Vladimir Salnikov
email: vladimir.salnikov@univ-lr.fr

Aziz Hamdouni
email: aziz.hamdouni@univ-lr.fr

Daria Loziienko
email: daria.loziienko1@univ-lr.fr

Generalized and graded geometry for mechanics: a comprehensive introduction

Keywords: MSC 2020 classification: 70G45, 58A50, 70H45, 70H30 Dirac structures, differential graded manifolds, theoretical mechanics, geometric integrators

In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of classical differential geometric constructions in the context can be conveniently described using the language of Q-structures, and thus Q-structure preserving integrators are potentially of great use in mechanics. We give some hints how the latter can be constructed, and formulate some open problems.

Since the text is intended both to mathematics and mechanics communities, we tried to make it accessible to non-geometers as well.

I. INTRODUCTION

This paper is an attempt to put together several results of the authors, united under the same manifest: "geometry encodes the physics of the system for reliable modelling and efficient computer simulations". By this we mean that it is useful and sometimes even necessary to take into account the internal structure of the equations governing the dynamics of a mechanical system. This internal structure is often encoded using the concepts from differential geometry, preserving them in computations typically results in better quality, i.e. more reliable simulation results.

Disclaimer: This text is a recollection of several talks we have delivered at different editions of the CITV meeting 1 we will stick to Souriau's style -paying attention to conceptual ideas rather than to technicalities. We will however do our best to honestly introduce all the necessary notions, addressing both the specialists in geometry and mechanics. In the main text of this paper we will assume that the reader is familiar with basic notions of classical differential geometry, if this is not the case we have added the appendix A to make the presentation self-contained.

The following table shows the most commonly encountered properties of mechanical systems and the geometric structures that are appropriate to describe them. Those structures that are written in bold have discrete analogues, i.e. the numerical meth- 1 Colloque International de Théories Variationnelles, Souriau Colloquium.

Mechanical property

ods preserving them can be constructed. First we will walk the reader through the upper part of this table, devoted to finite dimensional mechanical systems, which we call classical (= not quantum) classical (= old and well established) mechanics. Afterwards, we give some hints on what can be done for the lower part, i.e. for modern (= continuous media) classical mechanics. Then we will present another even more general geometric formalism -the one of graded manifolds -and explain (or at least try to convince the reader) that it generalizes all the notions simultaneously. We thus believe that finding the discrete analogue of graded geometry is a fruitful direction that potentially permits to produce geometric integrators for generic mechanical systems.

II. STATE OF THE ART REVISITED

In this section we will consider several cases of geometric formulations of mechanical problems. We will follow the pattern physics of the system → geometry → numerical methods at least where all of the steps are known.

A. Conservation laws and symmetries

Historically, the first instance of so-called geometric integrators is related to symplectic structures. Consider a manifold M equipped with a symplectic form2 ω and a Hamiltonian vector field v with a Hamiltonian H: ι v ω = dH, and study the evolution equations governed by v. Because of non-degeneracy of ω, H defines v uniquely, one easily checks that preservation of ω by the flow of it is equivalent to conservation of the total energy H of the system. A straightforward idea is then to keep track of both in numerical simulations, i.e.

produce a discrete version of the equations, that "respects" the geometric internal structure. Such methods are naturally called symplectic integrators ( [START_REF] Verlet | Computer "Experiments" on Classical Fluids[END_REF][START_REF] Yoshida | Construction of higher order symplectic integrators[END_REF]) and are widely used in for example molecular simulations, where conservation of energy is extremely important.

While qualitatively the idea "preservation of the symplectic forms guarantees the energy Poisson structure; some partial results are given in [START_REF] Kotov | Dirac structures for Poisson geometry and Hamiltonian mechanics[END_REF], where the attempt to apply the techniques from systems with constraints to Poisson foliations is presented.

Going further in the formalism of theoretical mechanics one inevitably comes across the study of conserved quantities and integrability. Here again, given a first integral even in the symplectic/Hamiltonian case there is no general method of preserving (a discrete version of) it in the numerical computation. There is however one important situation which does work: the conserved quantities obtained from the Noether's theorem, i.e. associated to the symmetries of the system. Mathematically, symmetries are described by a group acting on the phase space of the system, this group, depending on the setup, leaves invariant the Lagrangian of the system, or equations of motion of it, or the space of solutions of them. To keep track of the symmetries numerically, one considers the infinitesimal version of the theorem, i.e. instead of the Lie group action the symmetries are parametrized by the generators of its Lie algebra. This permits to use the approach of Cartan's moving frames ( [START_REF] Olver | A Survey of Moving Frames[END_REF]): the flow of the system is in a sense equivariant with respect to the action of the vector fields describing these symmetries. It has been shown (for example in [START_REF] Chhay | Comparison of some Lie-symmetry-based integrators[END_REF]) that the numerical methods preserving the symmetries are more robust than those that neglect them.

Let us note here, that up to non-degeneracy assumptions the Noether's theorem works both ways3 , i.e. from a conserved quantity one should be able to construct back a generating (generalized) symmetry. So, in principal the method should be applicable to a very general situation of conservation laws, however this "back" direction is very rarely explicit.

B. Dirac structures

This short section is to define the main geometric construction used further for port-Hamiltonian and implicit Lagrangian systems -the Dirac structure. Since it goes beyond the framework of classical differential geometry and rather fits into what is called generalized geometry or higher structures, we are putting this section to the main text. But as mentioned before, for specialists this is merely to fix the notations, so may be safely skipped.

Let us consider the so-called Pontryagin bundle E = T M ⊕ T * M -the direct sum of the tangent a cotangent bundles (we recall the definition of these bundles in the appendix A).

On pairs of its sections, i.e. on two couples vector field -one form, one defines two natural operations:

1) symmetric pairing defined at each point of M :

< v ⊕ η, v ′ ⊕ η ′ >= ι v ′ η + ι v η ′ (1)
2) Courant-Dorfman bracket: The first condition related to (1) is basically studying the linear algebra of the fibers over each point of M . The second one is more involved, and is sometimes called the integrability condition for D -we will comment on it as well.

[v ⊕ η, v ′ ⊕ η ′ ] = [v, v ′ ] Lie ⊕ (L v η ′ -ι v ′ dη). (2)
As a historical remark, let us mention that the Dirac structures were introduced by T. Courant ([9]) in his PhD thesis with some initial motivation coming from mechanics:

roughly speaking this may be a way to treat simultaneously velocities and momenta of a system, which are obviously related. It turned out however, that a more appropriate description uses double bundles (see the works of W. Tulczyjew, e.g. [START_REF] Tulczyjew | The Legendre transformation[END_REF]), that is a Dirac structure is constructed on a manifold, which is a bundle itself.

C. Constraints

Now having introduced the geometric formalism of Dirac structures we can profit from it in several different ways. Recall the construction of variation integrators from above: in [START_REF] Marsden | Discrete mechanics and variational integrators[END_REF], the authors claim that the formalism can be extended to systems with constraints, i.e.

to mechanical systems with conditions on coordinates and velocities:

ϕ a (q, q) = 0, a = 1, . . . , m (3)
This idea has been made explicit in [START_REF] Yoshimura | Dirac Structures in Lagrangian Mechanics Part I: Implicit Lagrangian Systems[END_REF], and we have considered some examples and improvements in [START_REF] Salnikov | From modelling of systems with constraints to generalized geometry and back to numerics[END_REF][START_REF] Razafindralandy | Some robust integrators for large time dynamics[END_REF], all based on Dirac structures.

Let us mention here that geometrically the constraints are described by a distribution

∆ Q ⊂ T Q -i.
e. a set of subspaces of the tangent spaces to the configuration space of the system. One can actually make an elegant link between holonomic constraints and the integrability of the corresponding Dirac structure. In this paper we will only illustrate the use of this approach on a scholar example of non-holonomic constraints -the Chaplygin sleigh leaving the general discussion with proper mathematical details to [START_REF] Kotov | Dirac structures for Poisson geometry and Hamiltonian mechanics[END_REF] and the validity of equations to [START_REF] Loziienko | Dirac structure based numerical methods -accuracy and benchmarks[END_REF]. We just mention here that the approach permits to reproduce the usual equations of motion with Lagrange multipliers corresponding to so called ideal constraints4 .

Let us also note that the phenomenon of non-holonomic constraints that we observe here is an example of the existence of a contact structure, and may open a potentially rich direction in the context of jet spaces ( [START_REF] Kushner | Contact Geometry and Nonlinear Differential Equation[END_REF]).

The Chaplygin sleigh is a mechanical system on the plane that is allowed to move only in a direction given by its edge. The configuration space is then given by the position of its contact point (x, y) and the angle θ with respect to a fixed axis. In these coordinates the Lagrangian reads

L = m 2 ẋ2 + ẏ2 + I m + a 2 θ2 -2a sin θ ẋ θ + 2a cos θ ẏ θ ,
where m is the mass of the sleigh, and a is a geometric parameter representing the distance from the contact point to the center of mass. The condition of absence of orthogonal sliding translates into the constraint ẋ sin(θ)ẏcos(θ) = 0.

In the Dirac language, this means that the dynamics is governed by L and the constraint distribution ∆ Q = Ker(sin θdxcos θdy). We thus have all the necessary ingredients to apply the Dirac structure based numerical methods.

For this simple system one can actually choose the description in terms of the value of the velocity of the sleigh v = v 2 x + v 2 y and compute the components v x and v y afterwards, satisfying thus the constraint automatically. We will use this solution as a reference one to compare the accuracy of numerical methods. The figures (1.a -1.c) and the table I clearly show that the Dirac structure based integrators produce more satisfactory results at comparable resources, while for instance the symplecticity is no longer relevant. Here we obviously consider rather basic methods. A thorough comparison of more advanced (higher order) Dirac structure based numerical methods with the real-life tools will be provided in [START_REF] Loziienko | Dirac structure based numerical methods -accuracy and benchmarks[END_REF]. FIG. 1: Version of Euler method compared to Dirac-1 from [START_REF] Razafindralandy | Some robust integrators for large time dynamics[END_REF] Let us also mention that the dynamics defined in the original article [START_REF] Yoshimura | Dirac Structures in Lagrangian Mechanics Part I: Implicit Lagrangian Systems[END_REF] by the so-called partial vector field can be made intrinsic using the language of Lie algebroids ( [START_REF] Grabowska | Geometrical Mechanics on algebroids[END_REF]), and those are important examples of differential graded manifold we discuss in section (III).

