Lucas Menou

Martin Castelnovo

Mechanical stress relaxation in molecular self-assembly

Molecular self-assembly on a curved substrate leads to the spontaneous inclusion of topological defects in the growing bidimensional crystal, unlike assembly on a flat substrate. We propose in this work a quantitative mechanism for this phenomenon by using standard thin shell elasticity. The Gaussian curvature of the substrate induces large in-plane compressive stress as the surface grows, in particular at the rim of the assembly, and the addition of a single defect relaxes this mechanical stress. We found out that the value of azimutal stress at the rim of the assembly determines the prefered directions for defect nucleation. These results are also discussed as function of different defect combinations, like dislocations and grain boundaries or scars. In particular, the elastic model permits to compare quantitatively the ability of various defects to relax mechanical stress. Moreover, these findings allows to understand the progressive buidling-up of the typical disclinations and grain boundaries pattern observed for ground states of large 2D spherical crystals.

Individual molecules are able to acquire collective behaviors by self-assembling, therebye reducing significantly their free energy. This behavior is widely observed across biological systems, forming structures of different spatial dimensionality (from 1D to 3D) and different shapes [START_REF] Mateu | Assembly, stability and dynamics of virus capsids[END_REF][START_REF] Hinzpeter | Optimal compartmentalization strategies for metabolic microcompartments[END_REF]. In two dimensions, such structures are for example fluid bilayer membrane made of phospholipids, or viral capsids which are closed shells made of proteins. In this last example, icosahedral symmetry is mostly observed, especially for small viruses [START_REF] Caspar | Physical principles in the construction of regular viruses[END_REF][START_REF] Zandi | Origin of icosahedral symmetry in viruses[END_REF]. From a local point of view, proteins are packed hexagonally and form "hexamers", almost anywhere on the surface of the virus, except at twelve locations where they form pentamers. Within the context of topology, these pentamers are considered as particular topological defects, known as disclinations. Their topological charge is positive, and negative charges correspond to heptamers with seven subunits being packed locally [START_REF] Seung | Defects in flexible membranes with crystalline order[END_REF]. The presence and the net topological charge of these defects in a closed shell is mandatory (+12) from a global point of view, as it is shown by Euler relation in order to satisfy the correct balance in the number of vertices, edges and faces of the associated triangular network [START_REF] Bowick | Interacting topological defects on frozen topographies[END_REF][START_REF] Vitelli | Crystallography on curved surfaces[END_REF][START_REF] Giomi | Crystalline order on Riemannian manifolds with variable Gaussian curvature and boundary[END_REF]. However the physical mechanism associated to the progressive arising of these defects for the growth on a curved substrate is still far from clear.

From a purely local geometric point of view, using equilateral triangles to build a regular hexagon results in a flat surface, while using the same triangles to build a regular pentagon leads to a buckled structure. This shows how incorporation of pentamers in a growing surface made of hexamers amounts to produce localized net curvature for the surface, and therefore it definitely helps to curve and to close it, but it does not predict the precise location of the defects. The answer to this question is to be found by considering the elastic properties of the growing surface. Indeed, it has been observed that curved molecular surfaces are mainly produced either by using spontaneous curvature for the building blocks [START_REF] Hicks | Irreversible growth model for virus capsid assembly[END_REF][START_REF] Olson | Chemical mimicry of viral capsid self-assembly[END_REF][START_REF] Levandovsky | Non-equilibrium assembly, retroviruses and conical shape[END_REF][START_REF] Wagner | The robust assembly of small symmetric nanoshells[END_REF][START_REF] Grime | Coarse-grained simulation reveals key features of HIV-1 capsid self-assembly[END_REF][START_REF] Castelnovo | Viral self-assembly pathway and mechanical stress relaxation[END_REF], or by using an additional scaffold on which the growing sur-face is adsorbed [START_REF] Vitelli | Crystallography on curved surfaces[END_REF][START_REF] Giomi | Crystalline order on Riemannian manifolds with variable Gaussian curvature and boundary[END_REF][START_REF] Dinsmore | Colloidosomes: selectively permeable capsules composed of colloidal particles[END_REF][START_REF] Bausch | Grain boundary scars and spherical crystallography[END_REF][START_REF] Irvine | Pleats in crystals on curved surfaces[END_REF]. In a recent theoretical work, Li et al. have studied both analytically and numerically the energetics of fivefold defects for a surface growing onto a spherical scaffold [START_REF] Li | Why large icoshedral viruses need scaffolding proteins[END_REF]. They showed that, upon assuming irreversibility for the assembly, these defects arise as local minima of total elastic energy, and that these minima are localized at the vertices of an icosahedron, in perfect agreement with self-assembly simulations. Yet, this result is obtained from a global point of view, as all points of the surface contribute to the total energy. It is not clear from the single building block point of view what determines the favorable localization of the defects: at this length scale, the building blocks are sensible to local interactions or forces, rather than the total energy.

On the other hand, for large triangulated structures that are closed, it has been shown that the ground state configuration involves not only twelve isolated disclinations, but also a combination of fivefold and sevenfold defects, the net topological charge being conserved (+12) [START_REF] Bowick | Interacting topological defects on frozen topographies[END_REF]. More precisely, a single pair of fivefold-sevenfold defects forms a dipole-like type of topological defect called a dislocation, and alignments of several of these defects form grain boundaries or scars [START_REF] Bausch | Grain boundary scars and spherical crystallography[END_REF]. Ground state configuration has been shown to consist in twelve fivefold defects located on the vertices of an icosahedron, together with a large number of scars. In this case, it is not clear whether these configurations can be obtained through self-assembly.

We propose in this work a mechanism for defect nucleation based on the analysis of mechanical stress at the rim of the growing surface. We demonstrate in particular that the fivefold defects arise in regions where the azimuthal stress (or hoop stress) σ θθ becomes negative during self-assembly, forcing therefore the local closure of pentagons instead of hexagons. The presence of compressive hoop stress has already been used in the literature to explain the wrinkles observed in thin sheets forced to adhere on spheres [START_REF] Azadi | Emergent structure of multidislocation ground states in curved crystals[END_REF][START_REF] Azadi | Neutral versus charged defect patterns in curved crystals[END_REF][START_REF] Grason | Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets[END_REF]. In these works, a compressionfree model was proposed, in which the hoop stress van-ishes in the outer regions of the surface thanks to some defect distributions. Notice that both compression free relaxing axisymmetry and numerical model without axisymmetry assumption have been used in order to reach conclusions about the optimal bulk defect distribution relaxing mechanical stress. In our work, the physics of defect nucleation through stress relaxation is similar. In particular, we do not assume any axisymmetry, allowing therefore to have a local comparison of stress relaxation patterns. In the next section, the elastic model is presented, and it is used to compute pattern of stress relaxation for different configurations. The comparison of these patterns is discussed in the last part of the article. Moreover, we provided additional datas in a supplemental material file.

ELASTIC MODEL FOR GROWTH ON A CURVED SUBSTRATE

In order to describe the elastic properties of a molecular surface growing on a curved scaffold, we have to estimate the elastic strain and stress within this surface as compared to equivalent flat geometry. We consider therefore as a reference state a flat disc of radius R and thickness h << R. We then force this disc to cover part of a sphere of radius R 0 . Following the classical framework of plate elasticity, which we briefly summarize below, such a deformation can be reached by the use of external loads [START_REF] Landau | Theory of elasticity[END_REF][START_REF] Ventsel | Thin plates and shells: theory, analysis and applications[END_REF]. Using polar coordinates {r, θ, z}, the deformation is first characterized by the displacement of each point from the reference state {u r , u θ , w}, which is also associated with elastic strains u ij and stresses σ ij . For large deflection of the disc, we assume the following non linear strain-displacement relationships (von-Karman strain)

u ij = 1 2
∂u i ∂x j + ∂u j ∂x i + 1 2 ∂w ∂x i ∂w ∂x j (1)
which allows coupling of in-plane and out-of-plane deformations. For an isotropic material, the relation between strain and stress tensors is given by Hooke's law. Next, the state of mechanical equilibrium of the system is described using local force balance equations involving the stress tensor ∂ j σ ij = 0. Finally, a compatibility condition between the strain tensor elements has to be fulfilled in order to guarantee a unique relation between the displacements and the strain components. Introducing Airy stress function χ(r, θ) defined by σ ij = ik jl ∂ k ∂ l χ with ij is the antisymetric unit tensor, and focusing on vertical external loads only p(r, θ), all these equations are rewritten in the form of the two famous non-linear Föpplvon Karman equations (FvK) for the Airy stress function χ(r, θ) and the vertical deformation w(r, θ) [START_REF] Ventsel | Thin plates and shells: theory, analysis and applications[END_REF]:

∇ 4 χ Y = - 1 2 L(w, w) (2)
D∇ 4 w = L(w, χ) + p(r, θ)

where Y is the 2D young modulus, D is bending rigidity, and ν is the Poisson's ratio. In these equations, the operator L is defined as

L(w, χ) = ∂ 2 w ∂r 2 1 r ∂χ ∂r + 1 r 2 ∂ 2 χ ∂θ 2 + ∂ 2 χ ∂r 2 1 r ∂w ∂r + 1 r 2 ∂ 2 w ∂θ 2 -2 ∂ ∂r 1 r ∂χ ∂θ ∂ ∂r 1 r ∂w ∂θ (5)
The bilaplacian is such that ∇ 4 w = ∇ 2 ∇ 2 w, and finally the Laplacian in polar coordinate is

∇ 2 w = 1 r ∂ ∂r r ∂w ∂r + 1 r 2 ∂ 2 w ∂θ 2 .
Notice that in the stress equation Eq.2, the operator L(w, w) is often assumed to be equal to the Gaussian curvature of the surface characterized by w(r, θ), which is a true statement only in the limit where | dw dr | is small [START_REF] Seung | Defects in flexible membranes with crystalline order[END_REF]. Once the Airy stress function is found as a solution of FvK equations, the planar stress components are computed in polar coordinate as

σ rr = 1 r ∂χ ∂r + 1 r 2 ∂ 2 χ ∂θ 2 (6)
σ θθ = ∂ 2 χ ∂r 2 (7)
σ rθ = - ∂ ∂r 1 r ∂χ ∂θ (8)
Finally, the planar strain components are also computed inverting Hooke's law as

u rr = 1 Y [σ rr -νσ θθ ] (9)
u θθ = 1 Y [σ θθ -νσ rr ] (10)
with ν the Poisson ratio. These equations have been widely used in the litterature in order to address various problems of thin plates elasticity. However, since these equations are based on the mechanics of continuous media, the configuration of building blocks making up the shell is not taken into account. This lack has been cured by Seung and Nelson in a seminal work, where they show that the presence of topological defects is included as additional source terms in the compatibility equation [START_REF] Seung | Defects in flexible membranes with crystalline order[END_REF]. In this case, the equation Eq.2 has to be modified

∇ 4 χ Y = s(r, θ) - 1 2 L(w, w) (11)
where s(r, θ) is the defect density. Within the present work, we will mainly consider two types of defects: disclinations (positive and negative) and dislocations. In triangular lattices, such as those formed by many selfassembling proteins like viral capsid proteins, positive disclinations are associated with proteins having five nearest neighbours, while negative disclinations are associated with seven neighbours. Within the context of continuous elasticity, the respective topological charges of these defects are +π/3 and -π/3. On the other hand, a single disclocation is associated to a bound pair of positive and negative disclinations, forming a structure similar to an electric dipole [START_REF] Vitelli | Defect generation and deconfinement on corrugated topographies[END_REF], and it is mainly characterized by its Burgers vector b, whose amplitude is proportional to the triangular lattice step and its orientation is orthogonal to the line joining the two opposite disclinations. Following Seung et al., the defect density is written as

s(r) = α q α π 3 δ (2) (r -r α ) + β ij b β j ∂ j δ (2) (r -r β ) (12
) where q α = ±1 are the topological charges of the set of disclinations considered, and b β are the Burgers vectors associated to the dislocation set.

