Francesca Poggiolesi

Three Solutions to the Knower Paradox

Keywords: hierarchy solutions, Knower Paradox, provability logic, self-reference

, is developed in the framework of modal logic and it is based on the idea of interpreting one of the basic systems of the modal logic of provability in an epistemic way. I shall give arguments in support of this solution. The third solution, proposed by P. Egrèé [8] attempts to connect the first and the second solutions. I will show that this attempt fails for philosophical and formal reasons.

Two versions of the Knower Paradox

The Knower Paradox was introduced in 1960 by Kaplan and Montague [START_REF] Kaplan | A Paradox Regained[END_REF] and basically is the epistemological counterpart of the Liar Paradox. I will start by presenting it in its original version, the one which treats knowledge as a predicate attached to names of sentences and uses the framework of first order arithmetic.

Let then T be a theory which extends Robinson Arithmetic (see, for example, [START_REF] Boolos | Computability and Logic[END_REF], pp. 110) and satisfies the following three conditions.

First Condition. T can prove the fixed point lemma.

The fixed point lemma. For every formula P(x) of L(T), with one free variable x, there exists a sentence α of L(T) such that T α ↔ P (α) (where the overlining of a sentence stands in for Gödel numeral of that sentence).

Proof. See [START_REF] Boolos | Computability and Logic[END_REF], pp. 167.

Second Condition. T contains an arithmetical predicate I(x, y) meaning that the sentence with Gödel number x is derivable from the sentence with Gödel number y, the Gödel numbering being given in advance. Note that in T we can formalize the proof predicate Bew(x), normally defined as ∃y.P roof T (y, x), which satisfies the rule (for the proof see [START_REF] Boolos | The Unprovability of Consistency[END_REF], pp. 43):

α ⇒ Bew(α)
From this fact it follows that if T satisfies the deduction theorem, then I(α, β) is expressible as Bew(α → β).

Third Condition. T contains a unary predicate K(x) meaning that the truth of the sentence with Gödel number x is known and satisfying, for every sentence α and β of L(T): K(α) → α, the reflection principle (RP), K(K(α) → α), assumption number two (AT), K(α) ∧ I(α, β) → K(β), epistemic closure (EC).

Note that if T satisfies the deduction theorem and it contains the rule R.K.:

α ⇒ K(α)
then the principle of epistemic closure can be expressed in the following alternative form:

K(α → β) → (K(α) → K(β)), epistemic closure (EC 1 ).
The Knower Paradox. If the theory T satisfies these three conditions, then T is inconsistent.

Proof. By the fixed point lemma, there is a sentence ¬p of L(T) such that:

(1) T ¬p ↔ ¬K(¬p)

(2) T ¬¬p ↔ ¬¬K(¬p), by contraposition from ( 1

) (AT ) T K(RP ), (9) (EC) T K(RP ) ∧ I(RP , ¬p) → K(¬p), 8
(EC), (AT ) T K(¬p), by M. P. form [START_REF] Cross | More on the Paradox of the Knower without Epistemic Closure[END_REF], ( 8), ( 9), [START_REF] Montague | Syntactical Treatments of Modality, with Corollaries on Reflection Principle and Finite Axiomatizability[END_REF] (EC), (AT ) T p, by logic from [START_REF] Kripke | An Outline of a Theory of Truth[END_REF] and (3).

(12) (EC), (AT ), (RP ) T p ∧ ¬p, combining [START_REF] Montague | Syntactical Treatments of Modality, with Corollaries on Reflection Principle and Finite Axiomatizability[END_REF] with [START_REF] Cross | The Paradox of the Knower without Epistemic Closure[END_REF].

(q.e.d.)

Even if the Knower Paradox in its original formulation was presented as a Paradox affecting the framework of first-order arithmetic, it can be easily reproduced in the framework of modal logic.

Let L be a normal system of modal logic in the modal language L: so L contains all tautologies, all instances of the distribution axiom schemata K [2(α → β) → (2α → 2β)], it is closed under modus ponens and the rule of necessitation R.N.

[ α ⇒ 2α] and it satisfies the deduction theorem. Let L * a modal system in the modal language L * that satisfies the following two conditions.

First condition. L * extends L by proving a fixed point lemma.

The modal fixed point lemma. For every L -formula α(p, q 1 , ..., q n ), in which every occurrence of p is in the scope of some occurrence of 2, there exists a sentence β(q 1 , ..., q n ), where β contains no sentence letter except q 1 , ..., q n and p is distinct from the q i 's, such that:

L * β(q 1 , ..., q n ) ↔ α(β(q 1 , ..., q n ), q 1 , ..., q n ).
Proof. See, [START_REF] Boolos | The Unprovability of Consistency[END_REF], pp. 141-145.

Second Condition. L * contains the axiom schemata, T: 2α → α Let us make three observations. Firstly, T is quite obviously the modal counterpart of the reflection principle; it suffices to substitute in the reflection principle K by 2, to obtain T:

K(α) → α 2α → α
Secondly, if L * contains T and the rule of necessitation, then L * also contains the axiom schemata U: 2(2α → α), which is the modal counterpart of the assumption number two. Again it suffices to substitute in the assumption number two K by 2, to obtain U:

K(K(α) → α) 2(2α → α)
Thirdly, as L * contains the distribution axiom schemata and the rule of necessitation, then L * contains the modal counterpart of the principle of epistemic closure in his alternative form (EC 1 ). It suffices to substitute in the rule R.K. and in EC 1 K by 2, to obtain the rule R.N. and the distribution axiom schemata K:

α ⇒ K(α) α ⇒ 2α K(α → β) → (K(α) → K(β)) 2(α → β) → (2α → 2β)
The Knower Paradox in the Framework of Modal Logic. If L * satisfies these two conditions, then L * is inconsistent.

Proof. This result follows immediately from the inconsistency result just derived in the first-order arithmetic theory.

Two important points common to the three solutions

The aim of this article is to present three solutions to the Knower Paradox. These solutions agree on the following two points. Firstly, they agree on the source of the Paradox. In order to solve the Paradox, we must find an actual mistake in the argument. This can be: a fallacious assumption, an invalid inference or an inconsistent theory (or system). Given that the theory T (or the system L * ) is supposed to be consistent, or at least possibly consistent, and the proof is trivially valid, then one of the three assumptions, namely RP, AT and EC (or their modal counterparts, namely T, U and K) must be at fault and consequently, to avoid the Paradox, we have to renounce to one of them. The three solutions agree that the assumption number two is the source of the contradiction and I want to give three reasons which support this claim.

Firstly, the other two assumptions are relatively difficult to doubt, if we attentively analyze them. A recent and really interesting result of C.B. Cross [START_REF] Cross | The Paradox of the Knower without Epistemic Closure[END_REF] and [START_REF] Cross | More on the Paradox of the Knower without Epistemic Closure[END_REF] tells us that we can prove the Knower Paradox without an explicit epistemic closure. If we can derive the Knower Paradox without the epistemic closure, then we can draw two conclusions: the first is that the principle of epistemic closure can't be the source of the Paradox, the second is that the culprit of the contradiction has to be the reflection principle or the assumption number two. Let us then analyze these two assumptions. The reflection principle is the most natural and intuitive one. More, it's supported by our standard conception of knowledge, the one which defines knowledge as justified true belief , which has been firstly introduced by Plato in the Theaetetus (see, e.g., [START_REF] Campbell | Theaetetus of Plato[END_REF]). As we do want to preserve this conception of knowledge, we can't refuse the reflection principle. So, it is useless to reject the principle of epistemic closure because the contradiction can be proved without it; it is implausible to reject the reflection principle without rejecting knowledge itself; hence the only assumption we can reject, to avoid the Knower Paradox, is the assumption number two.

