Gaël Giraud
email: gael.giraud@parisschoolofeconomics.fr

Cécile Renouard

Camille Sutter

Coherent Multidimensional Poverty Measurement *

Keywords: multidimensional poverty, geometric mean, Maximin solution, utilitarian solution, endogenous identification, coherence, continuity, decreasing marginal rate of substitution, cardinal date, ordinality, relative weights JEL Classification Numbers : I3, I32, D31, D63, O1

This paper presents a family of multidimensional poverty indices that measure poverty as a function of the extent and the intensity of poverty. I provide a unique axiomatics from which both extent and intensity of poverty can be derived, as well as the poor be endogenously identified. This axiomatics gives rise to a family of multidimensional indices whose extremal points are the geometric mean and the Maximin solution. I show that, in addition to all the standard features studied in the literature, these indices are continuous (a must for cardinal poverty measures) and ordinal, in the sense that they do not depend upon the units in which dimensions of achievements are computed. Moreover, they verify the decreasing rate marginal substitution property: the higher one's deprivation (or the extent of poverty) in one dimension, the smaller the increase of achievement in that dimension that suffices to compensate for a decrease of achievement in another dimension.

Introduction

As acknowledged by [START_REF] Villar | A new approach to multidimensional poverty measurement[END_REF], "defining a poverty measure in a truly multidimensional context involves a number of subtle and difficult issues: choosing the appropriate poverty dimensions beyond income or wealth, deciding on whether they all are equally important, fixing sensible thresholds in those dimensions and setting criteria to identify as poor those individuals whose achievements lie partially below them, defining an overall measure of poverty intensity, etc. Those difficulties 2 anticipate that many compromises are required and, indirectly, that the axiomatic approach may be the best way to deal with this type of problem as it makes explicit all those compromises." 1 Here, we provide an axiomatization for a family of Multidimensional Poverty Indices. This is part of a larger research programme devoted to a Relational Capabilty Index applied as a new poverty measure in Nigeria, Indonesia and India.

Each index can be characterized as lying somewhere between the two extremal points of our family of indices: the geometric mean [START_REF] Villar | A new approach to multidimensional poverty measurement[END_REF]) and the Rawlsian Maximin. Although both social choice correspondences have been thoroughly studied from the social choice theoretical viewpoint, 2 we are not aware of any attempt to link these two major concepts of justice with the concerns involved in the literature devoted to poverty measurement. This paper is a first attempt to fill this gap.

We suggest that the geometric mean can be interpreted as being a (hyperbolic) version of the "utilitarian" viewpoint. With this interpretation in mind, our family of indices builds a bridge between celebrated theories of justice and poverty measurements. An alternative standpoint enables us to characterize each one of our indices as being the supremum of the weighted geometric averages, the sup being taken over some collection of weights over dimensions and people. When the collection of weights reduces to the uniform vector, we are back to the standard geometric mean (this is the "utilitarian" solution). When the underlying collection of weights includes the whole unit simplex over dimensions and people, then we get the Maximin solution. One possible interpretation is as follows: suppose that the economist who is in charge of measuring poverty in a given population reflects as if she were in Rawls' original position. Beyond the veil of ignorance, the point that is ignored is not which role one will endorse (as in the standard, political interpretation of Rawl's theory of justice) but in which dimension one will get some talent (or some endowment, or some "social capital"). So uncertainty bears on dimensions rather than on persons. In addition, from the viewpoint of the economist standing beyond the veil, there might be some ambiguity concerning the probability according to which talents and deprivations will be distributed. As a result, if the economist has no prejudice about the distribution of talents and deprivations will be distributed, she might opt for the Maximin solution as a way to measure multidimensional poverty. if, on the contrary, she has good reasons to believe that the distribution will be uniform, she may choose the "utilitarian"' solution (i.e., in our context, the geometric average). Else, she might choose an index in our family which lies somewhere in between the first two. If one wishes so, it is also possible to include ambiguity about the persons (and not only dimensions) in the non-symmetric version of our family of indices.

To the best of our knowledge, this is the fist attempt to formally fill the gap between theories of justice and poverty measurements. As we take inspiration from [START_REF] Artzner | Coherent measures of risk[END_REF] (where an additive version of a similar axiomatization was introduced in order to measure the risk position of a portfolio), we call coherent a multidimensional poverty index belonging to our family. We finally show that such indices satisfy the following properties that are considered as desirable in the literature: 3 (i) Each index is continuous: slight changes in the achievements of certain persons only induces slight changes in the poverty measurement ;

(ii) Each index is ordinal, in the sense that it does not depend upon the choice of the specific units in which dimensions of achievements are measured. This property deserves some comment. In [START_REF] Alkire | Counting and multidimensional poverty measurement[END_REF] it is argued that data describing capabilities and functionings in the spirit of Sen's Multidimensional Human Index, are ordinal in nature. They therefore may lack a basis for comparisons across dimensions. This, of course, is a challenge for Mutldimensional Poverty measurement. In the above quoted paper, indeed, only one kind of measures is shows to be ordinal in that sense (the M 0 measure in their parlance) while they others don't. At the same time, this ordinal measure fails to satisfy a number of other properties. In particular, it cannot capture the intensity of poverty -a failure that can be viewed as arising from its being a piecewise constant (hence discontinuous) measure. Here, we prove that coherent Poverty measures are ordinal in the following sense: If one multiplies any dimension by λ > 0 (both for achievements and for the poverty cut-off), then the set of poor is unaffected while the Index, P , is simply multiplied by λ. As a consequence, a normalized version of the index, Q, is independent of such changes in the dimensions' unit.

(iii) it yields a criterion for "relative poverty" that depends upon the whole population under scrutiny ;

(iv) the marginal rate of substitution among subjects or among dimensions is decreasing. The reduction in the deprivation4 of dimension k for poor individual i required to compensate an increase in the deprivation of dimension k for individual h is larger the higher the initial level of deprivation in i.

(v) As in [START_REF] Villar | A new approach to multidimensional poverty measurement[END_REF], it is multiplicatively decomposable by population subgroups (but it does not satisfy Subgroup Decomposability in the additive form given in Bourguignon and Chakravarty (2003)). This property says the following: If the population is partitioned into subgroups, the overall poverty index corresponds to the weighted average of subgroup poverty values, where the weights correspond to population shares.

(vi) In certain circumstances, we may have additional informations that allow us to regard certain dimensions and/or certain subgroups of the population as meriting a greater relative weight than others. Each index can be adjusted so as to capture this kind of requirements. Of course, if one wishes so, it can as well be made symmetric among persons.

(vii) It verifies the transfer principle [START_REF] Villar | A new approach to multidimensional poverty measurement[END_REF]) : a reduction of size δ > 0 in the deprivation with respect to dimension k of a poor person i who is worse off in this dimension than another poor person, j, more than compensates an increase of the same size, δ, in the deprivation of j, provided their relative positions remain unaltered.

(viii) Principle of population: a replica of the population does not change the poverty measure.

To the best of our knowledge, coherent poverty measures are the first examples of continuous and ordinal Multidimensional Poverty measure that are sensitive to inequality. To take but alternative examples, the measure M 0 introduced in ? is ordinal but discontinuous and inequality-insensitive. On the other hand, the measures M 1 and M 2 are inequality-sensitive and continuous but no more ordinal.

The paper is organized as follows. The next section provides the model and makes the link between the "utilitarian" standpoint and the geometric mean explicit. Section 3 deals with the axiomatization of coherent multidimensional poverty indices. The last section provides the main properties of this family of indices.

The model

Let N = {1, ..., N } denote a society consisting of n individuals and let K = {1, ..., K} be a set of dimensions.

A social state is a matrix, y = y ij ij ∈ M N ×K (R ++ ), with N rows, one for each individual, and K columns, one for each dimension. The entry y ij ∈ R ++ describes the value of variable j for individual i. Since we are going to deal with ordinal Poverty measures, there is little loss of generality in imposing from the outset that all variables be strictly positive.5 A vector z ∈ R K ++ of reference values describes the poverty thresholds for all dimensions. How these thresholds are fixed is definitely an important issue, but we leave it aside here. We denote by N (y; z) ⊂ N the set of poor that results from a social state matrix y and a vector z of reference values. The number of poor people is n(y; z) := |N (y; z)|. As we shall see, N (y; z) (hence n(y; z)) will be determined endogenously by our multidimensional poverty index).

The utilitarian index

A poverty index is a mapping P : R KN ++ → R + . We begin with three axioms that unambiguously characterize the "utilitarian" Poverty index.

The first one, anonymity, says that all agents and all dimensions are equally important:

Anonymity. Let x ∈ R KN
++ and let π ∈ S KN denote a permutation over its components {1, ..., kn}. Then, P (s) = P (π(s)).

The second Axiom implies that P reduces to the identity mapping on the diagonal of R KN + :

Normalization. Let s ∈ R KN ++ be such that s ij = a ∀i, j. Then P (s) = a.
The last property requires that the difference between the new and the initial values of P when subject i's achievement relative to dimension j changes from s ij to t ij , be a monotone function of the difference between s ij and t ij .

Difference Monotonicity

Let s, t ∈ R KN + be such that s hq = t hq ∀(h, q) = (i, j). Then P (s) -P (t) = g ij (s ij -t ij ),
for some increasing function g ij : R + → R. Since g ij (0) = 0, it follows that g ij (x) ≥ 0 if, and only if, x ≥ 0. Proposition 2.1 An index P (•) satisfies Anonymity, Normalization and Difference Monotonicity if, and only if, it takes the form

P (s) = 1 kn i∈N,j∈K s ij . Proof. Let s ∈ R KN + .
By difference monotonicity and normalization, P (s 11 , 0, ..., 0) -P (0, ..., 0) = g 11 (s 11 ) P (s 11 , s 12 , ..., 0) -P (s 11 , 0, ..., 0) = g 12 (s 12 ) so that

P (s) = P (0) + i,j g ij (s ij ).
By anonymity, g ij (•) = g(•) ∀i, j. The Normalization axiom yields: P (0, ..., 0) = 0. Moreover,

P (a, ..., a) = kng(a) = a.
Therefore, g(a) = a kn . The conclusion follows.