D. Interaction / dissipation

In all the cases discussed above the systems were isolated, it is though very natural to consider interacting systems or introduce some external forces. One possible approach to treating those is called port-Hamiltonian ( [START_REF] Maschke | An intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators[END_REF][START_REF] Van Der Schaft | Port-Hamiltonian systems: an introductory survey[END_REF]). To the Hamiltonian equations v = π ♯ dH one adds all possible internal and external forces obtaining thus a system of the following

form ẋ = (J(x) -R(x)) ∂H ∂x + g(x)f , (4)
where J(x) -an antisymmetic matrix (morally reproducing π), while R(x) and f are new terms responsible for interaction and dissipation. It is important to understand that this approach is not really a new formalism generalizing Hamiltonian systems, it is rather a way to "order" the system and [START_REF] Maschke | An intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators[END_REF][START_REF] Van Der Schaft | Port-Hamiltonian systems: an introductory survey[END_REF] give a classification of the terms one can add together with their physical meaning. This permits to decompose the system into simple blocks, interacting by transparent rules (via ports), which in turn is useful for implementing the simulation algorithms (see e.g. [START_REF] Falaize | Passive simulation of the nonlinear port-hamiltonian modeling of a rhodes piano[END_REF]).

The reason why we are interested in this approach is that apparently ( [START_REF] Van Der Schaft | Port-Hamiltonian systems: an introductory survey[END_REF]) port-Hamiltonian systems may also be described using Dirac geometry, namely the defining property of (almost) Dirac structures serves an avatar of power balance. It is in general not known how to preserve an arbitrary Dirac structure in a discretization, however for a large class of port-Hamiltonian systems the structure is given by a graph of a skew-symmetric operator, so is rather similar to the Poisson bivector case. For those we will suggest a promising technique in section (III).

E. Control

We conclude the overview of finite dimensional systems by briefly mentioning a work in progress related to geometic aspects of control theory. A control system is a system of differential equations of the form

ẋ = f (x, u),
where u are control inputs that can be chosen freely. The case of linear controls is rather well studied and gives rise to the theory of foliations. We expect that the theory of singular foliations can be of use for non-linear case as well.

Not going into actual definitions, to give a rough 5 idea, a regular foliation may be though of as a lasagna: a decomposition of a 3-dimensional space into 2-dimensional layers (leaves). 5 We have heard this beautiful explanation from Camille Laurent-Gengoux.

of rice -those will be singular leaves.

For control systems the first natural question to ask is whether every couple of points in the phase space can be connected by an appropriate choice of controls u. If so, the system is called controllable, and the associated foliation is trivial containing only one leaf.

Otherwise the phase space decomposes into attainability leaves. The idea ( [START_REF] Laurent-Gengoux | Singular foliations for control theory -possible graded descriptions[END_REF]) is then to study transversal directions to a singular leaf to figure out what to add to make a system controllable.

F. And what for continuous media?

The above constructions, as mentioned, concern finite dimensional mechanical systems, contemporary continuous media mechanics however relies heavily on partial differential equations, the phase space of which is essentially infinite dimensional. The results concerning our preferred pattern of geometrization and discretization are only partial there.

For conservative systems the analogue of Hamiltonian formalism is related to multisymplectic geometry, and the appropriate numerical methods do exist. They are mentioned for example in [START_REF] Marsden | Discrete mechanics and variational integrators[END_REF], although not many thorough benchmarking results can be found. Moreover spelling out the multisymplectic form and the n-Hamiltonians of a given system is less straightforward than in the finite dimensional case.

For the symmetries, however, the machinery is rather well developped and is still based on Cartan's moving frames. The same robustness result as before is typically observed ( [START_REF] Chhay | Comparison of some Lie-symmetry-based integrators[END_REF])

for symmetry preserving integrators for PDEs.

For other more complicated systems, like the ones with interaction, the results are rather case-by-case. Two important classes are worth mentioning though. The first one is an attempt to apply the port-Hamiltonian logic, where the interaction is done through the common boundary of the domains in the continuous media. This produces the so-called Stokes-Dirac structures, which is however a slight abuse of notations, since the proper infinite dimensional analogue of Dirac structures is not defined. The second one is when the equations permit a geometric formulation from the very beginning: in terms of differential forms and (co)boundary operators. This results in Discrete Exterior Calculus (DEC, [START_REF] Desbrun | Discrete Exterior Calculus[END_REF]), which is a state of the art tool in many important situations (see [START_REF] Ayoub | A new Hodge operator in Discrete Exterior Calculus. Application to fluid mechanics[END_REF] and references therein).

III. GRADED WORLD

In this section we will describe the geometric construction which is even more general than Dirac structures above -Q-structures on graded manifolds. The motivation to do that is twofold: first -it potentially permits to treat all the finite dimensional cases from section (II) in a uniform way, second -in contrast to most of the above constructions it does have a well defined infinite dimensional analogue. We will still stick to the local description, morally replacing manifolds by vector spaces, although in the graded setting this simplification may produce interesting consequences ( [START_REF] Kotov | On the structure of graded Lie groups[END_REF]).

A. Idea of the definition

Let us start with an important idea from classical differential geometry. The usual way of defining manifolds is basically saying that one can define local coordinate systems and then glue them globally in a consistent way. Alternatively, one can say that knowing all the functions on a manifold gives all the information of the manifold: coordinates are functions after all. The correct mathematical justification for that would come from the sheaf theory, but for "hands on manipulations" those details are not needed. In the graded world the situation is not much more difficult, just the sheaves will be of a different nature.

One says that a manifold M is graded if it is equipped with a grading, i.e. to each coordinate x i on M one can assign a label deg(x i ) ∈ Z, called its degree, and this can be done globally in a consistent way. There is an essential difference between degree zero coordinates, that behave as on usual manifolds, and all the others. One may think of a graded manifold locally as a usual (degree zero) manifold with a bunch of vector spaces attached to each point, labelled by the gradings. For the purpose of this paper it is enough to assume deg(x i ) to be non-negative for all the coordinates, and restrict the discussion to polynomials in non-zero degree coordinates.

To understand this restricted functional space one needs to make sense of products of non-trivially graded coordinates -those can be defined formally as multiplets satisfying two unusual properties with respect to the grading:

-compatibility with algebraic operations:

deg(x i ⋅ x j ) = deg(x i ) + deg(x j ). x i ⋅ x j = (-1) deg(x i )deg(x j ) x j ⋅ x i .
The gradings can thus for the moment be treated formally. The most general possible function on such a graded manifold would be a formal power series depending on all the variables and then the functional space becomes more complicated (see appendix in [START_REF] Jubin | Differential graded Lie groups and their differential graded Lie algebras[END_REF] and [START_REF] Kotov | On the structure of graded Lie groups[END_REF]). But as mentioned, for interesting cases it is enough to consider polynomials in nonzero degree variables with smooth coefficients depending on zero-degree ones. For them the notion of homogeneous functions is well-defined, which produces the commutation relations:

f ⋅ g = (-1) deg(f )deg(g) g ⋅ f.
A lot of objects and operations from classical differential geometry can be defined on graded manifolds "out of the box", meaning that they work almost as usual with the only essential difference: every time when graded quantities are being permuted a sign may appear non-trivially depending on their degrees. For example a graded vector field v of even degree deg(v) is automatically self-commuting:

[v, v] ≡ vv -(-1) deg(v)deg(v) vv = 0.
Meanwhile self-commuting of a vector field Q of an odd degree produces a non-trivial condition:

[Q, Q] ≡ QQ -(-1) deg(Q)deg(Q) QQ = 2Q 2 .
On a graded manifold, a self-commuting vector field of degree 1 is called a Q-structure. A graded manifold equipped with such a vector field is called a differential graded manifold, or a Q-manifold, the latter name often appears in physics literature.

Below we describe two classical examples of differential graded manifolds.

B. Differential forms

Consider a tangent bundle to a smooth manifold Σ, and declare the fiber linear coordinates θ to be of degree 1 -the usual notation for such a shifted tangent bundle is T [1]Σ.

Since deg(σ µ ) = 0, they commute: σ µ 1 σ µ 2 = σ µ 2 σ µ 1 and the degree of any function of them is zero: deg(h(σ 1 , . . . , σ d )) = 0. According to our assumption deg(θ µ ) = 1, i.e. θ's anticommute:

θ µ 1 θ µ 2 = -θ µ 2 θ µ 1 . A generic homogeneous degree p function on T [1]Σ reads f = f µ 1 ...µp (σ 1 , . . . , σ d )θ µ 1 . . . θ µp .
And as above,

f ⋅ g = (-1) deg(f )deg(g) g ⋅ f.
The reader has certainly noticed the resemblance of this construction with the one of differential forms from the appendix A:

f ↔ α = f µ 1 ...µp dσ µ 1 ∧ ⋅ ⋅ ⋅ ∧ dσ µp ∈ Ω(Σ).
Consider now a vector field

Q = ∑ θ µ ∂ ∂σ µ defined on T [1]Σ, it is easy to see that deg Q = 1. Its graded commutator with itself reads [Q, Q] ≡ 2Q 2 ≡ 2Q ○ Q, applying the right-hand- side to any graded function f on T [1]Σ, it is possible to convince oneself that it vanishes,
hence this is a Q-structure, mimicking exactly the properties of the exterior differential. This explains the name of differential graded manifolds. This also provides an alternative (certainly, not the easiest) way to define differential forms from scratch.

C. Poisson bivector

Consider a Poisson manifold M (see the appendix (A) to recall the notations), and consider now the shifted cotangent bundle T * [1]M -the cotangent bundle to M again with fiber linear coordinates of degree 1, i.e. deg(

x i ) = 0, deg(p i ) = 1. It carries a canonical (graded) symplectic form ω = ∑ i dp i ∧ dx i . Recall that the Poisson structure on M can be encoded in a bivector field ∑ π ij ∂ i ∧ ∂ j , which in turns defines a function on T * [1]M : H = 1 2 π ij p i p j .
The (graded) Hamiltonian vector field corresponding to this H reads

Q π = π ij (x)p j ∂ ∂x i - 1 2 ∂π jk ∂x i p j p k ∂ ∂p i (5)
It is instructive to check that deg(Q π ) = 1, and that the condition Q 2 π = 0 is equivalent to the Jacobi identity (A1).

