Tro Ng Biên Hoàng

William Pasillas-Lépine

Alexandre De Bernardinis
email: alexandre.de-bernardinis@ifsttar.fr

Mariana Netto
email: mariana.netto@ifsttar.fr

Extended braking stiffness estimation based on a switched observer, with an application to wheel-acceleration control *

Keywords: Automotive control, Anti-lock brake systems, Observer design, Switched linear systems, Lyapunov stability

In the context of hybrid anti-lock brake systems (ABS), a closed-loop wheel-acceleration controller based on the observation of the extended braking stiffness (XBS) is provided. Its objective is to improve the system's robustness with respect to changes in the environment (as changes in road conditions, brake properties, etc.). The observer design is based on Burckhardt's tyre model, which provides a wheel acceleration dynamics that is linear up to time-scaling. The XBS is one of the state variables of this model. The paper's main result is an observer that estimates this unmeasured variable. When the road conditions are known, a three-dimensional observer solves the problem. But, for unknown road conditions, a more complex four-dimensional observer must be used instead. In both cases, the observer's convergence is analysed using tools for switched linear systems that ensure uniform exponential stability (provided that a dwelltime condition is satisfied). Both experiments and simulations confirm the convergence properties predicted by the theoretical analysis.

I. INTRODUCTION

The anti-lock brake system (ABS) is now a standard equipment on most new passenger cars, in order to prevent wheel lock-up and limit the risk of skidding. With this system, the car maintains its steerability and reduces its braking distance, even in the case of an emergency braking. Historically, the first commercial ABS systems were designed using logicbased switching controllers, in which the mode changes are determined by the evolution of the wheel's angular acceleration (see, e.g., [START_REF] Leiber | Antiskid system for passenger cars with a digital electronic control unit[END_REF], [START_REF] Kuo | A four-phase control scheme of an antiskid brake system for all road conditions[END_REF], [START_REF] Kiencke | Automotive Control Systems[END_REF], [START_REF] Gmbh | Automotive Handbook[END_REF]). The main force of these controllers is that they avoid the use of the (unmeasured) wheel slip and of its (unknown) optimal value. They are therefore quite robust with respect to changes in tyre parameters and road conditions. Their main drawback is, however, that they were derived from purely heuristic arguments and are, as a consequence, difficult to tune. Despite of this, the ABS controllers present on today's commercial vehicles mainly belong to this category. More recently, mainly in an academic context, several wheel slip controllers have been proposed in the literature (see, e.g., [START_REF] Unsal | Sliding mode measurement feedback control for antilock braking systems[END_REF], [START_REF] Johansen | Gainscheduled wheel slip control in automotive brake systems[END_REF], [START_REF] Savaresi | Mixed slip-deceleration control in automotive braking systems[END_REF], [START_REF] Choi | Antilock brake system with a continuous wheel slip control to maximize the braking performance and the ride quality[END_REF], [START_REF] Pasillas-Lépine | Design and experimental validation of a nonlinear wheel slip control algorithm[END_REF]). The main interest of these controllers is that they apply a brake torque that converges to a specific value, which avoids the typical limit cycles generated by logic-switched algorithms. This leads to shorter braking distances, at least on standard road conditions. Unfortunately, these approaches assume (implicitly) that the wheel slip is measured (or estimated) and that its optimal value is known, two requirements that are often difficult to meet. Even if such algorithms might not be robust enough to be implemented on commercial ABS, they are still useful for some specific applications [START_REF] Van Zanten | Evolution of electronic control systems for improving the vehicle dynamic behavior[END_REF], like the electronic stability program (ESP).

In addition to hybrid and continuous approaches for ABS, which both have their pros and cons [START_REF] Shida | A study on effects of tire characteristics on stop distance of ABS braking with simplified model[END_REF], one can find a different research line (see, e.g., [START_REF] Gustafsson | Monitoring tire-road friction using the wheel slip[END_REF], [START_REF] Ono | Estimation of automotive tire force characteristics using wheel velocity[END_REF], [START_REF] Sugai | New control technique for maximizing braking force on antilock braking system[END_REF]) based on the concept of extended braking stiffness (XBS). The XBS is the slope µ ′ (λ) of the tyre characteristic µ(λ). For additional details, the reader is referred to Section II. In standard conditions, there exists an (unknown) value of the wheel slip λ * for which the curve µ reaches its maximum. That is, such that µ ′ (λ * ) = 0. The main theoretical interest of XBS for braking strategies is hence clear: unlike wheel slip, that has an unknown optimal target value λ * , the optimal value of XBS is always the same (zero). An intuitive approach for ABS control is thus to regulate the value of XBS around zero. But, actually, the XBS appears also in other contexts related to braking systems. A first example is wheel acceleration control. In this context, the XBS can be seen as a disturbance that must be compensated in order to increase either the controller's bandwidth or its delay margin (see, e.g., [START_REF] Corno | Hybrid ABS control using force measurement[END_REF], [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF], [START_REF] Hoang | Reducing the impact of wheel-frequency oscillations in continuous and hybrid ABS strategies[END_REF]). A second (related) example is wheel slip control. Indeed, since the wheel acceleration is closely related to the derivative of the wheel slip (see Section II-B), the XBS appears also naturally in this domain [START_REF] Pasillas-Lépine | Design and experimental validation of a nonlinear wheel slip control algorithm[END_REF]. One should stress, however, that the XBS cannot be measured directly using standard sensors. In order to use it in a control algorithm one must therefore address first its real-time estimation, which is the main objective of this paper. Because of the diversity of control problems in which the XBS appears, it would have been difficult to treat all of them here. The choice of the authors was thus to emphasize the contributions associated to the estimation problem, and to consider the control issues only for illustration purposes.

The simplest approach to estimate XBS is probably to consider this variable as a constant parameter, which allows the use of online least squares methods [START_REF] Ono | Estimation of automotive tire force characteristics using wheel velocity[END_REF]. Other approaches analyze the tyre/carcass resonance in the frequency domain [START_REF] Sugai | New control technique for maximizing braking force on antilock braking system[END_REF] or use algebraic methods [START_REF] Villagra | A diagnosisbased approach for tire-road forces and maximum friction estimation[END_REF]. Solutions based on wheel slip measurements are also available [START_REF] Tanelli | Real-time identification of tire-road friction conditions[END_REF]. Nevertheless, to the author's knowledge, the idea of exploiting the nonlinear XBS dynamics in a model-based observer has not been considered before in the literature, at least in the case of the longitudinal stiffness. The approach proposed in this paper is based on a new model for the wheel acceleration dynamics. In this model, the extended braking stiffness enters as one of the state variables. When the road conditions are known this model is three-dimensional. Otherwise a fourth order dynamics is obtained. In both cases an observer can be constructed using a copy of the system's dynamics and adding a nonlinear correction term that is proportional to the observation error. After a suitable time-rescaling, the observer error is reduced to a linear switched system that can be analyzed using standard methods [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF]. When the observer switches admit a strictly positive dwell-time, the observer's convergence is global, uniform, and exponential. Compared to previous works, the authors believe that the main interest of this method comes from its simplicity and from the fact that the parameters of the tyre model are not needed by the proposed algorithm.

In order to illustrate on a concrete example the interest of this observer, the case of a simple academic ABS strategy [START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF] is considered. In their standard form, this kind of algorithms might fail to cycle correctly [START_REF] Ait-Hammouda | Jumps and synchronization in anti-lock brake algorithms[END_REF] when there are significant changes in the environment (as changes in road conditions, brake properties, etc.). In a recent work [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF], it has been shown that adding closed-loop wheel acceleration control during the phases for which the brake pressure is modified can compensate this lack of robustness. But, in order to reach the bandwidth required by this kind of controllers, an XBS estimate is necessary. The combination of such control laws with the proposed XBS observer has been tested both on simulations (with changes of road conditions) and experimentally (with constant road conditions, imposed by the test-rig characteristics).

This paper is organized as follows. First, the system's dynamics is described in Section 2. Then, the main contributions of the paper (the design and the stability analysis of two switched observers) are presented in Sections 3 and 4, with the corresponding experimental and simulation results. An academic five-phase hybrid ABS and a closed-loop wheelacceleration control law are briefly described in Section 5, in order to exhibit a potential application for these observers. Finally, concluding remarks and perspectives for future research are presented in Section 6.

II. SYSTEM MODELLING

The basic dynamics of the wheel, which is central to this study, can be analyzed using a single-wheel model (see, e.g., [START_REF] Olson | Nonlinear dynamics of vehicle traction[END_REF] and [START_REF] Ono | Estimation of automotive tire force characteristics using wheel velocity[END_REF]). The main reason for using this model is that, despite of its simplicity, all the basic phenomena related to ABS control appear in it [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF].

A. Wheel dynamics

The angular velocity ω of the wheel has the following dynamics:

I dω dt = -RF x + T w , (1)
where I denotes the inertia of the wheel, R its radius, F x the longitudinal tyre force, and T w the torque applied to the The longitudinal tyre force F x is often modelled by the relation

F x = µ(λ)F z , (2)
where F z denotes the vertical load and

λ = Rω -v x v x (3)
denotes the wheel slip [START_REF] Kiencke | Automotive Control Systems[END_REF]. The longitudinal speed of the vehicle v x , which is considered as an external variable of the model, is assumed to be strictly positive. In a braking manoeuvre, this implies λ < 0 and F x < 0. The tyre characteristic µ(•) is a function that is both smooth and odd. It satisfies µ(0) = 0 and µ ′ (0) > 0 (see Figure 1), where µ ′ (λ) denotes the derivative of µ with respect to λ. Several mathematical descriptions are available in order to describe this curve. Two of them are considered in Sections III-A and IV-A.

B. Wheel acceleration dynamics

The state variables of the model are

x 1 = λ and x 2 = R dω dt -a x (t),
where a x (t) = dv x (t)/dt denotes the vehicle's longitudinal acceleration. The state x 1 is the wheel slip. The state x 2 is the wheel acceleration offset, that is, the difference between the acceleration of the wheel and that of the vehicle. These variables evolve with the following dynamics

dx 1 dt = 1 v x (t) (-a x (t)x 1 + x 2 ) dx 2 dt = - aµ ′ (x 1 ) v x (t) (-a x (t)x 1 + x 2 ) + R I dT w dt - da x (t) dt ,
where a = (R 2 /I)F z and the extended braking stiffness µ ′ (.) is defined as the derivative of the tyre characteristic µ(.) with respect to λ.

