic waves with the same frequency as the incident wave, which propagate in two directions [START_REF]The Corpuscular Theory of Matter[END_REF]. For simplicity, we defined the torsional harmonic as rotational polarization. In our analogue, the subunit cell is regarded as the electron; as shown in Fig. 3(b) (Please refer to the results of the time domain simulation in Supplementary S7 for more details), if the input is translational polarization, the subunit cell will vibrate in linear polarization but radiate another component wave which vibrates in rotational polarization; conversely, when the input is the rotational polarization, the radiative wave will vibrate in translational polarization.

Mechanism analysis

One can see some similarities between the process of the classical Thomson scattering (Fig. 3(a)) and the wave propagation in the models (Fig. 3(b) and Fig. 3(c)). In terms of the function, the compression-torsion feature of the subunit cell is similar to the charged property of the electron in Thomson scattering. In terms of the scattering process, the forced vibration of the electrons driven by the incident electromagnetic wave is similar to the vibration of the subunit cell driven by the incident elastic wave.

From the results, the translational radiated waves in Fig. 3(b) and the rotational radiated waves in Fig. 3(c) are dipoles, which resembles the phenomenon of classical Thomson scattering shown in Fig. 3(a).

The change of the wavefront in classical Thomson scattering is a superimposing effect of scattered waves propagating in multiple directions [START_REF] Pfeiffer | Metamaterial Huygens' surfaces: tailoring wave fronts with reflectionless sheets[END_REF][START_REF] Huygens | Traite´ de la Lumie`re[END_REF]. However, limited by the layout of the chiral subunit cell, the scattered waves cannot diverge to infinite space like the classical Thomson scattering but can only propagate in the direction of the periodical structure. These similarities enable us to draw an analogy with Thomson scattering to clarify the process of wave propagation in chiral subunit cells.

Therefore, for simplicity, we name the wave propagation process in the chiral subunit cell as Thomson scattering.

Furthermore, the process that these scattered waves (𝑢 # and 𝜑 # ) pass through the second subunit cell is equivalent to undergoing a second Thomson scattering. The distinction from the first is that the second scattering has two incident waves (𝑢 # and 𝜑 # ). Consequently, after the second scattering, it will produce four scattered waves, i.e., 𝑢 $" , 𝑢 $# , 𝜑 $" , and 𝜑 $# . Among them, 𝑢 $" and 𝑢 $# vibrate in translation, but 𝜑 $" and 𝜑 $# vibrate in rotation.

In ATR, by the first scattering, the initial direction of the scattered waves 𝑢 # and 𝜑 # are +𝑧 and clockwise around the +𝑧 axis (from the +z-axis perspective), respectively. After the second scattering, regarding 𝑢 # as the incident wave, the initial direction of the scattered wave 𝑢 $" is +𝑧, and the initial direction of the scattered wave 𝜑 $" is in the clockwise direction. Meanwhile, regarding 𝜑 # as the incident wave, the initial direction of 𝑢 $# is +𝑧, and the initial direction of 𝜑 $# is also in the clockwise direction. Since the two translations 𝑢 $" and 𝑢 $# , and the two rotations 𝜑 $" and 𝜑 $# have the same direction, 𝑢 $ and 𝜑 $ of 𝑚 $ in ATR can be written as Eq. ( 1) and Eq. ( 2).

𝑢 $ = 𝑢 $" + 𝑢 $# . ( 1
)
𝜑 $ = 𝜑 $" + 𝜑 $# . (2)
Eventually, the absolute angle 𝜑 $ of 𝑚 $ in ATT can be determined as

𝜑 $ = 𝑞(𝑢 " -𝑢 $ ), (3)
where 𝑞 is the rotational angle of translation per unit (see Supplementary S2 for details).

Conversely, in MTR, the initial direction of the scattered wave 𝑢 $" is +𝑧, and the initial direction of the scattered wave 𝜑 $" is in the clockwise direction; nevertheless, the initial direction of 𝑢 $# is -𝑧, and the initial direction of 𝜑 $# is in the counterclockwise direction. The 𝑢 $ and 𝜑 $ of 𝑚 $ in MTR can be written as

𝑢 $ = 𝑢 $" -𝑢 $# (4)
and

𝜑 $ = 𝜑 $" -𝜑 $# . (5)
Then, in MTR, the absolute angle 𝜑 $ can be determined as

𝜑 $ = 𝑞(𝑢 " + 𝑢 $ -2𝑢 # ). (6)
The difference in the absolute angle for 𝑚 $ in ATR and MTR leads to the immense variability of the elements in inertial matrixes 𝑴 and 𝑴 % in Eq. ( 7) (more details in Supplementary S2). Detailly, for instance, for the inertial matrix for ATR, 𝑴 is written as However, for the inertial matrix for MTR, 𝑴 is written as

In brief, after twice Thomson scattering, the property that the scattered waves vibrating in the same modes have opposite initial vibration directions similar in MTR plays a decisive role in the presence of the bandgap.

𝑴𝒖̈+ 𝑲𝒖 = 𝑴 % 𝑢" + 𝑲 % 𝑢 " . (7)
The splitting process and wave propagating in the translation-translation coupling (ATT and MTT)

PnCs can also be an analogy to Thomson scattering, as shown in Fig. 2(c) and Fig. 2(d). In ATT, by the first scattering, the initial directions of the scattered waves 𝑢 # and 𝑣 # point in +𝑧 and +𝑥 axis. 𝑢 $" , 𝑢 $# and 𝑣 $" , 𝑣 $# point to +𝑧 and +𝑥 directions after the second scattering when 𝑢 # , 𝑣 # are considered as the incident wave. However, in MTT, 𝑢 $" and 𝑢 $# point to +𝑧 and -𝑧, 𝑣 $" and 𝑣 $# point to -𝑥 and + 𝑥 directions, respectively. The superposition of these waves in MTT with opposite vibrating directions enables the capability of energy cancellation and generating bandgap, coinciding with the findings in MTR. Therefore, the analogy results confirm the universality of the mechanism proposed in this work.

Theoretical, numerical, and experimental verification

To verify the reasonability of the wave phenomena and bandgap feature described by using the analogy of Thomson scattering in Fig. 2, we have built the movement relationship in formulas according to the scattering process (more details of the theory in Supplementary S2). Meanwhile, the theoretical frequency response functions (FRFs) of the 2 unit cells in Fig. 4 As illustrated in Fig. 4(c) and 4(d), a significant attenuation occurs from 450 Hz to 1280 Hz in the FRF of MTR, and it does not exist in that of ATR. In a nutshell, the bandgap does only exist in MTR, not in ATR. Significantly, the consistency of experimental, numerical, and theoretical results demonstrates the correctness of the analysis. Also, it verifies the validity of the analogy that the bandgap generation mechanism of this PnC is similar to Thomson scattering. Limited by the insufficient degrees of the freedom in the lumped mass method, the theoretical FRF at high frequency has a large discrepancy with numerical and experimental results, such as over 500 Hz for ATR and over 1000 Hz for MTR.

Nevertheless, the intuitive nature of this method in bandgap-mechanism explanation is irreplaceable than other precise methods, such as spectral element method [START_REF] Lee | Spectral Element Method in Structural Dynamics[END_REF]. Previous studies [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF] mentioned that when the inertia is amplified, there will be an anti-resonant notch in FRF, and the anti-resonant notch can be observed in complex band structures [START_REF] Foehr | Spiral-Based Phononic Plates: From Wave Beaming to Topological Insulators[END_REF][START_REF] Li | Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores[END_REF]. Although the inertial amplification is involved in the Thomson scattering-induced bandgap, the anti-resonant notch is unessential for forming the Thomson scattering-induced bandgap. To accurately catch the effect of the anti-resonant notches on the FRF, Fig. 5(e) shows the numerical FRFs of the finite structures with 4 unit cells. For comparison purposes, we consider the influence of the geometric configurations of the lattice constant and equivalent physical parameters on the FRF [START_REF] Delpero | Inertia Amplification in Phononic Crystals for Low Frequency Band Gaps[END_REF][START_REF] Phani | Wave propagation in two-dimensional periodic lattices[END_REF], so we use Eq. ( 8) to normalize the frequency in FRFs.