D. Potential applications

The two examples of differential graded manifolds given above look specific, but they are actually rather generic, and in particular here we can already explain how this can be useful for mechanics. So, this section is here to put together all the pieces of the puzzle.

It is now clear that graded geometry provides a uniform description of rather different objects such as differential forms and Poisson geometry. So, going through the table in the introduction we see that we obtain a description of symplectic and Poisson structures "for free".

An attentive reader might already have noticed that the construction goes beyond.

Namely for port-Hamiltonian systems the almost Dirac structure in the game is (often) a graph of a skew-symmetric operator (i.e. a bivector). Thus one can always consider an appropriate T * [1]M (maybe with a different M ) as a graded manifold, and produce a degree 1 vector field on it as above -equation ( 5), by some language abuse we call it almost Q-structure. This vector field squaring to zero will amount to the integrability condition, i.e. the almost Dirac structure being actually Dirac.

In fact for any Dirac structure one can produce a natural graded manifold equipped with a Q-structure. We will not describe this construction here, referring to a general statement in [START_REF] Kotov | Dirac Sigma Models[END_REF]. Note just that for an almost Dirac structure one again obtains an almost Q-structure, covering thus in particular the case of generic constraints. This is an honest Q structure when the structure is actually Dirac, i.e. constraints are holonomic. Moreover, one can associate Q-structures to singular foliations as well ( [START_REF] Laurent-Gengoux | The Lie-infinity algebroid of a singular foliation[END_REF]), hence covering the control theory problems.

The previous paragraph basically means that the internal geometry of systems from all the upper part of the table from the introduction can be naturally encoded using the language of differential graded manifolds -this is the main conceptual message of this paper. It is obviously legitimate to ask how this can be useful for designing numerical methods preserving this internal geometry. Two important remarks are in place here.

First, recall the Cartan's moving frame construction: the vector fields generating the symmetries were "interacting" with the vector field generating the dynamics. And recall that an almost Q-structure is merely a vector field, in the generic case on the shifted cotangent bundle T * [START_REF] Verlet | Computer "Experiments" on Classical Fluids[END_REF]M to the (eventually extended) phase space M of the mechanical system.

There is a canonical way to lift the dynamics from M to its cotangent bundle, using the Lie derivative along the vector field generating it. Hence the discretization procedure will look roughly like this (see Fig. Second, recall that an honest Q-structure, i.e. a vector field squaring to zero, resembles the exterior differential. And for it the Discrete Exterior Calculus is well developped.

Inspired from DEC, the natural idea is then to profit from the similar constructions of equivariant Q-cohomology ( [START_REF] Salnikov | Graded geometry in gauge theories and beyond[END_REF][START_REF] Kotov | Characteristic classes associated to Q-bundles[END_REF]) in order to take into account the squaring to zero condition where it is appropriate.

These two ideas are the main technical messages of this paper, they will be detailed soon in [START_REF] Salnikov | Discretization in the graded world[END_REF]. Last but not least, let us very briefly repeat that in contrast to most of the above classical geometric constructions, the Q-structures are also well defined on infinite dimensional graded manifolds. For example, the famous article [START_REF] Alexandrov | The geometry of the master equation and topological quantum field theory[END_REF] describes precisely the way to define the structure of a differential graded manifold on a space of mappings between differential graded manifolds. Such tools are widely used in high energy physics, and we expect the application to mechanics to be a regression of those.

IV. CONCLUSION

In this paper we have given an overview of geometric structures relevant for qualitative analysis of mechanical systems, including rather advanced "discoveries" from generalized and graded geometry. We hope that the level of details permits if not to start applying those, but at least to be convinced that they are not completely artificial. An interested reader is invited to follow the given references or even consider this text as a call for collaboration, since some of the described problems are open or at least not explored enough.

Acknowledgments.

This text is inspired by a lot of conversations during several CITV -Souriau Colloquium meetings, V.S. and A.H. would like to thank all the participants of them for creating this working atmosphere and especially Géry de Saxcé and Francesco dell'Isola for organizing the last event. V.S. personally thanks Wlodzimierz Tulczyjew for enlightening discussions at various stages of this work. We also thank the anonymous referees for their relevant questions and valuable suggestions to improve the manuscript.

Differential forms / multivector fields

On a vector space V one defines skew-symmetric k-linear forms. When this is done at each point of M , under some regularity assumptions, one obtains the object called a differential k-form, or a differential form of degree k, i.e. a skew-symmetric "function", the "arguments" of which are k vector fields on M . Usual functions on M can be viewed as 0-forms. The standard examples of 1-forms, i.e. covector fields, are the objects dual to vector fields ∂ i , they are denoted dx i . The duality is understood in the sense that dx i (∂ j ) = δ i j , where δ i j -is the Kronecker's symbol. In dimension 2 an example of a 2-form would be the oriented area.

Two natural operations are defined for differential form:

• Contraction of a vector field v with a form α(⋅, ⋅, . . . ) : ι v (α) ∶= α(v, ⋅, . . . ).

It lowers the form degree.

• Exterior differential of a form α: dα(v 0 , v 1 , . . . , v k ) ∶= i (-1) i v i α(v 0 , . . . , vi . . . , v k ) + i<j (-1) i+j α([v i , v j ], v 0 , . . . , vi , . . . vj , . . . , v k ), where v denotes omission of the argument. The differential raises the form degree.

The notation dx i above is not an abuse: the objects dual to ∂ i are actually differentials of coordinate functions. If dα ≡ 0, the form α is called closed, and when α is itself a differential of some other form β: α = dβ, then α is called exact, and the form β is sometimes called its integral. An interesting property of this differential is that it squares to zero: ddβ ≡ 0 for any from β. This means that any exact form is closed, but not necessarily vice versa, which results in the de Rham cohomology construction.

If a 2-form is non-degenerate (at each point, in the linear algebra sense), it is called almost symplectic; and if moreover it is closed, then symplectic. A vector field v, the contraction of which with a symplectic form ω is exact with an integral H (ι v ω = dH), is called a Hamiltonian vector field with the Hamiltonian function H.

Note that a similar (or better to say dual) construction is possible on a cotangent bundle -the result would be multivector fields. And the analogue (or in a sense the inverse) of a symplectic form would be the Poisson bivector. We use this description in the context of graded geometry in section (III).

An

  almost Dirac structure D is a maximally isotropic (Lagrangian) subbundle D of E, i.e. a subbundle of E on which the pairing (1) vanishes identically, and which is of maximal rank equal to dim(M ). If moreover the subbundle Γ(D) is closed with respect to the bracket (2), it is called a Dirac structure. It is easy to produce a trivial example of a Dirac structure: D = T M : the vector field part is governed by the commutator bracket, while nothing happens on the 1-form part. Some more interesting examples include a graph of differential 2-form or of a bivector; thus Dirac geometry describes uniformly symplectic and Poisson manifolds. It is also worth mentioning the examples coming from distributions on M -those will be important for systems with constraints.

FIG. 2 :

 2 FIG.2: Constructing a Q-structure preserving integrator -a schematic representation: start from a point x n at timestep n, lift it to (x, p) n on the graded manifold (dotted arrow facing up), make a step (dotted horizontal arrow) to (x, p) n+1 , "respecting" the Q vector field (symbolically depicted by thin arrows, which along the trajectory are parallel), project back to x n+1 (dotted arrow down).

  Let a manifold M be equipped with a Poisson bracket, i.e. a skew-symmetric operation on the space of functions on M: {⋅, ⋅}∶ C ∞ (M ) × C ∞ (M ) → C ∞ (M ),satisfying the Leibniz property: {f, gh} = {f, g}h + g{f, h} and the Jacobi identity {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 From the geometric point of view, the Poisson bracket can be rewritten as {f, g} = π(df, dg), where π ∈ Γ(Λ 2 T M ) -a bivector field with components π ij (x) = {x i , x j }. A bivector field π ij ∂ i ∧∂ j is a biderivation, so it satisfies the Leibniz property automatically.The Jacobi identity in components of π reads l ∂π ij (x) ∂x l π lk (x) + ∂π ki (x) ∂x l π lj (x) + ∂π jk (x) ∂x l π li (x) = 0. (A1)

  conservation" is true, the precise statement would be that what is conserved is a discrete analogue H d of H, which is eventually different for various symplectic methods. However the difference (H d -H) can be estimated, and by a proper choice of timestep kept bounded by a given parameter for exponentially long time ([3]).It is well known that for finite dimensional conservative mechanical systems the symplectic/Hamiltonian and the Lagrangian descriptions are equivalent (at least under some technical non-degeneracy assumptions). Naturally symplectic integrators have analogues in the Lagrangian description -they are called variational integrators ([START_REF] Marsden | Discrete mechanics and variational integrators[END_REF][START_REF] Buliga | A symplectic Brezis-Ekeland-Nayroles principle[END_REF]). The idea there is to produce a discrete version L d of the Lagrangian L used in the variational principal.Analyzing the extrema of it, one obtains the discrete version of Euler-Lagrange equations.

	And in order to recover (whenever possible) the symplectic picture, one needs to construct
	a discrete version of the Legendre transform.

Those two approaches are now state of the art for at least a couple of decades, but already here we can formulate an open question. The Hamiltonian formalism is defined for Poisson structures as well: given a Poisson bracket {⋅, ⋅} on functions on M , or a Poisson bivector field π, the Hamiltonian flow of H is given by v = {H, ⋅}, or equivalently by the condition v = π ♯ dH. To the best of our knowledge, neither the Lagrangian description nor the appropriate geometric integrator are known for this setting in the case of an arbitrary

TABLE I :

 I Versions of Euler method compared to Dirac-1 from[START_REF] Razafindralandy | Some robust integrators for large time dynamics[END_REF]

See appendix A for details and definitions of the geometric objects.

... and there is a huge confusion about that in literature

in contrast to the so-called vakonomic approach

This appendix recapitulates the minimal working knowledge from differential geometry needed to understand the constructions of the article. A trained geometer can obviously skip the whole section, for all the others we provide a concise description, eventually skipping or simplifying some details. In particular we will use the notions of manifolds and bundles without properly defining them -to get an idea one should just think (locally) of vector spaces of appropriate dimension.