Furthermore there are two arguments that suggest that the assumption number two is the source of the Paradox. The first argument relies on the analogy with provability and the Gödel's second theorem. Let's consider the three assumptions which are satisfied in the formal system T : the principle of reflection, the assumption number two and the principle of epistemic closure. If we substitute in every assumption the knowledge predicate K by the provability predicate Bew, we obtain:

Bew(α) → α, Bew(Bew(α) → α), Bew(α) ∧ Bew(α, β) → Bew(β)
But Bew(¬Bew(⊥)), which contradicts the Gödel's second theorem, is just an instance of Bew(Bew(α) → α). Therefore, quoting Anderson ([1], p. 350), the thing that's false for Q is the analogue of the assumption number two. Shouldn't we do the analogous thing for our case?

The second argument is based on the form of the three assumptions. The reflection principle and the epistemic closure describe some conditions and features of knowledge: the former says that if something is known, then it is so; the latter says that if β is derivable from α and α is known, then β is known. As we can trivially see, they do not say anything about what we really do know, whereas the assumption number two affirms knowledge of something (the reflection principle). This difference of shape could be a good sign for believing that the assumption we have to rule out is the second one. As Anderson says ( [START_REF] Anderson | The Paradox of the Knower[END_REF], p. 350), None of this is conclusive. Probably there's a more convincing argument here somewhere, but I can't find it .

Having presented the first point common to the three solutions, let us now pass to introduce the second one. Such point relates to the notion of knowledge. Indeed each of the three solutions to the Knower Paradox treats knowledge as if it were intimately connected with the notion of proof. They thus seem to deal with a more limited and abstract notion of knowledge than that which is generally accepted; so they appear as solutions to the the Paradox of a Mathematical Knower, and not the Paradox of an Ordinary Knower. As we will see in the fourth section the main difference between Anderson's solution and the one offered by the modal logic of provability is based on this point: while Anderson appears to confuse two different notions of proof by which we obtain knowledge, the logic of provability uses a unique and clear notion of proof.

The first solution: Anderson's solution

Anderson's solution, published in 1983 in the Journal of Philosophy, is, as we will see, a hierarchical solution expressed in the frame of first order arithmetic.

In order to be able to present Anderson's solution, we have, first of all, to consider a language Q * extending the language of Robinson Arithmetic with additional non-logical constants. Then we have to isolate K 0 , a recursively enumerable set of Q *formulas, containing things expressible in Q * and known by a certain person James. Now we can construct a formal system, call it Q1 , whose theorems are the sentences derivable from the axioms of Robinson Arithmetic together with the sentences of K 0 . Is Q 1 consistent? Yes it is: the axioms of Q (Robinson Arithmetic) are true in the standard model, the things contained in K 0 are known by James and hence they are true, the rules of inferences preserve truth, so every theorem of Q 1 is true. But in Q 1 , as it contains Robinson Arithmetic, we can formulate the Gödel sentence G ↔ ¬Bew(G). Is G provable? No, if it were, ¬Bew(G) would be provable, and so a falsity would be provable contrary to the truth of every theorem of Q 1 . Does James know G? No, G is not in K 0 , else it would be provable in Q 1 . But I have just given a proof that G is not provable, hence I know G, or G is contained in the set K 1 of the things that I know. If I were James, Anderson argues, it would seem that we have to distinguish different levels of knowledge: what I know at stage K 0 , and what I know at stage K 1 , after reflecting on Gödel's theorem.

The conclusion is that we have to distinguish levels of knowledge -K 0 , K 1 , K 2 , ... -and this idea of levels of knowledge has to be applied to the assumption number two, considered to be the source of the Paradox. Anderson formalizes such a conclusion with the following hierarchy of languages:

Languages. Let L = L(Q) be the language of Robinson Arithmetic, we have:

L 0 = L ∪ {K 0 , I 0 } 1 L n+1 = L n ∪ {K n+1 , I n+1 } L ω = n∈ω L n
So starting with the language of Robinson Arithmetic, Anderson constructs a hierarchy of theories (T n ) n∈ω satisfying:

T 0 = Q ∪    K 0 (α) → α K 0 (α) ∧ I 0 (α, β) → K 0 (β) T n+1 = T n ∪            K n+1 (α) → α K n+1 (K n (α) → α) K n+1 (α) ∧ I n+1 (α, β) → K n+1 (β)
where α and β ∈ L ω . As we can trivially see, the assumption number two has now the form: K n+1 (K n (α) → α) which is always true, while K n (K n (α) → α) is, in some cases, false. In this way, Anderson avoids the Paradox of The Knower.

My criticism to Anderson's solution

My criticism of Anderson's solution is based on two arguments: a principal and specific argument and a second general argument which is inspired by S. Kripke's article of 1975, [START_REF] Egré | The Knower Paradox in the Light of Provability Interpretations of Modal Logic[END_REF].

The main argument can be succinctly expressed by the following sentence: I argue that, in his solution, which relies on a certain relationship between knowledge and proof (see, section 2), Anderson confuses two different notions of proof: a syntactic one and an absolute one. So in order to be able to explain my criticism, firstly I have to introduce these two notions of proof. The syntactic notion of proof is well known and it is linked with the notion of formal system that is a set of sentences, called the axioms of the system, together with a set of relations on sentences, called the rules of inference of the system. A proof of a sentence in a system, or a syntactic proof, is a finite sequence of sentences, each of which either is an axiom of the system, or is immediately deducible from the earlier ones by one of the rules of inferences. One of the possible interpretations of Gödel famous theorem deals with this notion of proof: if a formal system satisfies certain conditions, there exists a formula p such that neither p nor ¬p is formally provable in that system. But in Myhill's opinion (see [START_REF] Myhill | Some Remarks on the Notion of Proof[END_REF]) there is another quite interesting and problematic interpretation of Gödel's theorem which is expressible in the following way: for any formal system S, adequate for the arithmetic of natural numbers, there are correct inferences which cannot be formalized in S. So the question is: what does Myhill mean when he uses the notion of correct inferences? Let's try to explain such a concept. When a person uses some methods of proof, he is committed to the validity of his methods of proof, otherwise he would be contradicting himself. Hence if someone uses the means furnished by the system of Peano Arithmetic (or Robinson Arithmetic) for proving arithmetical statements, and is able to make explicit those methods, he is ipso facto committed to the perfectly definite proposition that the use of his methods cannot lead to a false arithmetical statement ( [START_REF] Myhill | Some Remarks on the Notion of Proof[END_REF], p. 462). So let us consider a person who uses the means furnished by the system of Peano Arithmetic for proving arithmetical statements and who succeeds in making these means explicit. This person will be able to reason as follows: the axioms of Peano Arithmetic are true (he is committed to this, because he is committed to the validity of his proof methods), and the rules of inferences preserve truth (idem), therefore every theorem of Peano Arithmetic is true. Hence a false statement, such as 0 = 1, is not a theorem, so (the arithmetization of) the statement that says that 0 = 1 is not a theorem is true. However, such reasoning cannot be formalized in the system of Peano Arithmetic. It is thus not a syntactic proof. It can only be thought of as a proof in a different sense, which we will call, following Myhill's definition, absolute, and in this sense of proof the undecidable sentence p is demonstrable.

