Stanislav Aranovskiy

Rosane Ushirobira

AND Denis Efimov

G Éry Casiez

A SWITCHED DYNAMIC MODEL FOR POINTING TASKS WITH A COMPUTER MOUSE

Keywords: Pointing motion, modeling, human-computer interaction, switched model, pointing transfer function

In this paper, we study one of the most fundamental tasks in human-computer interaction, the pointing task. It can be described simply as reaching a target with a cursor starting from an initial position (e.g. executing a movement using a computer mouse to select an icon). In this paper, a switched dynamic model is proposed to handle cursor movements in indirect pointing tasks. The model contains a ballistic movement phase governed by a nonlinear model in Lurie form and a corrective movement phase described by a linear visual-feedback system. The stability of the model is first established and the derived model is then validated with experimental data acquired in a pointing task with a mouse. It is established that the measured data of pointing movements of different types can be fitted within the proposed model. Numerical comparison against pointing models available in the literature is also provided.

INTRODUCTION

The domain of human-computer interaction (HCI) covers a great deal of different tasks that can be performed by a user, the pointing task being one of the most frequently executed. In a graphical interface of any interactive system, the task of pointing for a user consists in displacing the pointer from a starting point to a final point in order to select a target (e.g. for instance, to select another window). It can be perceived and executed in different ways, depending on the used system, and the interaction may be direct or indirect. In direct interactions, users interact with a touch screen by moving hands and fingers, this input follows then different layers of hardware and software processes to produce the final visual response on the same screen. Indirect interactions deal with human inputs coming from an independent device, such as a computer mouse or a touchpad, based on visual information on a screen, that go through similar processes as in the direct interactions, ending with a visual response on the same screen. In the case of direct interactions the pointer is displayed on the same screen where the user interaction takes place while in the case of indirect interactions, the user controls the pointer displayed on a screen by the means of a different tool.

The pointing transfer function (PTF) corresponding to the pointing task is characterized by the ratio of the amount of movement of an input device (e.g. a mouse) and the controlled object (e.g. a cursor). In early graphical user interfaces, PTF were typically chosen as a constant control-to-display (CD) gain, while modern graphical user interfaces, such as Windows, OS X, and Xorg, track the input device velocity and dynamically adjust the CD gain over time [START_REF] Casiez | No more bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF]. The idea behind such an adjustment is to provide small amplification for low input velocities to improve pointing accuracy, and to provide high amplification for high input velocities to reduce traveling time. Experimental studies [START_REF] Casiez | The Impact of Control-Display Gain on User Performance in Pointing Tasks[END_REF] report that the choice of a PTF affects human-computer interaction performance, and users get better performance using switching PTF. However, despite all the research on evaluation and reverse-engineering of the existing PTF, the problem of PTF design and optimization continues to exist since there are no actual dynamical models for human pointing.

Models of pointing dynamics would also be of interest for endpoint prediction techniques [START_REF] Lank | Endpoint prediction using motion kinematics[END_REF], where the system attempts to predict the target cursor position from the beginning of the pointing movement. Such a prediction is further used to dynamically modify the visual interface (e.g. to reduce the distance to the target or increase its width). Some techniques and methods can be found in [START_REF] Balakrishnan | Beating" Fitts' law: Virtual enhancements for pointing facilitation[END_REF], and a toolbar with dynamically expanding icons represents an example of such an approach [START_REF] Mcguffin | Acquisition of Expanding Targets[END_REF]. Some other challenging applications where dynamic models of (pointing) movements can be relevant include analyses of mouse movements for user identification [START_REF] Cai | Mitigating Behavioral Variability for Mouse Dynamics: A Dimensionality-Reduction-Based Approach[END_REF], in robotics for motion planning [START_REF] Hersch | A biologically-inspired controller for reaching movements[END_REF][START_REF] Karimoddini | Hierarchical Hybrid Symbolic Robot Motion Planning and Control[END_REF] and for human movement estimation [START_REF] Zhang | Adaptive Information Fusion for Human Upper Limb Movement Estimation[END_REF], e.g. for manipulators teleoperation [START_REF] Santiago | Stable Delayed Bilateral Teleoperation of Mobile Manipulators[END_REF].

Models of pointing movements. Models used in the Human-Computer Interaction (HCI) community are mainly focused on static characteristics. They do not describe how exactly a movement is performed or what is its velocity profile, but rather describe static relationships between averaged values: average movement time, target width and traveled distance. The most popular model is the Fitts' law. It was presented in the '50s by Fitts in his seminal paper [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF] and claims the following:

(1)

M T = a + b ID,
where M T represents the movement time, a and b are positive constants and ID is the index of difficulty of the pointing task. Different formulations exist for the ID but the most commonly used was defined by MacKenzie [START_REF] Mackenzie | Fitts' law as a research and design tool in human-computer interaction[END_REF] as follows:

ID := log 2 D W + 1 ,
where D is the distance from the starting point to the target center, and W is the target width. In the next decades, it was found that the relation (1) holds in many experimental studies [START_REF] Mackenzie | Fitts' law as a research and design tool in human-computer interaction[END_REF]. One of the first attempts to explain the Fitt's law is the Impulse Variability model proposed by Schmidt et al. [START_REF] Schmidt Ra | Motor-output variability: a theory for the accuracy of rapid motor acts[END_REF] that credits almost entirely the motion to an initial muscle impulse.

On the other hand, the Iterative Corrections model proposed in [START_REF] Crossman | Feedback control of hand-movement and Fitts' Law. The Quarterly journal of experimental psychology Section A[END_REF] describes the pointing motion as controlled only by a visual feedback. Nevertheless, it was shown that neither of these models takes in consideration all different effects perceived in experiments. The hybrid Optimized Initial Impulse model developed by Meyer et al. [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF] is now accepted by the HCI community as the most well-established explanation for Fitts' law [START_REF] Balakrishnan | Beating" Fitts' law: Virtual enhancements for pointing facilitation[END_REF]. According to Meyer's model, pointing motion can be separated into two distinguishable stages: a rapid and large movement to bring the pointer close enough to the target without visual tracking (ballistic phase), and a slower corrective movement to reach the target under feedback control (corrective phase). Moreover, it should be stressed the models in [START_REF] Schmidt Ra | Motor-output variability: a theory for the accuracy of rapid motor acts[END_REF][START_REF] Crossman | Feedback control of hand-movement and Fitts' Law. The Quarterly journal of experimental psychology Section A[END_REF][START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF] do not use differential equations to describe the motion, but mainly determine verbally possible pointing scenarios.

In these days, Fitts' law and the Optimized Initial Impulse model are still the most broadly accepted models of pointing motions within a static point of view. In recent years, most progress in pointing model design are concentrated in extending these two models. For instance, the distribution of M T for computer mouse movements is studied in [START_REF] Chen | Structure of Hand/Mouse Movements[END_REF], targeted-tracking tasks are considered in [START_REF] Senanayake | Targeted-Tracking With Pointing Devices[END_REF], and a model for analyzing rapid point-and-click motions taking into consideration human effects has been proposed recently in [START_REF] Almanji | A Nonlinear Model for Mouse Pointing Task Movement Time Analysis Based on Both System and Human Effects[END_REF].

Concerning the ballistic phase, the authors in [START_REF] Gan | Geometrical conditions for ballistic and visually controlled movements[END_REF] have observed in experiments that for pointing tasks with low difficulty (i.e. with relatively large values of W ), the target may be selected during the ballistic phase only, and in such cases, the dependence of the movement time M T on the target width W is weak, and M T is mostly proportional to √ D. This observation also explains why Fitts' law (1) matches poorly experimental data for low values of ID [START_REF] Oel | Time prediction of mouse-based cursor movements[END_REF].

Many researchers tried to reconstruct the ballistic movement as a solution of an optimal control problem. A well-known result is the minimum jerk model proposed by Hogan in [START_REF] Hogan | An organizing principle for a class of voluntary movements[END_REF], which claims that the ballistic movement is performed in such a way that the total jerk cost along the trajectory is minimized. As it was shown in [START_REF] Richardson | Comparing smooth arm movements with the two-thirds power law and the related segmented-control hypothesis[END_REF], the minimization of the jerk cost leads to more plausible results than the use of acceleration or snap cost functions. The main drawback of the minimum jerk model is the implied symmetry of the velocity profile that is not often observed in experiments. To this end, a novel discontinuous absolute work cost function has been recently proposed in [START_REF] Berret | The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements[END_REF][START_REF] Berret | How humans control arm movements[END_REF][START_REF] Berret | Evidence for composite cost functions in arm movement planning: An inverse optimal control approach[END_REF] yielding asymmetrical velocity profile. However, this model is applied only to describe vertical arm movements in gravitation field, which is not the case for HCI pointing tasks.