Vector fields

Thus let M be a manifold of dimension n, denote T M -its tangent bundle (i.e. all the tangent vectors to it at each point) and T * M -its cotangent bundle. Locally T M can be viewed as

where V is a vector space of the same dimension as M , then

The set of sections of T M is denoted by Γ(T M ) -those are vector fields on M . One can think for example of a velocity field on a surface. On these sections a binary operation is defined: their commutator

In components it reads:

where ∂ j -is the derivative by the j-th coordinate. For the mechanical intuition, given two velocity fields, one can follow the flow of one of them, then the other one, or in the reversed order, the result is typically not the same -the commutator quantifies this difference infinitesimally.


Anais Laleve
email: anais.laleve@gmail.com

Cristina Panozzo

Inge Kühl

Alexa Bourand-Plantefol

Jelena Ostojic

Abdoulaye Sissoko

Déborah Tribouillard-Tanvier

David Cornu

Angélique Burg

Brigitte Meunier

Marc Blondel

Jerome Clain

Nathalie Bonnefoy

Romain Duval

Geneviève Dujardin
email: genevieve.dujardin@i2bc.paris-saclay.fr

Artemisinin and its derivatives target mitochondrial c-type cytochromes in yeast and human cells

Keywords: Mitochondria, c-Type cytochromes, Artemisinins, Fluorescent probe, Bcs1

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

Introduction

Artemisinin (ART) is a potent anti-malarial drug isolated from the plant Artemisia annua, and a series of clinical ART derivatives have been semi-synthesized including dihydroartemisinin (DHA), artemether (ARTMT), arteether (ARTET) and artesunate (ARTS) (for reviews, [START_REF] Ho | Artemisinins: Pharmacological actions beyond anti-malarial[END_REF][START_REF] Bhatt | The effect of malaria control on Plasmodium falciparum in Africa between 2000 and 2015[END_REF]).

Their anti-malarial action is a two-step process within the eukaryotic parasitic cell. First, artemisinins are activated by reductive cleavage of the endoperoxide pharmacophore bond to generate artemisinin radicals, and in Plasmodium falciparum heme would be predominantly responsible for this bioactivation. In a second step, the ART radicals irreversibly alkylate numerous parasite proteins, thereby compromising the parasite viability [START_REF] Yang | Alkylation of proteins by artemisinin. Effects of heme, pH, and drug structure[END_REF][START_REF] Meunier | Heme as trigger and target for trioxane-containing antimalarial drugs[END_REF][START_REF] Wang | Haem-activated promiscuous targeting of artemisinin in Plasmodium falciparum[END_REF][START_REF] Tilley | Artemisinin action and resistance in Plasmodium falciparum[END_REF][START_REF] Ismail | Artemisinin activity-based probes identify multiple molecular targets within the asexual stage of the malaria parasites Plasmodium falciparum 3D7[END_REF][START_REF] Bridgford | Artemisinin kills malaria parasites by damaging proteins and inhibiting the proteasome[END_REF]. Studies performed in the malaria parasite and in the yeast Saccharomyces cerevisiae have shown that mitochondria may also play an important role in the mode of action of ART derivatives. The respiratory complexes might deliver reducing equivalents for the activation of the endoperoxide bond of the drug, causing ROS production and depolarization of mitochondrial membranes ( [START_REF] Li | Yeast model uncovers dual roles of mitochondria in action of artemisinin[END_REF][START_REF] Wang | Artemisinin directly targets malarial mitochondria through its specific mitochondrial activation[END_REF][START_REF] Antoine | Rapid kill of malaria parasites by artemisinin and semi-synthetic endoperoxides involves ROS-dependent depolarization of the membrane potential[END_REF] and for review [START_REF] Sun | The molecular and cellular action properties of artemisinins: what has yeast told us?[END_REF]). In addition to their action against malaria, ART and derivatives are active against cancer cells in which they also target many proteins involved in critical biological pathways. In human cancer cells, it has been proposed that heme-dependent endoperoxide bioactivation would also take place in the mitochondria, generating oxidative stress and ROS production ( [START_REF] Zhang | Heme mediates cytotoxicity from artemisinin and serves as a general anti-proliferation target[END_REF][START_REF] Zhang | Artesunate-induced mitophagy alters cellular redox status[END_REF][START_REF] Mercer | The role of heme and the mitochondrion in the chemical and molecular mechanisms of mammalian cell death induced by the artemisinin antimalarials[END_REF][START_REF] Hamacher-Brady | Artesunate activates mitochondrial apoptosis in breast cancer cells via iron-catalyzed lysosomal reactive oxygen species production[END_REF][START_REF] Yang | Artesunate induces cell death in human cancer cells via enhancing lysosomal function and lysosomal degradation of ferritin[END_REF][START_REF] Wang | Mechanistic investigation of the specific anticancer property of artemisinin and its combination with aminolevulinic acid for enhanced anticolorectal cancer activity[END_REF] and for reviews [START_REF] Bhaw-Luximon | Artemisinin and its derivatives in cancer therapy: status of progress, mechanism of action, and future perspectives[END_REF][START_REF] Dai | The pharmacological activities and mechanisms of artemisinin and its derivatives: a systematic review[END_REF][START_REF] Konstat-Korzenny | Artemisinin and its synthetic derivatives as a possible therapy for cancer[END_REF]). Thus, both heme and mitochondria appear to play an important role in the action of ART and its derivatives in diverse eukaryotic cells.

Heme is an essential cofactor of mitochondrial cytochromes that are key components of the mitochondrial oxidative phosphorylation (OXPHOS) chain, which provides most of the ATP in non-plant cells. Cytochromes play the role of either mobile transporter like cytochrome c or catalytic membrane subunits of the two respiratory complexes III or IV (for review [START_REF] Kim | Structure, function, and assembly of heme centers in mitochondrial respiratory complexes[END_REF]). In these cytochromes, heme can be covalently attached like in c-type cytochromes, c and c1, or interact by electrostatic or hydrophobic interactions like in cytochromes b and a+a3 (for review [START_REF] Hamel | Biochemical requirements for the maturation of mitochondrial c-type cytochromes[END_REF]). Complex III contains the two types of cytochromes, c1 and b, and is assembled through a dynamic modular pathway (for review [START_REF] Smith | Biogenesis of the cytochrome bc(1) complex and role of assembly factors[END_REF]). The mitochondrial chaperon Bcs1 is required at a late step of assembly of complex III and its absence leads to the accumulation of an inactive pre-complex III [START_REF] Nobrega | BCS1, a novel gene required for the expression of functional Rieske iron-sulfur protein in Saccharomyces cerevisiae[END_REF][START_REF] Cruciat | Bcs1p, an AAA-family member, is a chaperone for the assembly of the cytochrome bc1 complex[END_REF][START_REF] Conte | Bcs1p can rescue a large and productive cytochrome bc1 complex assembly intermediate in the inner membrane of yeast mitochondria[END_REF][START_REF] Wagener | A pathway of protein translocation in mitochondria mediated by the AAA-ATPase Bcs1[END_REF][START_REF] Ostojić | The energetic state of mitochondria modulates complex III biogenesis through the ATP-dependent activity of Bcs1[END_REF]. Mutations in the human gene BCS1L are the most frequent nuclear mutations that lead to complex III-related pathologies and they are associated with very disparate clinical phenotypes (e.g. [START_REF] De Lonlay | A mutant mitochondrial respiratory chain assembly protein causes complex III deficiency in patients with tubulopathy, encephalopathy and liver failure[END_REF][START_REF] Visapää | GRACILE syndrome, a lethal metabolic disorder with iron overload, is caused by a point mutation in BCS1L[END_REF][START_REF] Hinson | Missense mutations in the BCS1L gene as a cause of the Björnstad syndrome[END_REF][START_REF] Fernandez-Vizarra | Impaired complex III assembly associated with BCS1L gene mutations in isolated mitochondrial encephalopathy[END_REF][START_REF] Morán | Cellular pathophysiological consequences of BCS1L mutations in mitochondrial complex III enzyme deficiency[END_REF][START_REF] Baker | Clinical spectrum of BCS1L Mitopathies and their underlying structural relationships[END_REF][START_REF] Oláhová | Molecular genetic investigations identify new clinical phenotypes associated with BCS1L-related mitochondrial disease[END_REF]). By screening the Prestwick ® and BIOMOL ® chemical libraries, we have identified several molecules able to compensate for the respiratory deficiency of bcs1 mutations in yeast. The first class of compensatory molecules contains two antibiotics that target the mitochondrial rRNA and lead to an imbalanced synthesis of respiratory subunits that would modify the equilibrium between illegitimate inactive and active supercomplexes III/IV [START_REF] Panozzo | Chemicals or mutations that target mitochondrial translation can rescue the respiratory deficiency of yeast bcs1 mutants[END_REF].

In this paper, we show that ART, ARTS, DHA, as well as a synthetic derivative of DHA covalently attached to a fluorophore (Fluo-DHA) are able to compensate for a yeast bcs1 defect and we identify c-type cytochromes as key elements that not only mediate the action of artemisinins in the bcs1 mutant but can also explain the anti-tumoral effect of these drugs in human cells.

Materials and methods

Chemicals

Artemisinin (ART, ref. Equine heart cytochrome c (Sigma Aldrich C7752) was prepared in 100 mM phosphate buffer at pH 7.4. This cytochrome c is mainly oxidized. To obtain the fully reduced form, 62 mg of cytochrome c were mixed with 4.5 mg of ascorbate, purified on a Sephadex G25 gel filtration column and aliquots were stored at -80°C. The level of reduction of each cytochrome c aliquot was systematically controlled after defrosting, by comparing the absorbance at 550 nm, before and after addition of ascorbate.

Yeast media and strain construction

Yeast media were prepared as in [START_REF] Dujardin | Long range control circuits within mitochondria and between nucleus and mitochondria. I. Methodology and phenomenology of suppressors[END_REF]. The fermentable medium used for all liquid cultures contains 2% galactose and a limited amount of glucose (0.1%) to avoid glucose repression of genes encoding mitochondrial proteins. Halo tests were performed as in [START_REF] Couplan | A yeast-based assay identifies drugs active against human mitochondrial disorders[END_REF] using solid respiratory medium containing 2% glycerol and 2% agar. Each single mutant was constructed by deleting the open reading frame (ORF) of the gene of interest in the wild-type strain CW252 (MATalpha ade2-1 ura3-1 his3-11,15 trp1-1 leu2-3,112 can1-100) [START_REF] Saint-Georges | A pathogenic cytochrome b mutation reveals new interactions between subunits of the mitochondrial bc1 complex[END_REF]. The bcs1-F401I mutant has been described previously [START_REF] Ostojić | The energetic state of mitochondria modulates complex III biogenesis through the ATP-dependent activity of Bcs1[END_REF]. The ∆cyt1 mutant is deprived of cytochrome c1 (∆c1). In S. cerevisiae, there are two two isoforms of cytochrome c (Cyc1 and Cyc7) and two heme lyases: the cytochrome c1 heme lyase Cyt2 and the cytochrome c heme lyase Cyc3; therefore, we generated the double mutants ∆cyc1∆cyc7 (named ∆c) and ∆cyt2∆cyc3 (named ∆hl). These were thus constructed by crossing the single mutants of opposite mating type and dissecting tetrads using a Singer MSM micromanipulator. Since the bcs1-F401I mutant spontaneously generated revertants that grew faster on respiratory medium, we systematically verified that the level of revertants was less than 1% in each liquid culture before validating any data.