Since I have clarified the meaning of the syntactic and the absolute notion of proof, I am now able to explain my criticism to Anderson's solution. At the beginning of his discussion, Anderson adopts a notion of knowledge based on a formal notion of proof: the set K 0 of the things known by James is recursively enumerable and contains Q, Robinson Arithmetic (as we will see in a moment, K 0 has to contain Q, otherwise James could never be me, as Anderson wants). Hence James knows by a syntactic notion of proof, and so, for what we have just said, he doesn't know the Gödel sentence G: nobody, who knows by this sense of knowledge, could never know the Gödel sentence G. So how could James know G? He has to use the proof I have utilized for knowing G, but that kind of proof is not at all a syntactic one: on the contrary is the sort of proof we have defined as an absolute notion of proof. So Anderson'solution invites two important questions. The first one is: how can Anderson define, at the beginning of his argument, James's knowledge as knowledge based on a syntactic notion of proof, and then adopt a definition of James's knowledge based on an absolute notion of proof ? The second one is: if Anderson had to give a definition of the concept of knowledge, on which of the two notions of proof would depend ? The trivial answer is that there is no reason for changing the notion of proof on which James's knowledge is based, and that, therefore, Anderson cannot give a definite and clear notion of knowledge.

This was the first argument against Anderson's solution. Let us now pass to the second and more general one. The second argument I want to propose is based, as I have already said, on the analogy with Kripke's criticism of the Tarski's idea of introducing a hierarchy of languages each with its own predicate of truth for the preceding language. In Kripke's opinion, Tarski's theory goes against our intuitive usage of the predicate "being true": in the natural language there is only one truth predicate and not many. In the same way, we can argue against Anderson's solution that in the natural language we have only one knowledge predicate. So the idea of introducing many levels of knowledge is not only based on a incorrect argument, but is also quite counterintuitive.

5.

The second solution: modal logic of provability.

The second solution is based on the idea of giving an epistemic interpretation of the modal logic of provability. Let us, first, explain what is the modal logic of provability.

The modal logic of provability, developed only in the early 70's and based on the two systems GL and GL * , is a logic that tries to answer the following question: can we capture the features of the provability predicate Bew(x) in the framework of modal logic? The positive answer to this question, given by R. Solovay in 1976 [START_REF] Solovay | Provability Interpretations of Modal Logic[END_REF], comes in the form of an equivalence between the system PA (Peano Arithmetic) and the two systems GL and GL * . In order to explain such an answer in a mathematically precise way, we have to present the axiomatization of GL and GL * and introduce the notions of realization and translation. GL is a normal system of modal logic characterized by the two axiom schemata:

4 : 2α → 22α G : 2(2α → α) → 2α
On the other hand, the axioms of GL * are the theorems of GL and all sentences of the form 2α → α (T), and the sole rule of inference is modus ponens. GL * is not a normal system of modal logic as it is not closed under the rule of necessitation.

Let us now introduce the two notions of realization and translation.

A realization is a function that assigns to each sentence letter a sentence of the language of Peano Arithmetic.

A translation α φ of a modal sentence α under a realization φ is defined inductively:

⊥ φ = ⊥ p φ = φ(p) (α → β) φ = (α φ → β φ ) (2α) φ = Bew(α φ )
These definitions allow us to express Solovay's result:

GL α if f, f or every φ, P A α φ GL * α if f, f or every φ, |= N α φ
The central idea of the second solution is to use the system GL * (which is a consistent system of modal logic even if we add the fixed point lemma) to resolve the Paradox of the Knower. In order to develop such an idea, we have to give an epistemic interpretation of GL * . Let us then consider GL * , not as a modal logic system, but as an idealized mathematician knower who has a fairly complete knowledge of mathematics. The theorems of GL * , axiomatized by the principles we have given on page 9, are exactly what our mathematician knows. Hence the set of formulas he knows: contains only, but not all, arithmetical truths (modulo φ), is closed under modus ponens.

Finally his knowledge has the following features: K : it is closed under modus ponens, T : it satisfies the reflection principle, 4 : if he knows something then he knows that he knows it, G : Our knower couldn't be more modest about his knowledge: he never knows to be sound with regard to an unknown sentence (see [START_REF] Smullyan | Gödel's Incompleteness Theorems[END_REF], pp. 350-351).

Remark that the modal logic of provability substitute the problematic assumption number two, whose modal counterpart we have designed with the axiom schemata U (see, p.3), with the new axiom G. G has a form which is suitable for the observations we have made before: it does not affirm knowledge of something, but it just describes a feature of knowledge as the reflection principle and the principle of epistemic closure do. So thanks to an idealized and abstract notion of knowledge and to the consistent and decidable system GL * , we menage to avoid the Paradox of the (Mathematical) Knower.

The third solution: Egré's solution

We arrive finally at the third and last solution, which tries to connect the first two. Indeed P.Egré in his paper [START_REF] Egré | The Knower Paradox in the Light of Provability Interpretations of Modal Logic[END_REF] arrives at the conclusion of the section 5 above and then, instead of stopping as he should, argues that there is a link between the modal logic of provability and the first system of Anderson's hierarchy. What sort of connection does he propose? First of all P. Egré affirms that GL * corresponds to P A + , where P A + contains as axioms the theorems of P A and all sentences of the form Bew(α) → α, and it is closed under modus ponens. Then he says that:

P A + = T 0
where T 0 is the first system of Anderson's hierarchy (see, p.6).

Hence in Egré's opinion, Anderson's solution and the solution offered by the modal logic of provability are not so different: Anderson's hierarchy is a generalization to all the finite degrees of the separation of axiom schemata reflected in Solovay's system GL * ([8], p. 45). However this conclusion is incorrect. For a start, it runs into the following technical problems. Namely, we will show, contra Egré, that:

P A + = T 0
In order to prove such a formula, we have to analyze the assertion "GL * corresponds to P A + " which is quite unclear since there is no notion of realization and translation, nor theorem as the Solovay's one. So there are two possible interpretations:

(1) (2α) φ = K(α φ ) Then P A + = T 0 , for two reasons. The first one is that P A + contains the translation of the axiom schemata G and 4, whereas T 0 does not. The second reason is that P A + contains the translation of the distribution axiom schemata K which is not equivalent to the principle of epistemic closure contained in T 0 . In fact K is only equivalent to K(α) ∧ I(α, β) → K(β), in the presence of the translation of the rule of necessitation, that is not, as we have already said, a rule of P A + . The other possible interpretation is:

(2) (2α) φ = Bew(α φ ) Then P A + = T 0 , for two reasons. The first one is that, as before, P A + does not contain a 'real epistemic closure' whereas T 0 does. The second, and maybe more important reason, relies on the tension between the following two facts. On the one hand, as we have already said in section 1, in any arithmetical theory where we can define the proof predicate Bew(x) as ∃y.P roof T (y, x), Bew(x) satisfies the rule α ⇒ Bew(α). On the other hand, we want the arithmetical theory P A + to contain a proof predicate Bew(x), defined as ∃y.P roof T (y, x), which does not satisfy the rule α ⇒ Bew(α), since P A + is the arithmetical counterpart of GL * . It is clear that these two facts are incompatible (it is no accident that the Solovay theorem hold between GL * and the standard model, and not, as for GL, between GL * and a formal system).