During an ABS-controlled braking manoeuvre, the vehicle's acceleration a x (t) stays almost constant and close to the maximal value a *

x allowed by the road's conditions [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF]. Moreover, the wheel slip λ remains relatively small. In such conditions, the control and observer designs can be simplified by considering that (-a x x 1 + x 2 ) ≃ x 2 . This approximation is exact only at constant speed, but it remains reasonable in the case of ABS manoeuvres [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF]. Its validity is checked a posteriori in Sections III and IV, by simulating the proposed observers on the original (non-simplified) model. This approximation leads to a simpler dynamics

dx 1 dt = 1 v x (t) x 2 dx 2 dt = - a v x (t) µ ′ (x 1 )x 2 + bu, (4)
where the control variable

III. OBSERVER DESIGN (KNOWN ROAD CONDITIONS)

A. Tyre characteristic

In the literature, one can find several mathematical formulas that have been used to describe the tyre characteristic µ(λ), such as trigonometric functions in [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF], second order rational fractions in [START_REF] Kiencke | Automotive Control Systems[END_REF] and [START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF], and exponentials in [START_REF] Burckhardt | Fahrwerktechnik: Radschlupf-Regelsysteme[END_REF]. This section is based on Burckhardt's [START_REF] Burckhardt | Fahrwerktechnik: Radschlupf-Regelsysteme[END_REF] model

µ(λ) = c 1 (1 -e -c2λ ) -c 3 λ, (5)
where the coefficients c i are constants depending on the road conditions, the tyre model, the tyre pressure, etc. Therefore, for the sake of robustness, the ABS algorithms should be able to handle the uncertainty associated with these coefficients. A typical tyre characteristic associated to this model is illustrated in Figure 1.

B. Extended braking stiffness dynamics

Burckhardt's tyre model is particularly interesting when it comes to estimate the value of the extended braking stiffness, which cannot be measured directly. Indeed, a simple mathematical formula for µ ′ (λ) can be obtained by differentiating [START_REF] Burckhardt | Fahrwerktechnik: Radschlupf-Regelsysteme[END_REF], with respect to λ. From this formula and from the second order derivative of (5), one can establish a relation between these variables:

µ ′′ (λ) + c 2 µ ′ (λ) + c 2 c 3 = 0. (6)
Now, defining the wheel acceleration offset z 1 = x 2 , the extended braking stiffness z 2 = µ ′ (x 1 ), and the unknown product of parameters z 3 = -c 2 c 3 as new variables, combining equations ( 4) and ( 6) gives

dz 1 dt = -a v x (t) z 1 z 2 + bu dz 2 dt = (cz 2 + z 3 ) z 1 v x (t) dz 3 dt = 0, (7)
where c = -c 2 is a constant that depends on road conditions. This model can be seen as a generalization of the model proposed in [START_REF] Ono | Estimation of automotive tire force characteristics using wheel velocity[END_REF] and as a particular case of ( 4), associated to Burckhardt's tyre model. Somehow, considering the unknown constant z 3 as a new state variable (and not as a parameter) is not optimal. Indeed, the adaptive observer approach [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF] could have been a more standard way to handle this problem. Nevertheless, that approach has not been followed here because (for the authors) it is not obvious how to combine it with the switchings introduced in the next section. While, using a representation of the form [START_REF] Choi | Antilock brake system with a continuous wheel slip control to maximize the braking performance and the ride quality[END_REF], the approach of [START_REF] Hoang | A switching observer for systems with linearizable error dynamics via singular timescaling[END_REF] is directly applicable. On the one hand, an interesting quality of this model is that the wheel slip (which cannot be measured) does not appear explicitly in it as a state variable. One might argue that this is not that interesting, since the velocity (which cannot be measured neither) appears instead in the system's dynamics. Nevertheless, at least at high speeds, it is much easier to estimate the vehicle's velocity than to estimate wheel slip [START_REF] Daiss | Estimation of vehicle speed: Fuzzy-estimation in comparison with Kalman-filtering[END_REF]. On the other hand, the main drawback of our model [START_REF] Choi | Antilock brake system with a continuous wheel slip control to maximize the braking performance and the ride quality[END_REF] is that it is assumed that the value of c is known, which is true only for a fixed type of road conditions (the more complex case of unknown road conditions is considered later, in Section IV).

C. Observer Design

Since, unlike the wheel acceleration offset z 1 , the extended braking stiffness z 2 is not directly measurable, it must be estimated using an observer. To that aim, one can start with a copy of the original system and add some terms proportional to the observation error, in order to ensure the convergence of the trajectories between both systems. As it is shown later, multiplying these terms by z 1 simplifies considerably the analysis. At the end, one obtains

d z 1 dt = -a v x z 1 z 2 + bu + k 1 (z 1 ) v x z 1 (z 1 -z 1 ) d z 2 dt = (c z 2 + z 3 ) z 1 v x + k 2 (z 1 ) v x z 1 (z 1 -z 1 ) d z 3 dt = k 3 (z 1 ) v x z 1 (z 1 -z 1 ), (8)
where z i are the observer states.

In [START_REF] Corno | Hybrid ABS control using force measurement[END_REF], the observer gains k i (z 1 ), for 1 ≤ i ≤ 3, must depend on the value of z 1 in order to ensure the observer's stability independently of the sign of z 1 . The simplest choice might be

k i (z 1 ) = k + i if z 1 > 0 k - i if z 1 < 0. (9)
Even if the gains k i (z 1 ) are discontinuous, it must be stressed that the observer gains k i (z 1 )z i are continuous, which ensures the existence and uniqueness of solutions for (8) when z 1 (t) is considered as an external input. Consider the observer errors e i := z i -z i , for 1 ≤ i ≤ 3. Subtracting equation ( 8) from equation ( 7) leads to

de dt = z 1 v x   -k 1 (z 1 ) -a 0 -k 2 (z 1 ) c 1 -k 3 (z 1 ) 0 0   e. ( 10
)
Notice that if the right hand side of ( 10) is divided by z 1 /v x then the observer error dynamics is transformed into a linear system. This leads to the idea of changing the time-scaling. Indeed, let

s(t) := t 0 |z 1 (τ )| v x (τ ) dτ, (11)
which ensures that dt/ds > 0, independently of the values of z 1 . Since, for any function ϕ : R → R n , one has

dϕ ds = dϕ dt dt ds = dϕ dt v x |z 1 (t)| . (12)
This implies

de ds =                    A + e =    -k + 1 -a 0 -k + 2 c 1 -k + 3 0 0    e if z 1 > 0 A -e =    k - 1 a 0 k - 2 -c -1 k - 3 0 0    e if z 1 < 0, (13)
which can be written using a more compact notation in the form de(s) ds = A σ(s) e(s),

where σ denotes a piecewise constant signal that selects, at each instant, a matrix from the pair {A + , A -}.

D. Stability Conditions

It results from the previous section that the analysis of the asymptotic convergence of the observer (8) can be derived from the stability analysis of the error equation [START_REF] Hildebrand | Introduction to numerical analysis -Second Edition[END_REF], which is an autonomous switched linear system. It appears that numerous stability results are available for that class of systems [START_REF] Liberzon | Switching in Systems and Control[END_REF]. Most of them are based on classical Lyapunov-functions. But some LaSalle-like results are also available [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF], for which the stability properties of the switched system are proved via regularity assumptions on the set of switching signals.

Define the switching signal σ(t) = sign(z 1 (t)), and assume that the solutions of ( 14) are such that e and σ are piecewise differentiable and piecewise constant, respectively. Following [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], define moreover the set S[τ D ], with τ D > 0, of switchings for which any two consecutive discontinuities of σ are separated by no less than τ D . The constant τ D is called the dwell-time. The origin of a switched system of the form ( 14) is said to be uniformly exponentially stable if there exists constants c 0 and λ 0 such that, for each t ≥ 0, we have e(t) ≤ c 0 exp(-λ 0 t) e(0) . In this definition, the word uniform refers to the fact that c 0 and λ 0 do not depend on the switching signal [START_REF] Angeli | A note on stability of arbitrarily switched homogeneous systems[END_REF].

Under a dwell-time condition, as a particular case of Theorem 4 of [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF], one can prove that a switched linear system is uniformly exponentially stable if there exists a symmetric positive definite matrix that satisfies simultaneously two nonstrict Lyapunov equations (more details on this point are given in the Appendix). The aim of Theorem 1 below is to show that, for the switched system [START_REF] Hildebrand | Introduction to numerical analysis -Second Edition[END_REF], it is always possible to find a pair of gains K + and K -such that this LaSalle-like condition is satisfied. To ensure the stability of ( 14), a first natural condition is to impose the matrices A + and A -to be Hurwitz. The corresponding conditions on the observer gains can be derived using Routh's criterion, which gives

k + 1 > c, k + 2 < - c a k + 1 , and
- (ck + 1 + ak + 2 )(c -k + 1 ) a < k + 3 < 0; (15)
and

k - 1 < c, k - 2 < - c a k - 1 , and
0 < k - 3 < - (ck - 1 + ak - 2 )(c -k - 1 ) a . (16)
From these conditions, with the help of Theorem 4 of [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF], one can obtain the following result (proved in the Appendix, at the end of the paper).

Theorem 1 Assume that the three following conditions are satisfied (i) The gain 16). (iii) The gains K + and K -satisfy

K + = k + 1 k + 2 k + 3 satisfies (15). (ii) The gain K -= k - 1 k - 2 k - 3 satisfies (
(c -k - 1 ) ak - 3 = (c -k + 1 ) ak + 3 > 0 and (ck + 1 + ak + 2 ) = (ck - 1 + ak - 2 ) < 0. ( 17
)
Then, the system ( 14) is uniformly exponentially stable, provided that the switching signal σ admits a strictly positive dwell-time.