Whole yeast cell respiratory analyses

O2 consumption was measured polarographically at room temperature using an oxygraph (Oxytherm, Hansatech) equipped with a Clark electrode, prepared according to the manufacturer's recommendations. 3x10 9 cells grown in galactose medium for 2 days were added to the chamber containing 1 mL of fresh galactose medium and O2 consumption was recorded for about 5 min. The rate of respiration was estimated by calculating the ratio of O2 consumption to the cell amount (nM O2/min/10 7 cells).

Yeast cytochrome spectra

Cytochrome spectra were recorded with a Cary 400 (Varian, San Fernando, CA) spectrophotometer at low temperature, from dried whole yeast cells supplemented with the reducing agent sodium hydrosulfite just before freezing in liquid nitrogen. For each spectrum, 10 10 cells (OD600=6) grown in liquid galactose medium without or with 1, 3 or 10 µM ARTS were used. A wt absorption spectra shows four main peaks corresponding to the cytochromes c, cytochromes c1 and b of complex III and cytochrome a+a3 of complex IV at 546, 552, 558 and 602 nm respectively.

Human cell culture

Unless otherwise indicated, human embryonic kidney cell lines (HEK293T/17, ATCC ® CRL-11268™) were cultured in Dulbecco's modified Eagle's medium containing 25 mM glucose (DMEM, GlutaMax, Thermo Fisher Scientific, 31966-047), supplemented with 10% fetal bovine serum (Thermo Fisher Scientific, 10270-106) and 1% Penicillin/Streptomycin (Thermo Fisher Scientific, 15070-063) at 37 °C under a 5% CO2 humidified atmosphere for a week, in six well plates with increasing concentrations of ARTS. Cells were tested for the presence of mycoplasma using the MycoAlert™ PLUS Mycoplasma Detection kit (LONZA, LT07-701).

To determine the IC50 of ARTS (the concentration of ARTS that inhibited growth by 50%) and its toxic dose, HEK293T cells were grown for a week in DMEM medium containing increasing concentrations of ARTS (from 0 to 10 µM). Cells were harvested and counted with an automated cell counter (Bio-Rad, TC20™). Under our conditions, the IC50 was between 3 and 5 µM and ARTS compromised cell survival above 8 µM.

Mitochondria purifications and total protein extractions

Yeast cells were grown for 24 h in galactose medium with or without ARTS and mitochondria were purified by differential centrifugation after digestion of the cell walls with Zymoliase-100T according to [START_REF] Lemaire | Preparation of respiratory chain complexes from Saccharomyces cerevisiae wild-type and mutant mitochondria[END_REF] except that a protease inhibitor cocktail (Roche) was added all along the purification to avoid the degradation of pre-complexes III/IV. Human mitochondria were isolated from cultured HEK293T cells harvested and collected by centrifugation (800 g, 7 min, 4 °C) before resuspension in buffer A (70 mM sucrose, 20 mM HEPES pH 7.6, 220 mM mannitol, 1 mM EDTA and 2 mg/ml BSA) supplemented with protease inhibitor (Roche). The samples were incubated on ice (20 min) and homogenized in a drill fitted pestle for 30 strokes. Homogenates were centrifuged (800 g, 7 min, 4 °C) and the supernatant was removed and subjected to centrifugation at 10,000 g for 10 min (4 °C). Crude mitochondrial pellets were resuspended in buffer B (70 mM sucrose, 20 mM HEPES pH 7.6, 220 mM mannitol, 1 mM EDTA) supplemented with protease inhibitor (Roche) [START_REF] Johnston | Insertion and assembly of human TOM7 into the preprotein translocase complex of the outer mitochondrial membrane[END_REF]. Total human proteins were extracted from cell pellets after a wash in cold DPBS (Thermo Fisher Scientific, 14190-250), lysis in SDS loading dye (100 mM Tris pH 6.8, 4% SDS, 20% glycerol, 200 mM DTT), incubation at 95°C for 5 min and sonication with a VibraCell™ (75042, Bioblock Scientific) apparatus (6 cycles of 5 s ON, 5 s OFF, at 20-30% amplitude).

SDS-PAGE, BN-PAGE and Western blot analyses

50 µg of mitochondrial proteins or total proteins extracted from 10 5 HEK cells were separated by SDS-PAGE (using 4-12% precast gels from Invitrogen) and then transferred, either directly or after fluorescence imaging (see below), onto a nitrocellulose membrane (GE Healthcare, Lifescience, Amersham). For BN-PAGE, the same amounts of mitochondria were lysed in a buffer containing 2% digitonin and mitochondrial complexes were separated on 3-12 BN-PAGE according to [START_REF] Lemaire | Preparation of respiratory chain complexes from Saccharomyces cerevisiae wild-type and mutant mitochondria[END_REF].

Immunodetection was performed according to standard techniques using enhanced chemiluminescence detection (Chemidoc, Bio Rad). Antibodies against yeast proteins were: [START_REF] Ho | Artemisinins: Pharmacological actions beyond anti-malarial[END_REF] the in-house polyclonal antibodies: anti-Cytc (1/30,000), anti-Cyt1 (1/30,000), anti-Cytb (1/5,000) and anti-Nam1 (1/5,000), (2) the polyclonal antibodies anti-Atp1 (1/10000) from J. Velours, (Bordeaux, France) and anti-Rip1 ( 1/3000) from N. Fisher (Liverpool, UK) and [START_REF] Yang | Alkylation of proteins by artemisinin. Effects of heme, pH, and drug structure[END_REF] the monoclonal antibodies anti-Cox1 and anti-Cox2 (Abcam, 1/5,000 or 1/2000, respectively).

For the detection of human proteins, the monoclonal anti-cytochrome c (CYCS, 1/1,000) and the polyclonal anti-cytochrome c1 (CYC1, 1/5,000) were from Abcam; the monoclonal anti-Tom40 (1/1,000) was from Santa Cruz Biotechnology and the monoclonal anti-Tubulin (1/10,000) was from Sigma Aldrich.

Fluorescence gel imaging and microscopy

Proteins were separated by SDS-PAGE and the gel was directly visualized with a Typhoon Trio fluorescence imager (excitation 488 nm emission 555 nm). The optimal amounts of Fluo-DHA for yeast or human cells or mitochondria as well as of purified cytochrome c were determined in function of the IC50 for in vivo experiments and in line with the sensitivity of the fluorescent imager (data not shown). The amount of fluorescence was quantified using the Fiji software.

Yeast cells grown overnight in galactose liquid medium were incubated with 100 nM MitoTracker ® Red CMXRos (Molecular Probes) for 30 min, washed and diluted in phosphate buffered saline (PBS) before addition of 3 µM Fluo-DHA and incubation for up to 30 min. Time-lapse fluorescence imaging of live cells was performed using a Nipkow Spinning Disk confocal device (Yokogawa CSU-X1-A1) mounted on Nikon Eclipse Ti E inverted microscope equipped with a 100 X APO TIRF oil immersion objective (Numerical Aperture (NA): 1.49).

Blue (488 nm Vortran laser, 150 mW) and yellow (561 nm Vortran laser, 100 mW) lasers were used for excitation of the Fluo-DHA and MitoTracker ® respectively. A quad band dichroic mirror 405/488/561/647 nm (Chroma) was used and band pass filters 525/545 nm and 607/636 nm (Semrock) allowed the detection of the Fluo-DHA and MitoTracker ® respectively.

For detection, a LiveSR module (GATACA systems) was placed above the Prime 95B sCMOS camera (Photometrics) to reach a two-fold increase in maximal resolution, and the whole system was driven by MetaMorph software version 7.10 (Molecular Devices).

HEK293T cells were grown for two days on a glass coverslip in low glucose DMEM medium, washed in PBS and incubated for 5 h with 5 µM Fluo-DHA. The cells were washed and incubated with 100 nM MitoTracker ® for 30 min in the dark at 37 °C. After washing, HEK293T cells were fixed using 4% paraformaldehyde in PBS, incubated for 30 min, washed and then the coverslip was placed on the slide using Vectashield mounting medium (Vector Labs) that preserves fluorescence and prevents rapid photoblenching of fluorescent molecules.

Images were acquired with a Leica SP8X inverted confocal laser scanning microscope, equipped with a 63x HC Plan Apochromat CS2 oil-immersion objective (NA: 1.4).

Mitotracker ® Red was excited at 578 nm and detected between 590/640 nm while Fluo-DHA was excited at 488 nm and detected between 500/550 nm. Samples were illuminated sequentially and the emission was collected on GaAsP Hybrid (Hamamatsu) detector in timegating mode (between 0.3 ns and 6 ns) to eliminate laser reflection. The whole system was driven by LAS X (Leica).

Mass spectrometry

Mass spectrometry measurements were performed with an electrospray Triple-TOF 4600 mass spectrometer (ABSciex) coupled to the nanoRSLC ultra performance liquid chromatography system (Thermo Scientific) equipped with a C4-desalting column. For ESI-MS measurements, the instrument was operated in positive and RF quadrupole modes with the TOF data being collected between m/z 400-2990. Collision energy was set to 10 eV and azote was used as collision gas. Mass spectra acquisition was performed after loading and desalting of protein samples on C4-column. The Analyst and Peakview softwares were used for acquisition and data processing, respectively. The protein average masses are calculated from the spectra with a mass accuracy of ±2 Da.