Discussion

Anti-resonant notches

𝑓 ) = *+ , ! "## $ " . (8)
The filling ratio can be determined as

𝜌 -= . / , (9)
where the parameter 𝑉 can be calculated by

𝑉 = 0 % 1+ 2 . ( 10
)
The equivalent stiffness 𝑘 -** can be determined by Eq. (S14).

According to the analysis, the filling ratio is 586 kg/m 3 for 𝑎 = 35 mm, and it is 346.9 kg/m 3 for 𝑎 = 60 mm; the equivalent stiffness is equal to 377630 N/m for 𝑎 = 35 mm, and that is equal to 256060 N/m for 𝑎 = 60 mm. The normalized FRFs are shown in Fig. 5(e).

It can be observed from Fig. 5(e) that the depth of 𝑎=35 mm is significantly weaker than that of 𝑎 = 60 mm when the anti-resonant notches disappear. When the number of periods is up to 4, the attenuation difference is even by more than two orders of magnitude. Most importantly, influenced by the resonant frequency at the upper boundary, the theoretical anti-resonant frequency is not reflected in the FRFs. Despite the disappearance of the anti-resonant notches, the bandgap is still reserved in the band structure. Undeniably, the existence of anti-resonant notches has significant implications. Fig. 6(a) gives the pattern of variation in the number of MTRs. As shown in Fig. 6(b), as the number of the unit cells increases, the converged value at high frequency will decrease, and the number of the anti-resonant notch will increase (These properties do not exist in ATR, please see Supplementary S4 for more details). The increase in anti-resonant gaps can accelerate bandgap formation. Notably, the antiresonant notches are located mainly at the lower boundary of the bandgap and are spread, which enhances the modulation capacity of low-frequency and broad elastic waves.

In fact, as shown in Fig. 6(b), the infinite extension of the attenuation will be cut off due to the higherfrequency resonance modes. These resonant modes will suppress the anti-resonant notches close to it.

Therefore, the anti-resonant notches at high frequency will be covered with the number of periods increases, thus weakening the attenuation, as illustrated by the green plot in Fig. 6(a) and 6(b).

To sum up, the bandgap formation does not depend on the anti-resonant notch. The anti-resonance frequency just has the function of enhancing the manipulation of low-frequency. Determining the type of the bandgap by the shape of the attenuation is inaccurate [START_REF] Liu | Wave Motion in Periodic Flexural Beams and Characterization of the Transition Between Bragg Scattering and Local Resonance[END_REF][START_REF] Matlack | Composite 3D-printed metastructures for low-frequency and broadband vibration absorption[END_REF][START_REF] Miniaci | Proof of Concept for an Ultrasensitive Technique to Detect and Localize Sources of Elastic Nonlinearity Using Phononic Crystals[END_REF][START_REF] Chang | Complex band structures of crystalline solids: An eigenvalue method[END_REF], especially for Bragg scatteringinduced and Thomson scattering-induced bandgaps. Bragg scattering bandgap due to the impedance mismatch between the lumped masses and the ligaments [START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF][START_REF] Kushwaha | Theory of acoustic band structure of periodic elastic composites[END_REF].

Thomson scattering-induced and Bragg scattering-induced bandgaps

For a fair comparison, all physical parameters are the same except that the elasticity modulus of the ligaments in the Bragg scattering unit cell is determined to be 1.05e8 Pa to maintain the same stiffness between both unit cells in Fig. 7. Consequently, the equivalent stiffness and lattice constants of the cells are identical. There is a slight variation in the equivalent density of the unit cells because of the variation in ligament length. The total mass of the six ligaments in the MTR unit cell is 2.3 g, while the one in the Bragg scattering unit cell is 1.9 g (The total mass of the MTR unit cell is 58.86 g).

Therefore, we named the condition quasi-equal density.

Under the condition of the same lattice constant, stiffness, and quasi-equal density, the Thomson scattering bandgap extends from 400 Hz to 1200 Hz with a relative width of 100% (as illustrated in Fig. 7(a)), while a Bragg scattering bandgap at 930 -2610 Hz with a relative width of 95% can be observed in Fig. 7(b). A shift in the starting frequency to a lower frequency is evident, reaching 57%

with 930 Hz as a reference.

From Fig. 7(c) and 7(d), significant variances in energy distributions of both types of bandgaps can be observed. In classical Thomson scattering, the forced vibration of the electron is the key to scattering.

In the bandgap induced by the analogous Thomson scattering, it requires that the scattered waves undergone the first scattering needs to undergo the Thomson scattering again. In addition, the second-round scattered waves polarized in the same mode must have opposite phases to achieve destructive interference. It can be determined from this process that the involvement of the lumped masses is crucial for achieving attenuation. Since the first lumped mass is the vibration source and the waves have destructive interference in the process of the second scattering, the energy within the bandgap should be mainly concentrated in the second lumped mass. Indeed, Fig. 7(c) verifies this reasoning.

However, in classical Bragg scattering, the interaction between the incident wave and the scatterer is for reflecting incident waves. In Bragg scattering PnCs, the function of the scatterers is to reflect the incident wave and thus produce destructive interference. Correspondingly, the energy of the Bragg scattering bandgap is mainly localized in the matrix medium. For the unit cell of Fig. 7(b), the lumped masses act as the scatterers and the ligaments act as the matrix. Therefore, the energy of the bandgap should be mainly localized to the ligaments as the matrix medium, as can be observed in Fig. 7(d). The analogy of Thomson scattering can reasonably explain several phenomena from the perspective of underlying physics. It is well known that Bragg scattering is the elastic collision between the waves and the atoms. The scattered waves in Bragg scattering rely on the heavy atoms to reflect the incident waves and thus to destructive interferences [START_REF] Thomas | The birth of X-ray crystallography[END_REF]. Thomson scattering depends on the polarization of electrons to generate divergent outgoing waves in different polarizations [START_REF] Jackson | Classical electrodynamics[END_REF]. In the Bragg scattering lattice, the propagation of the scattered wave does not depend on the vibration of the scatterers represented by the lumped masses, and thus the scattered waves propagate mainly as reflections [START_REF] Thomas | The birth of X-ray crystallography[END_REF] with the vibration concentrated on the ligaments [START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF][START_REF] Matlack | Composite 3D-printed metastructures for low-frequency and broadband vibration absorption[END_REF]. In Thomson scattering, however, the propagation of the scattered waves depends on the polarization of the electrons represented by the entire subunit cell. The orientation of the scattered waves is primarily in the forward direction of the incident wave. As a result, the attenuation in the bandgap is gradient rather than localized on ligaments (please refer to Fig. S8). Because the electrons are much lighter in mass than the atoms, the Thomson scattering will produce a more lightweight unit cell than a Bragg scattering unit cell for the same lattice, stiffness, and bandgap starting frequency [START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF][START_REF] Ding | 3D syndiotactic elastic metastructure with single-phase material[END_REF], which can also be drawn from Fig. 7. In addition,

Thomson scattering allows a smaller number of the periods to reflect the significant attenuation of the bandgap (please refer to Fig. 6).

Conclusion

In [ [START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF][START_REF] Li | Multiple band gaps for efficient wave attenuation by inertial amplification in periodic functionally graded beams[END_REF] and Eq. (S58)) occurs in theory. Although the quantitative characterizations demonstrate the anti-resonant frequencies in FRFs, the anti-resonant frequency is not essential for the bandgap generation. This work revealed that PnCs with inertial amplification only cannot ensure an extensive bandgap similar to MTR and MTT. Chirality is a virtual design element and one of the methods in realizing the Thomson scattering but not an indispensable condition for achieving such a bandgap.