The modeling of the ballistic movement as a sensorimotor feedback control for motor coordination in stochastic environment is discussed in [START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF][START_REF] Todorov | Optimality principles in sensorimotor control[END_REF]. As noted in [START_REF] Todorov | Optimality principles in sensorimotor control[END_REF], the magnitude of motor noise is proportional to muscle activation, so minimumvariance models adjust well with the minimum-jerk curves.

To model the tracking stage, the linear time-varying or -invariant models are typically applied. The crossover model [START_REF] Mcruer | A Review of Quasi-Linear Pilot Models[END_REF] is a classic model used in pilot behavior studies and is given by the transfer function

(2) C(s) := Ke -sT0 T 1 s + 1 (T 2 s + 1) (T 3 s + 1)
,

where K ≥ 0 and T i ≥ 0, i = 0, . . . , 3 are model parameters.

Another second-order model resembling a spring-mass-damper system (second order lag) was recently used in [START_REF] Müller | Dynamics of Pointing with Pointer Acceleration[END_REF] to examine pointing dynamics in reciprocal pointing with pointer acceleration.

The goal of the Vector Integration To Endpoint (VITE) model introduced in [START_REF] Bullock | Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation[END_REF] is to describe a motion controlled by an agonist-antagonist pair of muscles, e.g. wrist rotation. Considering only one muscle, the VITE model is given by

Ṗ (t) = G(t)[V (t)] + , (3) V (t) = γ (-V (t) + T -P (t)) , (4)
where P (t) is the pointer position, V (t) is the pointer velocity, T is the target (desired) position, γ > 0 is a constant, [•] + denotes a positive projection:

[V ] + = V for V ≥ 0, 0 for V < 0,
and the GO signal G(t) launching the movement execution is a positive constant or a step function.

Assuming that the considered two muscles are equivalent, the projection in (3) can be relaxed, and the VITE model yields a second-order linear time-invariant system, where the stable equilibrium point is P = T , V = 0. In [START_REF] Wolpert | An internal model for sensorimotor integration[END_REF], the authors used a modified version of the VITE model substituting G(t) ≡ 1 to analyze the sensorimotor integration in absence of a visual feedback. In robotics, the VITE model can be applied to design human-like pointing movements [START_REF] Hersch | A biologically-inspired controller for reaching movements[END_REF].

The VITE model has been extended quite recently in [START_REF] Varnell | Characteristics of Human Pointing Motions with Acceleration[END_REF] to study pointing transfer functions. In this paper, the GO signal G(t) in (3) has been substituted by a function of motion acceleration G( V ). In addition, with no feedback delay and a non-decreasing G( V ) function, the closed-loop stability has been proven.

Since for linear systems the transient time is logarithmically related with the traveled distance, then the asymptotic behavior of such models reproduces Fitts' law [START_REF] Casiez | No more bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF]. Nevertheless, these models are only able to describe the corrective phase of the pointing movement, and they cannot model the ballistic part, where the movement time is proportional to the square root of the traveled distance.

To the best of our knowledge, the first attempt to handle both ballistic and tracking phases with a single model was taken almost fifty years ago in [START_REF] Costello | The Surge Model of the Well-Trained Human Operator in Simple Manual Control[END_REF], and the result is referred to as the Surge model. This model describes the ballistic movement as a time-optimal bang-bang acceleration curve, and the tracking part is governed by the crossover model [START_REF] Casiez | The Impact of Control-Display Gain on User Performance in Pointing Tasks[END_REF]. Switching between these two models occurs when a linear combination of velocity and position error crosses a threshold. The Surge model was applied for a target following task with abrupt jumps of the target, where only qualitative comparison of the model's prediction with the measured data was given. It is also worth noting that the target following task considered in [START_REF] Costello | The Surge Model of the Well-Trained Human Operator in Simple Manual Control[END_REF] was not in the domain of HCI, thus no PTF was involved in the model design.

As discussed in [START_REF] Maggio | On the Use of Feedback Control in the Design of Computing System Components[END_REF], control theory is a powerful tool to empower the design of computer systems. Particularly, in [START_REF] Müller | Control Theoretic Models of Pointing[END_REF] an experimental comparison of some of the models described above is presented, confirming that control theory offers a promising complement to Fitts' law based approaches in HCI.

Problem statement. In this paper, our objective is the dynamic model identification of pointing motions based on an experimental data acquired with a computer mouse from [START_REF] Casiez | No more bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF], within a control theory framework. Inspired by the Optimized Initial Impulse model [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF] and the Surge model [START_REF] Costello | The Surge Model of the Well-Trained Human Operator in Simple Manual Control[END_REF], the obtained model would have hybrid or switching nature to represent both ballistic and visual tracking phases.

In addition, the model should consider the pointing transfer function (PTF) of the operating system as in [START_REF] Varnell | Characteristics of Human Pointing Motions with Acceleration[END_REF]. The resulting model can be used to design PTF that may improve the user performance in pointing tasks.

It is worth noting that some types of human movements may be considered as parts of unstable trajectories, e.g. throwing or hitting. However, for pointing motions considered in this paper it is natural to expect a trajectory with smooth velocity acceleration and deceleration at the start and the end of the movement, without abrupt changes. This natural considerations confirmed by experimental observations motivate us to search for a stable dynamical model of the pointing movement.

The abridged version of this paper appeared in [START_REF] Aranovskiy | Modeling pointing tasks in mouse-based human-computer interactions[END_REF]. The main added values of this submission are: 1) Proposition 1 of this manuscript is stronger than the one of [START_REF] Aranovskiy | Modeling pointing tasks in mouse-based human-computer interactions[END_REF]; 2) the proof of Proposition 3 is not given in [START_REF] Aranovskiy | Modeling pointing tasks in mouse-based human-computer interactions[END_REF]; 3) significantly extended experimental and numerical results; 4) a comparison against pointing models available in the literature.

The outline of this work has the following structure. In Section 2, the experimental data used in our studies is described. Then in Section 3, the proposed model and an analysis of its properties are given. The results of the experimental validation are illustrated in Section 4. Finally, in the results of experimental validation, conclusions and a discussion of future research directions are provided.

DESCRIPTION OF THE EXPERIMENTAL DATA

The model proposed in this paper is based on the experiment described in [START_REF] Casiez | No more bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF], where the authors considered a one-dimensional pointing task and compared the performance of pointing transfer functions used by default in modern operating systems (OS X, Windows and Xorg). To provide a baseline for comparison, a constant CD gain function was also considered by the authors. In this experiment, a 400 counts-per-inch corded USB Logitech mouse was used by the participants to select targets on a 23" display. The targets presented to the participants where solid vertical bars of 4 possible widths: 1 pixel (0.26mm), 3 pixels (0.77 mm), 6 pixels (1.55 mm), and 9 pixels (2.32 mm) with the distance between the bars equal to 1163 pixels (300 mm). More details about the hardware setup and experimental procedure can be found in [START_REF] Casiez | No more bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF].

To eliminate all uncertainties related to dynamically adjusted PTFs and to focus on modeling human movements, only the static gain PTF is chosen for this paper. Notice that such a static PTF is not used by default by modern operating systems [START_REF] Casiez | No more bricolage! Methods and Tools to Characterize, Replicate and Compare Pointing Transfer Functions[END_REF], and it has less motion acceleration than a PTF with dynamically adjusted gains. As a result, this choice of a PTF becomes important during the ballistic phase since this phase is not visually tracked. To bring the cursor close to the target position, some participants made a usual wrist-elbow gesture, but then they did not find the cursor where expected due to the less than usually perceived motion acceleration. Thus, they had to perform several ballistic motions before the corrective phase began. As participants adapted their gestures to the constant PTF, the number of repetitive ballistic motions reduced. An example of trajectories of non-trained, partially trained, and trained participants are given in Fig. 1, where multiple ballistic phases are observed for non-trained and partially trained participants. Further, only trained participants will be considered.

MODEL

3.1. Model overview. The proposed model structure is given in Fig. 2. The dynamics of the pointing device (here, a mouse) is described by the double integrator: where A m (t) is the mouse acceleration, V m (t) is the mouse velocity and P m (t) is the mouse position. The PTF G(•) maps the measured mouse velocity (counts/second) to the cursor velocity V c (t) (pixels/second) as

Ṗm (t) = V m (t), Vm (t) = A m (t),
V c = G(V m ).
The following assumption implies that the PTF G(•) belongs to a certain sector [k g , ∞), where k g > 0.