This result gives at least a certain degree of freedom: we can chose any K + that stabilizes the system. Once this choice has been made, it imposes however an almost unique choice for K -(in order to assign the same spectrum to A + and A -). We do not know, in general, if this constraint can be avoided, but this issue is discussed in [START_REF] Hoang | A switching observer for systems with linearizable error dynamics via singular timescaling[END_REF].

E. Experimental Results

The observer design proposed in this section has been validated on data coming from the tyre-in-the-loop setup of TU Delft, acquired in the context of ABS research [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF]. The test-rig consists in a large steel drum on top of which the tyre is rolling. The tyre is mounted on a wheel that is attached to a rotating axle, which has a rigidly constrained height. The axle is supported by two bearings on both sides of the wheel.

The bearing housings are connected to a fixed frame by means of piezo-electric force transducers. A hydraulic disk brake is mounted on one side of the axle. The pressure in the calliper is locally controlled by an analog electronic circuit connected to a servo-valve, in order to match the reference pressure. An illustration of this test-rig can be seen on Section 9.4 of [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF].

The setup has been used for several years, at TU Delft, for tyre modelling and identification (see [START_REF] Pacejka | Tyre and Vehicle Dynamics[END_REF]Section 9.4] and [37, Appendix A], and the references therein).

In order to satisfy the conditions imposed by Theorem 1, for positive z 1 's, the following observer gains are chosen

k + 1 = c + (β 1 + 2β 2 ) k + 2 = -(β 2 2 + 2β 1 β 2 + ck + 1 )/a k + 3 = -β 1 β 2 2 /a.
And, for negative z 1 's,

k - 1 = c -(β 1 + 2β 2 ) k - 2 = -(β 2 2 + 2β 1 β 2 + ck - 1 )/a k - 3 = β 1 β 2 2 /a,
where β 1 and β 2 are positive constants that assign the spectrum of the error's dynamics. More precisely, the error dynamics (13) will always have two real eigenvalues -β 1 (with multiplicity 1) and -β 2 (with multiplicity 2), independently of the sign of z 1 . The interest of assigning the same spectrum to A + and A -is explained in [START_REF] Hoang | A switching observer for systems with linearizable error dynamics via singular timescaling[END_REF].

The experimental results are shown on Figure 2, where it can be seen that the states of the system and of the observer remain close to each other. In this figure, the observer's variable ẑ1 is compared to the measure of z 1 , while the variable ẑ2 is compared to an estimation µ ′ (x 1 ) of the XBS obtained directly from the measure of wheel slip. One can observe in this figure a surprising phenomenon: the noise of the observed variable ẑ2 is bigger when the wheel acceleration is positive. A possible explanation for these oscillations might be that the norm of K + is bigger than that of K -, a constraint imposed by Theorem 1.

This phenomenon reduces the accuracy of the estimation, which is nevertheless good enough to detect whether the tyre is in its stable or unstable region. The proposed observer has however another weak point: it only works correctly when the parameter c 2 of Burckhardt's model is known, at least approximatively. The knowledge of this parameter is closely related to the knowledge of road conditions, a problem that is considered in the next section.

IV. OBSERVER DESIGN (UNKNOWN ROAD CONDITIONS)

In contrast to the simpler approach of Section III, it is now assumed that the observer does not have any information on the road conditions (and thus on the parameters of Burckhardt's model). This new context imposes the use a more complex four-dimensional observer, which can be considered as a generalization of the previous three-dimensional observer [START_REF] Corno | Hybrid ABS control using force measurement[END_REF].

A. Tyre characteristic

The main difficulty with Burckhardt's model ( 5) is that its parametrization is nonlinear. Recently, in [START_REF] Tanelli | Real-time identification of tire-road friction conditions[END_REF], an alternative parametrization of this model by exponentials has been proposed (see also [START_REF] Castro | Real-time estimation of tyreroad friction peak with optimal linear parameterisation[END_REF]). This kind of approximations can be traced back up to the work of Prony [START_REF] Prony | Essai experimental et analytique: sur les lois de la dilatabilite de fluides[END_REF] (see [START_REF] Hildebrand | Introduction to numerical analysis -Second Edition[END_REF] for a modern treatment). In this section, Burckhardt's model is approximated with a similar parametrization In this figure, it can be seen that the XBS observer is highly perturbed by the swift road transition but that, once the transition is over, it converges again in a fraction of a second towards the appropriate state. I. For different road conditions, the coefficients θ i can be identified using the Least Squares method (see Table I). In Figure 1, the tyre characteristics given by Burckhardt's model ( 5) is compared to its approximation [START_REF] Kiencke | Automotive Control Systems[END_REF].

µ(λ) = θ 0 λ + θ 1 e d1λ -1 d 1 + θ 2 e d2λ -1 d 2 , ( 18
)

B. Extended braking stiffness dynamics

Computing the first, second and third derivatives of the approximate model [START_REF] Kiencke | Automotive Control Systems[END_REF], with respect to λ, one can see that these derivatives satisfy the following relation

µ ′′′ (λ) = α 0 + α 1 µ ′ (λ) + α 2 µ ′′ (λ), (19)
where

α 0 = d 1 d 2 θ 0 , α 1 = -d 1 d 2 , α 2 = (d 1 + d 2 ).
Therefore, following the ideas of Section III, we take as state variables

z 1 = x 2 , z 2 = µ ′ (λ), z 3 = µ ′′ (λ)
, and z 4 = α 0 . Now, combining equations ( 4) and [START_REF] Kuo | A four-phase control scheme of an antiskid brake system for all road conditions[END_REF] gives

dz 1 dt = -a v x (t) z 1 z 2 + bu(t) dz 2 dt = z 1 v x (t) z 3 dz 3 dt = z 1 v x (t) (α 1 z 2 + α 2 z 3 + z 4 ) dz 4 dt = 0, (20)
where a and b are defined in Section II.

The most important property of this model is that the parameters α 1 and α 2 do not depend on road conditions. This leads to the possibility of observing the extended braking stiffness, using neither the wheel slip nor the parameters that describe the tyre characteristic. It should be stressed, however, that this model is only valid for constant road conditions. In the case of a change of road conditions (see, e.g., Figure 3), the validity of the model fails temporarily, which might induce a brief divergence between the system's and the observer states.

C. Observer design

For system [START_REF] Leiber | Antiskid system for passenger cars with a digital electronic control unit[END_REF], an observer with an error dynamics that is linearizable by a time-scaling can be constructed following the same approach as in Section III. This leads to a switching error dynamics [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF], with

A + =     -k + 1 -a 0 0 -k + 2 0 1 0 -k + 3 α 1 α 2 1 -k + 4 0 0 0     and A -=     k - 1 a 0 0 k - 2 0 -1 0 k - 3 -α 1 -α 2 -1 k - 4 0 0 0     .
Conditions for the stability of ( 14), in the case of these new matrices A + and A -, can be derived following the same approach as for Theorem 1 (see the Appendix).

D. Simulation results

In test-rigs like those of TU Delft, changes of road conditions are not possible. Nevertheless, numerical simulations can still be used to assess the performance of the proposed observer. This has been done considering the (non-simplified) model of Section II and using the observer's output to implement the control law of Section V. In order to ensure the observer's stability, for positive z 1 's, the following observer and4) is achieved using the observer of Section IV and the control design of Section V. The car runs on dry asphalt during three seconds, then on wet asphalt for one second, and finally on snow until the end of the simulation. When the road conditions change, the brake pressure is reduced and follows the available tyre force potential. gains are chosen:

k + 1 = α 2 + 2(β 1 + β 2 ) k + 2 = (-α 1 -k + 1 α 2 -(β 2 1 + β 2 2 + 4β 1 β 2 ))/a k + 3 = (-k + 1 α 1 + ak + 2 α 2 -2β 1 β 2 (β 1 + β 2 ))/a k + 4 = -β 2 1 β 2 2 /a. And, for negative z 1 's, k - 1 = α 2 -2(β 1 + β 2 ) k - 2 = (-α 1 -k - 1 α 2 -(β 2 1 + β 2 2 + 4β 1 β 2 ))/a k - 3 = (-k - 1 α 1 + ak - 2 α 2 + 2β 1 β 2 (β 1 + β 2 ))/a k - 4 = -β 2 1 β 2 2
/a, are positive constants that assign the spectrum of the error's dynamics, which has two double real eigenvalues -β 1 and -β 2 .

Figure 3 shows the obtained simulation results. The details of the braking scenario are given in the figure's caption. The observer estimates accurately the values of the XBS, for different road conditions. During transitions, (which last 25 ms), the estimated XBS values change abruptly. The observer cannot give good estimations during these transitions. Nevertheless, as soon as they are over, the observer error decreases in a relatively short period of time that, of course, depends on the choice of β 1 and β 2 .

V. CONTROL DESIGN

A five-phase hybrid ABS algorithm [START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF] is described in Figure 5. Each of the algorithm's phases defines either a constant or quickly changed brake pressure P b (t) that is applied to the brake. The switches between each phase are triggered when the value of the wheel acceleration offset x 2 crosses some predefined threshold. The main interest of such hybrid approaches is that they do not use any information on the unmeasured variable x 1 . Nevertheless, they are able to keep the wheel slip in a small neighborhood of its optimal value λ * , for which the longitudinal tyre force is maximal (with the aim of minimizing the braking distance), without using explicitly the value of the optimal setpoint. The reader can find in [START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF] more details about this five-phase hybrid ABS algorithm.

When the algorithm of Figure 5 is tested on an experimental setup [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF], it might fail to cycle correctly as soon as there are considerable changes in the environment. The main reason behind this lack of robustness is that, during the different phases, the wheel acceleration is controlled in open-loop, with a brake pressure increase that is independent of the wheel's acceleration. This shortcoming can be overcome [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF] by controlling the wheel acceleration x 2 in closed-loop (around a predefined trajectory x * 2 ), during the phases for which the brake torque changes quickly.