Results

Partial rescue of the respiratory defect of yeast bcs1-F401I mutant by ARTS

The yeast mutation bcs1-F401I is the equivalent of the human mutation BCS1L-F368I found in a patient with early-onset encephalopathy [START_REF] Fernandez-Vizarra | Impaired complex III assembly associated with BCS1L gene mutations in isolated mitochondrial encephalopathy[END_REF]. The yeast mutant is respiratory deficient and thus unable to grow on a non-fermentable medium containing glycerol as sole carbon source. When solid, non-fermentable medium was spread with a lawn of the mutant, a halo of growing cells appeared around the filters where ARTS was spotted, showing that this drug was able to partially restore the respiratory growth of the yeast mutant (Fig. 1A).

We next grew the bcs1-F401I mutant in liquid galactose medium containing various concentrations of ARTS (Fig. 1B). In this medium, yeast cells use both fermentation and respiration to proliferate. After 29 h of growth in drug-free medium, the bcs1-F401I mutant exhibited a significantly reduced growth compared to the wild type (wt) with an OD600 nearly three times lower (10 compared to 27). The addition of ARTS to the medium partially rescued the respiratory growth defect of the mutant, with the most effective ARTS concentrations ranging from 3 to 10 µM. For both the mutant and wt strains, high concentrations of ARTS inhibited growth, with the bcs1 mutant being less sensitive to the drug and exhibiting an IC50 (drug concentration inhibiting the cell growth by 50%) of around 30 µM compared to 10 µM for the wt. In parallel, we found that addition of 5 µM ARTS for two days raised the oxygen (O2) consumption of the mutant from about 15% to 50% of that of wt cells, showing that the respiratory capacity of the bcs1 mutant significantly increased in presence of ARTS (Fig. 1C).

Finally, we have performed BN-PAGE followed by western blot on mitochondria purified from wt as well as bcs1-F401I cells grown with or without 5 µM ARTS (Fig. 1D).

Under our conditions, the anti-Cyt1 antibodies mainly detected the wt complex III in the form of supercomplexes III/IV. Conversely, in the mutant, inactive pre-complexes III that were previously shown to contain cytochrome c1 but lack the catalytic subunit Rip1 [START_REF] Nobrega | BCS1, a novel gene required for the expression of functional Rieske iron-sulfur protein in Saccharomyces cerevisiae[END_REF][START_REF] Cruciat | Bcs1p, an AAA-family member, is a chaperone for the assembly of the cytochrome bc1 complex[END_REF][START_REF] Conte | Bcs1p can rescue a large and productive cytochrome bc1 complex assembly intermediate in the inner membrane of yeast mitochondria[END_REF][START_REF] Wagener | A pathway of protein translocation in mitochondria mediated by the AAA-ATPase Bcs1[END_REF][START_REF] Ostojić | The energetic state of mitochondria modulates complex III biogenesis through the ATP-dependent activity of Bcs1[END_REF] accumulated. In the presence of ARTS, the steady state level of these pre-complexes III slightly decreased by about 30%. Immunodetection of the complex IV subunit Cox2 in the same samples revealed only supercomplexes III/IV in the wild type while in the bcs1-F401I mutant there appeared to be a whole range of illegitimate complexes IV, already described in rip1 and bcs1 mutants [START_REF] Ostojić | The energetic state of mitochondria modulates complex III biogenesis through the ATP-dependent activity of Bcs1[END_REF][START_REF] Cruciat | The cytochrome bc1 and cytochrome c oxidase complexes associate to form a single supracomplex in yeast mitochondria[END_REF][START_REF] Cui | Modulation of the respiratory supercomplexes in yeast: enhanced formation of cytochrome oxidase increases the stability and abundance of respiratory supercomplexes[END_REF], that migrated from the position of pre-complexes III up to the position of the bona fide complex III/IV. Thus, using BN-PAGE, we could not confidently discriminate supercomplexes III/IV from the various illegitimate pre-III/IV present the bcs1-F401I mutant. Next, immunoprecipitation experiments of the early-assembled complex III subunit Qcr7 fused to a c-Myc tag retrieved Rip1 massively in the wild-type and only barely in the Bcs1-F401I mutant. However no increase of Rip1 level could be detected after ARTS treatment of the mutant, suggesting that the drug causes only very little, if any, increase in mature complex III level (see Fig. S1). Thus the oxygen uptake restoration upon ARTS treatment might be due either to a tiny increase of mature complex III too subtle to be detected in our experiments or to a decrease of pre-III or illegitimate pre-III/IV complexes, whose accumulation in the mutant is likely to alter respiration (see discussion).

ARTS affects the steady state level of respiratory complex cytochromes in yeast

As the complex III contains two cytochromes, cytochrome b and c1, and heme has been implicated as a key molecule activating ART (see introduction), we decided to determine the effect of ARTS on mitochondrial cytochromes in wt and mutant bcs1 cells. The association of the apoproteins Cytb and Cyt1 with their heme cofactor to form cytochromes b and c1 respectively is required at an early stage of complex III assembly. These two cytochromes, as well as cytochrome c and cytochrome a+a3 from complex IV, once fully assembled and reduced, show specific light absorption maxima that can be monitored and quantified by recording cytochrome spectra from whole yeast cells (Fig. 2A). Wt and bcs1-F401I cells were grown with increasing concentrations of ARTS and the cytochrome levels were quantified relative to the drug free control (Fig. 2B). For the wt at drug concentrations that did not affect cell growth under our conditions (i.e. 1 µM and 3 µM), the levels of cytochromes b, c1 and c decreased by 30% to 40% with 1 µM ARTS and by 40% to 50% with 3 µM ARTS while cytochrome a+a3 was less affected. Treatment of the wt with 10 µM ARTS, the drug concentration corresponding to the IC50, led to a significant decrease of all cytochrome levels by at least 60%. For bcs1-F401I, the levels of cytochromes b, c1 and c were only slightly decreased by treatment with 3 µM ARTS and lowered by 50 to 60% with 10 µM drug. Thus, ARTS treatment strongly lowers the absorption spectrum maxima of mitochondrial cytochromes, cytochromes c, c1 and b being more affected than cytochrome a+a3.

We next determined whether the ARTS-induced decrease in cytochrome spectra was associated with a lower level of the corresponding apoprotein. Mitochondria from wt and bcs1-F401I cells grown with increasing concentrations of ARTS were purified. The steady state level of Cytb and Cyt1 of complex III, Cytc, and the complex IV subunit Cox1 which binds the hemes a and a3 were analyzed by western blot (Fig. 2C). The results show that the steady state level of all mitochondrial cytochromes was reduced at high ARTS concentrations, starting from 3 µM ARTS for the wt and 10 µM for the mutant. With the anti-Cyt1 antibody, a band of lower mobility appeared first when wt or bcs1-F401I cells were grown with 1 µM or 10 µM ARTS respectively. In the wt, this lower mobility band became more intense when the concentration of ARTS increased. It is well known that yeast pre-Cyt1 is processed in two sequential steps [START_REF] Nicholson | Biogenesis of cytochrome c1. Role of cytochrome c1 heme lyase and of the two proteolytic processing steps during import into mitochondria[END_REF][START_REF] Arnold | Two distinct and independent mitochondrial targeting signals function in the sorting of an inner membrane protein, cytochrome c1[END_REF]. The first processing takes place in the mitochondrial matrix where the signal sequence addressing pre-Cyt1 to the mitochondria is removed to give the intermediate form of Cyt1 (i-Cyt1). Then heme c1 has to be covalently attached to the apoprotein i-Cyt1 by heme lyase before allowing the second processing step to occur in the intermembrane space, thus generating mature cytochrome Cyt1 (Fig. S2A). Thus, in the yeast mutant deprived of heme lyases (∆hl), the i-Cyt1 accumulates. This i-Cyt1 band appeared to co-migrate with the lower mobility band generated in presence of ARTS in both the wt and bcs1-F401I. This suggests that ARTS treatment disturbs the second step of yeast pre-Cyt1 maturation. Taken together, the results in Fig. 2 show that ARTS treatment decreased the steady state levels of OXPHOS cytochromes, and that the bcs1-F401I mutant was less sensitive to this effect.

A fluorescent version of DHA targets cytochromes c and c1 in yeast mitochondria

In mammals, ART-derived drugs are in vivo prodrugs of DHA, which is mainly produced by the action of the liver isoenzyme 3A4 of cytochrome P450 [START_REF] Grace | Metabolism of betaarteether to dihydroqinghaosu by human liver microsomes and recombinant cytochrome P450[END_REF][START_REF] Silamut | Artemether bioavailability after oral or intramuscular administration in uncomplicated falciparum malaria[END_REF][START_REF] Haynes | Artemisone, a highly active antimalarial drug of the artemisinin class[END_REF]. For these reasons, their biological action is believed to recapitulate that of DHA ( [START_REF] Melendez | Determination of the antimalarial arteether and its deethylated metabolite dihydroartemisinin in plasma by highperformance liquid chromatography with reductive electrochemical detection[END_REF] and for review [START_REF] Ho | Artemisinins: Pharmacological actions beyond anti-malarial[END_REF]). First, using the halo test, we showed that like for ARTS, the respiratory growth defect of the bcs1-F401I mutant was partially rescued by ART, DHA and Fluo-DHA (Fig. 3B). We found that the optimal rescuing concentrations in liquid cultures were around 1 µM for ART and DHA and 3 µM for Fluo-DHA (data not shown). Altogether these data indicate that Fluo-DHA is able to improve the respiratory growth of the bcs1-F401I mutant, thus validating its use as a fluorescent probe.

Second, to localize Fluo-DHA within live yeast, wt cells were incubated successively with MitoTracker ® , a red-fluorescent dye that stains mitochondria, and with Fluo-DHA, before fluorescence imaging using a confocal microscope (Fig. 3C). We observed that Fluo-DHA rapidly labeled the yeast mitochondria (in less than 5 min) since it colocalized with MitoTracker ® . Thus, we conducted in organello interaction experiments to identify the mitochondrial targets of Fluo-DHA. We incubated Fluo-DHA for 15 min with mitochondria purified from wt and mutants deprived of cytochrome c1 (∆c1), cytochrome c (∆c) or heme lyases (∆hl). Mitochondrial proteins were then resolved by SDS-PAGE and visualized using gel fluorescence imaging and Western blot analysis (Fig. 3D). Two main fluorescent bands were detected in wt mitochondria upon incubation with Fluo-DHA, consistent with the formation of covalent protein adducts resistant to denaturing conditions. These two bands had the same migration profile than Cyt1 and Cytc. This was further confirmed since with mitochondria from ∆c1 or ∆c only one band appeared, corresponding to the molecular weight of Cytc or Cyt1 respectively, moreover all fluorescent signal was lost in ∆hl, showing that Fluo-DHA does not interact with i-Cyt1 in absence of its heme moiety. Regardless of the strain tested, no fluorescent bands were detected at the size corresponding to Cytb and Cox1. This could be due to the non-covalent nature of the interactions existing between these apoproteins and their corresponding hemes, or to the fact that the heme moieties of Cytb and Cox1 are buried into the inner membrane, thereby preventing or delaying the activation and chemical attack by ART derivatives. Thus, Fluo-DHA appears to target c-type cytochromes in yeast mitochondria.