In order to describe the growth of self-assembling proteins for example on a spherical surface, we will use the FvK equilibrium equations, together with the constraint that the growth occurs on the surface of a sphere. This imposes a deformation w(r, θ) such that L(w, w)/2 = K = 1/R 2 0 is constant, corresponding to a constant Gaussian curvature. This condition is easily integrated to obtain dw dr = √ α + Kr 2 . One further simplification of this profile is obtained by letting α = 0, which imposes a vanishing slope for the profile at the origin dw dr = r/R 0 . With this imposed deformation, it is possible to solve the stress equation Eq.11, with free boundary conditions

σ rr (r = R, θ) = 0 = σ rθ (r = R, θ).
Due to the linearity of the equation, the solution is found by considering the two source terms separately. Considering first the case of a spherical surface without defects, the constant positive Gaussian curvature leads to the following classical solutions for the Airy stress function, radial and azimuthal (or hoop) stress:

χ 0 (r) = Y R 4 64R 2 0 2 r R 2 - r R 4 (13)
σ rr,0 = Y R 2 16R 2 0 1 - r R 2 (14)
σ θθ,0 = Y R 2 16R 2 0 1 -3 r R 2 (15)
It was observed by many authors [START_REF] Azadi | Emergent structure of multidislocation ground states in curved crystals[END_REF][START_REF] Azadi | Neutral versus charged defect patterns in curved crystals[END_REF][START_REF] Grason | Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets[END_REF] that, while the radial stress is always positive, the hoop stress is positive close the center of the hemisphere, and negative as r/R > 1/ √ 3. A positive stress is associated to a tensile region of the surface, while a negative one is associated to a compressed region. In the latter region, the molecules composing the surface are therefore subjected to azimuthal in-plane compression. This state of stress, which is also associated to a negative planar hoop strain, is prone to an out-of plane buckling instability [START_REF] Grason | Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets[END_REF], and it is at the origin of the wrinkling observed when a flat disk is forced to adhere on a sphere. The wrinkling instability is possible when the adhesion to the sphere is not strong enough, permitting therefore an out-of-plane deformation. Wrinkling has also been shown to relax the hoop stress both at a local and global scale. The nega-tive hoop stress region has been used also to discuss defect distributions in compression-free models, assuming further axisymmetry for the surface [START_REF] Azadi | Emergent structure of multidislocation ground states in curved crystals[END_REF][START_REF] Azadi | Neutral versus charged defect patterns in curved crystals[END_REF][START_REF] Grason | Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets[END_REF].

In the context of self-assembly, we anticipate that the negative hoop stress has very important consequences on the building up of local contacts between molecules at the rim line of the assembly (see figure 1). Indeed, considering a local arrangement of five subunits, awaiting for another subunit to eventually close into an hexagon, the negative hoop strain reduces strongly the space available for subunit addition, and closing the structure into a pentagon might become more likely, therefore leading to the nucleation of a fivefold defect of positive topological charge. Notice that the same argument could be reversed and applied to the nucleation of a sevenfold defect of negative charge: if the hoop stress is positive and it reaches a critical threshold, the large positive tension and strain on a local arrangement of five subunits might be enough to incorporate up to two subunits, therebye creating a sevenfold defect.

In order to estimate more quantitatively the mechanical influence of defect inclusion as the surface grows, we need to solve the Airy stress equation in the presence of defects for a positive constant Gaussian curvature. The derivation of the solution to this equation has been recently proposed by Grason [START_REF] Grason | Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations[END_REF][START_REF] Azadi | Defects in crystalline packings of twisted filament bundles. II. Dislocations and grain boundaries[END_REF] for an arbitrary defect distribution. In this case, the Airy stress function associated to each defect has two contributions: a direct contribution which would exist for an infinite system, and an indirect contribution associated to the presence of a finite-size boundary. The latter contribution is equivalent to the inclusion of an image charge outside of the boundary. Following this derivation, we obtained lengthy expressions for the exact Airy stress function, hoop stress and radial stress, that are reported in Appendix A. With these expressions, we can observe quantitatively the influence of fivefold defect inclusion on stress relaxation, as it is shown in figure 2. Focusing on hoop stress evaluated at the rim r = R, we obtain a simpler expression for a single disclination of charge s located at (ρ 1 , θ 1 ):

σ θθ,1 (R, θ) = Y s 8π 2(R 2 -ρ 2 1 ) 2 R 2 (R 2 + ρ 2 1 -2Rρ 1 cos(θ -θ 1 )) (16)
Assuming the existence of a negative threshold value σ 5c for the hoop stress represented as a red line in the figure, the inclusion of a single defect relieves the hoop stress at the rim in an anisotropic fashion: the effect is larger close the defect, where the stress becomes positive and therefore tensile, and it reduces for diametrically opposed points, where the initial negative stress is still present. In fact, defect inclusion changes the stress not only at the rim, but throughout the whole surface because of the long range nature of elastic potentials (see inset of figure 1a). However, as far as self-assembly is concerned, the modulation of hoop stress is mostly relevant at the rim of the growing surface, as it is the only location having degrees of freedom in order to relax the stress in an irreversible assembly. Increasing the size of the surface, it is observed that placing defects where the hoop stress is the most negative and beyond threshold σ 5c relaxes significantly the compressive hoop stress. In figure 2c for example, three defects relieve the stress efficiently for three quarters of the rim, but significant compressive hoop stress is still present in the last quarter, suggesting the optimal location of the next defect, in agreement with Li's analytical and numerical results [START_REF] Li | Why large icoshedral viruses need scaffolding proteins[END_REF]. Iterating this type of minimal hoop stress analysis, it is easy to realize that the azimutal location of defects is compatible with icosahedral symmetry. The elastic model is therefore able to predict the optimal directions for defect inclusion. Note that the radial position of the defect might however depend on further details of the model, like the contact energy between subunits.

Inspecting further the variation of hoop stress across the growing surface in the presence of fivefold defects, one realizes that other types of defect might also be nucleated in order to relax the mechanical stress. Indeed, as it is clearly observed in the inset of figure 1a, the presence of a single fivefold defect close to the rim induces a large positive hoop stress between the defect and the rim, along a radius of the disc. As a consequence, this burst of tensile (positive) hoop stress is prone to nucleate a sevenfold defect very close to the fivefold defect. In the continuous limit, this pair of defects is equivalent to a dislocation. Based on this observation, one expects that one or several dislocations might also be efficient at relieving the stress. Combining linearly the solution for the stresses associated to a positive and a negative disclination in the limit of vanishing separation, one obtains analytically the stresses for a single dislocation. The result is shown in Appendix A. Considering a single dislocation located at (ρ 1 , θ 1 ), with an azimutal Burgers vector b = be θ1 , the hoop stress evaluated at the rim is:

σ θθ,1d (R, θ) = (17) Y b 24
4(R 2 -ρ 2 1 )[ρ 1 (3R 2 + ρ 2 1 ) -R(R 2 + 3ρ 2 1 ) cos(θ -θ 1 )] R 2 (R 2 + ρ 2 1 -2Rρ 1 cos(θ -θ 1 )) 2
Within such a dislocation, the equivalent fivefold and sevenfold disclination are located along a radius for the disc, the fivefold defect being closer to the center of the disc.

In figure 3a, several defects were compared, with respect to their ability to relax hoop stress along the rim of the assembly: a single disclination, a single dislocation, a dislocation plus a disclination, a small grain boundary (three dislocations in a row) and a small grain boundary terminated by a disclination. For a given size of the disc, it is observed that all the configurations produce similar pattern of hoop stress relaxation at the rim of the surface: close to the defect, hoop stress is relieved and becomes positive, while this stress relief decreases while reaching positions diametrically opposed to the defect(s). On the other hand, keeping the same defect configurations, and increasing the size of the disc as in figure 3b, we observe that the range of stress relaxation presents strong variation among defects configuration. For example, the combination of a grain boundary and a disclination provides a more efficient long range stress relief at the rim than a single disclination. This observation is compatible with the presence of grain boundaries in the ground state energy of closed shell for large radii previously mentioned in several works [START_REF] Bowick | Interacting topological defects on frozen topographies[END_REF]. Our work is therefore able to provide an assembly pathway leading to the observed ground state for a closed shell.