These technical weaknesses of Egré'solution may not be unrelated to the philosophical points underlined in the preceding discussion. In fact the modal logic of provability and Anderson's solution employ different means: while Anderson uses, in an illegitimate way, two notions of proof (the absolute one and the syntactic one) to define knowledge, in the modal logic of provability it's clear that to know means to prove in a formal system.

), ( 3 )

 3 T p ↔ K(¬p), by logic from (2), (4) (RP ) T K(¬p) → ¬p, (5) (RP ) T p → ¬p, by transitivity from (3) and (4), (6) (RP ) T ¬p, by logic from (5), (7) T I(RP , ¬p), from what precedes, (

Note that Anderson adds inferences predicates In to his hierarchy, but he does it with minimal argument; this point is not, however, central to my discussion.


Marie-Julie Dejardin

Arnaud Hemmerle

Anaïs Sadoun

Yannick Hamon

Pierre-Henri Puech

Kheya Sengupta
email: sengupta@cinam.univ-mrs.fr

Laurent Limozin
email: laurent.limozin@inserm.fr

Lamellipod reconstruction by three dimensional reection interference contrast nanoscopy (3D-RICN)

Keywords: Super-resolution, Label-free imaging, Lamellipodium, Cell surface imaging, Interference imaging, Cell membrane topography

There are very few techniques to reconstruct the shape of a cell at nanometric resolution and those that exist are almost exclusively based on uorescence, implying limitations due to staining constraints and artefacts. Reection Interference Contrast Microscopy (RICM), a label-free technique, permits measurement of nanometric distances between refractive objects. However, its quantitative application to cells has been largely limited due to the complex interferometric pattern caused by multiple reections on internal or thin structures like lamellipodia. Here we introduce three dimensional reection interference contrast nanoscopy, 3D-RICN, which combines information from multiple illumination wavelengths and aperture angles to characterize the lamellipodial region of an adherent cell in terms of its distance from the surface and its thickness. We validate this new method by comparing data obtained on xed cells imaged with Atomic Force Microscopy and Quantitative Phase Imaging. We show that as expected, cells adhering to micropatterns exhibit a radial symmetry for the lamellipodial thickness. We demonstrate that the substrate-lamellipod distance may be as high as 100 nm. We also show how the method applies to living cells, opening the way for label-free dynamical study of cell structures with nanometric resolution.

Cell adhesion and spreading are central phenomena regulating cell function and behavior, ranging from activation to proliferation. [START_REF] Cretel | A new method for rapid detection of T lymphocyte decision to proliferate after encountering activating surfaces[END_REF] Cell spreading, as well as motility, is usually mediated by a at and dynamic protrusion, called the lamellipodium, [START_REF] Giannone | Lamellipodial actin mechanically links myosin activity with adhesion-site formation[END_REF][START_REF] Oakes | Lamellipodium is a myosin-independent mechanosensor[END_REF] which has been the object of numerous structural studies based on uorescence 46 and electronic microscopy. [START_REF] Small | Unravelling the structure of the lamellipodium[END_REF] The shape of a motile cell, especially the lamellipodium, is known to be crucial for spreading, migration and mechanosensing. [START_REF] Oakes | Lamellipodium is a myosin-independent mechanosensor[END_REF][START_REF] Mogilner | The shape of motile cells[END_REF][START_REF] Dillard | Ligandmediated friction determines morphodynamics of spreading T cells[END_REF] However, in cell adhesion studies the focus is often on the substrate-proximal surface, which is typically imaged using uorescence based techniques to reveal specic proteins and structures involved in adhesion and associated signaling. [START_REF] Geiger | Environmental sensing through focal adhesions[END_REF] An important aspect of adhesion is the quantication of the intimate contact of the cell membrane with the surface in terms of the cell to surface distance, usually achieved using dierent types of interference microscopy, [START_REF] Curtis | The mechanism of adhesion of cells to glass. A study by interference reection microscopy[END_REF][START_REF] Paszek | Scanning Angle Interference Microscopy Reveals Cell Dynamics at the Nano-scale[END_REF] allowing determination of distances between cell surfaces well below the optical resolution imposed by the limit of diraction, out of reach for classical or confocal uorescence microscosopy.

Dierent uorescence related strategies, based on superresolution or deconvolution, and combined with total internal reection uorescence microscopy or dual objective astigmatism, can be also used to provide nanometer resolution in the axial direction. [START_REF] Xu | Dual-objective STORM reveals three-dimensional lament organization in the actin cytoskeleton[END_REF]1317 While permitting molecular specicity, uo-based methods are limited by the necessity of labeling the sample, which is often inappropriate for fragile cells or medical samples. They are compromised by possible labeling or xation artifacts [START_REF] Christiansen | In Silico Labeling: Predicting Fluorescent Labels in Unlabeled Images[END_REF] and by photobleaching in case of long-term live or repeated observation.

In the context of quantication of adhesion, Reection Interference Contrast Microscopy (RICM) has often been the technique of choice. [START_REF] Curtis | The mechanism of adhesion of cells to glass. A study by interference reection microscopy[END_REF]1921 It consists in epi-illuminating the cell-surface region with a monochromatic light and collecting the interfering rays. This purely interferometric approach, showed great initial promise, [START_REF] Curtis | The mechanism of adhesion of cells to glass. A study by interference reection microscopy[END_REF][START_REF] Bereiter-Hahn | Quantitative reection contrast microscopy of living cells[END_REF] but was later mostly used in a semi-quantitative manner for cell analysis due to the complex optics of cells, though RICM does provide excellent resolution in the context of simple objects like colloids, model membranes or thin lms. 2023 Recent application to cells of interferometric scattering microscopy (iS-CAT [START_REF] Piliarik | Direct optical sensing of single unlabelled proteins and super-resolution imaging of their binding sites[END_REF] ), which relies on scattering from ne features and edges, provides semi-quantitative informations at high time and lateral resolution, but with limited condence in the nature of the structure imaged and limited resolution in vertical direction. [START_REF] Park | Label-free and live cell imaging by interferometric scattering microscopy[END_REF] Label-free techniques are currently undergoing a strong development, in particular to meet the needs of biomedical research, [START_REF] Bon | Optical detection and measurement of living cell morphometric features with single-shot quantitative phase microscopy[END_REF][START_REF] Bon | Label-free evanescent microscopy for membrane nano-tomography in living cells[END_REF] a striking example being the trawling of big databases of transmitted light images of cells using machine learning strategies. [START_REF] Christiansen | In Silico Labeling: Predicting Fluorescent Labels in Unlabeled Images[END_REF][START_REF] Pavillon | Noninvasive detection of macrophage activation with single-cell resolution through machine learning[END_REF] RICM of cells has known a revival in this context. 2931 RICM is sensitive to abrupt changes in the refractive index at the interface of two media, and in a rst approximation, the collected intensity is related to the cell-surface distance. 3236 This approach was exploited to estimate membrane uctuations. Two major problems associated with quantication of RICM images from cells are that the cell refractive index is not known and that extraneous reections from internal organelles or the dorsal membrane may corrupt the data and render interpretation dicult. [START_REF] Limozin | Quantitative reection interference contrast microscopy (RICM) in soft matter and cell adhesion[END_REF][START_REF] Sengupta | Spreading of neutrophils: from activation to migration[END_REF] However, reections from the dorsal membrane can provide an insight into the three dimensional shape of the lamellipod. To quantify this, we get inspiration from previous work that circumvented other limitations of RICM: multiple reections which occur in presence of more than one interface, was overcome using a Fresnel model of multiple interfaces; [START_REF] Theodoly | New modeling of reection interference contrast microscopy including polarization and numerical aperture eects: application to nanometric distance measurements and object prole reconstruction[END_REF]3840 and ambiguity due to the periodic relation between distance and phase shift, was overcome by dual wavelength illumina- tion. [START_REF] Schilling | Absolute interfacial distance measurements by dual-wavelength reection interference contrast microscopy[END_REF][START_REF] Sengupta | Adhesion of soft membranes controlled by tension and interfacial polymers[END_REF] Here we image adherent cells in RICM mode, with multiple illumination angles and wavelengths, and develop a tting procedure to reconstruct the three dimensional prole of the adhering lamellipodium with nano-metric resolution (3D-RICN).