Assumption 1. There exists k g > 0, such that for all σ = 0, G(0) = 0 and

(5) (G(σ) -k g σ) σ > 0.
Since k g can be chosen arbitrary small, Assumption 1 is not restrictive and is reasonable for practical applications. One possible choice of G is a constant-gain PTF given by ( 6)

G(V m ) := g 0 V m ,
where g 0 > 0 is a constant representing both counts-to-pixels scaling and velocity amplification. The differential equation satisfied by the cursor position P c (t) is Ṗc (t) = V c (t). The cursor position is visually observed yielding the perceived cursor velocity V per (t) and the perceived cursor position P per (t). The perception process can be described by the stable LTI first-order filter

(7) V per (t) = 1 τ per (P c (t) -P per (t)) , Ṗper (t) = V per (t),
which is equivalent to

P per (t) = 1 τ per p + 1 P c (t); V per (t) = p τ per p + 1 P c (t), 1 p 1 p 1 p G(•) V c P c V m P m Ballistics Tracking 0 A m P per V per T m (•) T m T A bal A tr

Mouse

Cursor

Visual perception

Sensorimotor feedback where p = d dt and τ per > 0 is the perception time. The observed position P per (t) is then used to determine a switching instance of time for the mouse acceleration A m (t). In the beginning of the pointing motion, the movement is in the ballistic phase, and the mouse acceleration A m (t) equals to the accelerations provided by the ballistic dynamics, A bal (t). When the cursor reaches some predefined position P sw and the participant visually marks it, the motion switches to the corrective phase governed by the visual tracking. This switch is not instantaneous, and there exists the commutation period δ sw > 0 when zero acceleration is applied. Define t sw > 0 as the first instant of time when P per (t) = P sw . Then

A m (t) =      A bal (t) for t < t sw , 0 for t sw ≤ t ≤ t sw + δ sw , A tr (t) for t sw + δ sw < t,
where A tr (t) is the acceleration provided by the tracking dynamics.

The desired cursor position is the input reference signal T (in pixels), where for pointing tasks it is assumed that T is a step function or a constant. Based on the given value of T , the desired mouse displacement (in mm) is estimated by the participant, T m = T m (T ). This mapping is based on the user's experience with the used PTF and it can be approximated as a constant gain, [START_REF] Karimoddini | Hierarchical Hybrid Symbolic Robot Motion Planning and Control[END_REF] T m (T ) := g m T, with g m > 0.

When the value T m is set, the ballistic movement is initiated to bring the mouse cursor sufficiently close to the desired position T . If the mapping T m (•) is poor, i.e. if the participant does not have sufficient experience with the given PTF, then the resulting ballistic movement may be inaccurate, see Fig. 1a. Next we will provide accelerations models for the tracking phase A tr and the ballistic phase A bal .

Remark 1. As it is described in Section 2, only trained participants are considered in this study. It is worth to notice that for trained users each motion phase is activated just once, thus, there is no stability issue related to switching for the considered scenario.

Tracking dynamics.

The tracking dynamics of the proposed model can be seen as an extension of the VITE model ( 3), (4) taking into account the visual perception [START_REF] Hersch | A biologically-inspired controller for reaching movements[END_REF]. Consider [START_REF] Zhang | Adaptive Information Fusion for Human Upper Limb Movement Estimation[END_REF] A tr (t

) := γ (-k v V per (t) + T -P per (t)) ,
with the model parameter k v > 0. Assume that the model is in the corrective stage, that is t > t sw +δ sw , and combine together the mouse dynamics, the perception, the cursor dynamics, and the tracking law [START_REF] Zhang | Adaptive Information Fusion for Human Upper Limb Movement Estimation[END_REF]. Then the resulting closed-loop system is given by ( 10)

Ṗper(t) = 1 τper (Pc(t) -Pper(t)) , Ṗc(t) = G(Vm(t)),
Vm(t) = γ - kv τper (Pc(t) -Pper(t)) + T -Pper(t) .
Proposition 1. Consider the system (10) with γ > 0, k v > τ per > 0, and G(•) satisfying Assumption 1. Then the system is globally asymptotically stable with the equilibrium point P per = P c = T and V m = 0.

Proof. Define Pper (t) := P per (t) -T and Pc (t) := P c (t) -T . Then the tracking error model is ( 11)

Ṗc (t) = G(V m (t)), Ṗper (t) = 1 τ per Pc (t) -Pper (t) , Vm (t) = -γ α 1 Pc (t) + α 2 Pper (t) ,
where α 1 := kv τper and α 2 := 1 -kv τper . Define x tr (t) := V m (t) -Pper (t) -Pc (t) and G † (σ) := G(σ)k g σ, and note that G † belongs to the [0, ∞) sector due to [START_REF] Mcguffin | Acquisition of Expanding Targets[END_REF]. The tracking error dynamics [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF] can be rewritten as ẋtr,1 (t) = γα 2 x tr,2 (t) + γα 1 x tr,3 (t), ẋtr,2 (t) = -

1 τ per x tr,2 (t) + 1 τ per x tr,3 (t), ẋtr,3 (t) = -G(x tr,1 (t)) = -k g x tr,1 (t) -G † (x tr,1 (t)),
that is a nonlinear system in Lurie form, ( 12)

ẋtr (t) = A ss x tr (t) + B ss u(t), y(t) = C ss x tr (t), u(t) = -G † (y(t)),
where

A ss :=   0 γα 2 γα 1 0 -1 τper 1 τper -k g 0 0   , B ss :=   0 0 1   , C ss :=   1 0 0   .
The transfer function of the linear part of the system ( 12) is

W (s) = C ss (sI -A ss ) -1 B ss = γ (k v s + 1) τ per s 3 + s 2 + k g γk v s + k g γ =: B W (s) A W (s) ,
and, applying the Routh-Hurwitz stability criterion, W (s) is Hurwitz for k v > τ per . Following the Popov criterion [START_REF] Khalil | Nonlinear Systems[END_REF], the system ( 12) is absolutely stable for all G † belonging to the [0, ∞) sector if there exists µ > 0, such that

• -1 µ is not an eigenvalue of A ss ; • the transfer function (µs + 1)W (s) is strictly positive real.

The latter condition is satisfied if for all ω ∈ R

(13) Re[W (iω)] -µωIm[W (iω)] > 0.
Since A W (s) is stable, it follows that |A W (iω)| 2 > 0 for all ω, and inequality ( 13) is replaced with

Re[B W (iω)A W (iω)] -µωIm[B W (iω)A W (iω)] > 0,
where (•) is the complex conjugate of (•). The inequality [START_REF] Schmidt Ra | Motor-output variability: a theory for the accuracy of rapid motor acts[END_REF]

is thus equivalent to γ (µ (k v -τ per ) -k v τ per ) ω 4 + γ γk g k 2 v -1 ω 2 + γ 2 k g > 0.
Since k vτ per > 0, there exists

µ 0 := 1 k v -τ per k v τ per + γk g k 2 v -1 2 4γk g ,
such that the inequality ( 13) is satisfied for all µ > µ 0 . Therefore there exists µ > µ 0 satisfying the conditions of the Popov criterion, and the origin x = 0 of the system ( 12) is globally asymptotically stable.

Remark 2. It can be verified that if G(•) is chosen as (6), then the system (10) is stable exponentially.

Ballistics.

In contrast with the tracking phase, the ballistic movement is performed without visual feedback, i.e. the cursor position is not observed by the user in the ballistic phase. However, due to proprioception and sensorimotor integration, users sense the wrist displacement and velocity closing the sensorimotor loop, see Fig. 2.

Study of the available experimental data (see Section 2) shows that a typical ballistic movement of a trained participant can be approximated with the curve given in Fig. 3, where both approximation and measured data are presented.

The acceleration profile of the approximating curve consists of two constant values with a linear transient between them, see Fig. 3c. Note that this ballistic movement is not finished, i.e. the participant's wrist did not stop and the mouse velocity did not reach zero. Indeed, as it is illustrated by Fig. 1a and Fig. 1c, trained participants can notice the cursor (and switch to visual tracking) before finishing the ballistic phase.