It should be emphasized, the analogue to classic Thomson scattering of electromagnetic waves was proposed to help us to understand the underlying physics of the bandgap. Indeed, the underlying physics of the bandgap in our work cannot be considered to be completely identical to the general mechanism (classic Thomson scattering). Both have some significant differences and are not limited by the fact that the extremely important aspect of the Thomson scattering is for the long-wavelength limit which is not considered in this analogue. Despite these, the materialization of the inertial amplification and bandgap generation carried out in this work enables to couple of any two or more orthogonal modes to generate a substantial inertial amplification (instead of by the law of leverage [START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF][START_REF] Zeng | Inertially amplified seismic metamaterial with an ultra-low-frequency bandgap[END_REF]), and thus generate an ultra-low-frequency bandgap. The works could shed new light on the physics of the elastodynamic wave manipulation in inertial amplification-induced PnCs, and offer an entirely exotic avenue for the diverse design and investigation of PnCs with remarkable properties, such as the bandgap with lower starting frequency, broadband, and extensive attenuation.

  Fig. 1 shows the schematics of the unit cells and calculated band structures. The lattices in Fig. 1(a) and Fig. 1(b) are unit cells with translation-rotation coupling, and in Fig. 1(c) and Fig. 1(d) are unit cells with translation-translation coupling. Each type of lattice consists of two subunit cells. As depicted in Fig. 1(a) and 1(b), the arrayed translation-rotation coupling unit cell named ATR consists of two subunit cells Ⅰ. The mirrored translation-rotation coupling unit cell called MTR consists of subunit cells Ⅰ and Ⅱ. The right panels in Figs. 1(a)-1(b) represent the corresponding band structure calculated by utilizing finite-element method-based software COMSOL Multiphysics. In terms of the bandgap, there is an extensive bandgap in MTR but not available in ATR.

Fig. 1 .

 1 Fig. 1. The lattices and the band structures. (a) Arrayed translation-rotation coupling unit cell (ATR); (b) Mirrored translation-rotation coupling unit cell (MTR); (c) Arrayed translation-translation coupling unit cell (ATT); (d) Mirrored translation-translation coupling unit cell (MTT). As denoted in (a), each disk represents an individual lumped mass 𝒎 𝒊 . The symbols "Ⅰ" and "Ⅱ" represent two different subunit cells of translation-rotation coupling lattices (More details of the geometry can refer to Fig. S1.). The symbol "Ⅰ" is left-handed, and the symbol "Ⅱ" is right-handed. The symbol "α" is the subunit cell of translation-rotation coupling lattices, and its geometry is illustrated in Fig. S7.). The right panels in (a)-(d) represent the corresponding band structure with the bandgaps highlighted by gray areas.

Fig. 2 .

 2 Fig. 2. The decomposition of the movement for (a) ATR, (b) MTR, (c) ATT, and (d) MTT. 𝒖 𝒊 and 𝝋 𝒊 is the corresponds to the 𝒊 𝒕𝒉 lumped mass, as denoted in Fig. 1(a). The abbreviation "Scat" refers to scattering.

Fig. 3 .

 3 Fig. 3. (a) Diagram of the classical electromagnetic Thomson scattering. (b) The scattering diagram of the translational polarization in chiral sub-unit cells when the input is rotational polarization. (c) The scattering diagram of the rotational polarization in chiral sub-unit cells when the input is translational polarization. The translational polarization in (b) and (c) strictly is longitudinal. For clarity, we use the shear linear polarization to replace it. (Refer to Supplementary S7 for more details on the numerical data).

  (b) are carried out, as displayed by the red solid lines in Fig.4(c) and Fig.4(d). From the theoretical results, at high frequencies, the FRFs of MTR and ATR are constant, but only the MTR exhibits significant attenuation, which allows the excellent potential for generating an ultra-broad bandgap in this lattice. Fig.4(d)shows two antiresonant notches, which facilitate enhancing the attenuation of bandgaps. Meanwhile, compared with the FRF in FigureS2, the theoretical result in Fig.4(d)shows one more anti-resonance peak when the finite structure has two unit cells, which means that the number of the anti-resonance notches can be increased with the increase of the number of the unit cell. This property will be further discussed in Section 5.To demonstrate the correction of the theory, the finite-period structures containing 2 unit cells of ATR and MTR are numerically calculated in COMSOL solid mechanics module. The material is one kind of Nylon with the elastic modular 𝐸 = 1.6 × 10 & Pa, the Poison's ratio 𝑣 = 0.4, and the density 𝜌 = 1000 kg/m # . In the simulation, only the z-axis translational freedom of the disk at the input end is reserved to match the boundary conditions of the theory. The surface displacement load is applied at the input end disk rather than a force load to ensure consistency with the analysis and experiment. The domain point probes record the acceleration signals 𝑎 ' at the center of the input end lumped mass and 𝑎 ( at the output end. The FRF can be calculated by taking 𝑎 ' as the reference and the numerical results are presented as blue dotted lines in Fig.4(c) and Fig.4(d).

Fig. 4 (

 4 Fig. 4(a) shows the details of the experimental configuration, and Fig. 4(b) shows the fabricated ATR and MTR samples through photopolymerization-based 3D printing technology. The sample input diskis bolted to a plexiglass with a thickness of 10 mm. Notably, the plexiglass has approximately ten times weight than the two unit cells, to limit the freedom of rotation of the disk around the z-axis as much as possible. The plexiglass is supported using elastic supports to avoid exceeding the allowable amplitude

Fig. 4 .

 4 Fig. 4. (a) The schematic of the experimental configuration. The insert in the upper right corner is the detail of the connection between the sample and the plexiglass. (b) Photograph of the investigated samples. ATR sample on the left, and MTR sample on the right. (c) Experimental (green solid with stars), numerical (blue dotted), and analytical (red solid) frequency response functions (FRFs) of ATR. (d) Experimental (green solid with stars), numerical (blue dotted), and analytical (red solid) FRFs of MTR.

Fig. 5 .

 5 Fig. 5. The comparison of the MTR unit cells with 𝒂 = 𝟔𝟎 𝐦𝐦 and 𝒂 = 𝟑𝟓 𝐦𝐦. (a) & (b) The band structures (left side) and the model schematics (right side) of the ones with 𝒂 = 𝟔𝟎 𝐦𝐦 and 𝒂 = 𝟑𝟓 𝐦𝐦.(c) & (d) The analytical FRFs of MTR finite structures with the lattice constant 𝒂 = 𝟔𝟎 𝐦𝐦 and 𝒂 = 𝟑𝟓 𝐦𝐦. (e) The numerical FRFs of MTR finite structures composed of 4 unit cells with the lattice constant 𝒂 = 𝟔𝟎 𝐦𝐦 and 𝒂 = 𝟑𝟓 𝐦𝐦. The black dashes are the bandgap boundaries of 𝒂 = 𝟔𝟎 𝐦𝐦 and green lines are that of 𝒂 = 𝟑𝟓 𝐦𝐦.

Fig. 5

 5 Fig. 5 shows the band structures of the MTRs with 𝑎 = 60 mm and 𝑎 = 35 mm. When the lattice constant is 60 mm, the bandgap is 480 Hz -1250 Hz; when 𝑎 =35 mm, it is 960 Hz -1940 Hz. Fig. 5(c) and Fig. 5(d) show the analytical FRFs of MTR with different numbers of the unit cell when the lattice 𝑎 = 35 mm (blue line) and 𝑎 = 60 mm (red line). Because the lattice constant varies from 60 mm to 35 mm, the stiffness increases from 256060 N/m to 377630 N/m, and the amplified inertia decreases from 0.0432 to 0.0062 (Both variables are obtained theoretically). One can observe from Fig. 5(c) that the anti-resonant frequency disappears when 𝑎 = 35 mm. As shown in Fig. 5(d), despite the anti-resonant frequency appears again when the number of the unit cell is 4, the reduction of the amplified inertia leads to the shift of the anti-resonant frequencies toward higher frequency and even beyond the range of the bandgap.

Fig. 6 .

 6 Fig. 6. (a) Schematics of the finite structure with different numbers of the MTR unit cells. (b) Theoretical and (c) numerical discussion about the variation of the anti-resonant notch with the number of the MTR unit cells.