The geometric average

The link between the (admittedly fairly classical) index, P U , and the geometric average is given by the following transformation : Consider the following Poverty index, G(•), defined on R KN ++ :

G(x) := k,h x k,h 1 kn .
Given a vector, x ∈ R KN ++ , let us denote by ln x the vector whose coordinates are ln x k,h , every h, k. Obviously,

G(x) = exp U (ln x).
(1)

From this very simple remark, one deduces the axiomatization provided by Villar (2010) that fully characterizes the geometric average as a Poverty index: Indeed, it follows from (1) that G must verify the anonymity and normalization Axioms together with the following ratio monotonicity:

Ratio Monotonicity Let s, t ∈ R KN ++ be such that s hq = t hq ∀(h, q) = (i, j). Then, G(s) G(t) = g ij s ij t ij ,
for some increasing function

g ij : R ++ → R. Since g ij (1) = 1, it follows that g ij (x) ≥ 1 if, and only if, x ≥ 1.
In other words, the geometric (or Cobb-Douglas) average may be viewed as the outcome of the Utilitarian rule after the transformation given by (1). In the following section, we show that G(•) is but one extremal point of a whole family of Poverty indices that can be constructed in quite a similar way. The other extremal index of this family turns out to be the Maximin rule.

Coherent Poverty Indices

In order to define a coherent Poverty index, we impose some axioms on the mapping P (•). For this purpose, we define a poverty exit set, E ⊂ R KN ++ . A population belongs to E, whenever it is not poor.

Axioms for E.

We take z ∈ R KN ++ as given. The vector z describes the poverty thresholds for all dimensions and every individual. Those reference values may be fixed externally (absolute poverty lines) or may depend on the data of the social state matrix itself (relative poverty lines, such as a fraction of the median or the mean value). The units in which achievements are measured are chosen so that z >> 1 := (1, ..., 1) ∈ R KN ++ . 6 When all individuals are attribtued the same cut-offs, z = (z, ..., z), for some reference vector z ∈ R k ++ . In this case, if x i ≥ z, then individual i can be said to be poor (the converse being false in general).

In order to build an ordinal index (i.e., an index that does not depend upon the choice of unities in which dimensions are measured), we consider only normalized achievements. That is, if x ∈ R KN ++ is a given achievement, we shall deal with

x := (x hk /z hk ) h,k . Axiom 1. 1 + R KN + ⊂ E.
Consider, now, a population such that all its individuals have achievements (before normalization) that are all strictly below the thresholds given by z. Obviously, such a population should be considered as poor. This is the content of the next Axiom. 7 6 Given two vectors x, y, x << y if the strict inequality holds coordinatewise. 7 Actually, Axiom 2 says a little bit more since it implies that, in the case of a single person population, this person will be poor if none of her achievements end up strictly above the cut-off, and at least one of her achievements stay strictly below this cut-off. We could replace Axiom 2 by the weaker E ∩ z + R KN --= ∅. But our other axioms would nevertheless strengthen it into Axiom 2 in most cases of interest for practical purposes.

Axiom 2. E ∩ 1 + R KN - = {1}.
For every x, y ∈ R KN ++ , let x z denote the vector in R KN ++ whose coordinates are x k,h z k,h , all k, h. Consequently, 1/y denotes the (unique) vector such that y 1/y = 1, while x λ is the vector with coordinates x λ k,h . The "box" product, • •, is to be interpreted as formalizing a change in the achievements of the population under scrutiny. For instance, x 1 = x stands for "no change". By contrast, x 0 = 0 represents a radical depletion of the population, etc. For an arbitrary vector, y ∈ R KN ++ , x y will represent a change that may be dimension-and individual-dependent.

A set F ⊂ R KN ++ is multiplicatively convex whenever, as soon as x, y ∈ F, then

x α y 1-α ∈ F ∀α ∈ [0, 1].
Axiom 3. The Poverty exit set, E, is multiplicatively convex Axiom 1 says that, if all the individuals of a population exhibit all their achievements above the threshold (i.e., if x ≥ 1), this population is not poor. Conversely, if x << 1, Axiom 2 implies that the population is poor. Ambiguity remains only whenever some individuals exhibit some achievements above the threshold, and others, not. Our last axiom deals with such ambiguous cases. Suppose that a population, x, is not poor. Take λ > 0 and consider the auxiliary population given by x λ . Axiom 4 says that this new population should not be considered as poor neither. Clearly, if x 1/z ≥ 1 (resp. x 1/z < 1), then (x λ 1/z) ≥ 1 (resp. < 1), so that the auxiliary population turns out, indeed, not to be poor (resp. to be poor). What the next Axiom says is that this property should not hold only for the extreme cases envisaged by Axioms 1 and 2 but also for the "ambiguous" cases.

A set F ⊂ R KN ++ is a multiplicative cone whenever, as soon as x ∈ F, then x λ ∈ F for any λ ≥ 0.

Axiom 4. The Poverty exit set, E, is a multiplicative cone.

Examples The two following sets verify all four axioms.

a) The "utilitarian case". Consider

E := {x ∈ R KN ++ | G(x) ≥ G(z)
}. E is the upper-set of the hyperbola {x : G(x) = λ}, and is closed and (additively) convex.

b) The "Rawlsian case". Consider

E := {x ≥ 1},
E is an affine closed, convex (additive) cone.

Although it is not necessary for the core of our theory, the next Axiom will prove handful.

Axiom 5. E verifies the following Anonymity property : Let x = (x 1 , .., x n ) ∈ R KN ++ and σ(x) = (x σ(1) , ..., x σ(N ) ) ∈ R KN ++ the vector obtained after having swapped its individuals with the permutation σ ∈ S N . Then,

x ∈ E ⇐⇒ σ(x) ∈ E ∀σ ∈ S N .

Axioms for P

Given a set E, the Poverty index, P E , is defined as a measure of the minimal additional "achievements" that have to be added to a given distribution so that the population can be considered as non-poor, i.e., so that the resulting distribution belongs to E. Obviously, P E will heavily depend upon the choice of E. This is why the axiomatic studied here only defines a family of poverty indices, each index being identified through its corresponding poverty exit set.

Given E, the mapping P E (•) is defined as:

P E (x) := inf τ ∈ R | x z τ ∈ E . (2)
Axiom 1 implies P E (1) = 0, and lim x→0 P (x) = +∞. Conversely, given a Poverty exit index, P (•), one defines the Poverty exit set, E P , as

E P := x ∈ R KN ++ | P (x) ≤ 0 . (3)
We now state several properties for P . As we shall see, they can be deduced from Axioms 1-4 on E via (2). Conversely, Axioms 1-4 can be deduced from the following properties of P , via (3).

Axiom HI (Homothetic invariance

) ∀x ∈ R KN ++ , α ∈ R, P (x z α ) = P (x) - α.
An easy consequence of Axiom HI is that P (x z P (x) ) = 0 -which is consistent with the interpretation of P (•) given by [START_REF] Atkinson | The comparison of multidimensional distribution of economic status[END_REF]. It follows that ∀x, x z P (x) ∈ E.

(4)

Axiom S (Sub-multiplicativity) ∀x, y, P (x y) ≤ P (x) + P (y).

Axiom S can be interpreted as saying: averaging (in the multiplicative sense) two populations does not magnify the extent of poverty (i.e., the share of the poor within the global population), nor its intensity (i.e., the individual deprivation suffered from each individual) above the sum of the indices of the subpopulations. Since x and y are vectors of the same dimension, they correspond to populations of the same size. Hence, this axiom is a weak version of the subgroup additive decomposability applied to populations of equal size.

Axiom PH (Positive Homogeneity) : ∀x ∈ R KN ++ , ∀λ ≥ 0, P (x λ ) ≤ λP (x). Axiom S implies that P (x n ) ≤ nP (x) for every x and every integer n. Axiom PH extends this property to any nonnegative number.

Axiom M (Monotonicity): P (x) ≤ P (y) ∀y ≤ x. The aim of the last axiom is to avoid trivial indices that would be constant. Artzner et al. (1998), a Poverty index that satisfies Axioms HI, S, PH, M and NT is said coherent.

By analogy with

Quite similarly to the anonymity axiom for E, the next one is not needed for the characterization of coherent poverty measures, but will prove useful.

Axiom A (Anonymity): Let x = (x 1 , .., x n ) ∈ R KN ++ and σ(x) = (x σ1 , ..., x σn ) ∈ R KN ++ the vector obtained after having permuted its individual through the permutation σ ∈ S n . Then, P (x) = P (σ(x)). (ii) Conversely, if a set F satisfies Axioms 1-4, then P F is coherent, and E P F = F. 8(iii) E verifies the Anonymity axiom if, and only if, P does.

Proof.

(i) 1) P E (1) = 0 and Monotonicity imply that E verifies Axiom 1.

2) If x << 1, Monotonicity implies P E (x) ≥ 0. However, we can find α > 0 such that x z α << 0, so that P E (x z α ) ≥ 0. HI then implies that α ≤ 0. Contradiction. Thus, E P verifies Axiom 2.

3) Axioms S and PH imply that E P is multiplicatively convex. 4) If x ∈ E P , one has: P (x λ ) ≤ λP (x) ≤ 0 for all λ ≥ 1. Consequently, E P is a multiplicative cone. 5) Axioms PH and S imply that the function x R KN ++ → P (exp(x)) is convex, hence continuous. Consequently, x → P (x) itself must be continuous, so that E P is closed.

(ii) 0) Axioms 2 and 3 ensure that

P F is well-defined. 1) inf{τ ∈ R | x z t+α ∈ E} = inf{τ ∈ R | x z t ∈ E} -α, which proves HI.
2) Suppose that x z λ and y z β both belong to E. Axiom 3 implies that x z λ 1 α and y z β 1 1-α also belong to E for every α ∈ [0, 1). Axiom 2 then implies that x y z α+β = x z λ y z β ∈ E. This proves the multiplicative convexity. 3) Suppose x ≤ y and x z λ ∈ E. Then, y z λ ≥ x z λ , so that, by Axiom 1, y z λ ∈ E. The monotonicity of P follows.