To establish the proof of the principle, we chose the broblast-like simian cell-line COS-7, which spreads easily on uncoated glass, making large lamellipods and having relatively slow dynamics post-spreading. Cells were allowed to spread on glass and, after xation, were imaged in RICM mode with multiple illumination conditions, obtaining 15 snapshots corresponding to combination of illumination wavelengths 436 nm, 546 nm, and 625 nm, and illumination numerical apertures (INA) of 0.5, 0.56, 0.68, 0.86, 1.05 (Fig. 1A). Reection of light from the sample is modeled assuming locally at lms oriented orthogonally to the optical axis, taking into account the s and p polarisation of light through the polarizer / quarter waveplate / analyzer, and using the Jones formalism to calculate the reectivity taking into account multiple reections and polarization [START_REF] Theodoly | New modeling of reection interference contrast microscopy including polarization and numerical aperture eects: application to nanometric distance measurements and object prole reconstruction[END_REF][START_REF] Wiegand | Studies of Structure and Local Wetting Properties on Heterogeneous, Micropatterned Solid Surfaces by Microinterferometry[END_REF][START_REF] Limozin | Modulation of vesicle adhesion and spreading kinetics by hyaluronan cushions[END_REF] (see SI for details). The lamellipodial region is modeled as a thin layer of cytoplasm of thickness d and refractive index nin, sandwiched between two membranes, the whole placed above the glass surface at a distance of h (Fig. 1B). The images are normalized with respect to the background intensity, for comparing with theoretically generated map of the theoretical normalized reectivity dened as r = I lam /I bgd with intensity I lam calculated for the lamellipod and I bgd calculated for the background. For each experimental value of λ, INA and for variable nin (in 0.005 steps), a reectivity map is generated for h between 0 and 500 nm in 2 nm steps and d between 0 and 1000 nm in 10 nm steps. Qualitatively, it accounts for an expected resolution better in h than d as well as a limit of sensitivity for large h and d. An example of such a map for λ=436 nm, INA=0.95 and nin=1.40 is shown on g. 1C.

To solve the inverse problem of nding parameters h, d and nin from the normalized multichannel data stack, a tting error χ 2 is calculated for each pixel of the experimental image (see SI), and the best solution (h, d, nin), corresponding to the minimal value of χ 2 , is identied. A graphical example of data and best-t theoretical reectivities for one randomly chosen pixel is shown on g. 1D * . Estimates of the error in h, d and nin are expressed, for each pixel, in terms of the condence interval δh, δd and δnin, which quanties the error introduced in tting due to acquisition noise (see Fig. S2). The reconstruction algorithm was applied on an area of ca. 30µm x 30µm chosen in the lamellipodial region.

The map of height h and its error δh are shown on Fig 2A,B. The distribution of h is represented on the histogram 2C, which exhibits a peak at 90 nm. The thickness of the lamellipodium is described by the map of d (Fig. 2D) and condence interval δd (Fig. 2E). It is seen to be rather at with the distribution of d peaking at 75 nm (Fig. 2F). The map of cytoplasmic refractive index nin and its interval of 3D surface plots of the basal and apical membranes were produced using Fiji, [START_REF] Schindelin | Fiji: an open-source platform for biologicalimage analysis[END_REF] and are represented on Fig. 2K,L,M. Sequences with azimuthal rotation are available in Supp. Mat. The correlation between the t parameters and their errors as well as the independence of the three tted parameters are mapped in Fig. S4. Noticeably, h and d exhibit a certain degree of correlation shown in Fig. S4. Correlation is also visible between d and nin. This indicates that for certain pixel, there is an underdetermination of the 3 parameters.

Since no other technique is able to image a lamellipodium in 3D and with the accuracy reported here, to validate 3D-RICN, we compared the reconstruction of the dorsal surface with an atomic force microscopy (AFM) image, and the optical thickness (expressed in terms of the optical path density) with quantitative phase image (QPI). The lamellipodial zone of the AFM image was oriented and scaled to correspond pixel by pixel to the 3D-RICN images, and was corrected for indentation of 40 nm † (Fig. 3A, raw image in Fig. S6) from which the apical membrane height (AMH) was calculated by summing h and d (Fig. 3B). Comparison of AMH images 3A and 3B is provided on the density map of Fig. 3C, which exhibits a high density along the x=y line, indicating a high overlap. Some points on the y axis reects errors due to incorrectly identied fringes in RICM. The proles of AMH along the blue line of Fig. 3A andB and show a good agreement. The standard height dierence of the two proles is less that 15 nm, while the lateral difference is of the order of 100 nm. A second test of 3D-RICN reconstruction was performed by comparison with Quantitative Phase Imaging (QPI). The same cell was imaged with QPI (Fig. 3E) and the zone of the lamellipod was croped, scaled and oriented to coincide pixel by pixel with 3D-RICN images. The optical path dierence (OPD) was calculated from 3D-RICN data as OPD= (nin -nout) × d (Fig. S7A). The OPD from QPI and 3D-RICN were compared as density maps (Fig. S7B) revealing a relatively poor agreement, likely because nin may be error-prone, as already suggested by the observed correlation between d and nin identied in Fig. S4. We therefore corrected OPD from 3D-RICN taking a uniform value of nin = 1.355, which corresponds to the peak of nin distribution (Fig. 2J). This corrected OPD is mapped in Fig. 3F. The comparison is shown as a prole on Fig. 3G and a density map on Fig. 3H. This shows a fair agreement between the two methods. However the comparison is impeded by the observed limit of sensitivity of QPI in the lamellipod region, as well as inhomogeneous background level correction.