Fig. 7 (

 7 Fig. 7(a) and 7(b) exhibit a Thomson scattering unit cell and a Bragg scattering unit cell to find the intrinsic distinction between the Bragg scattering and Thomson scattering bandgaps. The parameter 𝜃 of the MTR is set as 90° to invalidate the coupled motion and reserve the translation alone, thus allowing a Bragg scattering unit cell to be obtained, as shown in Fig. 7(b). Consequently, there is a

Fig. 7 .

 7 Fig. 7. The comparison of the Thomson scattering-induced and Bragg scattering-induced bandgaps. (a) The Thomson scattering unit cell and its band structure. (b) The Bragg scattering unit cell and its band structure. Both unit cells have the same lattice constant, lumped masses, and equivalent stiffness. The deformation profiles of the (c) Thomson scattering (at 800Hz) and (d) Bragg scattering finite structures (at 1400 Hz).

  conclusion, we have theoretically and experimentally demonstrated the wave propagation and the formation mechanism of bandgaps in compression-rotation coupling PnCs. The coupling motion and wave propagation profile present an analogy to Thomson scattering. The results revealed that several conditions need to be met in the analogue Thomson scattering-based PnCs to generate a bandgap. First, the orthogonal coupling motions are essential for producing the Thomson scattering, which is quantitatively characterized as inertial amplification in equations. Second, these initial scattered waves must undergo a minimum of twice Thomson scattering. Third, the secondary scattered waves vibrating in the same mode should have the opposite initial direction of the vibration, which is the cause of the superimposable attenuation, and thus generates the bandgap. Remarkably, the third condition is the nature of that the inertial coupling phenomena (i.e. a non-diagonal inertial matrix as mentioned in Refs.

  Thomson scattering and the outgoing waves vibrating in identical mode have opposite phases after the second Thomson scattering are required to generate a bandgap. The findings are verified in a kind of chiral PnCs with translation-

rotation coupling. For the universality of the underlying physics, another non-chiral lattice with the translation-translation coupling is proposed to demonstrate that this bandgap is not unique to chiral lattices but available in other lattices with Thomson scattering effects.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (No. 12002258), the State Key Lab of Digital Manufacturing Equipment & Technology of HUST (DMETKF2021014), the Natural Science Foundation of Shaanxi Province (No. 2020JQ-043). Wei Ding is grateful for the support of the China Scholarship Council (Grant No. 202206280170).


Xudong Fan
email: fanxudong@njust.edu.cn

Yifan Zhu

Ning Li

Chunsheng Weng

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Dragomir N Kravchenko

Yuri S Neshev

Kivshar

Wenyu Zhao

Bingyi Liu

Huan Jiang

Jie Song

Yanbo

Yongyuan Jiang

Bo Wang

Fengliang Dong

Qi-Tong Li

Dong Yang

Chenawei Sun

Jianjun Chen

Lihua Zhiwei Song

Qiu Wang

Xueqian Zhang

Yuehong Xu

Yanfeng Gu

Zhen Li

Ranjan Tian

Shuang Singh

Jiaguang Zhang

Weili Han

" Zhang

Broadband

Adam C Overvig

Sajan Shrestha

Stephanie C Malek

Chengjun Zou

Andrei Komar

Stefan Fasold

Justus Bohn

Alexander A Muravsky

Thomas Pertsch

Dragomir N Neshev

Xingbo Liu

Hua Li

, Quan

Xieyu Xu

Shaoxian Chen

Meng Li

Liu

Jianxiong Li

Simon Kamin

Guoxing Zheng

Ho-Seok Ee

Peixia Zheng

Qi Dai

Zile Li

Zhiyuan Ye

Lei Jin

Yao-Wei Huang

Zhongwei Jin

Zhaogang Dong

Menghua Shengtao Mei

Wei Ting Jiang

Zhun Chen

Hong Wei

Jinghua Teng

Aaron Danner

Shumin Xiangping Li

Shuang Xiao

Changyuan Zhang

Joel K W Yu

Federico Yang

Cheng-Wei Capasso

Qiu

Inki Kim

Heonyeong Jeong

Younghwan Joohoon Kim

Seyedeh Mahsa Kamali

Ehsan Arbabi

Amir Arbabi

Yu Horie

Mohammadsadegh Faraji-Dana

Acoustic Meta-holograms for Encrypted Information Transmission

We report on the theoretical, numerical and experimental investigation of secure acoustic holography. This secure holography is achieved via an encoded hologram generated by a transducer array decoded by a transmissive acoustic meta-key. An unreadable image is received if directly using the encoded hologram, while the desired image can be obtained only with the aid of a decoding metastructure, i.e., a correctly decoding meta-key. The proposed meta-key consists of a series of unit cells, each of them is composed of a straight channel decorated with several opening panels.

By adjusting the opening size of unit cells within the meta-key, its transmitted phase profile can be effectively controlled, hence, the desired decoding meta-key can be successfully constructed. The patterns of a butterfly, sunglasses and the letters of NJUST are chosen as examples to demonstrate the performance of the secure acoustic holography. Vulnerability tests of the meta-key further confirm the safety and reliability of secure acoustic holography. With the advantages of simple design, flexible functionality, and high quality for reconstructed images, our work opens a new avenue for acoustic holography and may find applications in secure acoustic communications based on holograms.

I. INTRODUCTION

Acoustic holograms, with the ability of arbitrary sound field reconstruction, have been broadly studied in the last few years, [1][START_REF] Xie | Acoustic holographic rendering 430 with two-dimensional metamaterial-based passive phased 431[END_REF][START_REF] Marzo | Holographic acoustic elements 435 for manipulation of levitated objects[END_REF][START_REF] Marzo | Holographic 438 acoustic tweezers[END_REF][5][6][7][8][9][10][11][12] offering new inspirations in a variety of applications such as ultrasonic therapy and particle manipulations. [13][14][15][START_REF] Ma | Acoustic holographic cell patterning in 493 a biocompatible hydrogel[END_REF][START_REF] Ma | Dynamic ultrasound projector controlled 497 by light[END_REF][START_REF] Ma | Spatial ultrasound modulation by digitally 501 controlling microbubble arrays[END_REF] Holograms are generally realized by either active transducer arrays [START_REF] Marzo | Holographic acoustic elements 435 for manipulation of levitated objects[END_REF][START_REF] Marzo | Holographic 438 acoustic tweezers[END_REF][5]9] or passive acoustic metamaterials [6-8, [START_REF] Xie | Acoustic holographic rendering 506 with two-dimensional metamaterial-based passive phased 507 array[END_REF][START_REF] Michael D Brown | Phase and amplitude modulation 509 with acoustic holograms[END_REF][START_REF] Zhang | Converged wireless 513 infrastructure with acoustic holographic array[END_REF], where the phase and/or amplitude response on each hologram pixel are engineered to re-construct the desired field at a certain spatial location.

For conventional acoustic meta-holograms, the phase plate containing phase information is used for the field reconstruction [1,[START_REF] Xie | Acoustic holographic rendering 430 with two-dimensional metamaterial-based passive phased 431[END_REF], and the corresponding image quality is, however, barely satisfactory due to the missing amplitude information of holograms. To overcome this limitation, acoustic meta-holograms could be realized based on both phase and amplitude information [7,[START_REF] Zhu | Systematic 516 design of multiplexed-acoustic-metasurface hologram 517 with simultaneous amplitude and phase modulations[END_REF] so that high-quality images can be re-constructed with no need for optimizations. This extraordinary success has broadened the potential applications for acoustic holograms.