The ballistic movement model is proposed as:

(14) Ṗm (t) = V m (t), Vm (t) = A bal (t) = f bal ( Pm (t), V m (t)),
A bal (t) =      a 1 for t < t 1 , a 1 + l(t -t 1 ) for t 1 ≤ t ≤ t 2 , a 2 for t > t 2 ,
where a 1 , a 2 and t 2 > t 1 > 0 are profile parameters, a 1 a 2 < 0, and l := a2-a1 t2-t1 is the slope of the linear part, a 1 l < 0. Define also [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF] η(t

) := -Pm (t) -k bal V m (t),
where the ballistic model parameter k bal > 0. It can be shown via some tiresome but straightforward computations that ( 16)

k bal = 2a 3 1 -6a 2 1 lt 1 + 3a 1 l 2 t 2 1 -6l 2 T m 3a 1 l(a 1 -2lt 1 )
ensures that for t ∈ [t 1 , t 2 ] the following holds:

(17) η(t) = A bal (t) p 2 A 2 bal (t) + p 1 A bal (t) + p 0 , where (18) p 0 := a 1 (a 1 -2lt 1 ) 2l 2 , p 1 := - k bal 2l
, p 2 := -1 6l 2 , and p 0 > 0, p 2 < 0. Note that there always exists a value g m , see [START_REF] Karimoddini | Hierarchical Hybrid Symbolic Robot Motion Planning and Control[END_REF], such that k bal computed as ( 16) is positive.

To construct the function f bal ( Pm , V m ) we want to inverse [START_REF] Senanayake | Targeted-Tracking With Pointing Devices[END_REF] on the interval [a l , a r ], where a l := min(a 1 , a 2 ) < 0 and a r := max(a 1 , a 2 ) > 0. Equality [START_REF] Senanayake | Targeted-Tracking With Pointing Devices[END_REF] defines the implicit function

h(A bal , η) = A bal (p 2 A 2 bal + p 1 A bal + p 0 ) -η = 0.
Applying the Implicit Function theorem, the desired inversion exists if the partial derivative [START_REF] Gan | Geometrical conditions for ballistic and visually controlled movements[END_REF] ∂h(A bal , η)

∂A bal = 3p 2 A 2 bal + 2p 1 A bal + p 0
has no root in the interval [a l , a r ]. The right-hand side of ( 19) is a quadratic polynomial with a negative leading coefficient. Its roots are

s l,r := -k bal l ± a 2 1 + k 2 bal l 2 -2t 1 a 1 l.
Recalling a 1 l < 0, it follows that for any value k bal the roots are real and different; without loss of generality we assume s l < s r . The following assumption is instrumental to proceed. Assumption 2. The parameters t 1 , a 1 , a 2 and l are such that the inequality

t 1 > a 2 1 -a 2 2 2a1l holds. Define (20) k := a 2 1 -a 2 2 -2t 1 a 1 l 2a 2 l ,
where k > 0 due to Assumption 2. It can be verified that any choice of 0 < k bal < k yields s l < a l < 0 < a r < s r , i.e. on the interval [a l , a r ] ⊂ [s l , s r ], ∂h(A bal ,η) ∂A bal > 0 and ( 17) is invertible. Define the inversion of ( 17) on the interval [a l , a r ] as A bal = φ(η), and denote by η l and η r the solutions of equations h(a l , η l ) = 0 and h(a r , η r ) = 0, respectively. Now we can formulate the following proposition, whose proof follows from the derivations above.

Proposition 2. Let Assumption 2 be satisfied and choose 0 < k bal < k, with k as in [START_REF] Oel | Time prediction of mouse-based cursor movements[END_REF]. Then φ(η) for η ∈ [η l , η r ] is a uniquely defined real-valued monotonic function with φ(η l ) = a l , φ(0) = 0 and φ(η r ) = a r .

Remark 3. From a practical point of view, both Assumption 2 and the inequality k bal < k are not restrictive. The Assumption 2 implies that the mouse velocity does not return to zero before t 2 . As it is illustrated in Fig. 3, this assumption is reasonable for practical ballistic movements and it was satisfied for all the studied experimental trajectories. The inequality k bal < k limits how large the ballistic movement can be made restricting the value of T m , see [START_REF] Chen | Structure of Hand/Mouse Movements[END_REF]. The practical meaning of this inequality is that given the values of a 1 , t 1 , and l, the ballistic phase cannot achieve an arbitrary far point. Now the desired function f bal ( Pm , V m ) can be constructed. Choosing k bal , η l , η r and φ(η) as suggested by Proposition 2 and recalling the definition [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF], define [START_REF] Hogan | An organizing principle for a class of voluntary movements[END_REF] f bal (η) :=

     a 2 for η < η l , φ(η) for η l ≤ η ≤ η r , a 1 for η > η r .
Note that the function f bal (η) has the following properties: i) it is a continuous nondecreasing function, ii) it is strictly monotonic on [η l , η r ], and iii) f bal (0) = 0. The function f bal (η) can be reconstructed based on an experimental trajectory data, and an example of such a function is given in Fig. 4.

To summarize, the model of the ballistic phase is given by ( 14), ( 15) and [START_REF] Hogan | An organizing principle for a class of voluntary movements[END_REF]. Stability of this model is established by the following proposition. Proposition 3. Let Assumption 2 be satisfied and consider the system (14), ( 15) and (21) with 0 < k bal < k. All the trajectories of the system are bounded, and the equilibrium point Pm = V m = 0 is globally asymptotically stable.

Proof. Define x(t) := [ Pm (t), V (t)] . The system ( 14), ( 15), ( 21) can be rewritten as ẋ1 (t) = x 2 (t),

ẋ2 (t) = f bal (η(t)), η(t) = -x 1 (t) -k bal x 2 (t),
which is a Lurie system. Note that due to the properties of f bal (η) the following holds ηf bal (η) > 0 for η = 0. Define a positive definite Lyapunov function

W (x) := 1 2 x 2 2 + η(x) 0 f bal (s)ds.
Its time derivative is (the argument of time is omitted)

dW (x) dt = x 2 ẋ2 + ηf bal (η) = x 2 f bal (η) + (-x 2 -k bal f bal (η)) f bal (η) = -k bal f 2 bal (η) ≤ 0.
Using standard Lyapunov arguments we conclude that η(t), x 2 (t), and so x 1 (t), are bounded, and η(t) → 0 as t → ∞. Note that the origin x = 0 is the only invariant solution in the set {x | η(x) = 0}. Applying LaSalle's Invariance Principle, the global asymptotic stability of the equilibrium point x = 0 follows. Propositions 1 and 3 ensure that as long as the model parameters are chosen to satisfy the assumptions, all trajectories generated by the proposed switched model will remain bounded and will converge to the target. From this point of view, these propositions may be considered as defining the set of feasible parameters.

EXPERIMENTAL VALIDATION

In this section, we present the experimental verification of the proposed switched model. From the considered dataset (see Section 2) we have chosen 7 trained participants and 3 trajectories for each of them, that is 21 trajectories in total. All the trajectories correspond to the constant-gain PTF (6) having 1.5 times velocity amplification. The mouse has 400 counts per inch, and the display admits resolution 98.5 pixels per inch. Therefore, the gain of the used PTF equals g 0 = 1.5 98.5 400 , providing both counts-to-pixels scaling and velocity amplification. The measurements are performed with the sampling frequency 125Hz.

For each of the chosen trajectories we have identified parameters of the proposed model; see Table 1 for a list of the parameters. The identification was performed as follows. First, we separate the full movement into ballistic, switching and tracking phases, where the switching phase is defined as an interval with zero acceleration. Second, for the ballistic part we tune the parameters listed in the first row of Table 1 with a nonlinear optimization routine in order to minimize a quadratic approximation error cost function. Next, for the tracking part we use classic linear system identification tools to tune the parameters listed in the last row of Table 1. Finally, we slightly adjust the switching time instance to match the velocity measured during the switching interval. Figures 5,6, and 7 present a comparison of the proposed model outputs with the measured data. Three types of trajectories are considered, where the first one corresponds to a typical pointing movement, the second trajectory type is a movement with an overshoot due to a long ballistic phase, and the third trajectory presents a relatively short ballistic movement. As it can be seen from the figures, all these different types of pointing motions are handled by the proposed model fitting well the measurements.

The same 21 trajectories are also used to identify and evaluate two alternative models: the ballistic model given by the minimum jerk principle [START_REF] Hogan | An organizing principle for a class of voluntary movements[END_REF], and a linear tracking model.

The minimum jerk principle claims that the ballistic cursor movement can be described with the curve P c (t) minimizing the total jerk cost functional [START_REF] Richardson | Comparing smooth arm movements with the two-thirds power law and the related segmented-control hypothesis[END_REF] J [P c ] := where the coefficients a 0 , a 1 and a 2 are such that the transfer function is stable, and a 3 ≥ 0. These coefficients are tuned with standard linear system identification tools.