ART derivatives also primarily target human mitochondrial cytochromes c and c1

Since ART derivatives are used in clinics against the malaria parasite and have more recently been tested as a possible therapy for cancers (see Introduction), we investigated the mode of action of ARTS in human cells by testing if it also targeted mitochondrial c-type cytochromes.

First, when adding Fluo-DHA to whole human cells we observed that it rapidly labeled mitochondria as shown by colocalization with MitoTracker ® (Fig. 4A). Then, HEK293T human cells were grown for a week with different concentrations of ARTS and whole cell protein extracts were resolved by SDS-PAGE and analyzed by Western blot analysis (Fig. 4B). When ARTS concentration increased, the abundance of mature cytochrome c1 decreased while a lower mobility band appeared, consistent with the accumulation of partially processed cytochrome c1. The sequence comparison between the yeast Cyt1 and human CYC1 shows a good conservation of (i) the hydrophobic stretch important for the processing and insertion of yeast Cyt1 and (ii) the position of the proteolytic processing site generating the mature cytochrome c1 between yeast and human cells (Fig. S2B). Thus it is tempting to speculate that the processing steps are conserved. In yeast, heme is required for the second step of processing [START_REF] Nicholson | Biogenesis of cytochrome c1. Role of cytochrome c1 heme lyase and of the two proteolytic processing steps during import into mitochondria[END_REF][START_REF] Arnold | Two distinct and independent mitochondrial targeting signals function in the sorting of an inner membrane protein, cytochrome c1[END_REF] and our results suggest that heme is also probably required for the processing of human CYC1. Similarly, the cytochrome c signal tended to decrease as the drug concentration increased. Thus, ARTS treatment appears to interfere with the maturation of cytochrome c1 and the stability of cytochrome c in human cells.

To investigate the molecular targets of ART derivatives in humans, in organello and whole cell interaction experiments were conducted (Fig. 4C). Purified human mitochondria were incubated for 15 min and whole cells were incubated for 3 h with Fluo-DHA, proteins were extracted and analyzed. In presence of Fluo-DHA, two main bands appeared at the size corresponding to cytochromes c1 and c in both the mitochondrial and total protein samples in presence of Fluo-DHA, implying the formation of covalent adducts between DHA and the cytochromes. The fact that total proteins exhibited the same fluorescence profile as mitochondrial proteins strongly suggests that mitochondrial c-type cytochromes are protein targets of Fluo-DHA in human cells.

Fluo-DHA selectively interacts with reduced cytochrome c

To analyze the mechanism of interaction between Fluo-DHA and cytochrome c, we incubated purified cytochrome c from equine heart with Fluo-DHA under various conditions and ran SDS-PAGE gels to resolve Fluo-DHA and cytochrome c (Fig. 5A).

Beforehand we determined by recording its 550 nm absorption that 65% of the purified cytochrome c was in its oxidized form. This purified cytochrome c spontaneously presented a low level of fluorescence, that doubled upon a 15 min incubation with Fluo-DHA. Upon full oxidation of cytochrome c by a pre-incubation with H2O2, the intrinsic fluorescence signal of cytochrome c became barely detectable, independently of the addition of Fluo-DHA, suggesting that the reduction state of cytochrome c was critical for its interaction with Fluo-DHA. When the purified cytochrome c was fully reduced prior to incubation with Fluo-DHA, the fluorescence signal increased about four times, and up to five times when the loading buffer was supplemented with the reducing agent DTT. This was likely explained by the capability of DTT to maintain a fully reduced cytochrome c (of which a small proportion otherwise undergoes re-oxidation during sample preparation) allowing continuation of its reaction with Fluo-DHA in the loading buffer. As controls, we showed that (i) pre-incubating reduced cytochrome c with excess DHA decreased the subsequent binding of Fluo-DHA by about 30%, indicating that DHA and Fluo-DHA competed for identical binding site(s) on cytochrome c, and (ii) increasing the concentration of Fluo-DHA enhanced the fluorescence signal in a doseresponse manner (Fig. 5B). Taken together, our results show that Fluo-DHA selectively interacts with reduced cytochrome c.

Characterization of DHA covalent binding to the heme of reduced cytochrome c

The mechanism of free heme alkylation by ART upon reductive cleavage of its endoperoxide pharmacophore has been precisely described [START_REF] Robert | Characterization of the first covalent adduct between artemisinin and a heme model[END_REF][START_REF] Robert | Characterization of the alkylation product of heme by the antimalarial drug artemisinin[END_REF][START_REF] Robert | The antimalarial drug artemisinin alkylates heme in infected mice[END_REF][START_REF] Accardo | Interaction of iron(II)-heme and artemisinin with a peptide mimic of Plasmodium falciparum HRP-II[END_REF] and found to be a common feature of endoperoxide-containing antimalarial drugs [START_REF] Laurent | C10-modified artemisinin derivatives: Efficient heme-alkylating agents[END_REF][START_REF] Garah | The antimalarial artemisone is an efficient heme alkylating agent[END_REF]. Based on our observation of stable adducts between Fluo-DHA and reduced cytochrome c in SDS-PAGE, we performed electrospray ionization mass spectrometry to test if a covalent reaction could be definitely established between cytochrome c and DHA.

Fully-reduced cytochrome c was incubated for 15 min with increasing amounts of DHA and submitted to mass spectrometry analysis (Fig. 6). The molecular masses of the experimentally detected species were compared to the predicted ones (Fig. S3). When reduced cytochrome c was incubated with DMSO only, a main peak compatible with the predicted mass was observed, together with a minor peak corresponding to a form with a methionine oxidation (Fig. 6, upper panels, and Table S1). In presence of increasing amounts of DHA, three new peaks appeared (Fig. 6, bottom panels). The peak at m/15z ~ 843.8 corresponded to the predicted value for covalent DHA-cytochrome c adduct. The peak at about m/15z ~ 839.8 was consistent with a rearranged product following C-alkylation of cytochrome c by DHA and loss of an acetate fragment. The peak at about m/15z 841.0 likely corresponded to the same species than the m/15z 839.8 but with a methionine-oxidized form of the cytochrome c. The total relative abundance of all DHA-cytochrome c adducts tended to increase with the amount of DHA added, to reach 20% of bio-incorporation of DHA into cytochrome c at a 10:1 ratio (Table S1). No difference in the DHA incorporation could be observed when increasing the incubation time (from 15 min to 60 min, data not shown) indicating that cytochrome c alkylation by DHA was a very fast process, in accordance with our in vitro and in cellulo data. Altogether, these results strongly suggest that the porphyrin rings of hemes rather than the apoproteins are the actual sites of alkylation by ART derivatives at the level of yeast and mammalian mitochondrial c-type cytochromes.

Discussion-conclusion

In this work, we combined the use of in cellulo, in organello and in vitro analyses to gain insights into the molecular targets and biological effects of ART and its derivatives, in yeast and human mitochondria. All the data that we gathered by these different approaches are fully consistent with an interaction of ART derivatives with mitochondrial cytochromes that are catalytic components of the respiratory chain, and particularly with c-type cytochromes.

The effect of ART derivatives on yeast cytochromes b and a+a3 might be either the result of the direct binding of artemisinins to these cytochromes or a secondary effect of the loss of mature cytochromes c and c1. Indeed, it is well known that the steady state levels of Cytb and Cox subunits closely depend on that of Cyt1 and Cytc respectively (e.g. [START_REF] Crivellone | Assembly of the mitochondrial membrane system. Analysis of structural mutants of the yeast coenzyme QH2-cytochrome c reductase complex[END_REF][START_REF] Pearce | Degradation of cytochrome oxidase subunits in mutants of yeast lacking cytochrome c and suppression of the degradation by mutation of yme1[END_REF]).

Low concentrations of ART and its derivatives are able to phenotypically rescue a respiration-defective yeast mutant harboring a mutation in the Bcs1 protein, a key chaperone in mitochondrial complex III assembly. The interference of ART derivatives with the maturation of the complex III subunit cytochrome c1 might directly explain this compensatory effect.

Assembly of complex III is indeed delayed in the bcs1-F401I yeast mutant and a large amount of inactive cytochrome c1-containing pre-complex III accumulates. This is deleterious because complex IV is trapped in an inactive preIII/IV supercomplex [START_REF] Ostojić | The energetic state of mitochondria modulates complex III biogenesis through the ATP-dependent activity of Bcs1[END_REF]; in addition, this pre-complex III that accumulates in the inner-membrane could also be toxic by itself. Thus, it is likely that decreasing the quantity of complex IV [START_REF] Ostojić | The energetic state of mitochondria modulates complex III biogenesis through the ATP-dependent activity of Bcs1[END_REF] or mature cytochrome c1 (this work) limits the accumulation of pre-complex III and frees some complex IV, which can then associate with the small amount of complex III that slowly assembles despite the bcs1-F401I mutation. Together, this could restore some respiratory growth thanks to a very subtle equilibrium taking place between the rate of assembly of individual complexes and supercomplex association.

Interestingly, the complex III deficiency due to the corresponding BCS1L mutation in the patient muscle or liver is also associated with the accumulation of a cytochrome c1-containing pre-complex III [START_REF] Fernandez-Vizarra | Impaired complex III assembly associated with BCS1L gene mutations in isolated mitochondrial encephalopathy[END_REF], thus it would be highly relevant in the future to test whether ARTS might also decrease the accumulation of this aberrant pre-complex in patient cells and improve their respiratory capacity.