At the same time, remarkable progress has been made to achieve holographic images with high efficiency and quality using electromagnetic waves in terahertz, infrared, and visible regimes. [START_REF] Butt | Carbon 522 nanotube based high resolution holograms[END_REF][START_REF] Larouche | Infrared metamaterial 526 phase holograms[END_REF][25][START_REF] Ni | Metasurface holograms for visible light[END_REF][START_REF] Huang | Three-dimensional optical holography using a plasmonic 541 metasurface[END_REF][START_REF] Zheng | Metasurface holograms reaching 80% efficiency[END_REF][29][START_REF] Huang | Broadband hybrid holographic multiplexing[END_REF][31][32][33][34][35] The electromagnetic meta-holograms can be effectively tuned by controlling either metasurfaces composed of active materials or incident light sources. For the

A E i,j = A i,j , (1)
122 however, the encoded phase information ϕ E i,j is 123 modulated as

124 ϕ E i,j = ϕ i,j -ϕ R i,j . (2
151 p i,j = A i,j e ϕi,j = M m=1 A m r m exp (ikr m ), (3)
r 2 m = (x i,j -x m ) 2 + (y i,j -y m ) 2 + (z i,j -z m ) 2 .
157

Inversely, the acoustic field on the image plane can 158 be also calculated analytically based on the information 159 contained in the hologram, i.e., 160 where N × N is the total number of hologram pixels, and the distance r i,j between the spatial point (x, y, z)

P m = N i,j=1 A i,j r i,j exp [-i(kr i,j -ϕ i,j )], (4
and the hologram pixel (x i,j , y i,j , z i,j ) is calculated by

r 2 i,j = (x -x i,j ) 2 + (y -y i,j ) 2 + (z -z i,j ) 2 .

III. DESIGN OF ACOUSTIC META-KEY

The acoustic meta-key is composed of a series of Eq. ( 4), the numerical simulation based on the finite

347

To further test the security of the system, another 348 situation, when the specific information on some of its 349 random locations is deciphered, is considered (Fig. 7).

350

The deciphered parts of the meta-key are illustrated by

FIG. 1 .

 1 FIG. 1. Illustration of secure acoustic holography. (a) Encoded hologram contains the secure amplitude and enciphered phase information of the transmitted acoustic field, and an unreadable acoustic field will be generated directly using the encoded hologram (left panel). The acoustic meta-key contains the decoding phase, and the desired field can be recovered with the aid of the meta-key. (b) Encoded hologram, decoding phase provided by the meta-key and decoded hologram for the acoustic field of a butterfly.

) 125

 125 Fig. 1(a)].129

152

  where k = 2πf /c is the wavenumber, c = 343 m • s -1 is 153 the sound speed of the background medium, air. M is 154 the total number of image pixels, and r m is the spatial 155 distance between the image pixel and hologram pixel, i.e., 156

) 161 FIG. 2 .

 1612 FIG. 2. Illustration of the acoustic meta-key and experiment setup for the measurements of acoustic fields. (a) three-dimensional illustration and the corresponding two-dimensional cross-section of an element of the meta-key. (b) Phase shift and the transmission rate of the sound field when varying the geometry parameter of the element. Inset: simulation setup for the unit of acoustic meta-key. (c) The emitter array consisting of 16 × 16 elements is connected to a computer through an Arduino Nano board. An ultrasonic receiver is mounted on a translation stage, which is directly controlled by the computer through a Matlab script. Received signals are captured by the oscilloscope and then transmitted to the computer. (d) Photos of a meta-key sample and the emitter array.

16 229FIG. 3 .FIG. 4 .

 1634 FIG. 3. Results for the field of a butterfly. (a) Desired field of a butterfly. (b) Encoded amplitude and (c) phase information for the desired field. (d) Additional phase provided by the meta-key. (e) Unreadable field generated directly by the encoded hologram. (f-h) Analytical, numerical, and experimental results for the field generated by the decoded hologram.

264Fig. 4 (Figures 4 FIG. 5 .

 445 Fig.4(d), which is the same as that in Fig.3(d). Here, 280

307

  FIG. 6.The vulnerability test of the meta-key.(a) Match ratio between the holographic images using the partial meta-key and the entire meta-key. Inset: partial meta-key (marked by the shadow) containing n × n effective phases. (b) Intensity fields using partial meta-keys with different n.

351

  shadows in the inset in Fig.7(a), where the area ratio 352 between the deciphered part and the whole meta-key 353 is characterized by the decryption percentage. The 354 match ratio between the meta-holographic images using 355 the partial meta-key and the entire meta-key is shown 356 in Fig. 7(a), where the match ratio increases with the 357 decryption percentage. The reconstructed image is still 358 fuzzy even when 85% of the meta-key is deciphered 359 [Fig. 7(b)], and the image is barely recognized when 360 the decryption percentage reaches 90%, which again 361 confirms the security and reliability of the secure acoustic 362 holography proposed here; see Video 1 for the dynamic 363 evolution of the reconstructed holographic images when 364 the information of the meta-key is deciphered [59]. 365 We have also examined the effect of the image location 366 on the quality of the reconstructed hologram.

Figure

3678

  FIG. 7.The vulnerability test of the meta-key when the information at random locations is deciphered. (a) Match ratio between the holographic images using the partial meta-key and the entire meta-key.Inset: the partially-deciphered meta-key is illustrated by shadows. (b) Intensity fields using partially-deciphered meta-keys.

558[ 31 ]

 31 Lei Wang, Sergey Kruk, Hanzhi Tang, Tao Li, Ivan 559

"Holograms for acoustics," Nature 537, 518-522 (2016).

array," Scientific reports 6, 35437 (2016).

distribution on the image plane for the analytical results

315

[Eq. [START_REF] Marzo | Holographic 438 acoustic tweezers[END_REF]], the numerical simulation, and the experimental 316 measurements, respectively.

From the results, the 317 pre-designed pattern of letters NJUST is well recovered 318 and clearly observed. accessed by others (Fig. 6). The match ratio between 330 the meta-holographic images using the partial meta-key


Haitham Hassan
email: ibnel.haitham.hassan@gmail.com

Jeanne Chiavaralli

Afnan Hassan

Loay Bedda

Tim Krischuns

Kuang-Yu Chen

Alice Shi

Ming Li

Adrien Delpal

Etienne Decroly

Masoud Vedadi

Nadia Naffakh

Fabrice Agou

Sergio Mallart

Reem K Arafa

Paola B Arimondo
email: paola.arimondo@cnrs.fr

Design and synthesis of naturally-inspired SARS-CoV-2 inhibitors

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The late 2019 coronavirus disease outbreak turned out into the global COVID-19 pandemic that is still on-going. 1 The causative agent of the COVID-19 disease was identified as the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). [2][3][4] This highly transmissible airborne virus is a positive-sense single-stranded RNA virus, whose genome encodes 29 proteins including four structural proteins. 5 Although there are currently different vaccines and treatments available to tackle this pandemic, research efforts should continue to find therapeutic agents against the virus, as well as variants of concern or other zoonotic coronaviruses that might emerge in the future [14][15][16] . There are now several treatments for COVID19. For example, Remdesivir, which has been granted emergency-use authorisation, showed modest benefit to the patients with COVID19. [6][7][8][9] Paxlovid, the FDA approved oral SARS-CoV-2 main protease M pro inhibitor, showed benefit for the treatment of COVID-19 patients in severe conditions. 9b Although the structure of the active site of M pro is highly conserved among coronaviruses 9c , the emergence of drug-resistant variants cannot be excluded. Furthermore, the massive vaccination campaign currently on-going does not guarantee the herd immunity and that vaccinated people with SARS-CoV-2 antibodies will be immune to reinfection. [10][START_REF] Forni | On behalf of the COVID-19 Commission of Accademia Nazionale dei Lincei[END_REF][12][13] Therefore, designing small molecule libraries from new unexplored chemical space could open an avenue towards new antiviral drugs. To get better hit rates in drug discovery programs, a beneficial factor is natural product likeness. 17 In addition, enhancing the 3-D character and increasing the fraction of sp 3 carbons in the designed chemical libraries increase the opportunities to discover hits and consequently potential drug candidates. 18,19 Herein, we designed a naturally-inspired high 3-D fused-bridged dodecahydro-2a,6epoxyazepino [3,4,5-c,d]indole-based chemical library, with up to three points of diversity and six controlled chiral centres, two of them are all-carbon quaternary, accessible in one step. The synthesised 25 compounds were assayed in lung cell-lines infected with SARS-CoV-2 and screened for cytotoxicity leading to the confirmation of two hits promising for further optimisation. Furthermore, to identify the possible target, high-throughput virtual screening was carried out on four main protein targets in SARS-CoV-2: (i) the main protease (M pro ) that is a non-structural cysteine protease and plays a key role in the release of 16 nonstructural proteins involved in the virus replication; 20 (ii) the nucleocapsid phosphoprotein that packages the viral RNA into a helical ribonucleocapsid (RNP); 21 (iii) the non-structural protein nsp10/nsp16 complex; 22, 30 and (iv) the RBD/ACE2 membrane glycoprotein complex, responsible for the entry in the host cells. 23 The computational data led to the possible binding of the best two hits 3b and 9e to either M pro or the nsp10/nsp16 complex. A cellular and a biochemical assay for M pro protease activity confirmed the anti-M pro activity of 3b, while no activity was validated for 9e. These findings open a promising starting point for the identification of new antiviral lead candidates.