We next imaged a cell spread on a circular patch of diameter 65 µm, coated with the extracellular protein bronectin. In this case, the normalization of the 3D-RICN raw images was performed by dividing the cell image by the aligned image of an empty bronectin patch. The reconstruction was performed as described above for the non-patterned substrates. Results are shown on Fig. ??. The central region of the cell has been masked since the proposed reconstruction does not apply in this zone due to the presence of the nuclear membrane just above the proximal cell membrane. We next applied the method to live cell imaging, reducing the acquisition time by a factor of 5 to make sure that it was short enough for the cell to be considered quasi-static during acquisition. This could be achieved by taking a single frame in each channel, rather that 5 in the static reconstruction detailed before. The reconstruction of h, d are shown on Fig. S9. A thinning of the cell (upper left corner, see arrow) upon retraction can be observed. The close contact of stress bers is visible. Surprisingly, the extent of close contact is very limited, hinting at weak adhesion of the whole cell in the lamellipodial region. This conrms that the similar heights obtained on a xed cell were not a mere artifact of xation.

Future use of the technique for live imaging would require an accelerated acquisition. This could be realized by using a lter wheel and a piezo-mounted objective to adjust automatically the position of the focus depending on the wavelength, as well as a reduced set of INAs for imaging the cell. We have performed reconstructions of a close-up of the lamellipod for dierent sets of INAs to quantify how the h and d maps are inuenced by the values and the number of INAs (Fig. S8). Taking the results obtained for the full set of ve INAs as the reference, we rst see that using only two INAs leads to wrong reconstruction of most pixels, in particular for d and h. Taking three INAs largely improves the results, as shown also by the quantitative analysis of a patch of 40 x 40 pixels in an area of small roughness in the lamellipod. We conclude that the values of the INAs chosen have a strong inuence on the quality of the reconstruction, and that taking extreme values for the INAs leads to a better reconstruction than taking closely spaced ones.

We also observe that the roughness of the patch, as dened by the Root Mean Square (rms) of height h, i.e. of the bottom membrane, is a good indicator for the quality of the reconstruction. Indeed, a wrong reconstruction of a pixel often leads to the convergence of the t to extreme values far from the correct ones, leading to an articial roughness of the image. Through a careful analysis, we determined that the set of 3 INAs [0.5, 0.68, 1.05], taking the two extreme values of the full set and an intermediate one, gives almost identical results as the reconstruction with the full set. This shows that a carefully chosen, smaller set of INA can be used in further studies for live imaging. Moreover, this convergence also ensures that our method is robust and uses enough INAs for a correct reconstruction of the lamellipod.

We also tested the reconstruction of the zones out of the lamellipod, including that under the nucleus, using appropriate layer-models. This however proved to be unsatisfactory, possibly because we were forced to limit the range of values searched due to the time required by the minimization algorithm. An obvious improvement will be the implementation of faster algorithms like progressive step renement. One inherent problem of RICM that partly remains here is ambiguity linked to periodic fringes. Furthermore, the sensitivity of the technique to measure the refractive index is limited. Use of an additional wavelength may help to overcome both these problems. In addition to the reconstruction of the basal nuclear membrane, perspectives of applications for 3D-RICN include monitoring of apical membrane rue or mitochondrial membranes, as well as cell volume dynamics. Compared to wideeld uorescence, we can expect lower phototoxicity due to the alternance of illumination wavelengths in 3D-RICN.

To conclude, we have reconstructed the 3D shape of a lamellipod, with 10 nm axial resolution and 100 nm lateral resolution, of a label-free cell adhering to a functionalized substrate. We thus provide, for the rst time to our knowledge, a validated reconstruction of the absolute height of the basal and apical membrane of a cell based on interference contrast microscopy principle. We have demonstrated the reconstruction of a live cell with time resolution of ca. one minute per frame. For both xed and live cells, the lamellipod is relatively at and with a thickness of about 100 nm, as could have been expected from previous studies with AFM or with state-of-art super-resolution microscopy of the actin cytoskeleteton using double objective astigmatism. [START_REF] Xu | Dual-objective STORM reveals three-dimensional lament organization in the actin cytoskeleton[END_REF] The substrate proximal surface, inaccessible to AFM, of a cell adhering non-specically to glass was also imaged. The cell-surface distance was dis-tributed around 80 nm and reduced to about 40 nm when specic adhesion was present, which is compatible with uorescence data for COS cells. [START_REF] Xu | Dual-objective STORM reveals three-dimensional lament organization in the actin cytoskeleton[END_REF] Such ultra-ne topographical maps of the under-surface of cells are of interest not only for fundamental cell biophysics but also for applications like building cell-on-chip connections, for example in the context of neurotechnology. [START_REF] Toma | Label-free measurement of cell-electrode cleft gap distance with high spatial resolution surface plasmon microscopy[END_REF] The method presented here opens the way to high throughput characterization of cell shape at nanometer resolution without labelling.

FOR TOC ONLY

Material & Methods

Sample preparation: Fibroblast-like cells of the COS-7 cell line were cultivated in DMEMc medium with 5% CO2. 24h before observation, cells were detached with 5% EDTA, and were deposited in a petri dish with glass bottom (Fluorodish FD-35, e=170µm, WPI) at 37 • C. Alternatively, the glass bottom was rst coated by incubating during 10 min a 10:1 mixture of bronectine (50µg/ml, Sigma) and uorescently labeled brinogen (5µg/ml, Thermosher).

Before some observations, cells were xed with 2% paraformaldehyde during 15 min and rinsed with PBS. The microscope chamber was lled up with about 1 ml of PBS, resulting in a liquid height of about 2 mm above the cells.

Patterning of the substrate with bronectin was performed by using Micro-Contact Printing Method (Thery M and Piel M, Cold Spring Harb Protoc. 2009 Jul;2009 (7):pdb.prot5255). PDMS stamp exposing disks of 65 µm diameter were inked with a mixture of bronectin and uorescent brinogen for 45 minutes at room temperature in the dark. Then, the ink was removed and the PDMS stamp gently dried with N2. The stamp was transferred to the petri dish glass surface for 10 minutes at 37 • C 5% CO2 and then removed. The glass surface was extensively washed with PBS 1X and backlled with pluronics F127 2% (Sigma) for 1 hour at room temperature in the dark, in order to avoid non-specic adhesion.

Optical setup: The light source is a Lumen 200 (Prior) connected to the microscope via a liquid optical ber. The inverted microscope AxioObserver (Zeiss) is equiped with: three passband lters (436/20, 546/10, 625/26 nm), protected with heat lters and disposed on a slider; a motorized aperture diaphragm; a uorescence cube containing a polarizer, a 50/50 semi reector mirror and a crossed analyzer; an objective Antiex 63x NA=1.25 devoided of phase ring (Carl Zeiss). Camera is a 1004x1002 pixels, 16 bits Andor iXon+. Optionally, an additional 1.6x magnication integrated to the microscope was used (Optovar). Andor Camera and microscope are controlled by MicroManager, allowing semi-automated image acquisition. For measurements with the phase camera Sid4Bio (Phasics, Palaiseau, France), a dual port (Carl Zeiss) is used to connect both cameras simultaneously to the sideport of the microscope. The phase camera was controlled by a dedicated computer using the Sid4Bio software (Phasics).