It is possible to test the predictions of the elastic analytical model by using a standard numerical triangulated surface [START_REF] Seung | Defects in flexible membranes with crystalline order[END_REF]. In this numerical computation, the structure is constructed using a triangular network reproducing the topological properties of the protein surface: most vertices are shared among six triangles, and at some locations they are shared among five or seven triangles, reproducing respectively positive and negative disclinations. A single dislocation is obtained by having two vertices of coordination five and seven nearest neighbours. The elastic energy is composed of two terms: an in-plane energy, associated with compression or stretching of triangles within their own plane, and out-of-plane bending energy. The total elastic energy is therefore written as

E tot = k e 2 α,β (d α,β -d 0 ) 2 +k b i,j
(1-cos(θ i,j -θ 0 )) [START_REF] Li | Why large icoshedral viruses need scaffolding proteins[END_REF] where d α,β is the length of the edge relating vertices α and β, and θ i,j is the angle between the normal vectors to triangles i and j. The remaining set of parameters {k e , k b , d 0 , θ 0 } defines the intrinsic elastic properties of the building blocks. In particular, the spontaneous angle θ 0 is associated with a radius of spontaneous curvature of order R 0 ∼ d 0 /θ 0 . The continuous limit of this discrete model corresponds to the analytical model discussed so far provided that the elastic constants k e , k b are related to Young modulus and flexural rigidity with the following relations Y = 2k e / √ 3 and D = √ 3k b /2 [START_REF] Seung | Defects in flexible membranes with crystalline order[END_REF]. The triangular lattice with appropriate defect locations is constructed by putting each vertex on the surface of a sphere of radius R 0 , and the final structure is relaxed by minimizing the energy using a conjugate gradient method. Notice that we added a radial Morse potential between the vertices and the surface of the sphere, in order to force the triangular structure to adhere on the sphere during the energy minimization. Provided that the adhesion is strong enough, we checked that the purely elastic energy of this discrete hemisphere do not depend on the parameters used for Morse potential.

� � � � � � � -���� -���� ���� ���� ���� ���� ������ ������+� ������+(�-�) ������+(�-�)+� ������+��(�-�) ������+��(�-�)+� � � � � � � � -���� -���� ���� ���� ���� ���� ������ ������+� ������+(�-�) ������+(�-�)+� ������+��(�-�) ������+��(�-�)+� � � � � � � � -���� -���� ���� ���� ���� ���� ������ ������+� ������+(�-�) ������+(�-�)+� ������+��(�-�) ������+��(�-�)+�
We chose to compare the elastic energy density at the rim of the surface within analytical and numerical models. In both cases, we omit the bending energy since it is expected to be homogeneous. In the analytical case, it is known that the stretching energy of a surface element dS is given by dS(σ θθ + σ rr ) 2 /(2Y ) = dSσ 2 θθ /(2Y ), when evaluated at the rim of the surface. In the numerical case, the energy density e local is estimated by averaging, for each vertex of the rim, the stretching energy of edges connected to that vertex. The result of the quantitative comparison is shown in figure 4. In the case of a single defect (disclination or dislocation), the analytical and numerical energy densities show a satisfactory quantitative agreement. Remarkably, no parameter adjustment has been used to obtain this agreement. When the comparison is performed on a multi-disclination configuration, the agreement is also remarkable. Overall, these exam-ples of comparison validate the analytical result obtained in this work.

DISCUSSION

Curved 2D crystals are known to exhibit geometric frustration, i.e. the long range configuration imposed by the curvature is not compatible with the short range arrangement of its building blocks. This frustration is usually accommodated by having topological defects embedded in their structure. Raising the question of how and where these defects will arise in self-assembled systems where each building blocks are identical, we proposed in this work an analytical elastic model in order to quantify the mechanical stress and the assembly pathway on a spherical geometry. Our analysis led us to identify the in-plane hoop stress evaluated at the rim of the hemispherical cap as the most relevant quantity in order to explain defect nucleation. Indeed, in the scenario considered, the rim of the assembly is the only degree of freedom being able to reduce the geometric frustration

π 0 π 0 π 0 � � � � � � � -���� -���� ���� ���� ���� ���� ������ ������+� ������+(�-�) ������+(�-�)+� ������+��(�-�) ������+��(�-�)+� � � � � � � � -���� -���� ���� ���� ���� ���� ������ ������+� ������+(�-�) ������+(�-�)+� ������+��(�-�) ������+��(�-�)+� � � � � � � � -���� -����
= 0.88Ra, ρ (a) 5 = ρ (a) 57
= 0.88Rc.

of the curved crystal. This inherent frustration is associated to a transition of the hoop stress from a tensile to a compressive state, the latter being prone to buckling (out of plane) phenomena. Computing analytically this value of hoop stress for different disclination configurations, we showed that defect nucleation is able to relax locally the compressive stress at the rim, and therefore it might ease further assembly. We also showed that this stress relief is anisotropic, suggesting preferred directions for successive defect nucleations. These results are complementary to those of compression-free models [START_REF] Azadi | Emergent structure of multidislocation ground states in curved crystals[END_REF][START_REF] Azadi | Neutral versus charged defect patterns in curved crystals[END_REF][START_REF] Grason | Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets[END_REF], in which symmetric defect distributions are seeked in order to vanish the hoop stress in an entire region. Within our work, we focused on the self-assembly process and therefore we investigated the modulation of hoop stress at the rim. Non-axisymmetric hoop stress calculations allowed us to go beyond compression-free models at the rim in that case. Moreover, we computed the stress relaxation pattern at the rim for more complex defect configurations. In particular, our results confirm that radial lines of dislocations, known as scars, are also efficient at relaxing mechanical stress. This quantitative observation suggests for the first time that the ground state of large closed spherical surfaces with crystalline order can be reached in self-assembling systems thanks to the proposed me-chanical stress relaxation mechanism. Indeed, it has been shown for these systems, both experimentally and theoretically, that the ground state is a superimposition of twelve fivefold disclinations decorated by multiple scars. This particular geometry can be therefore reached as the system is self-assembled, rather than by reorganizing locally the structure in a system with an initial idealized icosahedral symmetry, which might be a process that would require more energy.

All the defect configurations that we tested in this work have a common trait: defects are arranged along a single radial direction. This simplifying assumption is discussed and justified in the supplemental material file by comparing single to multiple directions for the defect distributions. Indeed, we observed that choosing multiple directions in order to relax the stress has an amplifying effect, but its does not change the picture drawn from single direction defect distribution. Moreover, we analyzed also in the supplemental material file the scar length dependence of stress relaxation. In this case, it is observed that best relaxation efficiency is obtained with increasing scar length. Combining the stress relaxation ability of scar length and scar numbers observed in figure 1 and 2 of supplemental material, we draw the following conclusion: different defect configurations are able to induce similar stress relaxation pattern, and in particular if a larger number of scars is used in order to relax the stress, their lengths have to be reduced in order to reach a similar level of stress relaxation. Within our model, which is based on the analysis of stress distribution, we did not find some objective way to determine the actual defect distribution. Rather, it is likely that the precise distribution is determined by computing the elastic energy for each configurations and optimizing it, like in the work of Li et al. [START_REF] Li | Why large icoshedral viruses need scaffolding proteins[END_REF]. Therefore analyzing the stress pattern allows mainly to determine the different mechanisms of defect nucleation.

The mechanism of defect nucleation through hoop stress relaxation at the rim proposed in this work introduces two hoop stress thresholds σ 5c and σ 7c for pentamer and heptamer nucleation respectively. Although their scaling with physical parameters of the elastic model is expected to be similar, the precise values of these threshold is expected to be model-dependent. We provided for example in an appendix a scaling estimate of σ 5c , which mainly depends on the adhesion energy between assembling subunits, Young modulus and shell thickness. In order to test the influence of these threshold parameters on the nucleation of defects, we provided additional simulation results in the supplemental material file. Within these simulations, it is observed that decreasing the heptamer threshold σ 7c at fixed pentamer threshold σ 5c allows to modify the early steps of defect nucleation: for large σ 7c , the first defect to be nucleated are pentamers. This result is not really new as it has been observed by many groups [START_REF] Hicks | Irreversible growth model for virus capsid assembly[END_REF][START_REF] Li | Why large icoshedral viruses need scaffolding proteins[END_REF]. However, if σ 7c is reduced, the first defects to be nucleated are dislocations or a mix of disclination and dislocation. This is a new result, as previous simulations were mainly designed to monitor the appearance of pentamers. Reducing further the threshold shifts the defect nucleation to scar-dominated regime. Our model and algorithm is therefore able to take into account more complex defect distributions.

In order to highlight the role of mechanical stress in the nucleation of defects, we chose a spherical surface as an idealized geometry for the sake calculation tractability. However, the qualitative features of the results demonstrated analytically for this particular geometry are expected to be valid for various scaffolding surface. The main reason is that the Gaussian curvature of the growing surface is a source for in-plane stress. A given distribution of Gaussian curvature will therefore generate a particular stress pattern at the rim where the assembly is effectively proceeding. Nucleating a fivefold defect will reduce locally along the rim the mechanical stress. Depending on the particular Gaussian curvature profile, we expect to generate an appropriate distribution of defects, including fivefold and sevenfold defects and/or one or several dislocations. For example, for geometries with negative Gaussian curvature we anticipate that sevenfold defects are expected to be most relevant in relaxing mechanical stress, which should now be tensile in excess at the rim [START_REF] Irvine | Pleats in crystals on curved surfaces[END_REF].

One important assumption in the present model is that the growth of the structure is irreversible, in the sense that the local structure of the surface is determined by the rim of the assembly. This neglects any further rearrangement away from the rim in the bulk of structure that has been assembled. These rearrangements represent another degree of freedom that could be used in order to relax the mechanical stress. However, their contribution might not be as important for two reasons. First, single disclination motion requires large scale rearrangement and are therefore unlikely [START_REF] Grason | Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations[END_REF]. On the other hand, dislocation motion are known to require mostly local structural changes, and this is the cost for these changes that will determine if the structure can relax using dislocation motion [START_REF] Falk | Molecular dynamics study of ductile and brittle fracture in model non crystalline solids[END_REF][START_REF] Nabarro | Theory of crystal dislocations[END_REF][START_REF] Hirth | Theory of dislocations[END_REF]. Roughly, if the interaction between subunits is strong enough, bulk relaxation becomes again unlikely because of the high cost for individual dislocation motion, leaving defect nucleation at the rim as the only effective way of reducing mechanical stress during the assembly. Notice that the particular geometrical configuration of the subunit at the rim of the assembly favors defect nucleation rather than bulk defect mobility, whatever the physical parameters. This qualitative analysis is consistent with the recent work by Panahandeh et al. who compared directly the phase diagram of irreversible and relaxed self-assembly [START_REF] Panahandeh | The equilibrium structure of self-assembled protein nano-cages[END_REF]. They found that for most of the phase diagram, the structures are very similar. This confirms that the main structural features, including defect position is indeed determined at the rim of the assembly.