Results and Discussion

Chemistry. Tuning the right combination between 3-D dimensionality, natural product likeness, C-sp 3 fraction, ClogP and number of heteroaromatic rings bearing H-donor/acceptor atoms in one molecule is a challenging task. Finding the right blend leads to a better hit rate. Inspired by the synthesis by Shi and his co-workers, compound 1 can be accessed in three steps in large scale from the commercially available p-anisidine (Scheme 1).

24a This hydro-epoxybenzo [c,d]indole skeleton is a natural product-like and has a high 3-D character and high C-sp 3 fraction along with optimal ClogP value that is known to favour bioactivity. Modulation of 1 by introducing relevant groups bearing different H-donor/acceptor atoms can increase its biological activity potential. Compound 1 has two double bonds (highlighted in colours in Scheme 1) and a ketone function allowing further chemical modification. Because of the diverse stereo electronic environment, the two double bonds could be differentiated. The non-conjugated double bond (in magenta) was selectively di-hydroxylated by potassium osmate(VI) to give diol 2 (Scheme 1). This intermediate could be transformed into decahydro epoxy-azepino indoles (3a'-c'), which incorporate a constrained morpholine ring present in several bioactive molecules.

24b Ring expansion was achieved by oxidation of 2 with sodium periodate to give the intermediate bisaldehyde, that was reacted without isolation with different primary amines under reductive amination conditions to give 3a'-c'. Reduction of the ketone with sodium borohydride afforded alcohols 3a-c as single diastereoisomers These transformations allowed to introduce a new accessible point of diversity at the nitrogen atom. The enone moiety in 1 or 2 was resistant to reductive amination conditions. Indeed, in the preparation of 3a'-c' no reaction at the ketone function was observed and several attempts to transform 1 into the corresponding secondary amines using primary amines under diverse reductive amination conditions failed to produce the desired secondary amines. Thus, further modulation was conducted by reducing the ketone of compound 1 to the corresponding alcohol and then reacted with electrophiles such a 2-chloropyrimidine to afford 4 (Scheme 1).

To introduce other relevant groups bearing different H-donor/acceptor atoms, scaffold 3a' was hydrogenated in MeOH to give the deprotected secondary amine 5 with concomitant reduction of the double bond (Scheme 2). B by-product 6 was also isolated as the result of the reductive amination of the newly formed secondary amine 5 with adventitious formaldehyde, formed in-situ from the reaction of MeOH with Pd/C and hydrogen. 25 Changing the solvent to ethyl acetate slowed down the reaction and gave 7 resulting from the reduction of the double bond only without removal of the benzyl group (Scheme 2, panel A). Secondary amine 5 allowed to prepare a diverse set of compounds, showing the reactivity of this position. Reaction with aldehydes under reductive conditions gave tertiary amines and reaction with acyl chloride under basic conditions, or with carboxylic acids under peptide coupling conditions gave the corresponding tertiary amides. As reactive functional groups are not desirable, all the intermediate ketones were directly subjected to reduction with NaBH 4 to give 8a-g. In parallel, the secondary The third round of functionalisation was conducted on 6, 7, 8a, e, g (Scheme 3). Reaction of the free alcohol with halogenated heterocycles or alkanes gave compounds 9a-e (Scheme 3, panel A). Similarly, compound 3b reacted with 2-chloropyrimidine to give compound 10. Surprisingly, the di-hydroxylation of the left-hand site (LHS) double bond in compound 3b by potassium osmate(VI) was not successful. The isolated product was ketone 3b', resulting from the oxidation of the allylic alcohol catalysed by osmium salts 26 (Supplementary information at page S18). Molecular properties. The 3-D dimensionality and molecular properties of the 25 molecules were analysed using the web-free tool LLAMA (Lead-Likeness And Molecular Analysis; llama.leeds.ac.uk). 27 The chemical library showed average C-sp 3 fraction of 0.5-0.7, average ClogP of 1.0-5.0 and average molecular weight of 400-600, which are all features consistent with lead-likeness factors (Figure 1, panel A). It also provided a high 3-D character (Figure 1, panel B), a property which was proven to give better hit rates. 19 The natural product likeness was assessed using Natural Product Likeness Score calculator (NaPLeS) web-free tool and it indicated that all the synthesised compounds fall in the natural product likeness space with an average score of 0.2-1.3 (Figure 1, panel B). Biological evaluation. The 25 compounds were tested for inhibition of SARS-CoV-2 replication of the Beta variant in the lung cancer A549 cell line stably transduced with a lentiviral construct bearing the human ACE2 receptor. 40 The measurement of viral replication was carried out by quantitative RT-PCR as described in the Supplemental Methods section (Table 1). The key scaffolds 1 and its analogue 4 were not active, indicating the importance of the right-hand side (RHS) diversity point. Introducing dihydroxyl groups on compound 1 showed to be beneficial with an antiviral activity for compound 2 of EC 50 = 14 µM. Structure Activity Relationship (SAR) revealed that the presence of large aromatic groups in R 1 position on the RHS is important for the antiviral activity as compounds 5, 6 and 8f were inactive. It was also found that the presence of the ketone did not significantly change the viral activity when comparing compounds 3a‵ and 8d‵ to 3a and 8d, respectively. Substitution in R 2 with aromatic groups did not increase the activity; the aliphatic chain gave better activity. In parallel, the cytotoxicity activity of the compounds was assessed as the cytotoxicity could interfere with the antiviral activity of the compound (Supplemental Methods section). Several hits showed a favourable ratio between the antiviral activity and cytotoxicity. The most active compounds with favourable antiviral activity to cytotoxicity ratio and do not have electrophilic groups, which could favour covalent binding are 3b and 9e (Figure 2). In addition, 3b and 9e remained active against cells infected with the SARS-CoV2 Omicron BA1 variant with an EC 50 of 3.5 and 4.4 µM respectively, with 3b showing no cytotoxicity, while 9e giving cytotoxicity above 20 µM (Table 1 and Supporting information Figure S1).

Table 1.

Anti-SARS-CoV-2 Activity on A549-ACE2 cells (RT-qPCR) and cell cytotoxicity of the 22 compounds, expressed as EC 50 in µM, the concentration at which is observed 50% of inhibition of the maximal signal and its 95% confidential interval. In Silico studies. To identify potential target(s) behind the observed activity against SARS-CoV-2 of compounds 3b and 9e, a docking and virtual screening study followed by a molecular dynamics simulation (MD) was conducted against four SARS-CoV-2 proteins. The first selected potential target was the SARS-CoV-2 main protease (M pro ), a non-structural cysteine protease that is one of the key players in the release of 16 non-structural proteins involved in the virus replication 20 . Second was the nucleocapsid phosphoprotein that packages the viral RNA into a helical ribonucleocapsid 21 . The third selected target was the non-structural nsp10/nsp16 complex that has a 2'-O-methyltransferase activity (2'-O-MTase) activity involved in the methylation of RNA cap structure 22, 30 . Finally, the receptor-binding domain (RBD) of the viral spike protein responsible for the viral entry in the host cell was also virtually screened as the fourth plausible target 23 . 3D structures of the four target proteins were obtained from the protein data bank (PDB) as follows: the SARS-CoV-2 main protease M pro (PDB ID: 6LU7)

Entry

31

, the 2'-O-methyltransferase nsp10/nsp16 complex (PDB ID: 6W4H) 30 , the nucleocapsid phosphoprotein RNA binding domain (PDB IDs: 6VYO) 29 , and the RBD domain/ACE2/B(0)AT1 complex (PDB ID: 6M17) 33 . These 3D structures were selected based on their high resolution and lack of mutations.