In order to do this, define τ := t-t 0 , where t 0 is the instant at which a given phase begins. Consider the time T needed by the reference trajectory x * 2 to go from the previous threshold ǫ i to the next one ǫ j . Ideally, the duration T should be as small as possible but, due to the physical limitations of the brake actuator, there exists a lower bound on the achievable T 's. If Ṗ M b is defined as the maximum brake pressure derivative that the actuator can deliver (in absolute value), then the choice of the reference trajectory x * 2 must guarantee that | Ṗ | ≤ Ṗ M b . Furthermore, in order to minimize the system's sensitivity to actuator delays, it is natural to require a zero derivative for x * 2 at the beginning and at the end of each phase [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF]. A possible choice for a reference trajectory x * 2 that goes from ǫ i to ǫ j is therefore

x * 2 (τ ) = a 0 + a 1 τ + a 2 τ 2 + a 3 τ 3 , (21)
where a 0 = ǫ i , a 1 = 0, a 2 = -3(ǫ i -ǫ j )/T 2 , and a 3 = 2(ǫ i -ǫ j )/T 3 . By imposing, additionally, the constraint T ≥ (3/2b)|ǫ i -ǫ j |/ Ṗ M b , one can ensure that the reference trajectory respects the brake actuator's limitations described above. Now, define the tracking error ξ = x 2 -x * 2 and the control law

u(t) = 1 b a v x μ′ (t)x 2 + dx * 2 (t) dt -αξ(t) , (22)
where α > 0 is the control gain and μ′ (t) is an estimation of the extended braking stiffness µ ′ (x 1 (t)). In the absence of estimation error, the tracking error converges exponentially to zero, provided that the control gain α is taken big enough. Observe, however, that the gain α is limited by the delay margin of the system [START_REF] Hoang | Reducing the impact of wheel-frequency oscillations in continuous and hybrid ABS strategies[END_REF]. In this approach, the choice of controlling only the variable x 2 might be surprising. But it appears that the stability of all other variables actually comes from the fact that they are bounded functions of the wheel slip x 1 , which remains bounded both for hybrid [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF] and continuous [START_REF] Pasillas-Lépine | Design and experimental validation of a nonlinear wheel slip control algorithm[END_REF] control designs, provided that the wheel acceleration offset x 2 follows its reference.

In the simulation of Figure 4, the control uses the XBS estimation given by the observer. Thanks to the observer performance, the control law ( 22) ensures a good tracking performance of the wheel acceleration x 2 to its pre-defined reference. As a consequence of the robustness added by the closed-loop wheel acceleration control, the brake pressure is automatically increased or decreased to match road conditions.

VI. CONCLUSION

In the context of anti-lock brake systems (ABS), this paper presented a new approach to estimate the extended braking stiffness. The first contribution of this work is a new nonlinear wheel acceleration model in which the XBS enters as one of the state variables. This model is obtained using either Burckhardt's model or its linearly parametrized approximation. The second contribution is the design of two stable XBS observers. When the road conditions are known, a threedimensional observer solves the problem. But, for unknown road conditions, a more complex four-dimensional observer should be used instead. In both cases, the stability of the observers is proved via time-rescaling and LaSalle-like theorems for linear switched systems.

The three-dimensional observer has been tested on experimental data coming from TU-Delft's test-rig [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF]. In such tests, the parameters associated to the mounted tyre are known. The experimental results show the effectiveness of this observer. The four-dimensional observer has been tested in simulations in a scenario that includes unkown changes of road conditions. The simulation results show a precise estimation of the XBS even in the case of discontinuous jumps of road conditions.

The proposed method has nevertheless several limitations. First, it needs a (rough) estimation of the vehicle's speed (see, e.g., [START_REF] Daiss | Estimation of vehicle speed: Fuzzy-estimation in comparison with Kalman-filtering[END_REF] and [START_REF] Corno | Traction-control-oriented state estimation for motorcycles[END_REF] for works that consider this problem). Second, the combined convergence of the observer and of the control law has not been proved. One could expect, however, that such a proof is obtainable via cascaded design arguments [START_REF] Loría | Cascaded nonlinear time-varying systems: Analysis and design[END_REF]. Third, the vertical load F z has been considered to be both known and constant. It is true that F z can be "reconstructed" using the longitudinal and lateral accelerations as inputs. It is also known that hybrid ABS strategies have a Fig. 5. The academic five-phase hybrid ABS strategy proposed in [START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF]. The wheel acceleration thresholds ǫ i and the brake pressure increase and decrease rates u i must be tuned in order to obtain an asymptotically stable limit cycle (see, e.g., [START_REF] Ait-Hammouda | Jumps and synchronization in anti-lock brake algorithms[END_REF] and [START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF]). certain degree of robustness with respect to vertical load uncertainties (see, e.g., the Appendix A.1 of [START_REF] Gerard | Improvements to a five-phase ABS algorithm for experimental validation[END_REF]). Nevertheless, the impact of a time-varying vertical load on the proposed design is clearly a topic that deserves further investigations. APPENDIX This section includes a sketch of the proof of Theorem 1. The interested reader can find in [START_REF] Hoang | A switching observer for systems with linearizable error dynamics via singular timescaling[END_REF] an approach that generalizes this method to a more general class of switched systems that contains, as particular cases, the three and fourdimensional observers proposed in this paper.

The characteristic polynomial of the matrix A + is given by

η 3 + (k + 1 -c)η 2 -(ck + 1 + ak + 2 )η -ak + 3 = 0. ( 23
)
Using the Routh criterion for (23) leads directly to condition [START_REF] Hoang | Reducing the impact of wheel-frequency oscillations in continuous and hybrid ABS strategies[END_REF]. The same argument, but applied to A -, gives condition [START_REF] Hoang | A switching observer for systems with linearizable error dynamics via singular timescaling[END_REF]. Assume that the observer gains K + and K -satisfy, respectively, the conditions (15) and [START_REF] Hoang | A switching observer for systems with linearizable error dynamics via singular timescaling[END_REF]. For additional details concerning the following steps, the reader is referred to [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF]Theorem 4]. The objective is to show that there exists a pair {P + , P -}, of symmetric positive definite matrices satisfying all the conditions required by that theorem, for an appropriately defined pair of matrices {C + , C -}.

Define C + = c + 1 0 0 and C -= c - 1 0 0 , where c + 1 , c - 1 = 0. It is easy to check that the pairs (A + , C + ) and (A -, C -) are observable. In order to satisfy the conditions of [START_REF] Hespanha | Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle[END_REF]Theorem 4], one must find a matrix P that satisfies simultaneously the equations A T + P + P A + = -C

The term in the square root is positive because of (15) and p 22 > 0.

Similarly, since P has to satisfy the condition A T -P + P A -= -C T -C -, it follows that the elements of P are also of the form (26) From ( 24) and ( 26), additional conditions on the observer gains K + and K -can be obtained:

(c -k-1 )
ak -

(ck + 1 + ak + 2 ) = (ck - 1 + ak - 2 ) < 0. (27)
The element c - 1 of C -is also different from zero and, because of ( 16) and p 22 > 0, one obtains

c - 1 = ± -2 (c -k - 1 )(ck - 1 + ak - 2 ) + ak - 3 a 2 p 22 , (28)
which ends the proof.

  wheel. The torque T w = T e -T b is composed of the engine torque T e and the brake torque T b . It is assumed that during ABS braking the clutch is open and thus the engine torque is neglected. In other words, T b = γ b P b , where P b > 0 denotes the brake pressure and γ b > 0 the brake efficiency.

  u = dP b (t)/dt is the derivative of the brake pressure and b = -Rγ b /I. Indeed, we have T w = -γ b P b (see Section II-A), and thus (R/I)dT w /dt = b dP b /dt.

Fig. 2 .

 2 Fig.2. Comparison between experimental measurements and the estimated states given by the observer[START_REF] Corno | Hybrid ABS control using force measurement[END_REF], during an ABS test[START_REF] Pasillas-Lépine | Hybrid modeling and limit cycle analysis for a class of five-phase anti-lock brake algorithms[END_REF]. The parameters of the test-rig's tyre characteristics are: c 1 = -1.24, c 2 = -34, and c 3 = 0.65. The system's parameters are: I = 1.2kg.m 2 , R = 0.3m, Fz = 2500N , and γ b = 17.5N.m/bar. The speed of the drum is 65 km/h. The XBS estimated by the observer is compared to the theoretical value obtained from the wheel slip and the derivative of the tyre characteristic.

Fig. 3 .

 3 Fig.3. Simulation of a braking ABS scenario with changes of road conditions: The car runs on dry asphalt during three seconds and on wet asphalt afterwards. In this figure, it can be seen that the XBS observer is highly perturbed by the swift road transition but that, once the transition is over, it converges again in a fraction of a second towards the appropriate state.

  defined for λ ≤ 0. The constants d 1 and d 2 must be chosen in such a way that -c 2 ∈ [d 1 , d 2 ]. Since, for negative wheel slip, the parameter c 2 varies in the range [-50, -24], one can take d 1 = 22 and d 2 = 52. The parameters of Burckhardt's model are shown in Table

Fig. 4 .

 4 Fig.4. Wheel acceleration tracking (during phases 1, 3, and 4) is achieved using the observer of Section IV and the control design of Section V. The car runs on dry asphalt during three seconds, then on wet asphalt for one second, and finally on snow until the end of the simulation. When the road conditions change, the brake pressure is reduced and follows the available tyre force potential.

Table I -

 I Tyre parameters for Burckhardt's model.

		Burckhardt's model	Approximate model
		c 1	c 2	c 3	θ 0	θ 1	θ 2
	Dry road	1.28	24	0.52	-0.53	25.22	7.2
	Wet road	0.86	34	0.35	-0.36	8.86	24
	Snow	0.28	50	0.05	-0.05	0.24	14

  T + C + and A T -P + P A -= -C T -C -. Observe that P defines a nonstrict Lyapunov function only, because the symmetric matrices C T i C i are not definite positive. Denote by (p ij ) the elements of P . One can easily deduce from A T + P + P A + = -C T + C + that where p 22 > 0. With the elements of P computed as in (24), one obtains

	c + 1 = ± 2	(c -k + 1 )(ck + 1 + ak + 2 ) + ak + 3 a 2	p 22 = 0.
			p 11 = p 13 =	c 2 -(ck + 1 + ak + 2 ) a 2 1 a p 22 , p 23 = 0, and p 33 = p 22 , p 12 =	c a (c -k + p 22 , 3 ak + 1 )	p 22 ,	(24)

* The research leading to these results has been supported by the French region Ile-de-France, through the Regeneo project (RTRA Digiteo and DIM LSC). It has also received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement no 257462 HYCON2 Network of excellence.