We note also that the azimutal position of defects as the surface grows predicted by searching the most negative hoop stress values at the rim is consistent with results obtained by minimizing the global energy of the surface as the surface is assembled [START_REF] Li | Why large icoshedral viruses need scaffolding proteins[END_REF], at least for the nucleation of the firsts defects. The two models might differ at larger scale though because Li's model is solved in true spherical coordinates, while ours is solved using polar coordinates for the sake of tractibility and it is therefore expected to only apply for structures with moderate deviations from planar configuration, but the physical ingredients are essentially the same.

We hope that the present work will motivate future investigations, extending the present model in order to describe defect nucleation until full completion of the spherical surface, or to describe the interplay between defect nucleation and rearrangement. Additional data are provided as supplemental material file.

APPENDIX: SOLUTION OF AIRY STRESS EQUATION

The derivation of the solution to the Airy stress equation in the presence of a single defect of topological charge s = ±π/3 located at (r d = ρ, θ d = 0) has been provided in Grason's work [START_REF] Grason | Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations[END_REF]. We present here only the result for the complete Airy stress function, hoop and radial stress, which have not been provided in that reference. The idea of the calculation is to decompose the stress into two parts: the first part is the solution of the equation without boundary, and the second part takes into account the stress induced by the boundary. It is formally equivalent to use an image charge outside the boundary in order to enforce the free boundary conditions. Defining the distance between the point where the stress function is evaluated (r, θ) and the defect location (ρ, 0) as r2 = r 2 + ρ 2 -2rρ cos θ, the direct and indirect contributions are written respectively as

χ 1d Y s/(8π) = r2 log ρ - ∞ n=1 1 n r ρ n cos(nθ) if r < ρ = r2 log r - ∞ n=1 1 n ρ r n cos(nθ) if r > ρ χ 1i Y s/(8π) = ∞ n=0 C n r n + D n r n+2 cos(nθ) (19)
The coefficients C n and D n are chosen in order to satisfy the boundary conditions σ rr (r = R, θ) = 0 = σ rθ (r = R, θ). Once these coefficients have been found, exact summations are used to obtain an exact expression of Airy stress function. The image charge is located outside the boundary at a radial distance ρ such that ρρ = R 2 .

Introducing the distance between the point of observation and the image charge ri 2 = r 2 + ρ2 -2r ρ cos θ, the result can be rewritten as:

χ 1 Y s/(8π) = r2 2 log r2 R 4 ri 2 ρ 2 (20)
r(2ρ

3 cos θ + r(R 2 -ρ 2 + R 2 log R 2 )) 2R 2
Using the definition of Airy stress function in polar coordinate, we obtain the hoop and radial stress as: Using these exact expressions, it is possible to investigate the relaxation of hoop stress for various defect configurations, and in particular its anisotropic character. The solution of Airy stress equation in the presence of a single dislocation can be obtained by superimposing the Airy stress function of a fivefold and sevenfold defect. In this work, we will focus on dislocation with fivefold and sevenfold defects aligned along a radius for the sake of simplicity. The Airy stress function of a dislocation located at (r d = ρ, φ d = 0) is therefore computed as We provide in this appendix a scaling estimate of threshold value for hoop stress for defect nucleation. As the molecular surface grows by monomer addition at the rim of the hemisphere, we showed in the main text of this work that the hoop stress becomes eventually negative. In other words, a compressive stress develops within the surface, allowing therefore to close local fivefold configuration at the rim of the surface instead of the regular and defectless sixfold configuration. Within such a scenario, the energetic cost of local fivefold closure relative to sixfold closure is written as:

σ θθ,1 Y s/(8π) = -1 + ρ 2 R 2 + log r2 R 2 ri 2 ρ 2 (21)
∆E 5 -∆E 6 = -1 - σ 2 θθ a 2 h Y -2 1 = 1 - σ 2 θθ a 2 h Y < 0
(24) where ∆E 5 and ∆E 6 are respectively the energetic cost of fivefold and sixfold environment (h is the thickness of the surface, which does not play a role in this work). This balance includes in particular the elastic cost of adding a subunit ( a is the size of the subunit), and -1 is the subunit-subunit contact adhesion energy. In the estimation of elastic energy, we used the boundary condition for σ rr = 0. The hoop stress threshold therefore scales as

σ 5c Y ∼ - 1/2 1 a(Y h) 1/2 (25)
In particular, for increasing subunit-subunit adhesion energy, the critical hoop stress is more negative, reflecting larger resistance to in-plane compression.

FIG. 1 .

 1 FIG. 1. Stresses during the self-assembly. The arrows indicate the direction of in-plane forces associated to the stress. (a) Variation of the hoop and radial stress for a self-assembling surface without defect. (b) Illustration of hoop stress variation across the hemisphere. (c) Proposed mechanism for defect nucleation illustrated on triangulated surfaces, based on the value of hoop stress. Threshold values for the hoop stress are labeled as σ5c and σ7c. Parameters: R/R0 = 0.6

FIG. 2 .

 2 FIG. 2. Relaxation of hoop stress by defect nucleation. The hoop stress at the rim of the assembly is computed with (various dashed lines, see legend on figure) and without defects (line) as the surface grows (a to c). The size of the surface are shown on top of each graph. (a) Comparison of hoop stress without and with a single disclination (dashed line, ρ1 = 0.95Ra) as function of azimuthal angle. Inset: Hoop stress as function of radius. (b) A second disclination is added at the azimuthal location of smallest hoop stress, the radial position is ρ2 = 0.8R b (c) A third disclination is added at the azimuthal location of smallest hoop stress, the radial position is ρ3 = 0.85Rc. The defect radial locations are chosen in order to highlight the generic nature of anisotropic stress relaxation, but their precise location does alter the conclusion from this figure. The red line indicates a possible hoop stress threshold σ5c for defect nucleation.

FIG. 3 .

 3 FIG. 3. Short and long range relaxation of hoop stress using different defect configurations. These configurations are used to relax partially mechanical stress at a given size Ra (a), and their effect is shown if growth persists until R b without further inclusion of defects (b). The size of the surface are shown on top of each graph. All defects are arranged along a single radius of the disc. Their radial positions are as follows: isolated defects ρ5 = ρ57 = 0.95Ra. Combined dislocation (5-7) and disclination (+5) are arranged as follows ρ57 = 0.95Ra and ρ5 = 0.97Ra. Combined dislocations are arranged as follows: ρ (1) 57 = 0.95Ra, ρ (2) 57 = 0.97Ra, ρ(3) 57 = 0.99Ra, and final disclination is at ρ5 = 0.996Ra. Although the defect configurations seem equivalent in order to relax mechanical stress for Ra, strong modulation of relaxing effect is observed at large scale (b). (c) Typical configurations at the rim of the triangulated surface: a single dislocation (5-7) and a single scar made of two dislocations. The red links are connected either to fivefold or sevenfold vertices.

2 )FIG. 4 .

 24 FIG. 4. Comparison of energy densities at the rim between analytical (lines) and numerical (symbols) calculations. The case of a single disclination (star and dashed line) or a dislocation (dot and thin line) is shown for two sizes: (a) Ra = 0.55R0 and (b) Ra = 0.75R0. (c) Three disclinations configuration with 90 • relative orientation. Other parameters: Rc = 0.52R0, ρ

χ

  57 (r, θ) = b lim e→0 χ 1,(ρ,θ) (r, θ) -χ 1,(ρ+e,θ) (r, θ) e(23)APPENDIX: ESTIMATION OF STRESS THRESHOLD

  ) 5r 2 ρ + ρ 3 -2r cos θ(2R 2 + 3ρ 2 ) +4R 2 ρ(cos θ) 2 + 2ρ 2 r2 (rρ -R 2 cos θ) 2 A = 2R 6 + r 2 ρ 2 (r 2 + 4R 2 ) + r 2 ρ 4 -rρ cos θ 3R 4 + 2R 2 ρ 2 + 2r 2 (R 2 + ρ 2 )

	+	3r 2 + ρ 2 + 2ρ cos θ(ρ cos θ -3r) r2	-	B ri 4 ρ 4
	B = ( ri 2 ρ 3 σ rr,1 Y s/(8π) = ρ 2 R 2 + log	r2 R 2 ri 2 ρ 2 +	2ρ 2 (sin θ) 2 r2	-	A ri 4 ρ 2 (22)

+R 4 cos(2θ)(r 2 -2R 2 + ρ 2 ) + rρ cos(3θ)

Acknowledgments Stimulating discussions with R. Zandi, B. Dragnea, B. Davidovitch and M. Adda-Bedia are greatfully acknowledged at various steps of the work. This work was partly funded through the CNRS-PEPS program entitled "Modeling infectious processes".


Ph.D Pavel Müller
email: pavel.muller@i2bc.paris-saclay.fr

Dr Carsten Schultz

Dr Haw Yang

Elisabeth Ignatz

Stephan Kiontke

Klaus Brettel

Lars-Oliver O Essen
email: essen@chemie.uni-marburg.de

Chemical Science

DNA photolyases are flavoproteins evolutionarily and structurally related to photoreceptors of the cryptochrome type. Photolyases utilize blue light (absorbed by FAD or by an antenna cofactor) to repair UV-induced damage in DNA. Based on the differences in their structure, CPD photolyases specifically repairing the most common UV-induced lesions in DNA (the cyclobutane pyrimidine dimers, CPDs) can be divided into three classes: I, II and III. While photolyases from classes I and III occur only in microbes, class II photolyases are also present in most eukaryotic cells including multicellular organisms (plants and animals). With the exception of some higher mammals, these enzymes are indispensable for all sun-exposed forms of life.

 in the main text).

Our present manuscript represents the first endeavor to thoroughly characterize the first photoactivation step in a class II DNA photolyase, i.e., the primary electron-and proton-transfer

processes induced by absorption of blue light by FADox leading to a formation of FADH  via FAD -. Applying transient absorption spectroscopy to the Methanosarcina mazei CPDII photolyase (MmCPDII), we could (1) confirm the active role of the 'particular' triad of tryptophan residues (Trp381-Trp360-Trp388) in electron transfer to the photoexcited flavin and (2) show that the functionality of this triad is further enhanced by a fourth auxiliary tyrosine residue (Tyr345). The yield of the photoinduced radical pair of flavin with the most distal tryptophan (FAD -Trp388H + ) was found to be as high as 65%. This value is comparable to the bacterial (E. coli) class I CPD photolyase but it is much higher than that in many other members of the photolyase/cryptochrome-superfamily (e.g., 20% in plant cryptochromes). The 'particular' Trp triad in MmCPDII was hence shown to be as efficient as the bestperforming known 'standard' electron transfer chain in the whole protein superfamily.