4) If m ≥ P E (x), then, x z m ∈ E, hence, ∀λ > 0, x λ z λm = x z m λ ∈ E. Therefore, P E (x λ ) ≤ λm.

5) ∀x ∈ F, P (x) ≤ 0. Thus, F ⊂ E P F .

Properties of coherent multidimensional poverty measures

A representation theorem and ordinality

We now provide a full characterization of the whole family of coherent Poverty exit indices. For this purpose, let us define a weighted geometric average. Given any vector in the unit simplex, π ∈ ∆ KN

+ := {p ∈ R KN + | k,j p k,j = 1}, the π-geometric average, G π (•), is defined by: G π (x) := k,h x π k,h kh .
Proposition 4.1 The index P is coherent if, and only if, there exists a family, P ⊂ ∆ KN + , of weight vectors, such that

P (x) = -inf ln G π (x) ln G π (z) | π ∈ P .

Proof.

The "if" part is immediate. The "only if" part can be deduced from Proposition 2.1 in [START_REF] Huber | Robust Statistics[END_REF], and can be stated as a consequence of the bipolar theorem in linear duality theory. Consider the set

C := {x ∈ R KN | x hk = ln(y hk ) for some y ∈ E}.
It follows from Axiom 3 and 4 together withe the closedness of E that C is a convex and closed cone in R KN . Thus, its polar cone

C • := {α ∈ R KN + | hk α hk x hk ≥ 0 ∀x ∈ C}
is also a convex and closed cone in R KN + . The bipolar theorem implies that

C = {x ∈ R KN | hk α hk x hk ≥ 0 ∀π ∈ P},
where P := ∆ KN + ∩ C • . We deduce from (4) that ln x + P (x) ln z ∈ C, for every x ∈ R KN ++ . Thus, ∀π ∈ P, h,k π hk (ln x hk + P (x) ln z hk ) ≥ 0. Therefore,

P (x) h,k π h,k ln z hk ≥ - h,k
α hk ln x hk ∀π ∈ P.

Hence,

P (x) ≥ sup π - ln hk x π hk hk ln hk z π hk hk = -inf ln G π (x) ln G π (z) | π ∈ P .
Conversely, we deduce from Axiom 2 that ln x + P (x) ln z + ln ε / ∈ C for every x ∈ R KN ++ and every 0 < < ε < < 1. Therefore, ∀π ∈ P, h,k π hk (ln x hk + P (x) ln z hk + ln ε) < 0. It follows that

P (x ε) < -inf ln G π (x) ln G π (z) | π ∈ P .
The equality follows by continuity of P (•).

Examples a)

The "utilitarian case" corresponds to P = {((1/KN, ..., 1/KN )}.

b) The "Rawlsian" case corresponds to P = ∆ KN + . The next figure provides an illustration of the typical geometry of E. Observe that, in general, the frontier of the set E need not be smooth, as there is typically a kink at 1. The "utilitarian" case corresponds to the situation where the two branches of hyperbola coincide with the hypercurve : h,k x h,k = cst. It is the unique case where the boundary of E is a smooth submanifold. The larger the set P, the smaller the subset E. Finally, the Rawlsian case corresponds with the situation where E coincides with the affine nonnegative orthant

E = 1 + R KN + .
Notice that weights in P can differ across both individuals and dimensions. When P (or, equivalently, E) verifies Anonymity, the set of weights, P, reduces to weights over dimensions. The weighted geometric average now becomes:

G π(x) := k,h x πk N kh ∀π ∈ P ⊂ ∆ K + .
Corollary 4.1 The index P is coherent and anonymous if, and only if, there exists a family, P ⊂ ∆ K + , of weights over dimensions such that

P (x) = -inf ln G π (x) ln G π (z) | π ∈ P .
Thanks to Theorem 4.1, whether it is anonymous or not, a coherent index P can also easily be shown to be ordinal in the following sense.

Ordinality. A measure, Q, is said to be ordinal if the following holds. Given some diagonal matrix Λ = (λ jj ) j=1,...,KN with positive entries (λ jj > 0), given also a social status matrix y ∈ M N ×K (R ++ ), and a cut-off vector, z ∈ R KN ++ , one has:

Q(y; z) = Q(yΛ; zΛ).
An example will easily illustrate how this abstract property solves most of the problems related to ordinal data. Consider the question: "Which kind of toilet facility does your household have ?", together with three possible answers: a. "Open defacation field". b. "Shared flush". c. "Private flush". Of course, the metric between each one of these answers does not have any sensible meaning. To circumvent this issue, it suffices to capture this question through two dimensions, each of them accepting two answers, {a, b} and {a, c}, each captured by two variables {0, α} ⊂ R and {0, β} ⊂ R respectively. Ordinality then ensures that the choice of α and β does not matter.

Going back to coherent poverty measures, it is straightforward that, for any x ∈ R KN ++ and any Λ as above, xΛ/zΛ = x/z. Thus, as we only deal with normalized achievements, any Multidimensional Poverty Index is ordinal.

Who is poor ?

In this subsection, we confine ourselves to the subfamily of anonymous coherent Poverty indices. Consequently, P is associated with a set, P ⊂ ∆ k + , of K-dimensional vector of weights, π = (π k ) k -one for each dimension-, belonging to the unit simplex.

We now provide an answer to the question: "who is poor" ? Regarding this issue, two kinds of approach have been explored in the literature. 9 The "union" approach regards a person who is deprived in one dimension as being poor at the multidimensional level. This is usually acknowledged to be overly inclusive and lead to exaggerate estimates of poverty. By contrast, the "intersection" approach requires a person to be deprived in all dimensions before getting considered as poor. This is often considered too constricting, and may lead to untenably low estimates of poverty. We now show that the natural definition of a poor person that follows from our approach leads to an endogenous determination that is always strictly less inclusive than the "union" approach and weakly more inclusive than the "intersection" approach. Therefore it lies somewhere between these two extremes, and in fact, it turns out that only the Rawlsian case coincides with the "intersection" viewpoint.

Two examples will help identify how the determination of poor persons occurs in our setting. Consider the case where N = 1, i.e., the population consists of a single person. Then, clearly, this single person, i, will be poor whenever the population is so, i.e., when P (x i ) < 0. Next, suppose that the population is made of n identical people. Again, each person will be poor if the population is so, i.e., if, and only if,

k x π k i < k z π k i
for every π ∈ P.10 It is this latter condition that we adopt as a definition. Indeed, a simple continuity argument explains why now other choice can be made: Take 0 < < ε < < 1; one has x ε poor and G π (x) < 1 for any π. However, lim ε→1 G π (x) = 1 -. Thus, no population such that G π (x) < 1 can be considered as non-poor. Definition 4.1 Given a coherent Poverty index, P , associated with a set P ⊂ ∆ KN + of weights, a person, i, is poor whenever

k x π k i,k < k z π k k ∀π ∈ P
or, equivalently, when

sup π∈P G π (x i ) < 1.
In the "utilitarian" case (where P reduces to the uniform singleton), this definition coincides with the one introduced by [START_REF] Villar | A new approach to multidimensional poverty measurement[END_REF].

As an illustration, let us consider a society with two dimensions. The poor are all the individual strictly below the two branches of hyperbola:

q A B C D Dimension k Dimension j E 1 Fig 2.
Who is poor ?

The set E is always larger than the one defined by the intersection approach, and is always contained in the one provided by the union approach. The Rawlsian case, here, coincides with the intersection approach.

Other properties

Here are the properties verified by coherent Poverty indices. When they are evident, proofs are left to the reader.

1. Multiplicative decomposability : Suppose that x 1 (resp. x 2 ) is a population of size n 1 (resp. n 2 ). Let us denote by x 1 , x 2 the population of size n = n 1 +n 2 , obtained by merging the first two. One has:

G π ( x 1 , x 2 ) = G π (x 1 ) n 1 n G π (x 2 ) n 2 n ∀π ∈ P, so that P x 1 , x 2 = P (x n 1 n 1 x n 2 n
2 ).

The next property is a special case of multiplicative decomposability:

Replication invariance : For any population, x, P x, x = P (x).

3. Path independence: One can aggregate individual unidimensional values first across dimensions and then across agents, or viceversa, obtaining the same result. 14 4. The next property asks that a reduction of size δ > 0 in the deprivation with respect to dimension k of a poor person i who is worse off in this dimension than another poor person, j, more than compensates an increase of the same size, δ, in the deprivation of j, provided their relative positions remain unaltered. Formally, if x jk -x ik ≥ 2δ, y ik = x ik +δ and y jk = x jk -δ, while y h = x h ∀(h, ) / ∈ {(i, k), (j, k)}.

Transfer principle. P (y) ≤ P (x). Indeed,

x ik x jk < (x ik + δ)(x jk -δ) = y ik y jk .
It follows that x ik x jk α k ≤ y ik y jk α k , for every α k ≥ 0. The conclusion follows.

Also observe that the geometric mean is a distribution sensitive measure that penalizes the dispersion of the individual values, relative to the arithmetic mean. In particular, for two distributions with identical mean values it assigns higher value of the intensity of the poverty to that in which the distribution of the yij values is more disperse.

5. The reduction in the deprivation of dimension k required to compensate an increase in the deprivation of dimension is smaller the smaller the initial level of achievement in . This feature simply follows from the decreasing marginal rate of substitution of the individual poverty index across achievement dimensions). Obviously, this property cannot be satisfied by any (weighted) arithmetic measure.