Optical imaging: For 3D-RICN imaging, image acquisition was performed by exposing the sample successively to green, blue and red light using the passband lters. The focus was carefully adjusted between each wavelength in order to compensate for chromatic aberrations of the objective. For each wavelength λ, ve dierent openings of the aperture diaphragm were automatically and successively selected, corresponding to Illumination Numerical Apertures INA=0.5, 0.56, 0.68, 0.86, 1.05. The exposure time, typically comprised between 0.1 and 1s, was adapted in order to obtain in average about 3000 counts per pixel on the camera for each illumination (λ, INA). 5 images were taken and averaged at each illumination. Finally, raw RICM images of the cells were compiled in a test stack of 15 images corresponding to 15 (λ, INA) combinations or channels. A reference stack was taken in identical conditions in a neighbouring region of the sample devoided of cells. Finally, the value of the camera oset was subtracted for test and reference stack, and a stack of images with background normalized to 1 was obtained using image by image division of the test stack by the reference stack. An example of normalized multichannel stack is shown on the montage of g. 1A. For live cell imaging, only one frame was taken in each illumination. A calibration of the INAs was performed with a simpler system consisting in a glass shard forming an edge with the glass substrate in PBS (see Fig. S1).

For quantitative phase imaging, the halogen lamp was set to 7 V and the condenser for transmission bright eld microscopy was adjusted for Köhler illumination. The recorded interferograms were transformed automatically in OPD images by the Sid4Bio software. A reference image was rst taken in a cell free zone of the sample. Then a series of 5 images with exposure ca. 1 s each was taken and averaged to produce one phase image representing the map of optical thickness. [START_REF] Bon | Optical detection and measurement of living cell morphometric features with single-shot quantitative phase microscopy[END_REF] In case of patterns, an empty (cell-free) motif was taken as reference.

Atomic force microscopy (AFM) imaging and force mapping: AFM was performed using a JPK Nanowizard 1 (z range 15µm), mounted on an Zeiss Axiovert 200M. Gold coated MLCT levers (Veeco, nominal spring constant 10pN/nm) were used. Contact mode was used for imaging. Set point and gains were manually adjusted to keep the applied force the lower possible ie around 100 pN, while scanning at 0.3-0.5Hz areas as large as 100µm x 100µm, and down to 15µm x 15µm. Areas of interest were chosen using transmision and/or uorescence microscopy, in order to localize and nd the exact same cells as the ones examined using other optical techniques performed on a dierent microscope. At the end of imaging, the cantilever was calibrated using built-in procedures based on the analysis of its thermal motion. Sensitivity was determined using contact with cell-free glass surface. This allowed us to determined that our imaging forces were as low as 100pN in regular conditions.

For force mapping, 16 pts x 16 pts grids were set onto the AFM imaged zone. Typical contact force was 100 to 150 pN, indentation and retraction speeds were 1 -2µm/s for a motion range of up to 3µm. Rather at parts of the cells were selected in order to be able to gain complete force curves, regarding our 15µm piezo range. One force curve was collected per force pixel, and was analysed using a homemade python set of functions, allowing to t the pressing curve, after tip-sample correction, with a linear model for the contact zone to estimate the indentation. Indentation maps were then rebuilt and an average indentation estimated, since the force pixels were, in order to keep the experiments doable and the data tractable, by far larger than the optical ones (Fig. S5) (Franz CM, Puech PH 2008 Cell. Molec. Bioengin. 1(4) :1865.) Modeling 3D-RICN: Reection of light from the sample is modeled assuming locally at lms oriented orthogonally to the optical axis. Taking into account the s and p polarisation of light through the polarizer / quarter waveplate / analyzer, the reected intensity is equal to: 22

Figure 1 .

 1 Figure 1. Principle of 3D-RICN. A. Montage of a multichannel reectivity image stack. RICM images are taken sequentially with 3 illumination wavelengths (λ) and 5 illumination numerical apertures (INA). Image background intensity is normalized to 1, so that cell reectivity is measured with respect to the background. The width of each image is ca 30µm. B. Thin lm model to calculate the reectivity of a lamellipod with 3 parameters to determine: cell surface distance h, cytoplasm thickness d, cytoplasmic refractive index n in . C. Example of calculated reectivity map (λ=436 nm, IN A=0.95, n in =1.40) for variable cell surface distance h and lamellipod thickness d. D. Example of measured reectivity for a pixel taken in the middle of images A (triangle) and tted theoretical reectivity (circle) for each illumination condition (INA and wavelength).

  condence δnin are shown on Fig 2G and Fig 2H. The distribution of nin, shown in the histogram of Fig 2J, peaks at 1.355.

  are compared in Fig.3D

Figure 2 .

 2 Figure 2. Topography obtained from the reconstruction algorithm. A. Map of height h (cell-surface distance). B. Map of δh, condence interval for h. C. Distribution of height h on the entire image but excluding black pixels. D. Map of cell thickness d. E. Map of δd, condence interval for d. F. Distribution of thickness d. G. Map of cytoplasmic refractive index n in . H. Map of condence interval for n in . J. Distribution of n in . Note that only pixels that are deemed to be non-pathological (see SI) and correspond to d > 500 nm are represented. K, L, M: 3D representations of basal membrane (K: seen from under or L: seen from above) and of apical membrane (M:seen from top). Numbers on images axis indicate distance in µm for 2D maps A, B, D, E, G, H and in nm for 3D reconstructions K, L, M.

Figure 3 .

 3 Figure 3. Comparison of 3D-RICN with AFM and QPI. A. AFM image of height of the apical membrane after correction for an average indentation of 40 nm. B. 3D-RICN map of the apical membrane height, obtained as the sum h+d. C. Map density of the 3D-RICN map as a function of the corrected AFM image. D. Height prole taken along the vertical blue line in images A, and B. E. OPD obtained by QPI. F. OPD obtained with 3D-RICN using d and n in = 1.355. G. Map density of OPD from F vs OPD from E. H. Proles comparing OPD from E and F taken along the horizontal blue line. Scale bar for A, B, E, F: 10 µm.

Figure 4 .

 4 Figure 4. 3D-RICN reconstruction of a cell adhered on a patch of bronectin and comparison with AFM and QPI. For clarity of the representation, the central part of the cell has been removed and a 0.8µm radius median lter has been applied on the 3D-RICN reconstructions. A. Membrane height h. B. Cell thickness d. C. Distribution of h. D. Distribution of cytoplasmic refractive index n in . E. Distribution of d. F. Apical Membrane Height calculated from 3D-RICN data. G. AMH measured from AFM. H. Optical Path Distance calculated from 3D-RICN data. J. OPD measured from QPI. NB: Lateral scale for A,B,F,H is in µm.

Figure S4 .

 S4 Figure S4. Density maps illustrating the degree of correlation between the parameters h, d, n in and their condence interval. Data correspond to the reconstructions shown in Fig. 2. A. Density map of height h and height condence δh showing the absence of correlation between these quantities. B. Relation between thickness d and condence interval δh. C. The density map shows a certain degree of correlation between height h and thickness d. D-F. The dependence of h, d and δn in as a function of n in . A certain degree of correlation between d and n in is visible.

Figure S5 .