It is worth noting that the general structure [START_REF] Berret | How humans control arm movements[END_REF] can incorporate the tracking model ( 10), the crossover model [START_REF] Mcruer | A Review of Quasi-Linear Pilot Models[END_REF], the VITE model [START_REF] Bullock | Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation[END_REF] and the VITE-like model [START_REF] Varnell | Characteristics of Human Pointing Motions with Acceleration[END_REF] with a constant-gain PTF. Fig. 9 presents comparison of the measured data with the linear tracking model outputs for a typical pointing movement. The curve predicted with the linear model has smaller deviations from the measurements than the ballistic model [START_REF] Berret | The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements[END_REF]. However, as expected, the switching velocity profile cannot be reconstructed with the model [START_REF] Berret | How humans control arm movements[END_REF]. For quantitative evaluation of the models we use the following metrics:

• Mean Absolute Error (MAE) is computed as This value varies from zero to infinity, where zero means perfect fit, and if NRMSE equals to 1, then P c,pred matches P c,meas no better than a straight line. These metrics are computed for each of the chosen 21 trajectories, the minimum, maximum and average values among these trajectories are given in Table 2 for the ballistic model ( 23), linear tracking model [START_REF] Berret | How humans control arm movements[END_REF], and for the proposed switched model given in Fig. 2. The histograms of the MAE, ME and NRMSE metrics are shown in Figs. 101112. The results illustrate that the proposed switched model significantly outperforms ballistic and linear ones.

M AE := 1 N P c,meas -P c,pred 1 ,

CONCLUSIONS

In this work, the problem of modeling a pointing task with a computer mouse was studied. Based on the Optimized Initial Impulse model [START_REF] Meyer | Optimality in human motor performance: ideal control of rapid aimed movements[END_REF] and on the Surge model [START_REF] Costello | The Surge Model of the Well-Trained Human Operator in Simple Manual Control[END_REF], the pointing movement was separated into three distinct steps: a ballistic phase, followed by a commutation phase with no acceleration, finished with a tracking phase. The obtained model is then a switching system. In the first stage, there is no visual guidance to the user, only a sensorimotor feedback is available, and a nonlinear system of Lurie form is used to model this part. The commutation phase happening when the approach of the final cursor position is perceived by the user, it triggers the switching to the tracking phase. In this last phase, thanks to the visual perception of the cursor position by the user, an extended VITE model [START_REF] Bullock | Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation[END_REF] is used to model the tracking dynamics (a linear filter is applied to model the visual perception). We have shown that both ballistic and tracking dynamics are globally asymptotically stable under some established mild conditions. In this note, only trained users were considered, so each phase was activated only once, therefore the problem of instability caused by commutation was not analyzed here.

A series of experiments were performed to validate the model showing that it fits well different types of pointing movements and outperforms pure ballistic and tracking models. This first dynamical model for pointing tasks presented in our work, opens naturally new research directions. In the paper, the experimental validation has been performed for static-gain PTFs only. More realistic PTFs depending both on cursor velocity and position are an expected extension of our work. In addition, another extension would include the analysis of non-adapted users behavior that will naturally demand an extended stability analysis in the presence of switching. Moreover, the effect of delays on the closed-loop stability and the pointing task performance itself should be analyzed.

Another interesting research direction is to consider the proposed model from the biomechanical prospective and compare it with the dynamic models for wrist motions [START_REF] Ramsay | Muscle moment arm and normalized moment contributions as reference data for musculoskeletal elbow and wrist joint models[END_REF][START_REF] Lemay | A dynamic model for simulating movements of the elbow, forearm, and wrist[END_REF] to see possible connections, e.g. velocity and acceleration limitations.

Finally, an automatic tuning of the interface to match the user type should be studied in order to identify users in terms of their experience by applying the proposed model. Further works may also be directed towards the proposition of a somehow optimal PTF design given a user model, and the extension of the approach to other pointing devices.

Figure 1 .

 1 Figure 1. Pointing trajectories for differently trained participants, a constant-gain PTF. Cursor position (pixels) versus time (seconds).

Figure 2 .

 2 Figure 2. The proposed model structure.

Figure 3 .

 3 Figure 3. Approximation of the measured mouse trajectory for an experienced participant.

Figure 4 .

 4 Figure 4. The reconstructed f bal (η) function.

Figure 5 .

 5 Figure 5. Comparison of the proposed model output with the measured data, trajectory #1.

  t f is the time instance when the movement stops. The Euler-Lagrange equation for (22) yields d 6 dt 6 P c (t) ≡ 0 and given the boundary conditions P c (0) = P (1) c (0) = P (2) c (0) = 0 the desired curve is P c (t) = c 5 t 5 + c 4 t 4 + c 3 t 3 . The constants c 3 , c 4 and c 5 can be computed given the boundary conditions P

( 1 )

 1 c (t f ) = P

( 2 )

 2 c (t f ) = 0 and P c (t f ) = T , where the values t f and T are specific for the exact trajectory.

  Velocity (in pixels per second) versus time (in seconds).

Figure 6 .

 6 Figure 6. Comparison of the proposed model output with the measured data, trajectory #2.

Figure

  Figure 7. Comparison of the proposed model output with the measured data, trajectory #3 [36].

Figure 8 .

 8 Figure 8. Comparison of the ballistic model (23) output with the measured data, trajectory #1.

Figure 9 .

 9 Figure 9. Comparison of the linear tracking model (24) output with the measured data, trajectory #1.

Figure 10 .

 10 Figure 10. Histogram of the MAE metric.

Figure 11 .

 11 Figure 11. Histogram of the ME metric.

Figure 12 .

 12 Figure 12. Histogram of the NRMSE metric.

Table 1 .

 1 Model parameters.

				Ballistics a1, a2, t1, t2, gm, k bal
				Switching	Psw, δsw
				Tracking	τper, kv, γ
	1200				3500			Measurements
	1000				3000			Model output
	800				2500
					2000
	600				1500
	400				1000
	200			Measurements	500
	0 0	0.2	0.4	0.6 Model output 0.8	0 0	0.2	0.4	0.6	0.8
		(a) Position (in pixels)	(b) Velocity (in pixels
		versus time (in seconds).	per second) versus time
					(in seconds).

  7. Comparison of the proposed model output with the measured data, trajectory #3[START_REF] Aranovskiy | Modeling pointing tasks in mouse-based human-computer interactions[END_REF]. + a 0 s 3 + a 2 s 2 + a 1 s + a 0 ,

	1200		3500	Measurements
	1000		3000	Model output
	800		2500
			2000
	600
			1500
	400		1000
	200	Measurements	500
	Straightforward computations yield (23) P c (t) = T 6 t t f Fig. 8 presents comparison of the measured data with the ballistic model outputs 5 -15 t t f 4 + 10 t t f 3 . for a typical pointing movement. One can see that the curve predicted with the minimum jerk ballistic model has significant deviations from the measured one, and this model cannot reconstruct the measured velocity profile. As a linear tracking model we use the 3rd order transfer function from the input T to the output P c (24) W trc (s) := 0.2 0.4 0.6 0.8 0 Model output (a) Position (in pixels) versus time (in seconds). 0 0.2 0.4 0.6 0.8 0 (b) Velocity (in pixels per second) versus time a 3 s 0 (in seconds).

Table 2 .

 2 Models performance comparision. N the number of measurements, P c,meas the measured cursor position, and P c,pred the cursor position predicted with a model under evaluation; • Maximum Error (ME) is computed as M E := max |P c,meas -P c,pred |; • Normalized Root Mean Square Error (NRMSE) is computed as N RM SE := P c,meas -P c,pred 2 P c,meas -mean{P c,meas } 2 .

			MAE
		min. max. avg.
	Ballistic model	46.9 242.9 109.1
	Linear model	3.4	27.1	9.3
	Switched model	1.1	6.1	3.0
			ME
		min. max. avg.
	Ballistic model 119.8 621.7 280.0
	Linear model	9.0	74.4	26.7
	Switched model	4.3	26.5	11.1
			NRMSE
		min. max. avg.
	Ballistic model 0.142 0.737 0.334
	Linear model	0.010 0.083 0.029
	Switched model 0.004 0.022 0.010

where

This work was supported by ANR (TurboTouch, ANR-14-CE24-0009). Stanislav Aranovskiy is supported by the Russian Science Foundation grant (project 17-79-20341).