6. The Poverty focus requirement says that only changes within the population, N (y; z), of poor affect P . This property is not fulfilled, in general, by coherent indices as these capture some kind of substitutability among poor and non-poor. However, as long as the cut-off, z, is exogenous, one easy way to recover Poverty focus consists in censoring achievements as follows, before normalizing them:

11 xik := x ik if x ik < z ik z ik if x ik ≥ z ik .
7. The same censoring provides us with the Deprivation focus property, namely: only changes in dimensions where poor people are deprived affect P . 8. Following [START_REF] Kolm | Multidimensional egalitarianism[END_REF] and [START_REF] Alkire | Counting and multidimensional poverty measurement[END_REF], we can check how much a coherent poverty index, P , is sensitive to inequality in the distribution of achievements and deprivations. There are several ways to do this. One way consists in considering mean-preserving spreads, i.e., transformations of a given population that increase the spreads of the achievements with respect to their arithmetic mean without affecting the mean itself. (Such transformations are the reversal of the change considered above for the Transfer principle.) An inequality-sensitive Index should be decreasing with respect to such transformations. Formally, an increase of size δ > 0 in the deprivation with respect to dimension k of i should not compensate an decrease of the same size, δ, in the deprivation of j. Formally, if x jk -x ik ≥ 0, y ik = x ik -δ and y jk = x jk + δ, while y h = x h ∀(h, ) / ∈ {(i, k), (j, k)}, then, by the same argument as for the Transfer principle, we get: Mean-preserving spread sensitivity P (x) < P (y). 9. Following [START_REF] Atkinson | The comparison of multidimensional distribution of economic status[END_REF] and [START_REF] Alkire | Counting and multidimensional poverty measurement[END_REF], we say that x is obtained from y by a simple rearrangement among the poor if the achievements of two poor persons, i and j, have been reallocated so that, for each dimension k: (x ik , x jk ) = (y jk , y ik ) or (x ik , x jk ) = (y ik , y jk ), while the achievements of anyone else remain untouched. If, in addition, y i and y j are comparable but x i and x j are not, then x is said to be obtained from y by an association decreasing rearrangement among the poor. Reducing inequality this way does trivially decrease any coherent multidimensional Poverty Index:

P (y) ≤ P (x).
This property is called Weak arrangement.

10. Another way to test the sensitivity towards inequality of an Index consists in averaging the achievement vectors, y i and y j of two poor persons, i and j in such a way that i now exhibits x i := (1 -λ)y i + λy j (with λ ∈ (0, 1)) and x j := λy i + (1 -λ)y j . The new population (x i , x j ) is viewed as being unambiguously less unequal than the original one, (y i , y j ), which should result in a lower or equal value of the multidimensional poverty index. Here, we translate linear convex combinations in geometric combinations, so as to arrive at the following definition. We say that x ∈ M n×k (R ++ ) is obtained from y ∈ M n×k (R ++ ) by a geometric averaging of achievements among the poor if, for every poor i, there exist weights (α j ) j∈N (y;z) ∈ ∆ n(y;z) + such that x i = j∈N (y;z) y α j j , while non poor persons are not affected (i.e., x i = y i for i / ∈ N (y; z)). Multiplicative weak transfer. If x is obtained from y by a geometric averaging of achievements among the poor, then one should have P (x) ≤ P (y).

However, this property is not satisfied by a coherent measure, in general. Consider, for example, a population , (a, b), consisting in 2 persons and a single dimension (with a < b < 1). The population (a 1/3 b 2/3 , b) is obtained from (a, b) by a geometric averaging of achievements among the poor, but: G(a 1/3 b 2/3 , b) > G(ab).

  Documents de Travail du Centre d'Economie de laSorbonne -2012.95

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.95 Coherent Multidimensional Poverty Measurement 9 Proposition 3.1 (i) If a Poverty index, P (•), is coherent, then its Poverty exit set, E P , defined by 3, verifies Axioms 1-4 and is closed. Moreover, P (•) = P E P (•).

  Fig 1. A piecewise smooth poverty exit set

Recall that a person is said to be deprived in one dimension whenever her achievement falls below the cut-off or dimension-specific poverty line. Documents de Travail du Centre d'Economie de laSorbonne -2012.95

We cannot claim that this entails no loss of generality at all. Indeed, if some achievement is "naturally" given as being (strictly) positive, then, whether it is scored ε > 0 or λε > 0 does not matter. However, if the original achievement was 0, then, replacing it by ε > 0 might have an effect on the poverty measure. The same problem arises, e.g., in[START_REF] Seth | A class of association sensitive multidimensional welfare indices[END_REF].

Documents de Travail du Centre d'Economie de laSorbonne -2012.95

F is the topological closure of F. Documents de Travail du Centre d'Economie de laSorbonne -2012.95

See, e.g.,[START_REF] Alkire | Counting and multidimensional poverty measurement[END_REF] and[START_REF] Villar | A new approach to multidimensional poverty measurement[END_REF].

Notice that, here, x is not normalized. Documents de Travail du Centre d'Economie de la Sorbonne -2012.95 Coherent Multidimensional Poverty Measurement

This is standard practice, see[START_REF] Alkire | Counting and multidimensional poverty measurement[END_REF]. Documents de Travail du Centre d'Economie de la Sorbonne -2012.95 Coherent Multidimensional Poverty Measurement


Valentina Peschetola

Valérie M Laurent

Alain Duperray

Richard Michel

Davide Ambrosi

Luigi Preziosi

Claude Verdier
email: claude.verdier@ujf-grenoble.fr

Time-dependent traction force microscopy for cancer cells as a measure of invasiveness

Keywords: cancer cells, traction forces, invasiveness, adhesion, migration, MSD

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

Introduction

Cell migration has been studied since many years and it is well known that the shape of cells during motion obeys to multiscale processes (in time and space) [START_REF] Keren | Mechanism of shape determination in motile cells[END_REF]. The basic mechanism is the cytoskeleton remodelling driven by the actin machinery [START_REF] Pollard | Cellular motility driven by assembly and disassembly of actin filaments[END_REF], balanced by the traction forces exerted on the substrate [START_REF] Balaban | Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates[END_REF][START_REF] Dembo | Stresses at the cell-to-substrate interface during locomotion of fibroblasts[END_REF][START_REF] Fournier | Force transmission in migrating cells[END_REF]. These seem to correlate with the development of focal complexes [START_REF] Balaban | Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates[END_REF][START_REF] Galbraith | The relationship between force and focal complex development[END_REF][START_REF] Giannone | Lamellipodial actin mechanically links myosin activity with adhesion-site formation[END_REF]. The cell velocity of migration is therefore connected with its ability to form/break bonds on substrates and contract its own body to move forward. But it has also been shown [START_REF] Gupton | Spatiotemporal feedback between actomyosin and focal-adhesion systems optimizes rapid cell migration[END_REF]] that rapid migration does not necessarily correspond to an organized system between actin, myosin II and focal adhesions (FAs).

Although the arrival of new microscopy techniques [START_REF] Friedl | Amoeboid leukocyte crawling through extracellular matrix: lessons from the dictyostelium paradigm of cell movement[END_REF][START_REF] Iordan | Breakdown of cell-collagen networks through collagen remodeling[END_REF] makes it possible to determine traction forces in three dimensions [START_REF] Maskarinec | Quantifying cellular traction forces in three dimensions[END_REF][START_REF] Legant | Measurement of mechanical tractions exerted by cells in three-dimensional matrices[END_REF][START_REF] Delanoë-Ayari | 4D traction force microscopy reveals asymmetric cortical forces in migrating dictyostelium cells[END_REF], most studies have focused on cells adhering on plane polymeric substrates. PA (PolyAcrylamide) substrates properly coated with Extra-Cellular Matrix (ECM) components such as fibronectin or collagen are an effective adhesion surface.

Traction force methods couple inversion methods and microscopy in order to reconstruct the local values of forces/displacements developed by cells. Most methods use fluorescent beads embedded in PA substrates, and correlation techniques to reconstruct the displacement field. All methods need a regularization (usually the Tikhonov penalization) and adequate filtering of the data. Four main techniques are currently available: Traction Reconstruction with Point Forces (TRPF) [START_REF] Schwarz | Calculation of forces at focal adhesions from elastic substrate data: the effect of localized force and the need for regularization[END_REF], Fourier Transform Traction Cytometry (FTTC) [START_REF] Butler | Traction fields, moments, strain energy that cells exert on their surroundings[END_REF], the classical Boundary Element Method (BEM) [START_REF] Dembo | Stresses at the cell-to-substrate interface during locomotion of fibroblasts[END_REF] and the more recent Adjoint Method (AM) [START_REF] Verdier | Cellular traction as an inverse problem[END_REF][START_REF] Ambrosi | Traction patterns of tumor cells[END_REF]. A recent work [START_REF] Sabass | High resolution traction force microscopy based on experimental and computational advances[END_REF] gives a comparison between the first three methods and explains how to improve them by adequate filtering and regularization. Similarly, improvements in the resolution are also available by comparing FTTC and TRPF methods [START_REF] Stricker | Optimization of traction force microscopy for micron-sized focal adhesions[END_REF], or FTTC and AM [START_REF] Michel | Inverse problems for the determination of traction forces by cells on a substrate: a comparison of two methods[END_REF]. Earlier methods use wrinkled patterns on soft PDMS [START_REF] Harris | Silicone rubber substrata: a new wrinkle in the study of cell locomotion[END_REF][START_REF] Burton | Traction forces of cytokinesis measured with optically modified elastic substrata[END_REF] or the deformation of microposts [START_REF] Tan | Cells lying on a bed of microneedles: an approach to isolate mechanical force[END_REF]. However, wrinkled patterns do not provide traction maps or are difficult to analyze, while microposts use a direct method but provide a non-continuous environment, so we will rather focus here on the previous 2D methods on soft substrates, and the AM method in particular.

Several authors studied the role of substrate rigidity [START_REF] Lo | Cell movement is guided by the rigidity of the substrate[END_REF][START_REF] Discher | Tissue cells feel and respond to the stiffness of their substrate[END_REF][START_REF] Ghibaudo | Substrate topography induces a crossover from 2D to 3D behavior in fibroblast migration[END_REF], as well as the effect of ECM protein density [START_REF] Palecek | Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness[END_REF]] on cell migration. Cells are known to migrate more efficiently on rigid substrates [START_REF] Lo | Cell movement is guided by the rigidity of the substrate[END_REF], although this phenomenon is cell-dependent [START_REF] Discher | Tissue cells feel and respond to the stiffness of their substrate[END_REF], and the velocity of migration seems to find an optimum for a specific matrix protein concentration [START_REF] Palecek | Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness[END_REF]. This behavior is related to stiffening of the integrin-cytoskeleton bonds [START_REF] Choquet | Extracellular matrix rigidity causes strengthening of integrin-cytoskeleton linkages[END_REF] with matrix rigidity.