1 T. B. Hoàng (bien.


Pierre-Henri Puech
email: pierre-henri.puech@inserm.fr

Damien Nevoltris

Philippe Robert

Laurent Limozin

Claude Boyer

Pierre Bongrand

Force measurements of TCR / pMHC recognition at T cell surface

Keywords: T cell, molecular recognition, force, adhesion, atomic force microscopy

The rupture forces and adhesion frequencies of single recognition complexes between an affinity selected peptide/MHC complex and a TCR at a murine hybridoma surface were measured using Atomic Force Microscopy. When the CD8 coreceptor is absent, the adhesion frequency depends on the nature of the peptide but the rupture force does not. When CD8 is present, no effect of the nature of the peptide is observed. CD8 is proposed to act as a time and distance lock, enabling the shorter TCR molecule to bridge the pMHC and have time to finely read the peptide. Ultimately, such experiments could help the dissection of the sequential steps by which the TCR reads the peptide/MHC complex in order to control T cell activation.

Introduction

A key step of the immune response is the detection by T lymphocytes, thanks to their T cell receptor (TCR), of foreign peptides bound to major histocompatibility complex molecules (pMHC) on the surface of antigen presenting cells (APCs). In addition to its prominent physiological importance, the interaction of the TCR with pMHC raises enormous interest due to a number of extraordinary features. (i) Recognition is exquisitively specific since T lymphocytes have been reported to detect a single cognate pMHC complex on APCs exposing many tens of millions of proteins on their membranes [START_REF] Sykulev | Evidence that a single peptide-MHC complex on a target cell can elicit a cytolytic T cell response[END_REF][START_REF] Irvine | Direct observation of ligand recognition by T cells[END_REF]. (ii) The TCR repertoire must be rich enough to cope with many millions of potentially harmful structures and specific enough to avoid autoimmune phenomena. (iii) Recognition and subsequent activation must be rapid enough to occur during a typical contact ranging in duration from seconds to minutes between an APC and a T lymphocyte [START_REF] Miller | T cell repertoire scanning is promoted by dynamic dendritic cell behavior and random T cell motility in the lymph node[END_REF]. (iv) Recognition is not an all-or-none event since it may generate widely different outcomes, ranging from full lymphocyte activation to anergy following minute variations of the peptide antigen sequence [START_REF] Sloan-Lancaster | Altered peptide ligand-induced partial T cell activation: molecular mechanisms and role in T cell biology[END_REF].

The binding of a cognate pMHC by the TCR is thought to involve the participation of a coreceptor that may be CD4 or CD8 for class II and class I MHC respectively. While a well established role of CD4 or CD8 is to enhance signaling cascades [START_REF] Viola | T cell activation determined by T cell receptor number and tunable thresholds[END_REF][START_REF] Delon | Cd8 expression allows T cell signaling by monomeric peptide-mhc complexes[END_REF], these molecules also influence binding by acting as low affinity receptors [START_REF] Wyer | T cell receptor and coreceptor CD8 alpha alpha bind peptide-MHC independently and with distinct kinetics[END_REF], having a high association rate per se, increasing the association rate of soluble pMHC to T lymphocytes and decreasing the dissociation rate [START_REF] Gakamsky | CD8 kinetically promotes ligand binding to the t-cell antigen receptor[END_REF]. The interaction between CD8 and TCR is complex, since TCR engagement may activate CD8-mediated adhesion [START_REF] Varghese | TCR complex-activated CD8 adhesion function by human T cells[END_REF] and CD8 may modulate TCR avidity [START_REF] Luescher | CD8 modulation of T-cell antigen receptor-ligand interactions on living cytotoxic T lymphocytes[END_REF].

Signal generation as a consequence of pMHC/TCR interaction is difficult to explain on the basis of a conformational change [START_REF] Rudolph | How TCRs bind MHCs, peptides, and coreceptors[END_REF][START_REF] Smith-Garvin | T cell activation[END_REF]. Rather, interaction outcome was reported to depend on the lifetime of individual TCR / MHC complexes [START_REF] Matsui | Kinetics of t-cell receptor binding to peptide/i-ek complexes: correlation of the dissociation rate with T-cell responsiveness[END_REF][START_REF] Kersh | High-and low-potency ligands with similar affinities for the TCR: the importance of kinetics in TCR signaling[END_REF][START_REF] Aleksic | Dependence of T cell antigen recognition on t cell receptor-peptide mhc confinement time[END_REF]. Some experiments supported the intriguing hypothesis that signaling might also involve force generation at the lymphocyte / APC interface 3 / 40 [START_REF] Ma | The receptor deformation model of TCR triggering[END_REF][START_REF] Kim | The alpha beta T cell receptor is an anisotropic mechanosensor[END_REF].

In order to gain new insight into the mechanisms of TCR-mediated lymphocyte activation, it is essential to relate the outcome of the lymphocyte/APC interaction to the physical properties of TCR/pMHC interactions such as the kinetic parameters or the forces of the TCR / pMHC interaction. Much work has been done to measure these interactions using recombinant elements in soluble phase with surface plasmon resonance [START_REF] Matsui | Kinetics of t-cell receptor binding to peptide/i-ek complexes: correlation of the dissociation rate with T-cell responsiveness[END_REF][START_REF] Van Der Merwe | Molecular interactions mediating T cell antigen recognition[END_REF]. However, it is well recognized that molecular interactions between surface-bound, especially cell membrane bound, receptors are influenced by several parameters, e.g. force sensitivity, molecular flexibility or steric effects, that are not accounted for by measurements made in solution [START_REF] Robert | What is the biological relevance of the specific bond properties revealed by single-molecule studies?[END_REF] and that may be profoundly influenced by active cellular processes [START_REF] Huppa | TCR-peptide-MHC interactions in situ show accelerated kinetics and increased affinity[END_REF]. Therefore, the use of powerful biophysical tools such as atomic force microscopy applied to molecular studies is warranted to provide an accurate characterization of molecular interactions between TCRs and pMHCs on the cell surface.

An Atomic Force Microscope (AFM) in force mode uses cantilever deflection to measure the forces that are exerted on the lever extremity. The sensitivity of force determination is limited by the thermal noise of the system [START_REF] Franz | Atomic force microscopy : a versatile tool for studying cell morphology, adhesion and mechanics[END_REF]. AFM has proved to be a very well suited technique for measuring molecular interactions, from single molecule unbinding events [START_REF] Florin | Adhesion forces between individual ligand-receptor pairs[END_REF][START_REF] Hinterdorfer | Detection and localization of individual antibody-antigen recognition events by atomic force microscopy[END_REF] to single cell detachment [START_REF] Puech | Measuring cell adhesion forces of primary gastrulating cells from zebrafish using atomic force microscopy[END_REF][START_REF] Puech | A new technical approach to quantify cell-cell adhesion forces by afm[END_REF].

In this article, we assessed the respective roles of the TCR and CD8 co-receptor during the first hundred milliseconds following the contact of a model surface decorated with pMHC and a living T cell. We used two hybridoma lines expressing similar levels of BM3.3 TCR with (line C3.CD8) or without (line 4C8.98) the CD8 coreceptor. BM3.3 TCR recognizes the pBM1 peptide bound to allogenic MHC Class I [START_REF] Reiser | Crystal structure of a T cell receptor bound to an allogeneic MHC molecule[END_REF] and is powerful enough (EC 50 ~ 10 -14 -10 -11 M, K d ~ 2. 6 10 -6 M as measured by SPR), such that it can under certain conditions induce CD8 independent T-cell activation [START_REF] Guimezanes | Identification of endogenous peptides recognized by in vivo or in vitro generated alloreactive cytotoxic T lymphocytes: distinct characteristics correlated with cd8 dependence[END_REF][START_REF] Guimezanes | Structural and kinetic basis for low affinity cross-reactivity in T cell allorecognition[END_REF]. Thus, comparing the interaction between this TCR and its cognate ligand (pBM1 peptide presented by H-2K b ) or a non-activating peptide (OVA presented by H-2K b ) has relevance since currently available methods of studying single bond rupture may not be sensitive enough to analyze the interaction of "weak" TCRs with their ligands. Finally, the use of an hybridoma such as C3.CD8 expressing high levels of CD8 as compared to TCR molecules should be optimally suited to detect an additional effect of CD8 as compared to CD8-independent responses. This article is the first report to the authors's knowledge, of a direct monitoring of TCR/pMHC interaction at the single bond level and in terms of forces on the surface of living cells.

Results

Using flow cytometry, we verified that both cell lines employed in this study expressed the desired molecules on their surface (TCR, CD3 and CD8) and that TCR levels were similar (Fig. 1, first six panels). Using the recombinant fusion protein Dimer X to expose peptide loaded H-2K b , we showed that TCR binding was peptide specific (low binding for OVA, stronger binding for pBM1) and dose dependent. The presence of CD8 at the cell surface greatly enhanced the binding of the pMHCs to the cell surface (Fig. 1, last two panels). We measured by cytometry that 75% of C3.CD8 cells express both CD8α and CD8β on their surface (Suppl. Fig. S1). CD8 on C3.CD8 cell line is composed of both CD8αβ and CD8αα dimers. In view of previous reports [START_REF] Kern | Structural basis of CD8 coreceptor function revealed by crystallographic analysis of a murine CD8alphaalpha ectodomain fragment in complex with H-2Kb[END_REF], CD8α may be considered as responsible for the interaction between CD8 dimers and MHC molecules.