Docking validation and high-throughput virtual screening.

First, we validated the use of the method MOE 2019.01 for the virtual screening study by self-docking the co-crystallized ligands in the adopted crystal structures for the nsp10/nsp16 complex (PDB ID: 6W4H) 29 and the SARS-CoV-2 main protease M pro (PDB ID: 6LU7) 30 with the respective co-crystallized ligands S-adenosyl-Lmethionine (SAM) and the peptide-like inhibitor PRD_00221 (Supplementary Figure S2). The proteins were initially subjected to structure preparation and protonation state fixation, then London dG and GBVI/WSA dG scoring functions were used to assess the binding poses and binding interactions. The root mean square deviation (RMSD) of the best scoring poses was calculated using the DockRMSD online server (https://zhanggroup.org/DockRMSD/) 31 . Since the optimum RMSD for docking validation is conventionally ≤ 2 Å and the results obtained by MOE 2019.01 were within this range, the adopted docking protocol was considered reliable. The binding potential of the 25 synthesised compounds was assessed and their potential binding affinity was reported as Sscores (MOE internal scoring function) and compared to those of the co-crystallized ligands, when applicable, for each of the four target proteins (Supplementary Table S1). The 2D interactions of the molecules compared to the co-crystallized ligands showed that compounds 3b and 9e were among the best virtual hits, which is in line with the antiviral activity profile of these compounds (table 1). The 2D binding interactions of the two compounds with the four-screened SARS-CoV-2 proteins are shown in Figure 3. Docking studies showed that the peptide-like inhibitor PRD_002214 is covalently bound to Cys145 and forms seven other noncovalent interactions with the active pocket of M Pro (Figure 3.A and Supplementary Table S2). Compound 3b forms six interactions: four as a H-bond donor with distances 3.14 Å, 3.42 Å, 3.61 Å and 3.72 Å, respectively, and with energy scores of -0.6 Kcal/mol, -0.8 Kcal/mol, -0.8 Kcal/mol, and -1.6 Kcal/mol, respectively. It also indicated two pi-H interactions between the ligand's 6-membered ring and Thr25 and Thr26 in the pocket with distances 4.39 Å and 4.22 Å, respectively and with energy scores of -1.1 Kcal/mol, and -2.0 Kcal/mol respectively (Figure 3.A and Supplementary Table S2). This suggests that compound 3b might interact with this target protein. On contrary, compound 9e makes only two interactions with M Pro Figure 3.A and Supplementary Table S2). For Nsp10/nsp16 complex, we mapped strong 11 interactions with the co-crystallized ligand SAM (Figure 3.B and Supplementary Table S2), while compounds 3b and 9e formed only two interactions, H bonds and pi-H interactions. Concerning the nucleocapsid phosphoprotein, the docking suggests that both compounds 3b and 9e form two interactions only (Figure 3.C). Finally, with the spike protein RBD/ACE2 complex (Figure 3.D), only one Hbond was observed between O62 of compound 3b and Glu 165 and one hydrogen bond between O72 of compound 9e and Asp164 suggesting a reduced interaction with this target complex (Supplementary Table S1).

Molecular Dynamics Simulation for Compounds 3b and 9e. We then carried out a 100 ns MD simulation of 3b and 9e to assess their binding stability with the target M Pro and nsp10/nsp16 complex (Supplementary data and Figures S3, S4, S5 and S6). To measure how much the protein and ligand conformations change along the MD simulation trajectory of the target proteins-ligand complex, RMSD values were calculated for the proteins' Cα atomic coordinates and the screened ligands using GROMACS 2021.1 32, 34

. The complex of 3b with M pro formed a very stable complex reaching stability in the first nanoseconds of the simulation with very low perturbation (less than 1 Å) throughout the whole 100 ns simulation (figure 4.A). Additionally, the stability of the complex was confirmed by the RMSD of 3b itself, as displayed in figure 4.B, eliciting perturbations of less than 1 Å throughout the whole simulation. On the other hand, the RMSD of nsp10/nsp16 complex Cα backbone in its complex with 3b showed small perturbations, while the RMSD of the 3b itself gave very high fluctuations after 50 ns. Then the stability of 9e in complex with M pro and nsp10/nsp16 complex was assessed over 100 ns simulation (Figure 5). The RMSD of the protein referenced to the backbone was calculated. On one hand, the complex of 9e bound to M pro achieved stability at almost 5 ns, showing small perturbations (less than 1 Å) in the region from 65 ns to the end of the simulation (Figure 5.A, black curve). This indicates a stability of the protein upon interaction with 9e. On the other hand, the RMSD of 9e itself showed significant perturbations at the beginning (from 0 ns to 40 ns), representing the ligand jumping away from the binding pocket, and at the end (from 70 ns to 100 ns) (Figure 5.B, black curve). In parallel, nsp10/nsp16 complex exhibited convergence at 12 ns with small perturbations (~ 1 Å) thus indicating a stabilisation of the protein (Supplementary Figure S5). Besides that, the RMSD of 9e complexed to nsp10/nsp16 complex exhibited a very low RMSD (less than 6 Å) and RMSD perturbations, which reflect high 9e complex stability as shown in figure 5 Stable binding of 3b and 9e should reflect on the stability of the residues within the binding pocket. The rigidity and flexibility of such residues can be investigated through the root mean square fluctuation (RMSF) of the Cα atoms during the simulation (Supplementary Figure S5).. [START_REF] Ferreira | [END_REF] The RMSF of 3b and M pro complex showed low fluctuations., especially Cys145 and His41 that constitute the catalytic dyad. 9e exhibited fluctuations (~ 1.3-3.2 Å) in some residues, with higher fluctuations at residues (185:192) than 3b. In parallel, nsp10/nsp16 complex shows residues within 5 Å proximity from 3b and 9e, exhibiting lower fluctuations (~ 0.4-1.4 Å) (Supplementary Figure S5). The collective results for the in-silico studies suggest that the target of 3b is the main protease M pro , while 9e preferentially interacts with nsp10/nsp16 complex. To verify the in-silico results, compounds 3b and 9e were tested in in-vitro for inhibition of 2′-O-methyltransferase activity of nsp10-nsp16 complex, and nsp14 guanine-N7-methyltransferase activity, and in cellular assay involving M Pro .

A) B)

Biological targets validation:

Inhibition of nsp10-nsp16 complex. The nsp10-nsp16 complex is responsible for catalysing the final step of the coronaviral mRNA capping. Nsp16 is member of the (2′-O-MTase) family, which catalyse the transfer of methyl group to the RNA substrates from the methyl donor SAM. Nsp16 requires nsp10 for methyltransferase activity and stability. The in-vitro activity of SARS-CoV-2 nsp10-nsp16 complex was assessed by monitoring the transfer of 3 H-SAM to the biotinylated N7-meGpppACCCCC RNA substrate (Supplementary information). The subsequently methylated RNA was captured using scintillation proximity assay (SPA) beads followed by quantifying the level of incorporated 3 H-methyl by measuring the radioactivity level (counts per minute [CPM]). 41 In dose response experiments (Figure S7), commercially available S-adenosyl-L-homocystein (SAH) showed IC 50 values of 3.9 µM and 0.22 µM for nsp10-nsp16 complex and nsp14 methyltransferase activities, respectively (Supplementary Figure S7 and Table 2). However, 3b and 9e did not show any inhibitory effect on nsp10-nsp16 2'O MTase activity, which did not support the computational analysis prediction for this target. Although 3b showed no inhibitory effect on the methyltransferase activity of nsp14, 9e showed a weak activity against nsp14 at high concentrations (Table 2 and Supplementary Figure S7).