To our surprise, we found that the deprotonation of the terminal tryptophan cation radical (Trp388H + ) was unprecedentedly fast: it occurred in 0.4 ns in MmCPDII and was hence by 3 orders of magnitude faster than TrpH + deprotonation in aqueous solution or in other proteins from the photolyase/ cryptochrome family (typical values range from 200 ns to several μs under similar conditions). Given this remarkable difference, we first suspected that the proton has to be transferred directly to a neighboring amino acid residue, rather than to the bulk of buffer molecules (which is the case for other proteins of the family). Nevertheless, the mutation of the only plausible candidatethe glutamate Glu387had virtually no effect on the deprotonation kinetics. We hence conclude that the proton must be transferred to the unique cluster of protein-bound water molecules that is present in both known crystal structures of class II photolyases: (archeal) Methanosarcina mazei and (eukaryotic) Oryza sativa CPDII (PDB entries 2XRZ and 3UMV, respectively). According to our sequence alignment (with over 400 non-redundant sequences), most of the residues coordinating the water cluster seem to be conserved among CPDII photolyases but, to our knowledge, such clusters do not occur in any other proteins from the cryptochrome/photolyase superfamily. Protein-bound H2O clusters have been shown to play an essential role for the biological function of other proteins such as bacteriorhodopsins (K. Gerwert et al., Biochim. Biophys. Acta, 2014). Our findings imply that class II CPD photolyases add to these natural systems, in which proton conductors based on protein-bound networks of water molecules can be as important for the biological function as amino acids.

We are confident that our work represents a major advance in the field and will inspire future theoretical and experimental studies. We submit our manuscript to Chemical Science because we would like to reach a broad audience of readers that are interested in photoactive proteins, lightinduced electron and proton transfer reactions and/or in DNA repair. We would hence appreciate if you were to recognize the importance of our work and consider it for publication in Chemical Science.

Sincerely yours, Pavel Müller

TOC:

Light activation of class II DNA photolyases is enhanced by a unique cluster of protein-bound water molecules.

Introduction

The exposure of DNA to UV-light causes serious damage of the genetic code and eventually results in fatal mutations. The most prominent forms of UV-induced lesions are cyclobutane pyrimidine dimers (CPDs) and the pyrimidine(6-4)pyrimidone photoproducts ((6-4)PPs). Several mechanisms evolved to restore the integrity of DNA, like nucleotide excision repair, base excision repair and photorepair. 1,2 The latter is catalysed by photolyases, a class of flavoenzymes, which belong to the photolyase/cryptochrome-superfamily (PCSf). 3 Photolyases are substrate-specific and they are divided into (6-4) and CPD photolyases, which are further subdivided into classes I -III. Compared to other members of the PCSf, the class II CPD photolyases, which occur in plants, animals and many microbial organisms, are highly divergent in terms of their sequences, especially in functionally important parts of the catalytic C-terminal domain. [4][START_REF] Pokorny | Proc. Natl. Acad. Sci[END_REF][START_REF] Mees | [END_REF] For DNA repair, all photolyases depend on a fully reduced FAD cofactor, FADH -. Upon photoexcitation, FADH -transfers an electron to the damaged DNA and thereby catalyzes the repair of the lesion. Isolated photolyases often contain a semi-reduced (FADH  ) or even a fully oxidized flavin (FADox). 4,[7][8][9] Photocatalytically active FADH -is generated from oxidized flavin species by a second light-induced reaction, usually referred to as photoactivation. In the case of initially oxidized FADox, the FAD -resulting from the primary photoinduced electron transfer (ET) becomes protonated (on a timescale of a few hundred milliseconds to a few seconds) to form FADH  . 10,11 Further reduction of FADH  to FADH -can be achieved by absorption of another photon, inducing transfer of a second electron to the flavin cofactor.

The mechanism of photoactivation has been studied in detail in vitro for a class I CPD photolyase (from E. coli) and for a (6-4) photolyase (from X. laevis). [10][11][12][START_REF] Byrdin | Proc. Natl. Acad. Sci[END_REF][START_REF] Lukacs | [END_REF][15][16] Upon photoexcitation, FADox or FADH  abstract an electron from the first member of a chain of three 12 (or four 10 ) tryptophan residues to form a primary FAD -TrpH + radical pair or an FADH -TrpH + pair, respectively. The electron hole then migrates along the Trp chain from the flavin-nearest Trp towards the most distant one, stabilizing the pair thermodynamically and by its spatial separation within less than 100 picoseconds. [START_REF] Lukacs | [END_REF][15][16] Further stabilization of the pair is achieved by deprotonation of the exposed terminal TrpH + cation radical by the solvent, which typically occurs within a few hundreds of nanoseconds. 10,12,17 Finally, the resulting Trp  radical is scavenged by an extrinsic reducing agent.

Despite the low sequence identity with other photolyases and cryptochromes (<16%), 18 class II CPD photolyases such as the one studied here -MmCPDII from the archaeon Methanosarcina mazei -share the same overall structural fold, with an N-terminal domain comprising a Rossman-like fold and an α-helical C-terminus which harbors the catalytically active FAD cofactor. 4 Please do not adjust margins Please do not adjust margins from other branches of the PCSf by the localization of the tryptophan cascade and by the presence of auxiliary tyrosine residues (Fig. 1). A previous mutational study of this ET pathway revealed that the Trp triad can also be functional as a dyad including only the first two FAD-proximal tryptophans. 4 In this study, we investigated the photoreduction of FADox to FADH  via FAD -in MmCPDII photolyase using transient absorption spectroscopy. We identified a cluster of water molecules involved in an unprecedentedly fast deprotonation of the distal tryptophan, Trp388. Additionally, we demonstrated, that a conserved tyrosine, Tyr345, also participates in electron transfer to photoexcited FAD. This work represents the first endeavor to thoroughly characterize the initial photoactivation step in a class II CPD photolyase.

Experimental Multiple sequence alignment-based analysis of class II CPD photolyases

451 non-redundant class II sequences with a pairwise sequence identity of less than 90% were extracted from a previous sequence-similarity network analysis of the photolyase-cryptochrome superfamily (combined PFAM protein families PF00875 and PF04225) 19 using the Cytoscape suite. [START_REF] Killcoyne | Protein Networks and Pathway Analysis[END_REF] The multiple sequence alignment was done using Clustal omega. [START_REF] Sievers | [END_REF] WebLogo 22 was used for the visualization of the degree of position-specific conservation.

Protein Preparation

Cloning of MmCPDII mutants. The generation of the E387Q mutant was based on pET-28a-MmCPDII and done according to the phusion protocol (NEB) using the 5'-phosphorylated primers listed below. The preparation of the W388F and Y345F mutants was described earlier. 4 E387Q Primer sense: 5'-Pho-CAG TGG AGC GAA TCT CCC GAA AAA-3' reverse: 5'-Pho-CAG AAT TTT TTT TGC CCA GTA CAT GCG-3'

Overexpression and purification of MmCPDII and mutants.

Overexpression of MmCPDII and mutants was done as described previously using E. coli BL21-Gold(DE3) cells (Stratagene). 4 The cultivation was done in terrific broth medium for 24 hours at 25°C (20°C for W388F). The proteins were purified using a NiNTA column (MACHERY-NAGEL) with 50 mM NaH2PO4, 300 mM NaCl, pH 8.0 and SEC column with Superdex 200 material (GE-Healtcare) with 10 mM Tris-HCl of pH 8.0 and 100 mM NaCl.

Experimental Conditions

Unless otherwise stated, the solutions of wild-type (WT) and all mutant MmCPDII photolyases studied herein contained 10 mM Tris-HCl buffer at pH 8.0 (measured at room temperature), 100 mM NaCl and 10% (v/v) glycerol. For the experiment in D2O, 10 mM Tris-HCl buffer (at pH 8.0) with 100 mM NaCl was lyophilised to dry powder and the sublimated H2O was replaced by D2O, adding up to the original volume of the H2O buffer; note that both H2O and D2O samples in this experiment (Fig. 5) were hence glycerol-free. In the experiments where cysteine was added as an external reducing agent, cysteine was first dissolved in a more concentrated Tris-Cl buffer with NaCl. After titration by NaOH back to pH 8.0 and the addition of glycerol and water to the desired final volume, a stock of 500 mM cysteine solution was finally obtained in the standard 10 mM Tris-Cl buffer with 100 mM NaCl and 10% (v/v) glycerol. The addition of cysteine to the photolyase sample has hence diluted the protein but the concentrations of other components (buffer, salt and glycerol) and the pH were kept constant.

All samples were air-saturated and kept at 7°C during the measurements or on ice in between. Before each experiment, they were rid of free FAD and other low-molecular-weight impurities by filtration over size-exclusion columns (Micro Bio-Spin, Bio-Gel P-6). The UV/Vis spectrum was checked before and after each measurement to ensure that the sample was in a good shape, i.e., not aggregated and FAD was not released, but protein-bound and fully oxidized (> 95% FADox). UV/Vis spectra were recorded on a Uvikon XS spectrometer (Secomam).

Transient Absorption Spectroscopy

Transient absorption kinetics were measured on three different setups described in detail in Refs. 10,11,17,23 .

In experiments on ps/ns timescales, the photolyase samples were excited at 355 nm by a Nd:YAG laser (Continuum Leopard SS-10, pulse duration of 100 ps, repetition rate 1 or 2 Hz, and an energy in the order of 5 mJ per cm 2 ).

In all other experiments, the samples were excited at 470 nm by laser flashes of 5 ns duration and an energy ≤10 mJ per cm 2 , delivered by a Nd:YAG pumped optical parametric oscillator (OPO; Brillant B/Rainbow, Quantel, France).

Indicative values of excitation energies were obtained by measuring the laser pulse energy behind a cell filled with H2O using an energy meter (Gentec QE25SP-H-MB-D0).

For kinetic measurements on the ps/ns and ns/s timescales (with 2GHz and 100 MHz bandwidth limits, respectively), the monitoring light was provided by continuous-wave lasers as listed in Ref. 11 . 2 x 2 x 10 mm cells were used (excitation pulses entered the sample through the 2 x 10 mm window; monitoring light through the 2 x 2 mm window). The monitoring light beams were attenuated by neutral density filters and mechanically chopped to produce a rectangular light pulse of 140 s duration and energy in the order of 1 J at the entrance of the cell, thus avoiding significant actinic effects. This pulse was synchronized with the excitation laser flash (see Ref. 23 for more details).