Cantilever tips decorated with pMHC were used to investigate the molecular recognition at the cell surface. By bringing a tip and a cell into contact and separating them with controlled speed and force, the force exerted at the tip extremity can be assessed by measuring the bending of the cantilever (Fig. 2A,B and Suppl. Fig. S2). To ensure that the cantilever tip contacted the cell membrane and not only the cell glycocalyx, we performed micro-mechanical measurements on the hybridoma (Suppl. Fig. S3). We observed that the low contact force used here (50 pN) was sufficient to bring the molecules on the tip (pMHCs) in close contact with their partners on the cell membrane (TCR / CD8) since (i) the Young modulus of the cells was measured to be the same as for a tenfold higher contact force and (ii) reproductible contact between the tip and the cell was obtained for such a small contact force. A minimal contact time was chosen to have low adhesion frequencies and thus be able to investigate mainly single molecules interactions following statistical arguments : when adhesion frequency is lower of 30% it may be assumed that 80% of the binding events represent single molecule events [START_REF] Tees | A microcantilever device to assess the effect of force on the lifetime of selectin-carbohydrate bonds[END_REF]. Typical force curves, without or with an interaction, are shown on Fig. 2C andD respectively. A short contact time is highly relevant to dissecting the first steps of the molecular recognition between pMHC, TCR and/or CD8 (see Discussion). This contact time, t c , can be estimated, in average, as t c = t mech + t AFM , where t AFM is the AFM macroscopic experimental time, here set to 0 sec. t mech is the effective contact time imposed by the mechanical properties of the cells described by the Young modulus, E, the chosen contact force, F c , and cantilever speed, v. It can be calculated as The first striking observation was that the unbinding forces were low as compared to the forces measured with classical adhesion molecules, such as integrins and cadherins [START_REF] Franz | Atomic force microscopy : a versatile tool for studying cell morphology, adhesion and mechanics[END_REF][START_REF] Puech | Measuring cell adhesion forces of primary gastrulating cells from zebrafish using atomic force microscopy[END_REF][START_REF] Chen | Single-molecule force measurements[END_REF][START_REF] Taubenberger | Revealing early steps of alpha2beta1 integrin-mediated adhesion to collagen type I by using single-cell force spectroscopy[END_REF][START_REF] Franz | Studying integrin-mediated cell adhesion at the single-molecule level using afm force spectroscopy[END_REF].

t c = t mech = 2 d / v
The detachment forces measured here were of the same magnitude as the apparent noise of the force curves (measured to be on the order of 10 pN). This implied that averaging is necessary to precisely detect the force jumps (Fig. 2C, D, white line, see Methods).

Six combinations of interacting surfaces were studied with the same pulling speed (1μm/sec ; Table 1 and Fig. 3A). CD8-and CD8+ cells were contacted with cantilevers presenting one of two non activating, empty H-2K b or OVA:H-2K b , or one activating system, pBM1:H-2K b . In addition to the force of single de-adhesion jumps, we recorded the total number of force curves obtained, N curves , the number of force curves with at least one force jump, N adhesion , and the sum of the number of force jumps per force curve, N jumps .

To assess the specificity of the measured interaction with regards to the cytometry data, the adhesion frequency for each cell, AF=N adhesion /N curves , and the corresponding averages over the different cells of the same type were obtained (Fig. 3B). A multiplicity index of the unbinding events, namely MI=N jumps /N adhesion was calculated for each condition (all cells pooled, Fig. 3C). The presence or absence of a large amount of peptide in the solution while performing the force measurements did not significantly affect the results (not shown), excluding the possibility of a significant loss of the peptide for the time scale of the experiments and cantilever storage.

For the two cell lines, AF was measured to be the lower when the MHCs were presenting no peptide (13.7 +/-2.8 % and 10.5 +/-2.5 % for CD8-and CD8 + cells respectively) and was found of similar value (17.5 +/-3.0 %) for OVA:H-2K b presented to CD8-cells (Fig. 3B). AF was observed to be significantly higher for OVA:H-2K b presented to CD8+ cells (35.6 +/-4.3%).

Consistent with the known capacity of pBM1:H-2K b to activate BM3.3 T lymphocytes in absence of CD8 [START_REF] Guimezanes | Identification of endogenous peptides recognized by in vivo or in vitro generated alloreactive cytotoxic T lymphocytes: distinct characteristics correlated with cd8 dependence[END_REF], CD8-cells displayed higher AF for pBM1:H-2K b (28.8 +/-3.8 %) than for OVA:H-2K b . Interestingly, AF was found statistically similar for OVA:H-2K b and pBM1:H-2K b , (27.5 +/-3.3%) presented to CD8+ cells and of the same magnitude as for pBM1:H-2K b presented to CD8cells.

As a control, we performed experiments where BW cells, which lack TCR and CD8 (see Methods), were used. They lead to low AF, similar to those measured for the empty H-2K b situation described above : 17.1 +/-4.8 % for empty H-2K b , 10.6 +/-1.9 % for OVA:H-2K b and 14.4 +/-1.9 % for pBM1:H-2K b . Cantilevers bearing no H-2K b presented to CD8+ cells lead to similar AF (13.2 % for biotin-BSA, 13.5 % for streptavidin and 6 % for protein-G decorated levers). This allowed us to conclude that, for an AFM contact time of 0sec and a contact force of 50pN, a residual AF of 10-15% originated from non specific interactions.

In summary, AF were peptide dependent in the absence of CD8. In presence of CD8, the presentation of a peptide by the MHC was required to obtain high AF, but these AF did not discreminate between peptide antigens. When increasing the contact time from 0sec to 100 msec and 1sec, AF increases as expected for CD8-and CD8+ cells for the three peptides (Suppl. Fig. S5).

The ranking is similar as in the case of 0 sec contact (pBM1 > OVA > no peptide for a given cell type, and CD8+ ≥ CD8-for a given peptide). The increase in AF is monotoneous for CD8+ cells but not for CD8-.

MI was observed to be 1.00 and 1.06 when the empty H-2K b was presented to CD8-and 8 / 40 CD8+ cells respectively (Fig. 3C). For OVA:H-2K b , MI stayed closer to 1 for CD8-cells (1.10) in regard to CD8+ ones (1.16). MI increased when pBM1:H-2K b was used in comparison to OVA:H-2K b , slightly for CD8-cells (1.16) and strongly for CD8+ ones (1.37). This indicated that, on average, the number of detectable detachment events per force curve increased when a peptide was present, and this number was higher with an activating peptide. Surprisingly, the extracted average force values for single force jumps were not significantly dependent on either the cell line or the presented peptide, when present (Fig. 3D). The forces were similar to the ones measured using BW cells (not shown). Double jumps occuring in the same force curve for pBM1:H-2K b presented to CD8+ cells exhibited similar magnitude as the single jumps (Suppl. Fig. S4), independently of their separation in distance. Aside, forces of single jumps are not varying when the contact time is increased up to 1sec (Suppl. Fig. S5). Force and its variation relative to the presented peptide, as quantified here ie. at single molecule scale, has never been reported in literature for the T cell recognition machinery.

Discussion

In this report, atomic force microscopy (AFM) in force mode [START_REF] Franz | Atomic force microscopy : a versatile tool for studying cell morphology, adhesion and mechanics[END_REF] was used to measure the unbinding forces of single TCR / pMHC molecules on living murine T hybridoma cell surfaces.

MHCs bound to two peptides of known activity and "empty" MHCs were used to probe T cells expressing a TCR with or without its CD8 coreceptor. AFM allowed us to study interactions in a time short enough to minimize active cell phenomena that have been recently reported to profoundly influence TCR/pMHC interaction at the cell surface [START_REF] Huppa | TCR-peptide-MHC interactions in situ show accelerated kinetics and increased affinity[END_REF][START_REF] Huang | The kinetics of twodimensional TCR and pMHC interactions determine T-cell responsiveness[END_REF].

Flow cytometry experiments verified that the cell lines had similar TCR levels, together with the desired expression of CD8, and that the binding, at equilibrium, of pMHCs was peptide and dose dependent. The AFM experiments revealed that the frequency of adhesion events, AF, linked to the on rate, k on , of the recognition reaction, [START_REF] Hinterdorfer | Detection and localization of individual antibody-antigen recognition events by atomic force microscopy[END_REF][START_REF] Chesla | Measuring two-dimensional receptor-ligand binding kinetics by micropipette[END_REF], but not the rupture forces, related to its off rate, k off , [START_REF] Evans | Dynamic strength of molecular adhesion bonds[END_REF][START_REF] Merkel | Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy[END_REF] was dependent on the nature of the presented peptide. The case of CD8-cells allowed us to conclude that we observed specific and peptide dependent events. The low adhesion frequencies supported the hypothesis that these events were mainly due to single molecule recognition [START_REF] Tees | A microcantilever device to assess the effect of force on the lifetime of selectin-carbohydrate bonds[END_REF].

In line with previous reports and due to the very low forces measured, the rupture events recorded with the AFM were the latest formed (ie. the "youngest" bond), namely the TCR /pMHC interaction. The experimental procedure gave enough time to other non covalent bonds of the molecular construction used on the cantilever tips to mature, allowing them to reach energetically deeper, hence stronger, bound states [START_REF] Pincet | The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds[END_REF][START_REF] Marshall | Force history dependence of receptor-ligand dissociation[END_REF].

The time available for a TCR to detect a cognate ligand on the APC surface may be crudely estimated as follows : estimating x=10-25 nm to be the maximum distance compatible with molecular interaction, and D=10,000 nm 2 /s to be the diffusion coefficient of pMHC [START_REF] Bongrand | Quantitative aspects of T cell recognition: from within the antigen-presenting cell to within the T cell[END_REF], the interaction time between a TCR and a pMHC may be estimated to be approximatively x 2 /4D = 0.0025-0.0625 sec. Therefore, the study of very short contact times, such as the ones used here, that do not allow full bond maturation should be highly relevant to the biological problem we investigated.