Dr Guillaume Tcherkez
email: guillaume.tcherkez@u-psud.fr

Richard Bligny

Elizabeth Gout

Aline Mahé

Michael Hodges

Gabriel Cornic

Classification: Biological sciences/Plant biology Respiratory metabolism of illuminated leaves depends on CO 2 -and O 2 -conditions

Day respiration is the process by which non-photorespiratory CO 2 is produced by illuminated leaves. The biological function of day respiratory metabolism is a major conundrum of plant photosynthesis research: since the rate of CO 2 -evolution is partly inhibited in the light, it is viewed as either detrimental to plant carbon balance or necessary for photosynthesis operation (e.g. in providing cytoplasmic ATP for sucrose synthesis). Systematic variations in day respiration rate under contrasting environmental conditions have been used to elucidate the metabolic rationale of respiration in the light. Using isotopic techniques, we show that both glycolysis and the TCA cycle activities are inversely related to the ambient CO 2 /O 2 ratio: day respiratory metabolism is enhanced under high photorespiratory (low-CO 2 ) conditions. Such a relationship also correlates with the DHAP/Glc-6-P ratio, suggesting that photosynthetic products exert a control on day respiration. Thus, day respiration is normally inhibited by phosphoryl (ATP/ADP) and reductive (NADH/NAD) poise, but up-regulated by photorespiration. Such an effect may be related to the need for NH 2 -transfers during the recovery of photorespiratory cycle intermediates.

Introduction

It is now seventy years since Krebs and Johnson proposed the mechanism by which pyruvic acid is oxidised to CO 2 , and now called the 'Krebs cycle' or TCA cycle (1,2). While the basics of the metabolic reactions involved in leaf respiration are known, intense efforts are still currently devoted to elucidating the regulation of the TCA cycle (and, more generally, of day respiration) in illuminated leaves (for a recent review, see ref. 3).

Leaf day respiration (non-photorespiratory CO 2 evolution in the light) is an essential metabolic pathway that accompanies photosynthetic CO 2 assimilation and photorespiration. It is widely accepted that leaf respiration is partly inhibited in the light when compared to darkness (4). This acceptance is based on several strong lines of evidence, ranging from gasexchange to molecular studies (for a review, see ref. [START_REF] Atkin | Photosynthesis, Physiology and Metabolism[END_REF]): (i) the inhibition is thought to cause the light-enhanced dark respiration [START_REF] Cornic | [END_REF]; (ii) the pyruvate dehydrogenase (PDH) is downregulated in the light [START_REF] Budde | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Tovar-Mendez | [END_REF]; (iii) the metabolic flux through the TCA cycle in the light is reduced in both extracted mitochondria (8) and intact leaves (9,10); (iv) mitochondria experience high ATP/ADP and NADH/NAD + ratios in the light that inhibit NAD-dependent isocitrate dehydrogenase (11); and (v) carbohydrate molecules such as sucrose and glucose are prevented from entering glycolysis (9), due to a modification of phosphofructokinase activity by the allosteric effector Fru-2,6-bisphosphate (12). Nevertheless, not all leaf cells are photosynthetic (e.g. most epidermal cells, phloem and xylem) so that some 'heterotrophic' background respiration in the light is expected, but its contribution is minor.

Although inhibited by light, day respiration is critical for plant growth and leaf N assimilation, as it provides ATP for Suc synthesis and TCA cycle intermediates (e.g. 2oxoglutarate and oxaloacetate) for ammonium assimilation and amino-acid synthesis (13).

Thus for many years, the down-regulation of day respiration and of the TCA-cycle in the light has been viewed as a perplexing phenomenon. It may be argued that the partial inhibition of day respiration comes from a balance between two metabolic constraints: the energy requirement for Suc synthesis and the minimal competition between glycolysis and Suc synthesis for improved carbon gain. However, day respiration is additionally affected by other metabolic processes such as O 2 assimilation (photorespiration) (14), the rate of which depends on the internal CO 2 /O 2 ratio. The photorespiratory cycle leads to Gly oxidative decarboxylation in the mitochondrion that supposedly gives rise to a large NADH/NAD + ratio, that in turn inhibits certain respiratory mitochondrial enzymes in vivo (11).

Photorespiration is thus assumed to down-regulate day respiratory CO 2 -evolution.

Nevertheless, increased photorespiration rates could require more Glu cycling to provide amino groups for Gly synthesis in the peroxisomes. This higher demand might in turn require an increase in 2-oxoglutarate and Glu synthesis, and thus a higher day respiratory rate. The rationale of the metabolic homeostasis between day respiration and photorespiration are therefore currently uncertain.

To clarify the regulation of day respiration in illuminated leaves and its interactions with photorespiration, we have investigated the effect of the carboxylation-to-oxygenation ratio on day respiratory metabolic fluxes using isotopic 12 C/ 13 C spectrometry and 13 C and 31 P nuclear magnetic resonance (NMR). The results presented in this paper show that, while respiratory CO 2 evolution is always inhibited in the light when compared to the dark, the metabolic flux associated with the TCA cycle is inversely related to net CO 2 assimilation and this correlates with changes in phosphorylated metabolites levels. In addition, 13 C-distribution after labeling shows a larger commitment towards TCA intermediates and Glu as photorespiration increases. These findings, which are consistent with a role of day respiration in sustaining photorespiratory N cycling and perhaps nitrate assimilation, have important implications, ranging from the improvement of nitrogen use efficiency to the understanding of leaf and global ecosystem carbon budgets.

Results

In order to determine the amplitude and the steps of the leaf respiratory pathway that are inhibited in the light, detached leaves were fed with positionally labeled 13 C-enriched substrates (Pyr or Glc) and decarboxylation rates were measured by gas-exchange coupled to isotopic spectrometry. This method allowed us to calculate the decarboxylation rate in the light and, by comparing with the rate in darkness, the inhibition of decarboxylation in illuminated leaves. The positional labeling in Pyr allowed us to discriminate between the CO 2 produced by either the PDH ( 13 C-1-Pyr, 13 C-3-Glc) or the TCA cycle ( 13 C-2-Pyr, 13 C-1-Glc).

TCA-mediated decarboxylations are enhanced under low CO 2 /O 2

When leaves were fed with 13 C-enriched Pyr, the apparent carbon isotope discrimination ∆ obs increased, showing that 13 CO 2 was produced (Fig. 1A). Interestingly, the decarboxylation of 13 C-2-Pyr was low compared to that of 13 C-1-Pyr, showing the predominance of CO 2 produced by PDH as opposed to that produced by the TCA cycle. In addition, the smaller the CO 2 /O 2 ratio, the larger the ∆ obs associated with 13 C-1-Pyr. A similar but modest trend occurred with 13 C-2-Pyr (Fig. 1A). Dark-respired CO 2 was 13 C-enriched compared to the natural abundance (δ 13 C of -22.1±0.5‰) with both 13 C-1-and 13 C-2-Pyr, indicating that decarboxylation of Pyr was substantial (Fig. 1B). The decarboxylation of 13 C-1-Pyr in the dark was larger after a light period at low CO 2 (140 µL L -1 ) compared to a high CO 2 . Both 13 C-1-and 13 C-2-Pyr were more decarboxylated in the dark after leaves had been exposed to light under low-O 2 conditions (Fig. 1B), and this was sensitive to O 2 -conditions in the dark.

The isotopic data shown in Figure 1 were used to calculate the decarboxylation rates (Fig. 2) associated with PDH and the TCA cycle using mass-balance equations (see Materials and Methods). While the decarboxylation rate associated with PDH was inhibited by only 30% in the light (Fig. 2), TCA cycle-mediated decarboxylations were much lower in the light than in the dark, with an inhibition of around 80% under typical atmospheric conditions (400 µL L -1 CO 2 in 21% O 2 ). The inhibition of the PDH-mediated decarboxylation was relatively constant under the CO 2 and O 2 conditions investigated. In contrast, TCA-mediated decarboxylations were much more sensitive to low CO 2 conditions, with 35% inhibition only at 140 µL L -1 CO 2 in 21% O 2 (Fig. 2C). It should be noted that while the inhibition value associated with the TCA cycle was similar in 400 µL L -1 CO 2 2% O 2 and in 400 µL L -1 CO 2 21% O 2 , the absolute decarboxylation value was larger in 2% O 2 both in the dark and in the light (Fig. 2). This effect was due to a higher stomatal conductance in 2% O 2 and subsequently, larger transpiration rates that induced a higher absorption of labeled compounds (data not shown). Such a positive effect of low O 2 conditions on stomatal conductance has already been observed in Xanthium strumarium (15).