Please do not adjust margins

Please do not adjust margins

For experiments on millisecond to second timescales, the monitoring light was provided by a tungsten halogen lamp. For selection of a specific wavelength, an interference filter of 5 to 10 nm spectral bandwidth was inserted between the lamp and the sample. A similar filter was placed in front of the detector to block scattered light from the excitation flash and fluorescence. The bandwidth of the amplifier was limited to 30 or 100 kHz.

Signal Analysis

Multiple transient absorption signals were fitted globally with shared time constant(s) by using Origin 8.6.

Quantum Yield Determination

Quantum yields were determined by comparing the amplitudes of the 457 nm signals from the proteins (at t ≈ 10 ns; Figs. 3 and8) with those yielded by excitation of [Ru(bpy)3]Cl2 (99.95%) [START_REF] Thiagarajan | Proc. Natl. Acad. Sci[END_REF][START_REF] Byrdin | [END_REF][26] under the same geometry and excitation (by 5 ns laser pulses at  = 470nm). The quantum yield was then calculated using Equation (1):

𝚽 PL 𝚽 Ru = ∆𝑨 PL,457 ∆𝜺 Ru,457 (𝟏-𝟏𝟎 -𝒄 Ru 𝜺 Ru,470 𝒅 ) ∆𝑨 Ru,457 ∆𝜺 PL,457 (𝟏-𝟏𝟎 -𝒄 PL 𝜺 PL,470 𝒅 ) (1)
where A457 is the amplitude of the signals of the photolyase or the Ru complex at  = 457 nm, Ru,457 is the difference of molar absorption coefficients of the metal-to-ligand chargetransfer triplet ( 3 MLCT) of the [Ru(bpy)3] 2+ complex and its ground state ( -11 000 M -1 cm -1 ), c is the concentration (cRu = 27.4 × 10 -6 M), 470 stands for molar absorption coefficients at the excitation wavelength (= 470 nm) of the Ru complex (10 128 M -1 cm -1 ) or MmCPDII (9460 M -1 cm -1 ), and d is the optical path of the excitation beam (0.2 cm). PL,457 is the difference in the molar absorption coefficient of the respective photoinduced radical pair and FADox. The quantum yield of the 3 MLCT formation (Ru) equals one (100%).

For oxidized WT MmCPDII, we assume that the observed photoinduced radical pair (at t ≈ 10 ns) is predominantly FAD - Trp  , so PL,457 = 457(FAD -) + 457(Trp  ) -457(FADox) = (4740 + 940 -9205) = -3525 M -1 cm -1 . cPL = 40.0 × 10 -6 M, APL,457 = -0.023 and ARu,457 = -0.097 then yield PL,WT of 55.4%.

For the calculation of the quantum yield of FADox photoreduction in the W388F mutant MmCPDII, we assume the vastly prevailing radical pair (at t ≈ 10 ns) is FAD -Tyr  and PL,457 is hence equal to 457(FAD -) + 457(Tyr  ) -457(FADox) = (4740 + 150 -9205) = -4315 M -1 cm -1 . cPL= 38.5 × 10 -6 M, APL,457 = -0.002 and ARu,457 = -0.108 then yield PL,W388F of 3.7%.

Note that the used value of 457(FAD -) was obtained from the published spectrum 27 of FAD -in an insect cryptochrome. The spectrum of FAD -in MmCPDII is not available but it is likely not exactly identical to that in the insect cryptochrome, which could have a certain impact on the accuracy of the calculated quantum yields.

Results

The primary goal of this study was to characterize the first photoactivation step, i.e. the photoreduction of FADox to FAD - /FADH  , in the wild-type (WT) class II CPD photolyase (MmCPDII) and to find out how its class-specific tryptophan triad, Trp381-Trp360-Trp388, participates in this process. Our transient absorption spectroscopic data have confirmed the functionality of the tryptophan cascade but they have brought about an additional issue: it turned out that upon electron transfer (ET) to the excited FADox, one of the oxidized tryptophans (TrpH + ) in the triad undergoes an unprecedentedly fast deprotonation, that is by three orders of magnitude faster than the terminal tryptophans in other PCSf proteins (typically a few hundreds of nanoseconds). 10,12,17 This unusually fast rate indicated that the proton is probably transferred to a nearby, structurally-defined acceptor, rather than to disordered bulk solvent.

In order to identify the Trp undergoing the fast deprotonation and the proton acceptor, we have first examined the W388F mutant protein, in which the third (terminal) tryptophan, Trp388, was replaced by a redox-inactive phenylalanine, which cannot not participate in photoinduced electron transfer to the flavin cofactor. Experiments on this mutant protein unequivocally pointed to the terminal tryptophan Trp388 being the fast proton donor and we have hence searched for the likely proton acceptors in its vicinity.

For a better understanding of the following transient absorption data, absorption spectra of the species (FADox, FAD -, FADH  , TrpH + , Trp  and Tyr  ) that could contribute to changes in transient absorption are shown in Fig. 2.

Wild-type MmCPDII

Isolated MmCPDII photolyase (WT, as well as all mutants studied here) contains a fully oxidized FAD cofactor (FADox), which has two pronounced absorption maxima in the near-UV/Vis region (see Fig. 2): a double maximum centred around 370 nm and a triple band centred around 445 nm. FADox in MmCPDII absorbs up to 500 nm and can hence, in principle, be excited anywhere below this wavelength. For our initial experiments, we have chosen an excitation wavelength of 470 nm (provided by a Nd:Yag-pumped optical parametric oscillator, OPO; pulse duration of 5 ns) to avoid interference with our monitoring light sources and to minimize artifacts due to hydrated electrons that are formed upon excitation in the UV § .

Based on our previous experience with other members of the PCSf, we have first looked at timescales of a few ns to tens of s, where we anticipated to observe formation of an FAD - TrpH + pair, and deprotonation of TrpH All traces exhibit a step-like increase or decrease, the amplitudes of which (extrapolated to t = 0) are represented by black squares in Fig. 3b. If the anticipated FAD -TrpH + pair were formed, one would expect an absorbance increase below 415 nm and a bleaching between 415 and 485 nm (due to the transformation of FADox into FAD -) combined with an absorption increase between 450 and 650 nm (due to the formation of TrpH + )see red dashed line in Fig. 3b. Surprisingly, the signals at 562 and 594 nm (close to the absorption maxima of TrpH + ) and at longer wavelengths exhibited only very small amplitudes compared to the bleaching around 450 nm, excluding the presence of significant amounts of TrpH + beyond the given instrument response time (5 ns). On the other hand, the amplitudes of the absorption changes could be reasonably described by a difference spectrum reflecting the formation of FAD -and a deprotonated tryptophan radical Trp  (see Fig. 3b).

Assuming the signals are due to 100% FAD -Trp  pair, the quantum yield of FADox photoreduction in the first nanoseconds reaches 55% (see the Experimental section for quantum yield determination).

In order to see if we can detect the FAD -TrpH + pair (precursor of the FAD -Trp  pair), we have switched to another experimental setup enabling us to access faster timescales. 23,[START_REF] Thiagarajan | Proc. Natl. Acad. Sci[END_REF] To avoid time limitations due to the 5 ns long excitation pulse duration provided by the OPO, we have used a frequencytripled Nd:YAG laser at 355 nm with 100 ps pulses. The response time of this setup (200 ps) has allowed us to observe signals (Fig. 4a and Fig. S2 †), the initial amplitudes of which were indeed compatible with the presence of an FAD - TrpH + radical pair (Fig. 4b). The initial amplitudes partially decayed with a time constant of 350 ps to yield remaining amplitudes with a difference spectrum consistent with an FAD -Trp  radical pair (Fig. 4b). We hence attribute the 350 ps phase to deprotonation of TrpH + to Trp  (85%) in competition with FAD -TrpH + recombination (15%; the corresponding partial recovery of FADox is well visible, e.g., at 457 nm). Consistent with our attribution to a deprotonation reaction, the kinetics of this phase slows down from 350 ps in an H2O buffer to 800 ps in a D2O buffer (Fig. 5), corresponding to a kinetic isotope effect (KIE = kH/kD) of 2.3.

Within the 80 s experimental time window shown in Fig. 3a, one can observe the beginning of a virtually uniform decay of the radical pairs. Using a third setup adapted for monitoring of processes on milliseconds to seconds timescales, we have obtained signals containing the complete decay kinetics (Fig. 6a). It turned out that a biexponential decay function had to be used in order to obtain a good fit, which was an indication of recombination of two distinct pairs of radicals. Global fit of all signals yielded amplitudes for the two processes: those attributed to the faster process (with a time constant of 225 s and amounting to 70% of the total signal amplitude at 450 nm) were indeed consistent with the recombination of FAD -Trp  pairs but those attributed to the slower process (with a time constant of 1.1 ms and corresponding to the remaining 30% of A450 nm) exhibited a near-zero absorption change at 540 nm, which was incompatible with a tryptophan radical being the recombination partner of FAD -.

When looking at the structure of MmCPDII, 4 one can notice a tyrosine residue (Tyr345) in the vicinity of the terminal tryptophan of the ET chain (3.8 Å edge-to-edge distance from Trp388; Fig. 1). An involvement of this tyrosine residue in ET to the excited FADox was anticipated, since the mutation of Tyr345 to a red-ox inactive phenylalanine was previously shown to slow down the rate of FADox photoreduction in a steady-state experiment (compared to the WT protein) 4 . Indeed, the 1.1 ms phase nicely fits the difference spectrum for disappearance of an FAD -Tyr  pair (Fig. 6b).

Finally, in the absence of external reducing agents, all transient species are completely lost due to recombination within less than 5 milliseconds in the WT MmCPDII (Fig. 6a) and the initial state of the protein with fully oxidized FAD is restored.

In order for the photoactivation reaction to be efficient, the FAD -anion radical has to be stabilized by scavenging of its recombination partner (be it Trp  or Tyr  ) by extrinsic reductants. By adding sufficient amounts of cysteine to reduce transiently formed Trp  and/or Tyr  radicals, we could compete against the FAD -Trp  /Tyr  recombination (Fig. 7; the acceleration of the decay of ΔA540nm reflects the reduction of Trp  by cysteine) and obtain an isolated metastable FAD - radical, which got further stabilized by protonation to yield a neutral FADH  radical (see inset of Fig. 7; formation of FADH  from FAD -is accompanied by a pronounced absorption increase at 610 nm and decrease at 380 nm, see spectra in Fig. 2). Under our experimental conditions (0.01 M Tris-HCl buffer at pH 8.0, 0.1 M NaCl, 10% (v/v) glycerol, 7°C), this protonation occurred with a time constant of 630 ms. However, a closer look at the 540 nm signal with the highest cysteine concentration (0.3 M cysteine; Fig. 7) suggests that, in a small fraction of proteins, FAD -can be protonated at a much faster rate, which is reflected by the growth phase (with  2.2 ms) following the initial decay of the signal. Judging from the initial amplitude of the signal at 610 nm in the inset of Fig. 7, the fast protonation seems to be possible in 15% of proteins (note that only FADH  should absorb at 610 nm, contributions from FAD -or remaining Trp  /Tyr  radicals are expected to be zero at this wavelength -see Fig. 2).