We propose that the explanation of the observed results originates from the complex geometry of the recognition bridges that have to form (Fig. 4A) [START_REF] Rudolph | How TCRs bind MHCs, peptides, and coreceptors[END_REF][START_REF] Reiser | Crystal structure of a T cell receptor bound to an allogeneic MHC molecule[END_REF][START_REF] Housset | What do TCR-pMHC crystal structures teach us about MHC restriction and alloreactivity?[END_REF][START_REF] Mazza | How much can a T-cell antigen receptor adapt to structurally distinct antigenic peptides?[END_REF]. To recognize a given pMHC, a TCR has to contact both the MHC and the peptide in a very finely controlled way on a restricted set of amino-acids. This implies that the overall optimal geometry of the TCR/ pMHC complex is difficult to achieve, when the molecules are rare and presented in a membrane where they can diffuse and rotate [START_REF] Bongrand | Quantitative aspects of T cell recognition: from within the antigen-presenting cell to within the T cell[END_REF], thus decreasing the duration of efficient intermolecular contacts. All together, the molecules by themselves limit the access to the adequate geometry. The existence of a minimal contact time for an efficient TCR/pMHC recognition is consistent with the observed behavior for experimental contact times larger than 0sec. As a consequence, the capacity of our technique to detect the subtle differences between the peptides could be insufficient, leading to the observation of no force difference, but a difference in the adhesion frequency only. In other terms, the effective on-rate of the reaction, between a 2D, membrane bound TCR and a quasi 2D, cantilever bound pMHC could be the limiting parameter [START_REF] Robert | Biomolecule association rates do not provide a complete description of bond formation[END_REF].

AF could be linked to the effective k on of the recognition following the analysis proposed by [START_REF] Chesla | Measuring two-dimensional receptor-ligand binding kinetics by micropipette[END_REF], while the rupture force is linked to the lifetime of the bond under a given applied force ramp and relates to the off rate, k off , as dissected experimentally with Dynamic Force Spectroscopy [START_REF] Evans | Dynamic strength of molecular adhesion bonds[END_REF][START_REF] Merkel | Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy[END_REF]. If these hypotheses were accepted, our results would support the data from [START_REF] Huang | The kinetics of twodimensional TCR and pMHC interactions determine T-cell responsiveness[END_REF] where k off did not strongly vary as a function of the presented peptide, but A c k on varied over three orders of magnitude, A c being the contact area between a T cell and the model APC used. Morever, this study showed a stronger correlation of activity of the peptide with the on rate than with the off rate.

Similarly, other studies [START_REF] Hosseini | Immune synapse formation determines interaction forces between T cells and antigen-presenting cells measured by atomic force microscopy[END_REF][START_REF] Hoffmann | Single cell force spectroscopy of T cells recognizing a myelin-derived peptide on antigen presenting cells[END_REF] found no difference when measuring forces for multiple recognition pairs occuring between a T cell and an APC for contact times up to 10 minutes, but reported a difference in adhesive fraction, as a function of the presented peptide. Aside, in a separate set of preliminary experiments, we performed Dynamic Force Spectroscopy on CD8+ cells and we did not observe any variations of the off rate and position of the barrier as a function of the presented peptide (not shown).

Even if the forces were similar in all measured cases, the CD8-case showed that the adhesion frequency did depend on the peptide nature, indicating that our measurements were TCR-specific in that case, which proves their biological significance. The CD8+ case appeared to be more complex, since the system is no more tripartite (TCR/p/MHC) but quadripartite (with CD8 coreceptor) [START_REF] Wang | Structural basis of the CD8 alpha beta/MHC class I interaction: focused recognition orients CD8 beta to a T cell proximal position[END_REF].

Because CD8 could be responsible for certain level of adhesion with MHC bearing surfaces in absence of TCR involvment [START_REF] Norment | Cell-cell adhesion mediated by CD8 and MHC class I molecules[END_REF][START_REF] Huang | Kinetics of MHC-CD8 interaction at the T cell membrane[END_REF], the coreceptor presence could account for two principle points of our results : (i) AF did not vary strongly as a function of the presented peptide for CD8+ cells and (ii) AF was higher for OVA:H-2K b presented to CD8+ cells in comparison to CD8-ones.

Point (i) might originate from the size difference between the TCR (~ 10 nm) and the CD8 (~ 15 nm) [START_REF] Murphy | Janeway's Immunobiology 7th Edition[END_REF]: this simple geometrical consideration may explain an accessibility difference between both molecules and the pMHC, potentially accounting for the larger MI in the pBM1 case. Moreover, it has been observed that the ratio TCR:CD8 (on BM3.3 clones) was on the order of 1:10 (C. Boyer, unpublished results). Such a difference in surface densities of the molecules could decrease the probability of TCR/pMHC recognition events occuring, which may be merged with or masked by the more frequent and geometrically easier CD8/MHC interactions.

We propose to summarize our results in terms of the energy landscape of the TCR/pMHC recognition, as shown on Fig. 4C. Each well represent a degree of recognition, described in Fig. 4D.

The energy landscape is complex due to geometrical and environmental reasons. The TCR/pMHC recognition itself could be a double step-in situation, where the first, intermediate well might be due to the necessary close contact between TCR and MHC. Once this state would be optimally achieved, the bond could "mature" by reaching a deeper well due to the fine fitting of the TCR structure with the presented peptide [START_REF] Pincet | The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds[END_REF][START_REF] Boniface | Thermodynamics of T cell receptor binding to peptide-MHC: evidence for a general mechanism of molecular scanning[END_REF][START_REF] Wu | Two-step binding mechanism for Tcell receptor recognition of peptide MHC[END_REF][START_REF] Gakamsky | T cell receptor-ligand interactions: a conformational preequilibrium or an induced fit[END_REF]. Reaching this next state would be the mechanism allowing T cells to efficiently distinguish between self and non self peptides. The CD8 molecule, which has to interact with the MHC but has a longer extension than the TCR, could assist in reaching this state, by maintaining the TCR within close range of the MHC to which it is itself binding. This would introduce a "capture and guidance" supplementary well, giving time to the shorter TCR to find the MHC and mature (Fig. 4C,D).

An intriguing possibility would be that the TCR might test, using the cytoskeleton and active cell motion, the pMHC/TCR bond for a force of a few pN (that is below the force resolution of the reported experiments) shortly after formation to increase its sensitivity [START_REF] Kim | The alpha beta T cell receptor is an anisotropic mechanosensor[END_REF]. Such small forces have been demonstrated to be physiologically relevant [START_REF] Del Rio | Stretching single talin rod molecules activates vinculin binding[END_REF] and the observed behavior might be correlated to k on [START_REF] Robert | Biomolecule association rates do not provide a complete description of bond formation[END_REF].

Conclusion

Force measurements of TCR / pMHC recognition events at single molecule scale were performed using atomic force microscopy in force mode. When CD8 is absent, the nature of the peptide strongly influences the adhesion frequency. The presence of CD8 strongly modifies this behavior. Importantly, no effect of the peptide on the rupture forces was detected. The proposed explanation originates from the complex geometry and energetical pathway that the molecules have to follow for the peptide to be recognized efficiently, leading to relevant activation outcomes. CD8 could serve as a guidance and "time locking" molecule, to help the close fitting of the TCR and pMHC, by bringing and maintaining them in a sufficiently close range and for a sufficient time to interact. Further investigations, using T cell clones and/or recombinant proteins, could allow one to dissect the relative importance of the fine geometrical constraints (such as the peptide antigen sequence) and of the molecular environment (as exemplified by CD8) in the process of antigenic discremination, the first step in activating the powerful and robust mechanisms of the body protection by the adaptive immune system.

Material and methods

Commercial reagents

Chemicals for tip and glass surface functionalisation were obtained from Sigma : Bovine

Cell lines

Alloreactive TCR BM3.3 murine hybridomas were used. TCR BM3.3 recognizes its agonist pBM1/H2-Kb with high avidity in a CD8 co-receptor independent fashion for long term consequences of recognition [START_REF] Guimezanes | Identification of endogenous peptides recognized by in vivo or in vitro generated alloreactive cytotoxic T lymphocytes: distinct characteristics correlated with cd8 dependence[END_REF][START_REF] Guimezanes | Structural and kinetic basis for low affinity cross-reactivity in T cell allorecognition[END_REF][START_REF] Mazza | How much can a T-cell antigen receptor adapt to structurally distinct antigenic peptides?[END_REF][START_REF] Auphan-Anezin | Distinct orientation of the alloreactive monoclonal CD8 T cell activation program by three different peptide/MHC complexes[END_REF]. The 4C8.98 hybridoma was obtained by fusion between spleen cells from a Rag1-/-BM3.3-TCR-transgenic mice and BW-TCRα-/β-. This hybridoma was selected for expression of the BM3.3 clonotype mAb98 [START_REF] Buferne | Role of CD3d in surface expression of the TCR/CD3 complex and in activation for killing analyzed with a CD3dnegative cytotoxic T lymphocyte variant[END_REF], and was further transduced with a genetic construction of the CD8α cDNA chain inserted in the pHbAPr-1 neo vector [START_REF] Arcaro | Essential role of CD8 palmitoylation in CD8 coreceptor function[END_REF]. The C3.CD8 clone was selected for the 98b clonotype and CD8 surface expressions. It express mainly CD8α and CD8β at cell surface (Suppl. Fig. S1). These hybridoma are mentionned in the CGG for our group referred 2668. Cell lines were checked for CD3, CD8 and TCR expression and sorted by FACS. Resulting cells were used over one month before been sorted again and were passaged every three days.

Flow cytometry

Samples containing 10 5 cells were set in round bottom 96 wells plates with 40μl of various DimerX dilutions (μg/mL) in FACS Buffer (PBS, FCS 1.5%, EDTA 1mM, NaN3 0,02%, filtered at 0.22μm) and incubated 1h at 4°C under gentle shaking. DimerX were loaded with the desired peptide (OVA or pBM1) following the instructions of the provider (BD Biosciences) at a ratio of 1:200. After addition of 100μl FACS Buffer, the cells were centrifugated (1500rpm, 4min, 4°C). The 96 wells plate was then flicked and 50μl biotinylated anti-Mouse Ig (Chemicon International, AP181B) at 1/1000 in FACS Buffer was gently mixed for 30min at 4°C under gentle shaking. Rinsing was then performed with 100μl FACS buffer, followed by centrifugation and flicking before addition of 50μl

Streptavidin APC (E-Bioscience, diluted at 1/500 in FACS buffer) for 20min at 4°C. Following those steps, analysis was immediately performed on living cells. Alternatively, for the analysis of CD3, TCR and CD8, the first step was replaced by incubations of 30min with similar conditions with anti-CD3biotin (145.2C11, BD Pharmingen), anti-TCRβ (H57, BD Pharmingen or biotinylated anti clonotype 98, made in-house [START_REF] Buferne | Role of CD3d in surface expression of the TCR/CD3 complex and in activation for killing analyzed with a CD3dnegative cytotoxic T lymphocyte variant[END_REF] or anti-CD8 pacific blue (E-Bioscience).