The commitment to glycolysis is enhanced under low CO 2 /O 2

Similar experiments were carried out with 13 C-enriched Glc to determine whether the glycolytic carbon flow changes under varying CO 2 /O 2 conditions. Under typical conditions (400 µL L -1 CO 2 in 21% O 2 ), leaves fed with 13 C-1-Glc or 13 C-3-Glc did not produce significant amounts of 13 CO 2 in the light, as indicated by the very small deviation of the apparent carbon isotope discrimination ∆ obs (Table S1 of the Supporting Information). The same applied to high CO 2 -to-O 2 conditions (1000 µL L -1 CO 2 in 21% O 2 ), with a ∆ obs value of 24.5±0.5‰. In contrast, the decarboxylation of 13 C-1-Glc and 13 C-3-Glc became apparent in the light under low CO 2 conditions (140 µL L -1 in 21% O 2 ), with ∆ obs values increasing up to 106.4±13.8‰ (with 13 C-3-Glc, Table S1). This indicated that 13 C-Glc could be oxidised in the light by glycolysis and that PDH and TCA activities were both at the origin of the respired 13 CO 2 from 13 C-Glc. It should be noted that this increase was not an artefact caused by the low CO 2 mole fraction in the chamber (making decarboxylated CO 2 proportionally larger), because the CO 2 mole fraction was taken into account in the mass-balance-based calculations.

In darkness, the δ 13 C value of respired CO 2 increased to 365‰ (with 13 C-3-Glc) after photosynthesis at 400 µL L -1 CO 2 , and 719‰ (with 13 C-3-Glc) after photosynthesis at 140 µL S1). When these values were used to calculate decarboxylation values, it was found that both TCA-and PDH-mediated decarboxylations were inhibited by nearly 90% at 400 µL L -1 CO 2 , and 60% at 140 µL L -1 CO 2 in the light (Table S2 of the Supporting Information). These data show that leaf CO 2 levels modulate the entry of Glc molecules into the glycolytic and respiratory pathways both in the light and the dark; Glc is a better respiratory substrate at low CO 2 levels.

L -1 CO 2 in 21% O 2 (Table

Distribution of the 13 C-label in metabolites

To gain information on the changes in metabolic pathways under the different CO 2 /O 2 conditions in the light, the fate of the 13 C-atoms (from 13 C-substrate feeding) was determined in leaf metabolites by 13 C-NMR analyses. Leaves were fed with positionally-enriched (99% 13 C) substrates under either 140, 400, or 1000 µL L -1 CO 2 in 21% O 2 and the positional isotopic abundances of identified metabolites (in % of 13 C), measured by NMR, are displayed as an isotopomics array (Fig. 3). As expected, hexoses and Glc-or Fru-moieties of Suc were 13 C-labeled when 13 C-Glc was supplied to leaves so that several C-1 and C-6 positions formed clusters (these positions are redistributed by aldolase and triose-phosphates isomerase reactions). The C-4 and C-5 positions in hexoses clustered near to the C-3 positions, indicating that a redistribution of 13 C label occurred in the light through the pentosephosphate pathway.

The flux through the pentose-phosphate cycle may be estimated with both NMR and gas-exchange data. At 140 µL L -1 CO 2 , the 13 C-amount in the C-atom positions redistributed by the pentose-phosphate cycle (i.e. the 13 C-amount in C-2, C-3, C-4 and C-5 after 13 C-1-Glc labeling and the 13 C-amount in C-1, C-2, C-5 and C-6 after 13 C-3-Glc labeling), as found by NMR after labeling, corresponds to a 13 C flux of 0.03 µmol m -2 s -1 . With the gas-exchange data, the flux through the pentose-phosphate cycle can be estimated by the excess of calculated CO 2 production from the TCA with respect to the PDH, because the pentosephosphate cycle involves the decarboxylation of the C-1 atom of Glc. With 13 C-1-Glc labeling, The TCA-mediated decarboxylation rate was 0.11 µmol m -2 s -1 while the PDHmediated decarboxylation was 0.07 µmol m -2 s -1 (Table S2 of the Supporting information).

The CO 2 production by the pentose-phosphate cycle was thus of 0.04 µmol m -2 s -1 , a value that is very close to that obtained with NMR. Such a value is nevertheless not enough to explain the entire 13 CO 2 production from 13 C-Glc at low CO 2 , so that the enhancement of the commitment to glycolysis still holds under this condition.

The commitment to the TCA cycle can be assessed with 13 C-labeling in organic acids. Some organic-and amino-acids clustered on the upper part of Fig. 3, with positional 13 Cenrichment generally lower than 10%. However, the C-2 atom of malate and the C-2 and C-3 atoms of Glu were clearly labeled when 13 C-Pyr was supplied. This labeling was lower at high CO 2 , and the same applied to the C-2 and C-3 atoms of fumarate. Since these C-atom positions in malate, Glu and fumarate can only be labeled by the interplay (redistribution) of the TCA cycle, this labeling trend is indicative of an increase in TCA cycle activity. This view agrees with the increase of the 13 C-abundance in Glu, succinate and fumarate from high to low CO 2 /O 2 ratios as shown in Fig. 4 (where 13 C-abundances are relative to those found in malate in order to take into account variations in refixed decarboxylated 13 CO 2 by phosphoenolpyruvate carboxylase (PEPC) activity, as discussed below).

Nevertheless, the lower 13 C-abundance in organic acids under high CO 2 conditions may also partly be due to the diluting effect of assimilated 12 C-enriched carbon (inlet CO 2 has a δ 13 C value near to -50‰). This effect is expected in malate and fumarate, which are two major metabolites accumulated by Xanthium leaves. There was indeed a significant and negative correlation between the isotopic enrichment in fumarate C-2/3 and malate C-2 and the quantity of these organic acids (Fig. 4, inset). However, the correlation was reversed in the case of Glu C-2 (Fig. 4, inset) while there was no correlation at all with other organic acids or amino-acids (data not shown). Therefore, we conclude that the lower 13 C-enrichment in organic-and amino-acids at high CO 2 conditions was not only caused by an isotopic dilution but also by the decrease of the commitment of 13 C-Pyr to the TCA cycle.

CO 2 refixation

While the trend of 13 C-labeling in TCA cycle intermediates is clear, the rate of decarboxylation of 13 C-enriched substrates in the light and the 13 C-labeling of different metabolites may have been adulterated by 13 CO 2 refixation by either PEPC or Rubisco. Indeed, 13 CO 2 fixation by PEPC occurred under each CO 2 condition investigated in the present study, as revealed by the 13 C-enrichment in the C-4 of malate when 13 C-1-Pyr or 13 C-2-Pyr was supplied to leaves. However, the C-2 and C-3 positions in malate were also clearly 13 C-enriched under low CO 2 conditions when fed with 13 C-2-Pyr (Fig. 3), and this observation is consistent with an increased TCA cycle activity.

Metabolites could have been also 13 C-labeled via photosynthetic 13 CO 2 refixation. The labeling of hexoses and Suc in the C-3 position after Pyr feeding (refixation of 13 CO 2 decarboxylated from 13 C-1-Pyr by Rubisco) did occur but the maximum positional 13 Cabundance in C-3 indicates that the proportion of refixed 13 C in the whole molecule was about 0.7% only. Refixation of respired CO 2 into starch was also assessed after gas-exchange and on-line apparent ∆ obs measurements (Fig. 1): the carbon isotope composition (δ 13 C) of starch was between -36.5 (minimum value obtained at 1000 µL L -1 CO 2 , with 13 C-1-Pyr feeding) and -12.7‰ (maximum value obtained at 140 µL L -1 CO 2 with 13 C-1-Pyr feeding), indicating that the proportion of starch containing refixed CO 2 was between 0 and 0.4% only (see Table S3 of the Supporting Information). Therefore, although Rubisco-mediated refixation of decarboxylated 13 CO 2 did occur, it was always very low.

Correlations with phosphorylated metabolites

Phosphorylated metabolites, such as dihydroxyacetone phosphate (DHAP), are known to be important regulators of the primary carbon metabolism of plant leaves (12,16). Therefore, we measured the amounts of phosphorylated compounds by 31 P-NMR spectroscopy on the same samples used for the 13 C-labeling and 13 C-NMR analyses. There was a clear linear and positive correlation between the activity of the TCA cycle, as witnessed by gas-exchange (values from Fig. 2), and the DHAP-to-Glc-6-phosphate ratio (Fig. 5, dashed line). In contrast, there was no statistically significant correlation with the DHAP/Pi ratio although a positive trend was apparent (Fig. 5, dotted line).