 S5 Figure S5. Indentation mapping in AFM. A. Part of a lamellipodium imaged at a force of 100 pN in contact mode. B. Typical force curve obtained in this region, with the determination of the indentation at a force comparable as the force used for imaging. C. Reconstruction of the indentation map of a region of A, considering a particularly recognizable zone (*).

Acknowledgement We thank D. Marguet and S. Mailfert from the Picsel platform for lending of the Sid4bio camera, as well as S. Monneret for initiating us to quantitative phase measurement. We thank M. Biarnes-Pelicot and L. Borges (PMMC) for help with cell culture, O. Théodoly for stamps, and P. Dillard for preliminary experiments with Jurkat cells. PHP thanks JPK instruments for continuous support. This work was partially funded by GDR Imabio (MJD), AMIDEX (ANR-11-IDEX-0001-02) (AH and KS).

Supporting Information Available

The le SI.pdf is available free of charge and contains gures and details on materials and methods.

Supplementary information

Lamellipod reconstruction by three dimensional reection interference contrast nanoscopy (3D-RICN) Marie-Julie Dejardin 1*, Arnaud Hemmerle 2*, Anaïs Sadoun 1, Yannick Hamon 3, Pierre-Henri Puech 1, Kheya Sengupta 2 †, Laurent Limozin 1 † 1. Aix-Marseille University, Inserm, CNRS, Adhesion & Inammation, Marseille, France. 2. Aix Marseille University, CNRS, CINaM-UMR 7325, Marseille, 13288, France. 3. Aix-Marseille University, CNRS, Inserm , CIML Marseille, France. * These authors contributed equally to this work. † Corresponding authors: laurent.limozin@inserm.fr, sengupta@cinam.univ-mrs.fr

Contents:

-Material & Methods:

-Sample preparation -Optical setup -Optical imaging -Atomic force microscopy (AFM) imaging and force mapping -Modeling 3D-RICN -3D-RICN image reconstruction -Supplementary Movies 1-3.

-Supplementary Figures S1-S9.

with INA= n1 sin θ1, α a constant, and integration runs over illumination angle θ. Reectivity can be calculated using Jones matrices formalism, taking into account multiple reections and polarization: [START_REF] Theodoly | New modeling of reection interference contrast microscopy including polarization and numerical aperture eects: application to nanometric distance measurements and object prole reconstruction[END_REF][START_REF] Wiegand | Studies of Structure and Local Wetting Properties on Heterogeneous, Micropatterned Solid Surfaces by Microinterferometry[END_REF][START_REF] Limozin | Modulation of vesicle adhesion and spreading kinetics by hyaluronan cushions[END_REF]

Mj

with βj = 2πd j λ nj cos θj, λ the wavelength, nj and dj the refractive index and the thickness of layer j, θj the angle of incidence in layer j, and Nj the eective index in layer j dened as: Nj = N s j = nj/ cos θj for s polarization and Nj = N p j = nj cos θj

Rs,p = r s,p 0,m+1 =

In the case of the lamellipod, one considers for the model a thin layer of cytoplasm comprised between two membranes and hovering above the glass surface of the sample (Fig. 1B). The bottom membrane to glass distance is noted h, the lamellipod thickness d, with unknown refractive index nin comprised between 1.34 and 1.40 (g. 1B). The theoretical normalized reectivity is dened as r = I lam /I bgd with the intensity I lam calculated for the lamellipod and I bgd calculated for the background. For each experimental value of λ, INA and for variable nin (in 0.005 steps), a reectivity map is generated for h between 0 and 500 nm in 2 nm steps and h between 0 and 1000 nm in 10 nm steps. Example of such a map for λ=436 nm, INA=0.95 and nin=1.40 is shown on g. 1C.

3D-RICN image reconstruction

The inverse problem of nding parameters h, d and nin from the normalized multichannel data stack is performed as follows. For each pixel of the experimental image, a tting error χ 2 is calculated

where the sum is performed on i dierent illuminations. The best solution (h, d, nin) is obtained for the minimal value of χ 2 . A graphical example of data and theoretical reectivities for one given pixel is shown on Fig. 1D.

The reconstruction algorithm was implemented in Python 3 and conceived to run on a multicore processor by division of the image in the same number of subimages as the number of cores. The strategy consists in comparing, for each pixel, the series of measured illuminations with a 3D model matrix calculated for h, d and channel number. The coordinates of the minimum of the dierence matrix, corresponding to the best parameters, are recorded. This procedure is looped for every value of the refractive index tested. The current implementation running on a 24 core i7 computer with 64 Go RAM, with operating system Windows 7 pro, would reconstruct an image of 1000*1000 pixels in 1.5h, with 17 possible values for nin, 2 nm step for h, 10 nm step for d.

The detailed procedure for one example pixel is illustrated on Fig. S2. The χ 2 map shows periodic valleys in the (h,d) plane. To estimate a condence interval associated with the best solution, the next solutions (corresponding to the values of χ 2 < χ 2 max ) are also considered. χ 2 max was chosen from a measure of the noise in data, obtained by reconstructing individually the 5 frames recorded before averaging. For each pixel, χ 2 max is taken equal to the highest χ 2 out of the 5 reconstructions. This sets a range of acceptable solutions (h, d, nin), with respect to the data noise, on which ensemble statistics are calculated, like the range, the median and the interquartile range. We note δh, δd, δnin respectively the interquartile range of h, d, nin. Fig. S3 shows a comparison of reconstruction parameters between best solution and median solution.

To restrict the representation to the lamellipod in Fig. 2, only the pixels corresponding to a thickness between 1 and 500 nm are shown. Additionally, pixels were not represented when certain conditions, considered as pathological, were fullled: (a) d > 50 and δd < 15 nm, to prevent too low δd above a minimal d; (b) nin < 1.3475 and δnin > 0.0075 to prevent too high δnin under a minimal nin (c) h > 50 nm and d < 15nm to prevent low d at high h.

Surface plots of basal and apical membranes (Fig. 2K,L,M) were produced using the 3D surface plot plugin for ImageJ implemented in Fiji. [START_REF] Schindelin | Fiji: an open-source platform for biologicalimage analysis[END_REF] A preliminary median lter with a radius of 4 pixels was applied on images 2A, D.

Supplementary Movies

Supplementary Movie 1: Azimutal rotation of the 3D surface plot of the basal membrane seen from under, corresponding to Fig. 2K. Axis scale in nm. Supplementary Movie 2: Azimutal rotation of the 3D surface plot of the basal membrane seen from above, corresponding to Fig. 2L. Axis scale in nm. Supplementary Movie 3: Azimutal rotation of the 3D surface plot of the apical membrane seen from above, corresponding to Fig. 2M. Axis scale in nm. RICM images at various INA and wavelengths of a glass shard deposited in water on a glass substrate, forming an edge with an angle of 22 mrad. The bottom row shows ts of the intensity proles on a line perpendicular to the fringes for the three colors simultaneously (shown on the top left image), allowing for a determination of each INA with a precision of ±0.02. Blue and green proles are shifted vertically for convenience. Comparison of dierent methods for determining h, d and n in . This demonstrates a fairly close agreement between the 2 solutions, except the median thickness which is signicantly lower than the best thickness for d < 50 nm (Fig. S3B).

Supplementary Figures