Focal adhesions are indeed connected through integrin bounds, their size being proportional to the applied forces [START_REF] Balaban | Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates[END_REF]]. Nevertheless, it seems that this linear relation is only valid at short times [START_REF] Stricker | Spatiotemporal constraints on the forcedependent growth of focal adhesions[END_REF] when myosin-mediated contractions and focal maturation are at work, and that long term adhesion (with mature adhesions) does not exhibit this correlation.

Although these methods are quite promising and provide adequate tools for linking forces with cytoskeleton and focal complexes, more research is still needed to understand the precise mechanisms by which cells migrate in time. In particular, few studies involve cancer cells on 2Dsubstrates or lead to contradictory results [START_REF] Mierke | Contractile forces in tumor cell migration[END_REF][START_REF] Indra | An in vitro correlation of mechanical forces and metastatic capacity[END_REF][START_REF] Kraning-Rush | Cellular traction stresses increase with increasing metastatic potential[END_REF]. It is therefore interesting to determine in vitro methods for comparing quantitatively migration speed, applied forces, to the possibly correlated fluorescence images of the cytoskeleton and focal complexes in time. It is indeed possible that cancer cells develop particular forces and specific adhesion mechanisms to propel themselves through tissues or to extravasate. Such observations were initially proposed [START_REF] Munevar | Traction force microscopy of migrating normal and h-ras transformed 3T3 fibroblasts[END_REF]] by comparing fibroblasts with H-ras transformed fibroblast (known to exhibit mestastatic potential after injection of the H-ras oncogene [START_REF] Bondy | Experimental metastatic ability of H-ras-transformed NHI3T3 cells[END_REF]). The latter ones showed smaller traction forces, and a disorganized motile behavior. Recently other researchers [START_REF] Indra | An in vitro correlation of mechanical forces and metastatic capacity[END_REF] found inverse correlation between traction forces/adhesion strength and the metastatic potential of different cancer cells, but on the other hand another recent work reports opposite results [START_REF] Kraning-Rush | Cellular traction stresses increase with increasing metastatic potential[END_REF].

Here we provide a detailed analysis of the time-dependent behavior for two cancer cell lines (T24 and RT112, which are epithelial bladder cell lines) and measure the traction forces exerted on PA substrates coated with collagen. These features are correlated with data from complementary migration assays of migration velocities, while fluorescence microscopy enables us to quantify the cytoskeleton changes as well as focal adhesions. Finally traction force locations are investigated in relation with the actin dynamics. Section 2 presents the materials and methods used in this work.

In section 3, the main results are illustrated concerning traction forces, immunofluorescence and migration velocities. Finally, results are compared and discussed in section 4.

Materials and methods

Cell culture

Epithelial bladder cancer cell lines used in this study were obtained from ATTC (Rockville, USA) and were cultured in RPMI 1640 (Sigma Chemical Co.) supplemented with 10% fetal bovine serum and antibiotics. Cultures were kept at 37 • C in 5% CO 2 humidified atmosphere. In particular, two bladder carcinoma cell lines have been investigated: T24 and RT112. T24 cancer cells are poorly differentiated and have a cytological grade 3. The RT112 line is a moderately differentiated one and is characterized by cytological grade 2. In vitro, T24 cells display a high invasive capacity, whereas RT112 cells present a moderate invasive capacity [START_REF] Bindels | Influence of the microenvironment on invasiveness of human bladder carcinoma cell lines[END_REF][START_REF] Kawanishi | Secreted CXCL1 is a potential mediator and marker of the tumor invasion of bladder cancer[END_REF].

Cells are seeded on the Polyacrylamide gel (see below) and left overnight to spread. An image is acquired with a time interval of 60 sec for 2 hours.

Gel preparation

A protocol similar to the one reported in earlier works [START_REF] Dembo | Stresses at the cell-to-substrate interface during locomotion of fibroblasts[END_REF][START_REF] Pelham | High resolution detection of mechanical forces exerted by locomoting fibroblasts on the substrate[END_REF] has been used to produce 10 kPa PA gels. A square coverglass (22 mm × 22 mm) is treated with silane (Sigmacote SL-2) for 30 minutes. Round coverglasses (35 mm diameter) are washed with 0.1M NaOH, treated with APTMS for 10 minutes, washed with PBS twice and treated with 200 ñl of 0.5% glutaraldehyde for 30 minutes. A 0.5 ml gel solution is prepared, containing 30% acrylamide (166 ñl), 1% bis-acrylamide (14 ñl) in distilled water (314 ñl). Fluorescent beads (3 ñl, Molecular Probes, 0.2 ñm in diameter) are seeded into the solution before addition of the crosslinkers (2.5 ñl of APS and 0.5 ñl of TEMED). Overall, the gel contains 10% acrylamide and 0.03% bis-acrylamide. The adequate amount of solution (corresponding to a 70 ñm gel thickness) is set onto the square coverglass. The circular coverglass is brought carefully from the top to capture the gel solution by capillarity, thus avoiding to flip the preparation. The gel is left to polymerize for nearly 90 minutes. After polymerization, the square coverglass is gently removed. In this protocol, Sulfo-Sanpah is used to covalently bind the protein (collagen) to the polyacrylamide surface. The gel is covered with 200 ñl of a Sulfo-Sanpah solution (Sulfo-Sanpah 1mM, DMSO and HEPES at pH=8.5), then exposed to a UV lamp (2 × 15 Watts) for 2 mn: this enables to activate Sulfo-Sanpah molecules, so that they bind collagen. This process is repeated twice, then 200 ñl of the collagen solution (100 ñg/ml) are applied and let overnight to bind the activated Sulfo-Sanpah at 4 • C. To check the efficacy of the procedure, GFP-fibronectin was coated on the gels and it was shown that a uniform density was obtained (data not shown). Furthermore, T24 cell monolayers were cultured on these gels and were shown to cover the whole surface.

The mechanical gel properties have been determined previously [START_REF] Ambrosi | Traction patterns of tumor cells[END_REF] and the specific gel used throughout this study (10% acrylamide-0.03% bis-acrylamide) has an elastic

Young modulus E = 10 kPa as determined by rheometry, assuming that ν = 0.5 (Poisson coefficient). This was also checked using AFM measurements [START_REF] Peschetola | Traction forces exerted by cancer cells migrating on gels[END_REF]. The gel roughness was shown to be around 100 nm as shown using AFM contact mode. Note that the value of the gel rigidity (10 kPa) was selected as a compromise between stiffness of the physiological environment and the minimization of three-dimensional deformations that could occur for too soft gels. The value of 10 kPa appeared to meet these two conditions, and displacements were also easily detected (usually a few ñm).

Traction forces

The mechanical characterization of the gel is approximated as isotropic and elastic and the displacements are observed to be small enough so that linear elasticity applies. The gel depth, usually 70 ñm, is large enough (for the applied loads) so that it can be approximated by a half space. Since the response of the substrate is linear, the displacement u(r) at position r can be determined from the traction field t(r) at the boundary (x 3 = 0) of the 3D-gel via the integral:

u(r) = G(r -r ′ ) t(r ′ ) ds(r ′ ) (1)
where G is the Green's tensor for the Boussinesq-Cerruti problem [START_REF] Landau | Théorie de l'élasticité[END_REF],

and the integral is operated on the surface between gel and cell. In the particular case where (1) is used for r = (x 1 , x 2 , 0) and pure shear t = (t 1 , t 2 , 0) applies, G(r) takes the form:

G(r) = 1 + ν πE (1 -ν) 1 r + ν r ⊗ r r 3 (2)
where r is the magnitude of r, E is the elastic modulus, ν is the Poisson ratio, and 1 is the identity tensor.

In this work we use the adjoint method, whose details are given in previous works [START_REF] Verdier | Cellular traction as an inverse problem[END_REF][START_REF] Ambrosi | Traction patterns of tumor cells[END_REF]. We recall that first, the problem is reduced to a 2D-one by averaging along an effective thickness h (in the x 3 direction). Secondly, the tractions are determined by a minimization technique, following Lions [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF]. The solution method rewrites in solving the following partial differential equations and boundary conditions with unknowns u and the auxiliary unknown p:

-µ∆u -(µ + λ) ∇(∇.u) = - χ c ε (p -p) in Ω c , u = 0 on ∂Ω c (3) -µ∆p -(µ + λ) ∇(∇.p) = χ 0 (u -u 0 ) in Ω c , p = 0 on ∂Ω c (4)
In the above equations, p =1 |Ωc| Ωc pds, χ c and χ 0 are characteristic functions associated to Ω c (cell region) and Ω 0 (where beads displacements are known), u 0 is the known displacement, µ = hE 2(1+ν) and λ = hEν 1-ν 2 are the reduced 2D-Lamé coefficients (plane stress model). h is chosen as the height within which beads significantly move (typically h = 1.5 ñm). ε is the regularization parameter to be optimized using the L-curve technique [START_REF] Verdier | Cellular traction as an inverse problem[END_REF][START_REF] Peschetola | Traction forces exerted by cancer cells migrating on gels[END_REF]. We used ε = 6 × 10 -7 in our case.

Finally, the traction field t is given by t = -χc ε (p-p) [START_REF] Peschetola | Traction forces exerted by cancer cells migrating on gels[END_REF], which ensures that the traction forces are null outside the cell, and that the resulting force on the cell is zero: Ωc tds = 0.

The observed displacement u 0 is extracted from fluorescent images of the beads within the gel surface. The relaxed (i.e. Initial) beads positions are actually recovered at the end of the experiment by adding distilled water in the culture medium, which makes cells to detach and the gel eventually relaxes. Then the displacement u 0 can be calculated, thanks to the trajectory reconstruction routine 'Particle-Tracker' available in the ImageJ package [Rasband, 1997].

The system of equations (3-4) is solved using a finite element method with elements nodes matching the beads positions. Then linear basis functions are used on an unstructured mesh. A biconjugate gradient method has been used to solve the resulting linear system numerically.

The trajectories are determined by following of the center of mass, while the resultant force is computed by integration of the stress times the surface element area (from the finite element mesh) to obtain the resultant force. Time-dependent results have been obtained by following cell trajectories, and traction stresses are presented using boxplots. Then outliers 1 representing the highest forces are extracted, and the mean value is taken from this data.