Cell immobilisation

A scalpel-cut PDMS square well of 5mm×5mm×1mm was used to delimitate a zone on plasma activated clean microscope slides [START_REF] Puech | Measuring cell adhesion forces of primary gastrulating cells from zebrafish using atomic force microscopy[END_REF]. The obtained well was incubated with 100μL of 0.01% poly-L-lysine for 15 to 30 min. Before the experiment, substrates were gently rinsed with the cell culture medium used to perform the adhesion tests. The PDMS stamp was then removed and a plastic ring (diameter 25mm, height 10mm) was glued on the glass slide using vacuum grease. The experiment chamber was then filled with 1mL of Hepes-buffered culture medium. Diluted cell suspensions were then seeded onto the substrate, let to adhere during 15 to 30 min at room temperature, and gently washed with buffer to remove unbound cells. Using Trypan blue labelling, we observed that the fixation of the cells to the poly-L-lysine was keeping the fraction dead / alive cells to < 10%, comparable to what was measured in the cell suspension.

Atomic force microscope

Cell-tip recognition and mechanical measurements were conducted with an AFM (Nanowizard I, JPK Instruments, Berlin) mounted on an inverted fluorescence microscope (Zeiss Axiovert 200 equipped with 10x and 40x objectives). Bright field imaging was used to select cells and monitor their morphology during force measurements (Suppl. Fig. S2). The AFM head was equipped with a 15 μm z-range linearized piezoelectric ceramic scanner and an infrared laser. The setup was used in closed height feedback mode [START_REF] Puech | A new technical approach to quantify cell-cell adhesion forces by afm[END_REF]. We used Veeco MSCT cantilevers (nominal spring constant k = 10 mN/m, 320 μm long). The sensitivity of the optical lever system was calibrated and the cantilever spring constant were determined in situ using built-in routines of the JPK software before every experiment by using the thermal noise method [START_REF] Butt | Calculation of thermal noise in atomic force microscopy[END_REF]. The calibration procedure for each cantilever was repeated up to three times to rule out possible errors. Spring constants were found to be consistent with the manufacturer's nominal value (17-22pN/nm). The AFM and optical microscopes were isolated from ambiant acoustic and mechanical noises using acoustic foam and an active damping table (Halcyonics). All experiments were carried out at 25•C, for no more than an hour, before replacement of the substrate, cell suspension and cantilever.

Cantilever decoration

We adapted a previously developed protocol [START_REF] Franz | Studying integrin-mediated cell adhesion at the single-molecule level using afm force spectroscopy[END_REF] to the needs of the experiments.

Cantilevers were washed in 10% v/v Hellmanex / MQ water at 60•C, then rinced three times in alternating ethanol and water baths before air drying at 60•C, protected from dust. After residual air plasma activation for one minute, they were decorated sequentially using biotin-BSA (0.5mg/mL in NaHCO3 100mM, pH 8,6 ; overnight), streptavidin (0.5mg/mL in PBS w/o Ca 2+ /Mg 2+ , pH 7.4 ; 45min), biotin-protein G (0.5mg/mL in PBS w/o Ca 2+ /Mg 2+ , pH 7.4 ; 45min) and finally Dimer X (0.01mg/mL in PBS w/o Ca 2+ /Mg 2+ , pH 7.4 ; 3h). Between each step, the levers were washed intensively three times in PBS to remove unbound proteins. The functionalized levers were then incubated in an excess of peptide following the instructions of the provider (at least 200 to 2000 times more peptide than Dimer X, at 4°C, in PBS, overnight). The levers were kept up to three days in this solution until final rincing prior to use. This process ensures that all intermediate bonds can consolidate sufficiently for the measured rupture forces to be mainly attributed to the pMHC end of the molecular sandwich [START_REF] Pincet | The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds[END_REF]. To qualitatively assess that the molecular construction built up on the lever was present, several tests using either fluorescent proteins or antibody labelling were performed. Using a fluorescein labelled streptavidin, the fluorescence level was very weak without the biotin-BSA compared to the case where this preliminary layer was present. The presence of H-2K b dimers was checked by using an FITC labelled anti-MHC antibody (20.8.4, gift from A.

Guimezanes, CIML, Marseille). Compared to the case without MHC, the fluorescence level of the case with MHC was 3-fold higher, indicating the good functionalization of the levers with the desired "final" molecules (not shown). One has to note that it is technically difficult to dilute and measure precisely the density of molecules on the AFM sharp tip, hence no precise quantification of the number of molecules is provided here.

Adhesion measurements

Using the optical microscope, a calibrated cantilever is positioned over a chosen cell (Suppl. Fig. S2). The speed for bringing to or removing the tip from cell surface was set to v = 1μm/sec and the desired contact force to 50 pN. Contact force cannot be decreased to lower levels to minimize both the contact time and area without compromising a frequent and reproducible tip to cell contact.

At least 2048 deflection data points were collected over a pulling distance of 500nm to obtain a force curve. These parameters ensured that more than 90% of the acquired force curves will show a clear contact between the tip and the cell, and that this contact will be the gentlest possible. The time resolution (~ 1 msec) is sufficient to record the molecular unbinding events. The contact time was set to 0 sec before tip retraction, leading to an effective contact time between the tip and the cell surface, because of the deformability of this latter, on the order of 100 to 250 msec. For each condition, at least 17 cells and 154 force curves were examined over several days of culture (Table 1).

Data processing Each detachment curve was examined by eye and processing was performed using to the built-in JPK-IP software using force curve batch processing procedures : correcting for baseline shift and/or tilt, then applying a sliding average box of 15 to 45 points to detect the force jumps when present. Force jump magnitudes were recorded and pooled to calculate mean and SD. ANOVA + Tukey post test was used using Prism (GraphPad Software). In addition to the number of measured force curves, N curves , the number of the curves presenting at least one identified unbinding event, N adhesion , together with the number of unbinding events each presents, N jumps , were recorded.

The adhesion frequency, AF=N adhesion /N curves for each cell was calculated, and then the average and SEM over the different cells were obtained. t-tests were used to examine the significance of the observed AF differences using Prism (GraphPad Software). Additionaly, we calculated an index of multiplicity of the unbinding events, MI=N jumps /N adhesion . This index reveals the fraction of multiple adhesion events recorded for each condition.

  where d is the cell indentation. Using the Hertz model for a pyramidal tip of half angle α = 35° [31], we have d 2 = (4F c (1-υ 2 )) / (3E tan α). Assuming incompressibility (Poisson ratio υ =0.5) and taking F c = 50 pN, E ~ 3000 Pa and v = 1 μm/sec, one obtain a contact time t c ~ 300 msec.

  serum albumin (BSA), biotinamidocaproyl-labeled (# A6043) ; Streptavidin from Streptomyces avidinii (# S4762) ; Protein G-Biotin from Streptococcus sp. (# P8045) ; Poly-L-lysine 0.1% in water (# P8920) ; PBS (as tablets, w/o Ca/Mg). The H-2Kb-Ig recombinant fusion protein (Dimer X, # 550750) was obtained from BD Biosciences. The peptides (OVA : H-SIINFEKL-OH ; pBM1 : H-INFDFNTI-OH) were obtained from Schafer-n. The cell culture medium (RPMI 1640 + Lglutamine) and complements (7% FBS, 1% Hepes 1M, 1% Pen/Strep, 1% sodium pyruvate, 50μM β-mercaptoethanol) were obtained from Gibco.

Fig. 1

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Fig. 2

 2 Fig. 2

Fig. 3

 3 Fig. 3

Fig. 4

 4 Fig. 4

Aknowledgments

Claude Grégoire (CIML, Marseille) for his skilful help in the hybridoma preparation ; A.

Guimezanes (CIML, Marseille), Marion Seillier (ESIL, Marseille) and Thomas Biederman (ECM, Marseille) for providing materials and establishing experimental conditions ; Annemarie Lellouch (Inserm UMR600) for proofreading and helpful suggestions to the manuscript ; K. Kinoshita, A.

Leung and E. Evans for fruitful discussions about the complexity of rupture events of bio-molecular bonds and preliminary experiments using Biomembrane Force Probe; JPK Instruments company (Berlin, Germany) for its continuous support. The values are not significantly different (ANOVA + post-test, p > 0.05). For the number of cells and force curves that have been analysed, see Table 1.

Legends to the figures

Figure 4

Proposed mechanism. A : Schematics of the optimal configuration of TCR / pMHC recognition, following strutural data. This geometry is complex and may limit the formation of the recognition bridges. B : The cellular case, where the surface molecules are free to move and rotate due to their inclusion in a membrane or to their grafting spacer. This situation is rendered even complex by the presence of surrounding molecules, that can be larger that the TCR as exemplified by the case of CD8. C : The proposed energetical profile of the recognition E rec , without any force applied to the system, along a suitable reaction coordinate [START_REF] Robert | Biomolecule association rates do not provide a complete description of bond formation[END_REF]. Several wells may exist, and their access is limited by the exact "contact time" between the molecules eg. before entering the deeper well of TCR / pMHC close fitting. D. Proposed mechanism of CD8 enhancement of TCR/pMHC recognition. CD8 could help maintain the TCR and pMHC in close proximity to achieve (i) TCR / MHC binding, then (ii) TCR / p fitting, providing a time and distance guidance / locking mechanism, resulting in an addition intermediate well in the energetical pathway (panel C). Comparison with naive CD8 T cell from mouse lymph nodes suggest that at the cell surface of C3.CD8 the α chain is two times more abundant than at the surface of naive CD8 T cells. This suggests that the dimer CD8αβ coexists with CD8αα at the C3.CD8 cell surface.

Supplementary