In order to be a direct target of Fluo-DHA in vitro, purified cytochrome c is required to be under its reduced form. This observation is fully consistent with the fact that ART and its derivatives require reduced forms of iron (mainly heme from hemoglobin digestion in the malaria parasites) to become bioactivated under the form of protein-reactive radicals [START_REF] Yang | Alkylation of proteins by artemisinin. Effects of heme, pH, and drug structure[END_REF][START_REF] Meunier | Heme as trigger and target for trioxane-containing antimalarial drugs[END_REF][START_REF] Wang | Haem-activated promiscuous targeting of artemisinin in Plasmodium falciparum[END_REF][START_REF] Tilley | Artemisinin action and resistance in Plasmodium falciparum[END_REF][START_REF] Ismail | Artemisinin activity-based probes identify multiple molecular targets within the asexual stage of the malaria parasites Plasmodium falciparum 3D7[END_REF][START_REF] Bridgford | Artemisinin kills malaria parasites by damaging proteins and inhibiting the proteasome[END_REF]. The need of a reduced metal might also account for the limited sensitivity to ART and ARTS displayed by the respiratory-deficient bcs1-F401I cells (this paper) or other yeast respiratory mutants [START_REF] Li | Yeast model uncovers dual roles of mitochondria in action of artemisinin[END_REF][START_REF] Sun | The molecular and cellular action properties of artemisinins: what has yeast told us?[END_REF] compared to the wt since the reduction kinetics of cytochromes is slower in respiratory deficient mutants. Similarly, HeLa cells were previously shown to be less sensitive to ARTS when they are rho 0 , i.e. devoid of an active respiratory chain [START_REF] Mercer | The role of heme and the mitochondrion in the chemical and molecular mechanisms of mammalian cell death induced by the artemisinin antimalarials[END_REF]. In conclusion, we propose that reduced cytochromes per se might activate ART and its derivatives in mitochondria.

In the malaria parasite, using ART-based profiling probes, a hundred of ART covalent binding protein targets were identified upon heme activation [START_REF] Wang | Haem-activated promiscuous targeting of artemisinin in Plasmodium falciparum[END_REF][START_REF] Ismail | Artemisinin activity-based probes identify multiple molecular targets within the asexual stage of the malaria parasites Plasmodium falciparum 3D7[END_REF]. These protein targets of ART belong to multiple metabolic pathways, but no enrichment in cytochromes were found. This strikingly contrasts with our yeast and human cell data showing only a few dominant and specific protein targets (c-type cytochromes). This apparent discrepancy could be explained by the dramatically different sources and abundance of cellular heme between species, which in malaria parasites are constituted mainly by hemoglobin digestion products. Whether plasmodial c-type cytochromes are targeted by ART and participate to its antimalarial mode of action remains to be investigated knowing that a number of effective drugs target the parasite mitochondria (for review [START_REF] Goodman | Is the mitochondrion a good malaria drug target?[END_REF]).

In addition, ART and its derivatives are also active in cancer cells, mostly via arrest of cell cycle or apoptosis. The general oxidative stress appears to play a central role in these processes.

We propose that the decrease in membrane potential and increase of mitochondrial ROS production already described in yeast and human cancer cells upon ARTS treatment might be direct consequences of the effect on mitochondrial cytochromes that we uncovered in this work.

In cancer cells, this effect might be further enhanced by an increase of heme and lysosomal iron contents that could also be a source of ROS, thus synergistic pathways where iron (II)containing molecules play a central role might contribute to the anti-cancer activity of ART and its derivatives. grown with 5 µM ARTS or DMSO, were purified and solubilized with digitonin. Several BN-PAGE were performed in parallel using the same solubilization extracts and transferred onto membranes that were immunoblotted with antibodies against Cyt1 or Cox2 to reveal complex III or IV respectively, as well as against Atp1 (complex V) as control. The positions of the different complexes, super-complexes and protein molecular mass markers (in kDa) are indicated. The steady state level of pre-complexes III revealed with the anti-Cyt1 antibodies that accumulated in the bcs1-F401I mutant repetitively decreased by about 30% (mean 33 ± 9) when the mutant is treated with ARTS. In both panels, the fluorescence intensity is presented in the histograms as a percentage of the control value with cytochrome c only. Under our conditions of detection, cytochrome c spontaneously presents a low level of intrinsic fluorescence that depends on its oxidative state. S1 for quantifications). Arrows on the y axes correspond to the threshold of peak annotation.

Author contributions

AL both performed the experiments and analyzed the data to determine active/toxic doses, measure O2 consumption, record cytochrome spectra and study in gel interaction between cytochromes and Fluo-DHA, with the help of AB-P for O2 consumption measurements, mitochondrial preparations and western blots. CP constructed the yeast mutant strains and performed the microscopy experiments. IK purified the human mitochondria and brought her expertise in human cell culture and mitochondrial proteins. JO, AB-P and AB participated to the initial experiments following the drug screening. DC performed mass spectrometry analyses. BM brought her expertise in the modeling in yeast of anti-malaria drug, NB in mitochondrial biogenesis and JC in malaria. GD designed the experiments and participated to the analysis of data. DT-T and MB designed and performed the initial drug screening. AS synthesized the Fluo-DHA probe needed for these experiments under the supervision of RD. AL, GD, NB, RD wrote the initial draft and made revisions after the comments of the other authors. Relative abundance of Cytc-DHA adducts to Cytc was estimated for three charge states (z=+13, +14 and +15) by the ratios of the respective mass peak intensities. All different non-modified forms and DHA-protein adducts were considered to calculate total relative abundance. Oxide forms correspond to methionine oxidation. Means of total relative abundance for different DHA quantities were evaluated to represent 17.06 ± 0.77%, 19.38 ± 0.31% and 20.05 ± 0.29%, respectively for 1:2.5, 1:5 and 1:10 Cytc:DHA ratios. The relative increase observed in total relative abundance as treated with more DHA quantities is mainly explained by the increase of DHA-Cytc peak intensity.

  361593), dihydroartemisinin (DHA, ref. D7439), artesunate (ARTS, ref. A3731), hydrogen peroxide (ref. 216763) and dithiothreitol (DTT, ref. D9779) were purchased from Sigma Aldrich. Fluo-DHA (Sissoko et al., submitted to ACS Infectious Diseases) corresponds to an HPA-NBD (4-(1'-hydroxypropyl-3'-amino)-7-nitrobenzoxadiazole) spacerfluorophore moiety attached onto the C-10 center of DHA. ART and derivatives, including Fluo-DHA were all solubilized in DMSO.

  Thus, to investigate the compensation mechanisms by defining the targets of ART-derived drugs, we decided to use a fluorescent version of DHA, Fluo-DHA (Fig. 3A, Sissoko et al., submitted to ACS Infectious Diseases).

Figure legends Fig. 1 .

 legends1 Figure legends

.

Fig. 2 .

 2 Fig. 2. ARTS decreases the steady-state level of mitochondrial cytochromes.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Fluo-DHA rapidly enters yeast mitochondria where it binds cytochromes c and c1.

Fig. 5 .

 5 Fig. 5. Fluo-DHA binds purified reduced cytochrome c.

Fig. 6 .

 6 Fig. 6. Mass spectrometry analysis of DHA-cytochrome c adducts.

  Figure 1

Total relative abundance of protein-DHA adduct (%) 17.70 19.41 20.00 Table S1. Mass spectrometry-based relative quantification of Cytochrome c (Cytc) modification by DHA treated with different Cytc:DHA ratios.

			+ 2.5 DHA	+ 5 DHA	+ 10 DHA
	Cytc	824.91	320	330	330
	Cytc + oxidation	825.97	47	48	47
	Cytc + DHA -acetate	839.86	35	40	40
	Cytc + DHA -acetate + oxidation	841.02	20	25	25
	Cytc + DHA	843.88	16	24	28
	Total relative abundance of protein-DHA adduct (%)		16.21	19.06	19.79
			Peak intensity (z=+14)
		m/z	+ 2.5 DHA	+ 5 DHA	+ 10 DHA
	Cytc	883.75	290	310	310
	Cytc + oxidation	884.89	45	45	42
	Cytc + DHA -acetate	899.78	31	36	35
	Cytc + DHA -acetate + oxidation	901.03	20	25	25
	Cytc + DHA	904	19	26	30
	Total relative abundance of protein-DHA adduct (%)		17.28	19.68	20.36
			Peak intensity (z=+13)
		m/z	+ 2.5 DHA	+ 5 DHA	+ 10 DHA
	Cytc	951.66	240	260	260
	Cytc + oxidation	952.88	39	39	36
	Cytc + DHA -acetate	968.92	25	29	29
	Cytc + DHA -acetate + oxidation	970.26	18	21	21
	Cytc + DHA	973.55	17	22	24

Acknowledgements

We thank Laure Crabbe, Francesca Giordano, Pierrick Le Borgne and Anne-Marie Tassin for useful technical advices and the members of the MITOSCREEN consortium for fruitful discussions. We are grateful to Christopher J. Herbert for critical reading of the manuscript and for looking over the English. The present work has benefited from the light microscopy facility of Imagerie-Gif, member of IBiSA and supported by "France-BioImaging" and the Labex "Saclay Plant Sciences". We thank Laëtitia Besse and Romain Le Bars for their advice and help for the experiments and the presentation of the results. This work has benefited from the facilities and expertise of the I2BC proteomic platform SICaPS, supported by IBiSA, Ile de France Region, Plan Cancer, CNRS and Paris-Sud University. We thank Virginie Redeker for a fruitful initial discussion about mass spectrometry analyses. This work as well as AB and AL were supported by a grant from the "Association Française contre les Myopathies" (AFM, N°17122) within the MITOSCREEN consortium. DT-T was supported by a post-doctoral grant from the "Fondation pour la Recherche Médicale" (FRM N°DRM201001220457). JO was a PhD student from the University Evry-Val d'Essonne and was the recipient of a fellowship from the Ministère de la Recherche et de la Technologie.

Declaration of competing interest

No conflict of interest. Supplementary material

Comments:

Efficient co-immunoprecipitation of Rip1 with Qcr7-c-Myc in the wild-type indicated the presence of a fully assembled complex III. In the bcs1-F401I mutant, very little Rip1 was detected, suggesting that most of the complex III forms that are co-immuno-precipitated with Qcr7-c-Myc are not fully mature. No difference in Rip1 level was observed with or without ARTS treatment. There is a little less cytochrome b after ARTS treatment, which is consistent with the decrease of pre-complex III observed after western blotting of BN-PAGE since pre-III always contains Cytb (Fig. 1D), and with the decrease of spectral cytochrome b (Fig. 2B).