Table 2. Effect of 3b and 9e on the methyltransferase activities of nsp10-nsp16 complex and nsp14. S-adenosyl-L-homocystein (SAH) was used as control. 9b Encouraged by the importance of this target and guided by the results obtained from the computational analysis through docking and molecular dynamics simulations, we tested the two hits for their ability to inhibit M pro in a highly sensitive cell-based luciferase assay that we developed to monitor SARS-CoV-2 main protease activity. 36 This gain-of-function assay is based on a Reverse-Nanoluciferase (Rev-Nluc) reporter in which two Nanoluciferase domains are permuted and linked together by a cleavage site recognized by M pro . Coexpression with the wild-type SARS-CoV-2 M pro results in cleavage of the reporter and thereby a significant reduction in luciferase activity.

Compound

The addition of an inhibitor of M pro results in a dose-dependent restoration in luciferase activity. The assay is run in parallel on a wild-type and a catalytically inactive M pro , to assess the specificity of the drug (details are provided in the Supplemental Methods section). The commercially available GC376 compound was used as a positive control. The dose response curve showed that GC376 reduced M pro

Scheme 1 .

 1 Scheme 1. Synthetic routes to the first round of functionalisation. a.

2 amine 5 Scheme 2 .a

 252 Scheme 2. Synthetic routes to the second round of functionalisation. a

Scheme 3 .a

 3 Scheme 3. Synthetic routes to the third round of functionalisation. a

Figure 1 .

 1 Figure 1. Molecular Properties and diversity. Panel A: Molecular weight vs CLogP (CLogP was calculated by ChemDraw Version 15). Panel B: Principal Moment of Inertia PMI Plot. Panel C: Natural Product Likeness score, the green line indicates All Synthetic Products, the orange line the Natural Products and the red dots the here synthesised chemical library

Figure 2 .

 2 Figure 2. Anti-SARS-CoV-2 dose response curve (left) and cytotoxicity (right) in log of the concentration in µM of the compound. Panel A: Compound 3b. Panel B: Compound 9e. Error bars of triplicates are shown.

Figure 3 .

 3 Figure 3. 2D interactions of the best 2 hits, compounds 3b and 9e. (A) Binding of co-crystallized ligand (PRD_002214), 3b and 9e with M pro (PDB ID. 6LU7). (B) Binding of co-crystallized ligand (SAM), 3b and 9e with nsp10/nsp16 complex (PDB 6W4H). (C) Binding of 3b and 9e with the nucleocapsid phosphoprotein (PDB 6VYO). (D) Binding of 3b and 9e with the RBD/ACE2 complex (PDB 6M17).

Figure 4 .

 4 Figure 4. Structural dynamics of compound 3b bound to M pro (black curve) and nsp10/nsp16 complex (blue curve); RMSD of the Cα backbone of the proteins in nm (A) and RMSD of 3b in nm along the MD trajectory (B).

Figure 5 .

 5 Figure 5. Structural dynamics of compound 9e bound M pro (black curve) and nsp10/nsp16 complex (blue curve); RMSD of the Cα backbone of the proteins in nm (A) and RMSD of 9e in nm along the MD trajectory (B).

35

 35

  are shown in Figure S7. NI: No Inhibition, NA: Not applicable. Inhibition of SARS-Cov-2 main protease (M pro ). M pro is a validated target for treating COVID19 with the recently approval of the drug Paxlovid.

Acknowledgements

The authors thank Dr Benjamin BARDIAUX, Structural Bioinformatics Laboratory, Institute Pasteur, for constructive proofreading of the insilico studies part. The authors thank the Institut Pasteur and Fondation de France grants for the installation and the Drug pipeline against SARS-CoV-2 project. Sylvain Paisant and Marine Ghazarian are acknowledged for providing technical help for the M pro cell-based assay. TK, KYC and NN were funded by the «URGENCE COVID-19» fundraising campaign of Institut Pasteur and by the Agence Nationale de la Recherche (grants ANR-18-CE18-0026, ANR-18-CE18-0028 and ANR-10-LABX-62-IBEID).

Fundings

Fondation de France grant Programme Féderateur de Recherche sur SARS-CoV2 & COVID-19 PRF7 SARS-CoV2 drug pipeline project (to PBA). Vedadi's lab is funded by US NIH 1U19AI171110-01.

Notes and references

activity, as seen by a gradual increase of Nanoluciferase activity measured in the presence of M pro WT (Figure 6.A), while the luciferase signal measured in the presence of catalytically inactive mutant remained unchanged. Full inhibition of M pro was achieved at 10 µM GC376 and the estimated IC 50 was in the micromolar range. [37][38][39] Only compound 3b showed a specific activity against M pro in Rev-Nluc assay from 10 µM, as indicated by an increase in luciferase signal, which was not observed with the catalytically inactive M pro mutant (Figure 6.B). An in vitro enzymatic assay on the purified protein confirmed a weak inhibitory activity of compound 3b (Supplementary Figure S8), while compound 9e was inactive. This observed activity of 3b against SARS-CoV-2 M pro was consistent with the computational study.

Conclusions

In conclusion, to explore a new chemical space in the design of inhibitors of SARS-CoV-2, a chemical library of 25 fused-bridged dodecahydro-2a,6-epoxyazepino [3,4,5-cd]indole compounds guided by natural product likeness, 3-D dimensionality and lead likeness was synthesised. The screening in a cellular system of viral infection led to the identification of 2 hits against the SARS-CoV-2 virus, confirming that following lead likeness guidelines and enhancing 3-D dimensionality increases the hit rate in the design of chemical libraries. Although the compounds also showed cytotoxicity, the two hits were confirmed to have an antiviral activity against SARS-CoV-2 with EC 50 between 3.7-1.4 µM with acceptable cytotoxicity difference. They also showed anti-viral activity against SARS-CoV-2 variant Omicron BA1, which consolidate their inhibition profile. Computational analysis through docking and molecular dynamics simulations was carried out against four main target proteins of SARS-CoV-2: Main Protease M pro , nucleocapsid phosphoprotein, non-structural protein nsp10/nsp16 complex and RBD/ACE2 membrane glycoprotein complex. The analysis identified as possible binding targets either the main protease M pro or nsp10/nsp16 complex. A cellular assay using a reverse-Nanoluciferase (Rev-Nluc) reporter system confirmed that compound 3b shows some activity against M pro in cells and a weak activity in an in vitro enzymatic assay with purified M pro . In contrast, in-vitro inhibition assay of nsp10/nsp16 complex and nsp14 did not confirm the computational analysis suggesting the methyltransferases as a potential target. The confirmed hits 3b and 9e pave the way to the chemical optimisation of the fusedbridged dodecahydro-2a,6-epoxyazepino [3,4,5-cd] indole skeleton for the inhibition of SARS-CoV-2. This also confirms the interest in inspired natural product scaffolds with improved drug likeness properties for the identification of novel hits with antiviral activity. Finally, the chemical pathway was designed to favour modulation of the substituents and further chemical optimisation is underway to increase antiviral activity and, in particular, to improve the activity against the potential target, the main protease M pro .

Author Contributions

HH designed, synthesised the library and wrote the chemistry, molecular properties and biological activity parts, and revised the manuscript. JC, FA set-up and carried out the biological evaluation in cells and participated to the writing and revision of the manuscript. TK, KC and NN set-up and carried out the cell-based assay for M pro protease activity and NN proofread the manuscript. RKA conceptualized the in-silico screening. AH, LB carried out the in-silico study and primary writing of the relevant sections, both of them participated equally in the in-silico studies part. RKA, AH, and LB revised the results, data analysis and wrote the in-silico studies. ED and AD developed and performed M pro enzymatic assay), AL, AD, MV carried out the assay against the methyltransferase activity of nsp10/nsp16 and nsp14. SM synthetised key compounds and revised chemistry and biological activity parts. PBA supervised the project, obtained the grants, wrote and revised the whole manuscript.

Conflicts of interest

There are no conflicts to declare.