W388F mutant lacking the 3 rd tryptophan of the triad

In order to find out which of the three tryptophan residues undergoes the unusually fast deprotonation in the WT MmCPDII (Figs. 4 and5), we decided to examine the behaviour of its W388F mutant, in which the terminal Trp of the triad was replaced by a non-reducing phenylalanine. Our structural data for this and the Y345F mutant (PDB codes 5O86, 5O8D) 28 show that there are no compensatory structural changes of residues lining the ET pathway, which could complicate the following Please do not adjust margins Please do not adjust margins analysis and interpretation. The transient absorption signals obtained for the W388F mutant on the ns/s timescale upon excitation by 5 ns pulses at 470 nm (Fig. 8a) exhibit steep decays in the first nanoseconds, followed by plateaus, the spectral footprint of which corresponds neither to an FAD - TrpH + , nor to an FAD -Trp  pair, but is compatible with an FAD -Tyr  pair (Fig. 8b), formed with a quantum yield of mere 4% (see the Experimental section for quantum yield determination). The nearest tyrosine to the 2 nd Trp of the triad (Trp360), which could serve as electron donor to Trp360H + , is Tyr380 (4.6 Å edge-to-edge distance; Fig. 1). Tyr380 is situated in the vicinity (4.0 Å) of yet another tyrosine Tyr445, which is more exposed to solvent and could hence play the role of the terminal electron donor to FAD in the W388F mutant. Alternatively, Trp360H + could also be reduced by Tyr345 (7.1 Å edge-to-edge distance), which is involved in ET in the WT protein.

Analogously to the situation in the WT protein, we had to use a different experimental setup to resolve the fast process preceding the formation of the FAD -Tyr  pair. On the ps/ns timescale, we were able to resolve signals, which were decaying nearly completely with a time constant of 1.2 ns (Fig. 9a). The initial amplitudes spectrally fit an FAD -TrpH + pair (Fig. 9b). We tentatively assign the 1.2 ns decay in the W388F mutant protein to charge recombination in the pair FAD -Trp360H + . This recombination presumably competes with a much slower (in the order of 10 to 20 ns) side ET from Tyr380 (or Tyr445) to Trp360H + , yielding an FAD -Tyr  pair with a quantum yield of 4%.

In the absence of extrinsic reducing agents, the FAD -Tyr  pairs in W388F recombine with a time constant of 2.7 ms (Fig. 10), which is 2.5× slower than the FAD -Tyr  recombination in the WT (the observed Tyr  radical is hence most probably not Tyr345  in the W388F mutant MmCPDII). Like in the WT, the Tyr  radical could be scavenged in the presence of cysteine and the resulting metastable FAD -was further stabilized by protonation to form FADH  (see Fig. S3 †) at the same rate as in WT MmCPDII under the same conditions (i.e., in 630 ms).

E387Q mutant: searching for an intra-protein proton acceptor

According to structural analysis of the environment of the distal tryptophan Trp388 in MmCPDII (PDB entry 2XRZ), a straight-forward candidate for the intramolecular acceptor of the N1 proton from Trp388H + is deprotonated § § Glu387 (see Fig. 12): its carboxylic group is as close as 3.6 Å from the N1 atom of Trp388. We therefore prepared a mutant, in which the glutamate was replaced by glutamine, which could not serve as a proton acceptor. Fig. 11 compares signals recorded for the WT protein and the E387Q mutant at 562 nm, which is a suitable wavelength for direct monitoring of the deprotonation of TrpH + to Trp  (and/or its reduction by a tyrosine). The figure clearly shows that the kinetics of TrpH + disappearance was almost as fast as in the WT (500 ps in E387Q vs. 350 ps in WT), indicating that Glu387 is not the primary proton acceptor. As outlined in the Discussion section, we identified a protein-bound water cluster ideally positioned to serve as the primary proton acceptor.

Y345F mutant: identification of the tyrosine involved in ET

As mentioned above, there were indications that the tyrosine residue Tyr345 also participates in electron transfer to the photoexcited flavin in MmCPDII. We have hence investigated the involvement of this residue in ET by direct comparison of time-resolved spectroscopic signals of the WT and the Y345F mutant MmCPDII.

Signals in Fig. 13 show a decay of flash-induced absorption changes in the Y345F mutant on the millisecond timescale. Unlike in the WT protein, the signals decay monoexponentially and the slower (1.1 ms) component present in WT (attributed to the FAD -Tyr  pair and representing approx. 30% of the radical pairs present at the beginning of the ms time scale) is obviously missing. The decay rate of the remaining faster phase ( 250 s) is very close to the value obtained from the fit for the faster process in the WT protein. We conclude that indeed a fraction (30%) of tyrosine Tyr345 gets oxidized during photoactivation of WT MmCPDII and forms the longest-lived (1.1 ms) radical pair with FAD -.

With respect to the kinetics and pathway of formation of the tyrosyl radical Tyr345  , we considered two (mutually nonexclusive) possibilities: (i) fast ET from Tyr345 to Trp388H + (in competition with deprotonation of Trp388H + in 350 ps) and (ii) slow reduction by Tyr345 of the deprotonated Trp388  radical (by proton-coupled ET in competition with recombination of the FAD -Trp388  pair in 225 µs). The similarity of the initial signal amplitude ratios at different wavelengths in the Y345F mutant and the WT protein on the ms time scale (compare Figs. 6a and13) seems to contradict fast formation of a substantial amount of tyrosyl radical in WT. Nevertheless, we have also compared the signals at 408 nm (wavelength of Tyr  absorption maximum) obtained using our fastest experimental setup (Fig. S5 †). If the 30% of Tyr  radical were formed directly from TrpH + (in competition with its deprotonation and/or its recombination with FAD -), the amplitudes of the 350 ps decay in the 408 nm signal would have to be clearly different in the WT and in the mutant protein: we would expect that the decay due to 15% recombination would be visibly (almost fully) compensated by a growth due to the formation of the Tyr  radical in the WT protein. In the Y345F mutant, on the other hand, no Tyr  radical can be formed, and the phase should hence reflect pure and uncompensated FAD -TrpH + recombination (absorption changes due to TrpH + deprotonation are negligible at 408 nm; see Fig. 2). Given that there is no significant difference in the ps/ns signals for the two proteins, we can conclude that most of the Tyr  radicals are formed later -on the s timescale in competition with the recombination of FAD -with neutral Trp  .

ARTICLE

Discussion

Based on all the data obtained for WT and mutant proteins (in the absence of extrinsic reductants), we have constructed a reaction model (Scheme 1) of the primary processes following FADox photoexcitation in WT MmCPDII.

We observed formation of an FAD -TrpH + pair within our experimental time resolution of 200 ps, followed by deprotonation of TrpH + in 350 ps, concomitant with 15% charge recombination in the FAD -TrpH + pair (Fig. 4). Assuming direct competition between deprotonation and charge recombination, the intrinsic time constants would be 0.4 ns and 2 ns, respectively. We assign the deprotonation to Trp388, the third (most distant to FAD) member of the triad Trp381-Trp360-Trp388 (Fig. 1), because the deprotonation was lost (and replaced by charge recombination in 1.2 ns) in the W388F mutant (Fig. 9).

In analogy to the extensively studied class I CPD photolyase from E. coli (EcCPDI), we suppose that ET from Trp388 to *FADox occurs by ultrafast hopping along the Trp triad also in MmCPDII. Losses may occur due to competition of forward ET with charge recombination in the radical pairs. We have obtained a quantum yield of 55% for the formation of the FAD -Trp388  pair (see Results and Experimental Section). Taking into account charge recombination in competition with deprotonation of Trp388H + (see above), the quantum yield of formation of the FAD -Trp388H + pair should be 65%, i.e., the same as for the terminal FAD -TrpH + pair in EcCPDI, 11 but substantially higher than in a (6-4) photolyase (30%) 10 or in a plant cryptochrome (20%) 17 . The particular Trp triad in MmCPDII is hence as efficient in electron transfer as the best known "standard" Trp triad.

Our data (Fig. 13 and Fig. S5 †) imply that the tyrosine residue Tyr345 (situated 3.8 Å from the third tryptophan of the cascade, Trp388) acts as a fourth auxiliary member of the electron-transferring chain, though only about 30% of the Trp388H + /Trp388  radicals seem to eventually get reduced by Tyr345 under our experimental conditions. Because of spectral congestion, we were not able to monitor the kinetics of tyrosine oxidation directly, but from the lack of a significant effect of the Y345F mutation on the ps/ns signals at 408 nm (Fig. S5 †) we could conclude that most of Tyr345  was formed on a slower time scale, i.e., after the deprotonation of Trp388H + . Direct oxidation of Tyr345 by Trp388  seems highly unlikely because it would imply a hydrogen atom transfer over a distance of more than 6 Å. We rather suggest that Tyr345 is mostly oxidized by Trp388H + that is present in a very small amount in thermal protonation equilibrium with Trp388  . Charge recombination in the FAD -Trp  pair may also proceed via thermal reprotonation of the tryptophanyl radical. The observed biphasic recombination kinetics in WT MmCPDII (Fig. 6a) can be described assuming effective time constants of 300 s and 1 ms for the charge recombination and the oxidation of Tyr345, respectively, together with a 1.1 ms recombination of the FAD -Tyr345  pair. The µs/ms charge recombination reactions can be blocked (and FAD -hence stabilized) by extrinsic electron donors (which are abundant in living cells) that scavenge the Trp  /Tyr  radicals.