Cell migration on gels and Mean Squared Displacement (MSD)

Special Petri dishes with coverglass were used onto which PA gels (10 kPa) were prepared for the long-term migration experiments. Cells were seeded on the gels. Several cells were selected and followed in time for 2 hours with 5 mn time intervals. As previously, the center of mass was determined and recorded in time. Then the Mean Squared Displacement (MSD) was calculated according to the definition :

MSD(t) = < |r(t + τ ) -r(τ )| 2 > τ (5)
In this formula, r(τ ) is an initial vector position chosen along the center of mass' path, and r(t + τ ) is another vector, from which the squared norm is calculated. Then the averaging brackets <> τ are taken for all possible time lags τ (0 ≤ τ ≤ t), i.e. the different positions of the cell center of mass (see supplemental material S7 for more precise information).

The behavior of this MSD in time was analyzed by comparing it to t α , where α is the timeexponent or persistence parameter. It is known that values of α < 1 correspond to sub-diffusive motions (anomalous motion), α = 1 means diffusive motion (random walk), whereas α > 1 is a super-diffusive motion (due to possible active cellular transport). The case α = 2 is called the ballistic case (directional motion, cells encounter no obstacle and distance is proportional to time). Such exponents were measured during the migration assay and the final value of α was chosen as the time-average value.

The behavior of the MSD is an important feature of the cell migration process, as shown recently [START_REF] Dieterich | Anomalous dynamics of cell migration[END_REF] because it can allow to determine the signature of a particular cell to migrate, change direction or polarize over long periods of time.

Immunofluorescence and confocal microscopy

Briefly, after being washed with PBS, cells were fixed using PFA (paraformaldehyde 3% in PBS) for 10 min, permeabilized with PBS containing 0.5% Triton X100 for 10 min, and then washed with PBS containing 0.2% saponin and 2% BSA. Fixed cells were then incubated with rabbit antimyosin IIA (Biomedical Technologies Inc., USA) and mouse anti-human paxillin (clone 5H11, from Upstate Biotechnology, USA) for 30 min, and washed with PBS containing 0.2% saponin and 2% BSA. Cells were then stained with phalloidin Alexa Fluor-488 for actin, Hoechst 33342 for nuclei, Donkey anti-rabbit IgG (conjugated with Dylight 649, Jackson ImmunoResearch, USA), and Goat anti-Mouse IgG (labelled with Alexa Fluor-546, Molecular Probes, The Netherlands). The samples were imaged using a Zeiss LSM710 confocal microscope at the Institut Albert Bonniot. An X63 objective (oil, Plan-Apochromat DICIII, WD190m) was used for higher quality images. Images were 1024x1024, leading to a pixel resolution of 0.063 ñm and subsequent area resolution 0.004 ñm 2 . This insures good accuracy for measurements of the focal areas.

Live cell imaging was carried out on a confocal microscope (Zeiss LSM), also at the Institut Albert Bonniot. T24 and RT112 cells were transfected for actin-GFP and time acquisitions were taken every two minutes, simultaneously with the fluorescent beads (red) acquisition, so that actin localization and traction forces could be correlated. Co-localization analysis was produced using the ImageJ software [Rasband, 1997].

Results

Time dependence of traction forces

An image showing a T24 cell on a PA substrate is shown in Fig. 1. The cell develops a large lamellipodium and migrates from left to right (see also supplemental movie S1). A traction force map at this characteristic time (t = 6 mn) is also shown in Fig. 1. The force pattern (per unit surface) developed by this type of cancer cell reaches maximum values that are in the range of 150 Pa.

Cell migration is a time-dependent phenomenon [START_REF] Ambrosi | Traction patterns of tumor cells[END_REF][START_REF] Stricker | Spatiotemporal constraints on the forcedependent growth of focal adhesions[END_REF], as cells pull on their focal adhesion points and then relax. This dynamics is shown in Fig. 2 where a time-dependent locomotion is depicted over 40 minutes. Values of stress magnitudes are reported in the form of a boxplot for each time (the same information is in the supplemental movie S2).

As seen previously for this type of cell [START_REF] Ambrosi | Traction patterns of tumor cells[END_REF], T24 cells usually exhibit a periodic motion, pulling strongly and then relaxing stresses, repeatedly. This type of motion is the so called five-step process (mesenchymal motion): development of protrusions, formation of adhesion and pulling, cell contraction, release of bonds at the rear and recycling [START_REF] Sheetz | Cell migration: regulation of force on extracellular-matrix-integrin complexes[END_REF]]. Note that a large number of outliers are present, corresponding to the higher forces developed at focal sites. This will be studied below.

Analogous results are for less invasive RT112 cancer cells, shown in Fig. 3. The cell has a round shape, more likely resembling an ameboid type of motion (see also supplemental movie S3).

Tractions are more homogeneous, and the values are larger than for T24 cells (Fig. 4). The outliers over 78 minutes do not change much (see also supplemental movie S4).

To compare the motion of the two types of cancer cells, only higher values (i.e. outliers probably corresponding to focal complexes [START_REF] Balaban | Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates[END_REF]) are retained in Fig. 5. RT112 cells developing higher average values of outliers (around 170 Pa) are compared to T24 cells (120 Pa). This analysis was carried out for N = 9 cells of each type and by averaging results for all times. This result is in agreement with previous works [START_REF] Munevar | Traction force microscopy of migrating normal and h-ras transformed 3T3 fibroblasts[END_REF][START_REF] Indra | An in vitro correlation of mechanical forces and metastatic capacity[END_REF] and could describe a possible way for invasive cells to move rapidly without exerting large forces.

Global force exerted on half-cell

The total force developed by cancer cells on the PA substrate in the direction of migration is calculated integrating on half a cell only (indeed the resultant force over the cell domain is zero, i.e. Ωc t ds = 0) and is shown in Fig. 6. One can estimate the difference between T24 and RT112 cells. Forces are concentrated at the leading edge or within the lamellipodium for T24 cancer cells, whereas centripetal stresses are located all around the edges in the case of RT112 cells, a migration behavior similar to some keratocytes [START_REF] Burton | Keratocytes generate traction forces in two phases[END_REF].

Immunofluorescence: focal adhesions, actin and myosin

Immunofluorescence experiments are reported in figures 7-8. Several proteins involved during cell migration are selectively investigated: the actin structure (in green, using staining with Phalloidin Alexa 488), focal adhesion complexes (in red, using an anti-paxillin antibody) as well as myosin motors (in magenta, using an antibody for myosin IIA); finally the composite picture showing all sub-structures is presented, with the nuclei stained in blue (using Hoechst for DNA).

The two cell types exhibit different structural patterns. T24 cells show a thin cytoskeleton located close to the edges with interconnected stress fibers throughout the whole cell, whereas RT112 cells exhibit a wider actin network close to the edges. This phenomenon was also observed for other cells of the same lines (data not shown) with different shapes, and seems to be typedependent rather than shape-dependent.

T24 cells show larger focal complexes, as evidenced by the paxillin staining, thus allowing for a more efficient motion. This is confirmed by precise quantitative determination of the total area of focal complexes, which was made relevant by normalizing according to cell area. The contours of focal areas were determined for several cells (N=9 for each cell line, approximately 60-70 areas per cell), and only large focal areas (>0.5 ñm 2 , 30-40 per cell) were kept. It was shown that T24 cells present significantly (p<0.001) larger focal areas (1.13 ± 0.05 ñm 2 ) as compared to RT112 cells (0.86 ± 0.03 ñm 2 ), see also Table 1. Good accuracy is obtained thanks to the high resolution of the pixel area size (0.004 ñm 2 , see 'Materials and Methods'). As T24 cells exhibit a larger spreading area (2200 ñm 2 , as compared to 1600 ñm 2 for RT112 cells) on PA gels, it is therefore better to look at the average of the ratio F As/A(%) (ratio of total focal areas FAs over cell area A) which was plotted as a bar-plot in Fig. 9, and this average is slightly higher for T24 cells. Apparently RT112 cells develop a smaller amount of focal complexes on soft gels, and therefore cannot move efficiently: they are more likely to move slowly than undergo rapid migration. Note that similar focal areas, showing periodic spatial fingers at the edges, have been previously observed when applying mechanical stresses [START_REF] Paul | Propagation of mechanical stress through the actin cytoskeleton toward focal adhesions: model and experiment[END_REF] or under flow [START_REF] Couzon | A critical stress to detach cancer cells in microchannels[END_REF].

Actin and myosin are co-localized at the periphery of both cell lines (see Fig. 7 and Fig. 8).

Actin fibers organize in a cortical structure crossing the focal adhesions, and similarly for Myosin IIA, especially for RT112 cells (Fig. 8). This seems to correlate with the development of large forces. In the case of T24 invasive cells, the organization of the actin cortex shows interconnections over the whole cell body, as for Myosin IIA. There are larger adhesion sites than with RT112 cells. Regarding colocalization of actin and myosin, it is clear from Figure 8 that a larger cortex of acto-myosin is shown in the case of RT112 cells (thickness around 5 ñm), whereas T24 cells (Figure 7) only show thin acto-myosin structures (1-2 ñm) at the cell rims. This is to be discussed in the next section.

Cell migration and MSD

The migration process of the two cell types (T24 and RT112) can be described in terms of the MSD function, as well as the velocity of migration. The MSD was determined as explained in section 2.4, while the velocity of migration was chosen as the length of the trajectory over time, in order to detect cells going back and forth over long times. A clear difference between the two cell types appears, in terms of kinematics, i.e. the persistence parameter α equals 1.57 for T24 cells, whereas it is only 1.21 for the less invasive RT112 cells (N=9). The velocity of migration follows the same trend and shows that invasive T24 cancer cells (V = 0.38 ñm/mn) are faster than the slow RT112 cells (V = 0.17 ñm/mn), as could be expected.

Discussion

The main results reported above are summarized in Table 1: values of the cell cytological grade, the average magnitude of traction stresses outliers σ (in Pa), the average magnitude of the resultant force F (nN) developed on a half-cell, the ratio of focal adhesions to cell area F As/A(%), the focal adhesions average area F As (ñm 2 ), the migration velocity V (ñm/mn), and finally α, the exponent from the MSD relationship in Eq. ( 5). The value of the significance parameter p is also indicated.

Few studies have focused on the comparison between different cell lines behaviors on soft gels.

Previous authors [START_REF] Munevar | Traction force microscopy of migrating normal and h-ras transformed 3T3 fibroblasts[END_REF] considered normal and H-ras transformed 3T3 fibroblasts.

H-ras transformed cells were used previously [START_REF] Bondy | Experimental metastatic ability of H-ras-transformed NHI3T3 cells[END_REF]. The H-ras gene was shown to alter the cells by developping metastatic ability. Indeed, cells (NIH3T3) transfected with the T24 human H-ras oncogene (pT24-C3) were able to grow in vivo, within an immunodeficient embryonic chick tissue, whereas other control NIH3T3 cells did not proliferate. Munevar et al.

showed an unexpected behavior of H-ras fibroblasts which did not develop stable forces but rather disorganized values around the cell edges during motion. The same behavior is observed in this work with an increase of traction stresses with the less invasive cells (Fig. 56).

Another important aspect investigated here is the time-dependence of the traction forces for the two cell types observed in Fig. 2 and Fig. 4. Outliers represented in these figures contain the information on the large forces developed at focal sites. The dynamics of outlier forces correspond to a classical five-step process [START_REF] Sheetz | Cell migration: regulation of force on extracellular-matrix-integrin complexes[END_REF] in the case of T24 cancer cells, as already noticed [START_REF] Ambrosi | Traction patterns of tumor cells[END_REF]: forces peaks (as observed in Fig. 2) range from 80 Pa to 150 Pa roughly, with an average value around 120 Pa (Table 1). On the other hand, RT112 cells present outliers (in the time range considered) which remain rather constant (Fig. 4), equal to 171 Pa (Table 1), with very small fluctuations. This suggests that the motion described is continuous, where adhesions are rapidly formed and broken. This is also confirmed by inspection of Fig. 6, where local stresses on both sides of RT112 cells cancel out to give a resultant force in the direction of migration, as observed previously for keratocytes [START_REF] Burton | Keratocytes generate traction forces in two phases[END_REF]. Therefore, a distinct mechanism of migration was observed for these two cell types and quantified. To investigate this characterization further, we analyze the location, number and size of focal adhesion areas (Fig. 9); they are known to be linked to the cytoskeleton, in particular the actin filaments. We measure the total area of paxillin-rich domains, in particular those with size larger than 0.5 ñm 2 (mature adhesion sites). The percentage of focal areas with respect to cell size is shown and, interestingly, this fraction (around 1% for T24 cells, and 0.9% for RT112 cells, see Fig. 9) increases with cell invasivity. Moreover, significantly larger adhesion sites (1.13 ñm 2 ) are found for T24 cells as compared to RT112 cells (0.86 ñm 2 , p<0.001, see Table 1). In another work [START_REF] Indra | An in vitro correlation of mechanical forces and metastatic capacity[END_REF], adhesion centrifugation assays conducted on harder PA gels showed a smaller number of adhering cells when metastatic capacity increased, but focal adhesions were only shown on glass substrates, so results are therefore difficult to compare in size and number of focal adhesion sites. Nevertheless, our data confirm that traction forces are not proportional to focal adhesion areas, a framework previously described earlier [START_REF] Stricker | Spatiotemporal constraints on the forcedependent growth of focal adhesions[END_REF] in the early stages of myosin-mediated maturation of adhesions. This means that these data concern probably cells showing focal adhesions at a different level of maturation, probably more mature ones.

The role of the cytoskeleton is clarified by studying fluorescent actin-images, and their correlation with the myosin and paxillin sub-structures in Figs 78. The interplay between actin and myosin enables cell contraction and allows motility (see for example [START_REF] Gardel | Transient frictional slip between integrin and the ECM in focal adhesions under myosin II tension[END_REF]). Another correlation is between traction forces and the actin flow [START_REF] Gardel | Traction stress in focal adhesions correlates biphasically with actin retrograde flow speed[END_REF] in regions where large focal areas exist and where the actin flow is small (and conversely). The myosin activity is usually linked to focal adhesions formation in the actin-rich lamellipodium [START_REF] Giannone | Lamellipodial actin mechanically links myosin activity with adhesion-site formation[END_REF]. In the images showing myosin IIA and the actin network (Figs 78), it can be seen that the myosin IIA density is particularly enhanced at the leading edge of the cancer cells, and that it corresponds to the rich actin domains. On the other hand, the actin and myosin regions are thin in the case of T24 cells (usually 1-2 µm wide), whereas they are large for RT112 cells (around 5 µm) thus allowing stronger acto-myosin contractions. This feature was also observed in other cases (figures not shown) and is in agreement with the actin-localization in Figures 1011.

Focal areas located at the leading edge of T24 or RT112 cells differ: they are well connected to the actin and myosin networks for T24 cells, whereas they do not appear to be so well colocalized for RT112 cells. This difference could be explained as follows: focal complexes are not so important for less invasive RT112 cells, which mostly spread rather than migrate. On the contrary, T24 cells need to develop their mature adhesions rapidly and renew them faster, which makes them migrate more rapidly. This has been taken care of by using a precise criterion (focal area > 0.5 ñm 2 ) which insures quantification of mature focal adhesions, as opposed to unmature ones.

In terms of cell migration, the MSD analysis confirmed that T24 cells move more rapidly than less invasive RT112 cells and that their persistence exponent α is larger (1.57 for T24 cells vs. 1.21 for RT112 cells). The exponent of T24 cell is comparable to previous ones [START_REF] Dieterich | Anomalous dynamics of cell migration[END_REF] who measured super-diffusive behaviors as well (α ∼ 1.5). Therefore this gel stiffness is a good candidate to differentiate these cells.

Finally, traction force microscopy in time is compared with the evolution of the actin cytoskeleton [START_REF] Versaevel | Spatial coordination between cell and nuclear shape within micropatterned endothelial cells[END_REF]. This was achieved thanks to actin-GFP transfected T24 and RT112 cells, and observations of the migration behavior was followed in time as before. Figures 1011show images of both the actin filaments and the magnitude of the forces developed by the two cancer cells (see also supplemental movies S5 and S6 for a more detailed time-dependent process).

A good correlation appears between the actin density and the high levels of forces. In particular, Figure 11 shows a particularly large actin cortex developed by RT112 cells at the front, as seen before (Fig. 8): the larger forces are also located in that area. On the contrary, T24 cells as in Fig. 10 show a thinner cortex (same as previously, Fig. 7) with orientation following cell edges. This can be shown in the comparison of the intensity levels shown along the two lines in Figures 1011where a more defined actin intensity is seen at the cell edges for the T24 cell, whereas the actin cortex is wider for the RT112 cell. This result supports the idea that the two types of motion correspond to a clear different contractility: pulling at the front on adhesion sites for T24 cells, whereas RT112 cells pull on the sides thanks to larger acto-myosin complexes and use less adhesion. This effect is detailed in Figure 12 where colocalization of the actin density is compared with the magnitude of traction stresses. As can be shown in these two figures, colocalization of actin (in green) and force (in red) is strong at the leading edge of T24 cell, whereas it is located in wider areas in the case of the RT112 cell. As suggested before, this feature seems to indicate that RT112 cells move by using large lateral contractions due to the presence of a wide acto-myosin cortex, whereas T24 cells only use protrusions at the leading edge where the acto-myosin cortex is more efficient. This is a possible way to explain, at such cell levels, differences in the invasiveness of these two cell lines. Finally the ability to move faster might be controlled by the type of adhesive molecules involved, and their affinity to the extra-cellular-matrix; this work is presently under way.

Finally, despite the fact that the results obtained in this work can help to discrimate between more or less invasive cancer cells, the 3D aspects have been ignored here. Therefore one needs to be careful about generalizing these results in three dimensions. Indeed, some differences may appear. One should consider cells on 2D-substrates as well as cells embedded in a 3D environment, as previously described [START_REF] Kraning-Rush | Cellular traction stresses increase with increasing metastatic potential[END_REF][START_REF] Koch | 3D Traction forces in cancer cell invasion[END_REF]. Strictly speaking, all the latter methods are capable of giving information about cellular tractions but only qualitatively, except another recent work [START_REF] Legant | Measurement of mechanical tractions exerted by cells in three-dimensional matrices[END_REF], where a single case of cell migration is computed. Some interesting tendencies have also been observed by studying the velocity of migration of cells embedded in matrigels [START_REF] Zaman | Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell-matrix adhesion and proteolysis[END_REF] to show the effects of ECM density, adhesions and matrix rigidity. It was shown that the pore size of the matrigel plays a key role in cell migration and leads to an important proteolytic activity. Definitely, there remains a big challenge in transfering this work into the third dimension and compare the 3D-invasiveness or metastatic ability of such cells to invade another medium, while computing associated forces.

Conclusions

The motility of two types of epithelial bladder cancer cells has been investigated from a biophysical point of view. These cells present different invasive behavior, in particular T24 cells are more invasive than RT112 cells. Substrates of PolyAcrylamide (stiffness ∼ 10 kPa) have been chosen

to perform all studies. The results concern the relationship between focal areas, cytoskeleton structure, migration velocities and forces exerted by cells thanks to Traction Force Microscopy (TFM). The main conclusions are that invasive cells (T24) need less traction forces to migrate, and exhibit an organized cytoskeleton (actin and myosin structures) with large dynamic adhesion sites, resulting in larger migration velocities on this particular substrate. On the other hand, less invasive RT112 cells exert larger forces thanks to a wide acto-myosin cortex, with smaller focal adhesions. The applications of this method could be an interesting in vitro tool to test the invasivity of different cancer cells; in particular, the role of traction stress may provide a possible classification criterion.