In this situation, the isolated FAD -in MmCPDII becomes protonated in 630 ms to form a metastable FADH  (Fig. 7). This rate of protonation is slower than the reported 200 ms in the Xenopus laevis (6-4) photolyase 10 under similar conditions (0.05 M Tris-HCl buffer at pH 8.3, 0.05 M NaCl, 5% (v/v) glycerol, 10°C) but faster than the 4 seconds observed in the Escherichia coli class I CPD photolyase 11 (0.02 M phosphate buffer at pH 7.5, 0.2 M NaCl, 20% (v/v) glycerol, 7°C), in spite of a higher pH (8.0) used in the present experiment. The conversion of FADH  to the redox state active in DNA repair, i.e., the fully reduced FADH -, can follow directly after the absorption of another photon. However, in the presence of oxygen (and in the dark), both FADH  and FADH -in isolated MmCPDII spontaneously revert to FADox (within a few minutes in an air-saturated solution).

It is a matter of controversy 29,30 whether photoactivation of DNA photolyases through their respective Trp chains is a vital process in vivo or whether the FAD cofactor is naturally and always fully reduced in the living cell, but given that at least two such ET chains have evolved independently in the photolyase-cryptochrome superfamily and survived billions of years of evolution, we dare speculate that the efficient photoactivation of photolyases through the Trp (or Trp/Tyr) chains could be important, especially under intense solar irradiation (which causes simultaneously DNA damage and oxidative stress that could potentially deactivate DNA repair by photolyases by oxidation of their FADH -). In any case, the electron transfer cascade of three tryptophans and one tyrosine (Trp381-Trp360-Trp388-Tyr345 in MmCPDII) is almost strictly conserved within all class II photolyases (Fig. S6 †) including plant enzymes like OsCPDII from Oryza sativa 31 (Uniprot entry Q0E2Y1; Trp399-Trp378-Trp406-Tyr363), or animal orthologs like XlCPDII from Xenopus laevis (Q9I910; Trp470-Trp449-Trp477-Tyr434).

Previous observations in the context of our new data

MmCPDII mutant proteins, in which one of the tryptophans of the triad was replaced by non-reducing phenylalanines, were shown to exhibit slower rates of in vitro FADox photoreduction under steady-state irradiation in the presence of 25 mM dithiothreitol (DTT) as an extrinsic reducing agent. 4 Mutation of the first Trp of the triad (W381F) had the strongest inhibitive impact and essentially blocked the photoreduction of FADox. Mutation of the last Trp of the triad (W388F) had the smallest impact, slowing down the FADox photoreduction rate by a factor of 2 (with respect to the WT protein under the same conditions). 4 This could seem to be in discord with the quantum yield of the FAD -Tyr  radical pair estimated here (4%), which is substantially lower than the 55% of the FAD - Trp  pair in the WT, but one has to bear in mind that: a) the FAD -Tyr  pair in W388F is longer lived (2.7 ms) than the Please do not adjust margins Please do not adjust margins corresponding FAD -Trp  and FAD -Tyr  pairs in the WT (0.3 and 1.1 ms, respectively), which compensates for the lower yield by giving the extrinsic reducing agents more time to scavenge the Tyr  radical and stabilize FAD -in the W388F mutant, b) the accessibility of the different radicals to the extrinsic reductant and thereby also the efficiency of the productive encounter of the redox partners is likely to be different, and c) the systems in the steady-state experiment 4 were in dynamic equilibria, as the FADH  formed by illumination was continuously reoxidized back to FADox by molecular oxygen 32 present in the air-saturated samples. In any case, given the fast forward ET from Trp388H to Trp360H + in WT MmCPDII (<200 ps), the alternative electron transfer pathway involving Tyr380 (oxidized by Trp360H + with an intrinsic time constant of 10 -20 ns; see results on W388F), is essentially kinetically switched off and thereby unlikely to play any relevant role in the wild-type protein. By contrast and in spite of the fact, that it reduces only 30% of the Trp388  radicals, the tyrosine Tyr345 could noticeably increase the yield of metastable FADH  , because the lifetime of FAD -Tyr345  (1.1 ms) is almost 5× longer than that of the FAD -Trp388  pair (225 µs), which gives more time to the extrinsic reductants to act upon the recombination partner of FAD -, the precursor of FADH  .

A network of water molecules mediates fast deprotonation of Trp388H +

Deprotonation of the cation radical of the distal member of the Trp triad (Trp388) turned out to be three orders of magnitude faster than the corresponding reaction in other studied proteins of the PCSf (200 ns in plant cryptochrome, 17 300 ns in EcCPDI, 12 2.5 µs in X. laevis (6-4) photolyase 10 ) or in the case of free TrpH + in aqueous solution (700 ns) 33 or TrpH + attached to a ruthenium complex (400 ns) 34 .

The nearest and most plausible protein-derived candidate for the proton acceptor is Glu387, situated 3.6 Å from the deprotonating N1 atom of Trp388. Mutation of Glu387 had only a minor effect on the deprotonation kinetics, excluding it as the direct proton acceptor. A closer look at the crystal structure shows that N1 of Trp388 forms an H-bond to the water molecule 247 (N1 to O distance: 3.4 Å). This water is ideally positioned to function as initial proton acceptor from Trp388H + and it is part of a network of eight water molecules that are further coordinated by Tyr380, Lys384, Glu387, Ser342, His356 and Asp357 (Fig. 12). This network transfers the proton by a Grotthuss-like mechanism to bulk water or a final acceptor, the identity of which is yet to be established.

The superposition of the two available crystal structures of class II CPD photolyases, archeal MmCPDII and eukaryotic OsCPDII (PDB entries 2XRZ and 3UMV, respectively), revealed a similar arrangement of the water network (Fig. S6 †). Seven water molecules were found close to the distal OsCPDII Trp406 surrounded by a similar motif (Tyr398, Lys402, Glu405, Glu360, His361, His374 and Asp375). This water network and structural arrangement seems to be unique to class II CPD photolyases and it is absent in known crystal structures of proteins from other PCSf sub-families. When analysing the sequence divergence of these residues among class II photolyases (Fig. S6 †), it is appealing that over half of them (Trp388, Tyr380, Lys384, Glu387 and Asp357 in MmCPDII numbering) are highly conserved. These findings suggest that the unusually fast (water cluster-mediated) deprotonation of the terminal tryptophan is a general property of all class II CPD photolyases.

Protein-bound water molecules have been shown to be essential for proton transport and biological function of other proteins such as bacteriorhodopsins. 35 Based on our findings, class II photolyases add to biological systems, in which water clusters play an important role in proton transfer. 36 17 27, 37, 38 . 36 The FADH • spectrum was constructed as described previously 17 using the MmCPDII FADox spectrum and that of a mixture of FADox and FADH • in the same sample (obtained by partial photoreduction). The FAD •-spectrum (from an insect cryptochrome) and spectra of Trp and Tyr radicals are taken from the literature. 27,37,38 62x46mm (600 x 600 DPI)

  + to Trp  . Transient absorption changes at representative wavelengths are shown ARTICLE Journal Name 4 | J. Name., 2017, 00, 1-8 This journal is © The Royal Society of Chemistry 2017 Please do not adjust margins Please do not adjust margins in Fig. 3a, all signals are shown in Fig. S1 †.

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Fig. 1 Fig. 2

 12 Fig. 1 Superposition of the crystal structures from EcCPDI (PDB entry 1DNP) and MmCPDII (PDB entry 2XRZ) illustrating the different localization of the tryptophan cascades for electron transfer to FAD. The numbers in brackets indicate the shortest edge-to-edge distances in angstroms. 85x88mm (600 x 600 DPI)

Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 (Fig. 11 Fig. 12 AFig. 13

 345678910111213 Fig. 3 (a) Flash-induced absorption changes on a ns/µs time scale for 40 µM WT MmCPDII at seven characteristic wavelengths (see Fig. S1 † for signals at all measured wavelengths). (b) Signal amplitudes at t → 0 compared to difference spectra for the formation of FAD •-+ Trp • (black solid line) and of FAD •-+ TrpH •+ (red dashed line). The sample was excited at 470 nm by a 5 ns pulse of an energy E ∼5 mJ per cm 2 . Individual traces are averages of four single flash signals spaced by minute. 71x42mm (600 x 600 DPI)

Fig. S3

 S3 Fig. S3 Flash-induced absorption changes on a seconds time scale for 20.9 M W388F mutant MmCPDII (in the presence of 10 mM cysteine) reflecting the slow (630 ms) protonation of FAD -to FADH  . The sample was excited at 470 nm by a 5 ns pulse of E 10.0 mJ per cm 2 . The signals are averages of three single-flash experiments spaced by 1 minute.

Fig. S4

 S4 Fig. S4 Flash-induced absorption changes on a ms time scale for W388F mutant MmCPDII at all measured wavelengths. Experimental conditions are described in the legend of Fig. 10.

Fig. S5

 S5 Fig. S5 Comparison of the flash-induced absorption changes on a ps/ns time scale for a) WT MmCPDII (64 M) and b) its Y345F mutant (69 M). The samples were excited at 355 nm by 100 ps pulses of E 5 mJ per cm 2 (WT) or 4 mJ per cm 2 (Y345F).

Fig. S6

 S6 Fig. S6The immediate environment of the terminal tryptophan and the water network participating in its deprotonation during photoactivation: superposition of two crystal structures of class II CPD photolyases from the archaeon Methanosarcina mazei (PDB entry 2XRZ)1 and the plant Oryza sativa (PDB entry 3UMV)2 . The degree of conservation of the individual amino acids determined from 451 non-redundant class II photolyase sequences is proportional to the height of the corresponding symbols (W360, W381 and W388 are strictly conserved).

  Interestingly, class II CPD photolyases differ

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Acknowledgements

This work was supported by the French Agence Nationale de la Recherche (grant ANR-12-BSV8-0001) and by the French Infrastructure for Integrated Structural Biology (FRISBI; grant ANR-10-INSB-05-01), and by the Air Force Office of Scientific Research (AFOSR; grant FA9550-14-1-0409).

Conflicts of interest

There are no conflicts of interest to declare.

Notes and references §

Hydrated electrons [with max (720 nm) 18 000 M -1 cm -1 ] [START_REF] Spinks | An Introduction to Radiation Chemistry[END_REF] are formed upon absorption of a second photon of the excitation pulse by the rapidly formed FAD -anion radical. This artefact is much more pronounced in the fast experiments (Figs. 4, 5 and 9) using excitation in the UV because the molar absorption coefficient of FAD -at 355 nm is 15 000 M -1 cm -1 (vs. 5 000 M -1 cm -1 at 470 nm). § § In order to be a proton acceptor for Trp388H + , Glu387 would have to be deprotonated under the given experimental conditions (which is likely at pH 8) and, at the same time, have a higher pKa than Trp388H + ; note that a typical pKa value for a solvent-exposed TrpH + is 4.