Future studies could focus on the optimization of protocols (choice of substrate, functionnal-ization, topography) with high numbers of cells. Finally, promising current work is ongoing to relate the possible proteins involved in cancer migration of such cells [START_REF] Chotard-Ghodsnia | Morphological analysis of tumor cell/endothelial cell interactions under shear flow[END_REF][START_REF] Haddad | Tumor cell/endothelial cell tight contact upregulates endothelial adhesion molecule expression mediated by NF-κB: differential role of the shear stress[END_REF], like β1 integrins [START_REF] Indra | An in vitro correlation of mechanical forces and metastatic capacity[END_REF] or α5 -β1 [START_REF] Mierke | Integrin alpha5-beta1 facilitates cancer cell invasion through enhanced contractile forces[END_REF] and more precisely chemokine CXCL1 in the case of the two lines studied here [START_REF] Kawanishi | Secreted CXCL1 is a potential mediator and marker of the tumor invasion of bladder cancer[END_REF]. This would enhance the current results on the traction forces measured here by TFM. Possibly this could also enable models [START_REF] Mogilner | The shape of motile cells[END_REF][START_REF] Verdier | Review: Rheological properties of biological materials[END_REF][START_REF] Herant | Form and function in cell motility: from fibroblasts to keratocytes[END_REF][START_REF] Shao | Computational model for cell morphodynamics[END_REF][START_REF] Duperray | Initial dynamics of cell spreading are governed by dissipation in the actin cortex[END_REF] to predict cancer cell shapes during migration.

And last, but not the least, it is now necessary to compute traction forces in three dimensions [START_REF] Legant | Measurement of mechanical tractions exerted by cells in three-dimensional matrices[END_REF], in order to focus on the mechanisms and forces developped by cancer cells to migrate into tissues.

Table captions

Table 1. Summary of dynamic and kinematic quantities obtained for T24 and RT112 cells. Values of the cell cytological grade, the average magnitude of traction stresses outliers σ (in Pa), the average magnitude of the resultant force F (nN) developed on a half-cell, the ratio of focal adhesions to cell area F As/A(%), the focal adhesions average area F As (ñm 2 ), the migration velocity V (ñm/mn), and finally α, the exponent from the MSD relationship in Eq. ( 5). The value of the significance parameter p is also indicated. (*) p< 0.05, (**) p< 0.01 and (***) for p< 0.001. Values of the cell cytological grade, the average magnitude of traction stresses outliers σ (in Pa), the average magnitude of the resultant force F (nN) developed on a half-cell, the ratio of focal adhesions to cell area F As/A(%), the focal adhesions average area F As (ñm 2 ), the migration velocity V (ñm/mn), and finally α, the exponent from the MSD relationship in Eq. ( 5). The value of the significance parameter p is also indicated. (*) p< 0.05, (**) p< 0.01 and (***) for p< 0.001. ñm 2 ) divided by cell area (p = 0.25). See also Table 1 for details. Larger actin areas are found with the RT112 cell.

Figure captions

Figure 1 .

 1 Figure 1. Phase-contrast image of a T24 cancer cell and corresponding traction field. Time t = 6 min from Fig. 2. The traction field is represented as a color map. The color scale for stresses reads in Pascal (Pa).

Figure 2 .

 2 Figure 2. Time variation of the magnitude of traction stresses exerted by a T24 migrating cancer cell. The representation is made as a Boxplot. The lines in the Boxplot correspond to 25%, 50%, 75%, and the whiskers extend to the 1.5 interquartile range. Outliers are above. The mean value for each time step is indicated by the red dots.

Figure 3 .

 3 Figure 3. Phase-contrast image of a RT112 cancer cell and corresponding traction field. Time t = 18 min from Fig. 4. The traction field is represented as a color map. The color scale for stresses reads in Pascal (Pa).

Figure 4 .

 4 Figure 4. Time variation of the magnitude of traction stresses exerted by a RT112 migrating cancer cell. The representation is made as a Boxplot. The lines in the Boxplot correspond to 25%, 50%, 75%, and the whiskers extend to the 1.5 interquartile range. Outliers are above. The mean value for each time step is indicated by the red dots.

Figure 5 .

 5 Figure 5. Mean value of the outliers of traction stresses for T24 and RT112 cell lines. The data is shown as the mean + standard error about the mean (SEM). The difference among the mean value is significant according to the GEE test (p = 0.0069).

Figure 6 .

 6 Figure 6. Stresses in a half-cell and resultant. T24 cell (above), RT112 cell (middle) and bar-plot (below) showing time-average force in the direction of migration (p = 0.14). The arrow shows the resultant (direction and magnitude).

Figure 7 .

 7 Figure 7. Fluorescent images of T24 cancer cells adhering on PA gel. Gel stiffness is 10kPa. From left to right and top to bottom: cells are stained for actin with Phalloidin Alexa 488 (green), anti-paxillin for focal adhesion sites (red), anti-myosin antibody for myosin IIA (magenta) and Hoechst for nuclei (blue).

Figure 8 .

 8 Figure 8. Fluorescent images of RT112 cancer cells adhering on PA gel. Gel stiffness is 10kPa). From left to right and top to bottom: cells are stained for actin with Phalloidin Alexa 488 (green), anti-paxillin for focal adhesion sites (red), anti-myosin antibody for myosin IIA (magenta) and Hoechst for nuclei (blue).

Figure 9 .

 9 Figure 9. Focal adhesions. A: Microscopy image of focal adhesions. B: Details of segmentation procedure for focal adhesions calculation. Bar-plot representing total area of focal complexes (>0.5 ñm 2 ) divided by cell area (p = 0.25). See alsoTable 1 for details.

Figure 10 .

 10 Figure 10. Image of T24 cancer cell transfected for actin. PA gel substrate of stiffness 10kPa. Fluorescent image with insert indicating the level of actin expression for the line drawn. Corresponding traction force map (in Pa).

Figure 11 .

 11 Figure 11. Image of RT112 cancer cell transfected for actin. PA gel substrate of stiffness 10kPa. Fluorescent image with insert indicating the level of actin expression for the line drawn. Corresponding traction force map (in Pa).

Figure 12 .

 12 Figure 12. Colocalization of the actin and force levels. T24 cell (top) and RT112 cell (below). Larger actin areas are found with the RT112 cell.

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Time variation of the magnitude of traction stresses exerted by a T24 migrating cancer cell. The representation is made as a Boxplot. The lines in the Boxplot correspond to 25%, 50%, 75%, and the whiskers extend to the 1.5 interquartile range. Outliers are above. The mean value for each time step is indicated by the red dots.

Figure 3 :

 3 Figure 3: Phase-contrast image of a RT112 cancer cell and corresponding traction field. Time t = 18 min from Fig. 4. The traction field is represented as a color map. The color scale for stresses reads in Pascal (Pa).

Figure 4 :

 4 Figure 4: Time variation of the magnitude of traction stresses exerted by a RT112 migrating cancer cell. The representation is made as a Boxplot. The lines in the Boxplot correspond to 25%, 50%, 75%, and the whiskers extend to the 1.5 interquartile range. Outliers are above. The mean value for each time step is indicated by the red dots.

Figure 5 :

 5 Figure 5: Mean value of the outliers of traction stresses for T24 and RT112 cell lines.The data is shown as the mean + standard error about the mean (SEM). The difference among the mean value is significant according to the GEE test (p = 0.0069).

Figure 6 :

 6 Figure 6: Stresses in a half-cell and resultant. T24 cell (above), RT112 cell (middle) and bar-plot (below) showing time-average force in the direction of migration (p = 0.14). The arrow shows the resultant (direction and magnitude).

Figure 7 :

 7 Figure 7: Fluorescent images of T24 cancer cells adhering on PA gel. Gel stiffness is 10kPa. From left to right and top to bottom: cells are stained for actin with Phalloidin Alexa 488 (green), anti-paxillin for focal adhesion sites (red), anti-myosin antibody for myosin IIA (magenta) and Hoechst for nuclei (blue).

Figure 8 :

 8 Figure 8: Fluorescent images of RT112 cancer cells adhering on PA gel. Gel stiffness is 10kPa). From left to right and top to bottom: cells are stained for actin with Phalloidin Alexa 488 (green), anti-paxillin for focal adhesion sites (red), anti-myosin antibody for myosin IIA (magenta) and Hoechst for nuclei (blue).

Figure 9 :

 9 Figure 9: Focal adhesions. A: Microscopy image of focal adhesions. B: Details of segmentation procedure for focal adhesions calculation. Bar-plot representing total area of focal complexes (>0.5

Figure 10 :

 10 Figure 10: Image of T24 cancer cell transfected for actin. PA gel substrate of stiffness 10kPa. Fluorescent image with insert indicating the level of actin expression for the line drawn. Corresponding traction force map (in Pa).

Figure 11 :

 11 Figure 11: Image of RT112 cancer cell transfected for actin. PA gel substrate of stiffness 10kPa. Fluorescent image with insert indicating the level of actin expression for the line drawn. Corresponding traction force map (in Pa).

Figure 12 :

 12 Figure 12: Colocalization of the actin and force levels. T24 cell (top) and RT112 cell (below). Larger actin areas are found with the RT112 cell.

Table Cell type

 Cell Cyt. grade σ (Pa) F (nN) F As/A(%) F As(ñm 2 ) V (ñm/mn)

								α
	T24	G3	120	17.4	0.98	1.13	0.38	1.57
	RT112	G2	171	22.8	0.89	0.86	0.17	1.21
	p	-	(**)	0.14	0.4	(***)	(***)	(***)

Table 1 :

 1 Summary of dynamic and kinematic quantities obtained for T24 and RT112 cells.

values located outside the interval [Q 1 -α (Q 3 -Q 1 ); Q 3 + α (Q 3 -Q 1 )],where Q 1 is the lower quartile, Q 3 the higher quartile, and α = 1.5

Acknowledgments

We are thankful to the Université Franco-Italienne and to the European Commission (Marie Curie

Research Training Network, MRTN-CT-2004-503661). Confocal microscopy was performed at the microscopy facility of the Institut Albert Bonniot, thanks to grants "Association pour la Recherche sur le Cancer" (ARC, Villejuif, France) and the Nanobio program (Nanosciences Foundation, Grenoble). We thank Olivier Destaing for help with the transfection of actin-GFP.

Supplemental materials

Supplemental movie S1. Phase-contrast movie of a migrating T24 cancer cell. Time lapse is one image every 2 minutes.

Supplemental