Discussion

Based on data from gas-exchange analyses to enzymatic activities [START_REF] Budde | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Tovar-Mendez | [END_REF]8,9), it has become widely accepted that leaf day respiration (non photorespiratory CO 2 evolution in the light) is inhibited in the light (for a review, see 3). However, the rationale and the effect of environmental conditions on day respiration and its inhibition are uncertain and solving such an issue is critical to understand how leaves adapt carbon partitioning between export of photosynthates and respiration or N economy under varying natural conditions. CO 2 and O 2 levels are two parameters of fundamental importance because the leaf internal CO 2 /O 2 ratio changes when environmental conditions alter stomatal closure (e.g. drought). Here, we have developed isotopic methods to provide evidence that respiratory metabolism is up-regulated when the CO 2 /O 2 ratio decreases, and we argue that day respiration is an exquisite example of metabolic compromise between feedback inhibition by NADH and ATP, and 2-oxoglutarate precursor requirement for N metabolism.

The regulation of the TCA cycle

While no major effect on PDH-decarboxylation was observed, the carbon flow through the TCA cycle increased under low CO 2 /O 2 conditions, as evidenced by the larger decarboxylation rate of 13 C-2-Pyr, so that the inhibition by light of the TCA cycle was 40% at 140 µL L -1 CO 2 as compared to nearly 90% under typical conditions (400 µL L -1 CO 2 , Figs. 1 and2). Furthermore, the 13 C-labeling of succinate, citrate and Glu, as revealed by NMR tracing after 13 C-2-Pyr feeding, increased as the CO 2 mole fraction decreased (Fig. 4). In this context, the observed labeling in malate (Fig. 3), that indicated the simultaneous increase in PEPC activity, comes as no surprise: this is in agreement with the anapleurotic role of this enzyme, that compensates for 2-oxoglutarate consumption (for Glu synthesis) by feeding the TCA cycling with oxaloacetate molecules (17). The whole picture is thus consistent with an increased commitment to the TCA cycle and Glu production under low CO 2 conditions.

The up-regulation of the TCA metabolism under low CO 2 stems from a larger glycolytic carbon input, as evidenced by the enhancement of 13 C-3-Glc decarboxylation (see the Result section and Table S1 andS2) and the slight 13 C-labeling in malate C-2 after 13 C-Glc feeding (Fig. 3). In addition, as a consequence of the lower photosynthetic CO 2 fixation rate, the DHAP/Glc-6-P ratio decreased with CO 2 mole fraction, and importantly, this ratio was strongly correlated with the inhibition of the TCA cycle (Fig. 5). The relative abundance of triose phosphates is already known to be a metabolic parameter that controls carbon entry to glycolysis by the interplay of the effector Fru-2,6-bisphosphate (12), and promotes pyruvate kinase activity (18). It thus appears clear that it also controls the commitment of carbon molecules to the respiratory pathway. Uncertainty nevertheless remains about whether it acts indirectly on the TCA cycle (through the enhancement of glycolysis) or not.

Interactions with photorespiration

The higher carbon flow through the TCA cycle under low CO 2 may appear somewhat paradoxical as it has often been supposed that large photorespiratory rates inhibit mitochondrial respiratory enzymes (11,19), such as the NAD-dependent isocitrate dehydrogenase (11). Accordingly, Gly decarboxylase antisense lines of potato (Solanum tuberosum) have lower decarboxylation rates in the light (as revealed by 14 C-labeling experiments) and the ATP/ADP as well as the NADH/NAD + ratios are both higher than in the wild type (20). In addition, the predominance of NADH production by Gly oxidation over that by the TCA cycle has been shown using isolated mitochondria under ADP-limiting conditions (21). This might reduce the NAD + available for the mitochondrial dehydrogenase steps of the TCA cycle. Our results show that the inhibition of day respiration occurs whatever the CO 2 /O 2 ratio and furthermore, there is a lower inhibition value under very low-O 2 conditions (400 µL L -1 CO 2 , 2% O 2 ) as compared to high-CO 2 conditions (1000 µL L -1 CO 2 , 21% O 2 ) (Fig. 2). This would be consistent with a much reduced mitochondrial redox poise caused by the non-physiological, very low oxygen mole fraction. However, we show here that the inhibition of the TCA cycle is relaxed under low-CO 2 conditions in 21% O 2 (see above). Isocitrate production is probably not influenced by such photorespiratory conditions, as citrate synthase remains active because of the very large K i (ATP) (5 mmol L -1 ) and the absence of any NADH effect (22). As the mitochondrial NADdependent isocitrate dehydrogenase is believed to be inhibited in the light, we suggest that isocitrate is processed by the cytosolic or mitochondrial NADP-dependent isocitrate dehydrogenase (23). This bypass would allow to sustain the necessary Glu flow under photorespiratory conditions.

Possible rationale

The regulation of the TCA cycle by the CO 2 /O 2 conditions may be viewed as a side effect of the drop in the DHAP relative quantity on the commitment to glycolysis and therefore respiration (see above and Fig. 5). Unless there are other prevailing imperatives (such as the need of NADH to reduce photorespiration-derived hydroxypyruvate or H 2 O 2 ), it also reflects an increased need for Glu to feed photorespiratory N-recycling when conditions shift to low CO 2 mole fractions, simply because the production of Gly from glycolate would require a higher Glu flow. Unsurprisingly then, the production of the Glu-precursor 2-oxoglutarate by the TCA cycle is enhanced. The argument that photorespiration is beneficial for Glu synthesis is in agreement with the positive correlation between photorespiration and leaf nitrate reduction [START_REF] Rachmilevitch | Proc. Nat. Acad. Sci. USA[END_REF][START_REF] Bloom | Proc. Nat. Acad. Sci[END_REF]. This scenario is also consistent with the results obtained in the cytoplasmic male sterile CMSII mutant of tobacco (affected in the mitochondrial respiratory complex I) in which the day respiratory rate is similar or even higher than that of the wild type while both photorespiration and N-metabolism (amino-acid synthesis) are enhanced [START_REF] Priault | [END_REF]27). Therefore, we argue that day respiratory homeostasis in leaves is likely to be the result of a compromise between two opposing forces: (i) an inhibition of respiration and glycolysis due to high mitochondrial NADH levels generated by photorespiration (Gly decarboxylation) and elevated ATP/ADP and DHAP levels generated by photosynthetic activity; (ii) a stimulation of the TCA cycle in order to adjust 2-oxoglutarate production to photorespiratory Glu demand. Such a compromise should be very dynamic, adjusting to changes in environmental conditions that modify stomatal closure, thereby altering leaf internal CO 2 /O 2 balance. For example, water deficit, that leads to a low internal CO 2 mole fraction, presumably promotes day respiration and photorespiration. Therefore, in the summer months the quantitative significance of these metabolic changes should be evident in many C 3 crops and natural vegetation. However, it is probable that such a promoting effect may disappear on a long-term basis because of acclimation processes (28,29). Thus the extent to which the regulation of day respiration by CO 2 and O 2 conditions scale up to crop productivity and global carbon sequestration needs further experimental assessment.

Material and methods

Plant material

Cocklebur (Xanthium strumarium L., Asteraceae) plants were grown in the greenhouse from seed in 100 mL pots of potting mix and transferred to 3 L pots after two weeks. Minimum photosynthetic photon flux density during a 16-h photoperiod was kept at approximately 400 µmol m -2 s -1 by supplementary lighting. Temperature and vapor pressure deficit were maintained at approximately 25.5/18.5 o C and 1.4/1.2 kPa day/night, respectively. The carbon isotope composition (δ 13 C) of CO 2 in the greenhouse air was -9.5±0.3‰. The third or fourth leaves (from the apical bud) were used for all measurements.

Gas exchange measurements a-Closed system (dark respiration)

The respiration chamber was placed in a closed system, which was directly coupled to an elemental analyser (EA) NA-1500 (Carlo-Erba, Milan) through a 15-mL loop, as described in (30). After decarboxylating the system, respired CO 2 was accumulated until it reached nearly 300 µL L -1 . The loop was then shunted and the gas inside the loop was introduced into the EA with helium for gas chromatography. The connection valve between the EA and the isotope ratio mass spectrometer (VG Optima, Micromass, Villeurbanne, France) was opened when the CO 2 peak emerged from the EA. b-Open system (photosynthesis and on-line carbon isotope discrimination)

The photosynthesis system has already been described in (9). Briefly, a purpose-built assimilation chamber was connected in parallel to the sample air hose of the LI-6400 (Li-Cor Inc., Lincoln, NE). Leaf temperature was controlled at 21 o C with circulating water from a cooling water bath to the jacket of the leaf chamber, and was measured with a copperconstantan thermocouple plugged to the thermocouple sensor connector of the LI-6400 chamber/IRGA. Inlet air was adjusted to ca. 10 mmol mol -1 H 2 O and passed through the chamber at a rate of 30 L h -1 , monitored by the LI-6400. Light (400 µmol m -2 s -1 ) was supplied by a 500 W halogen lamp (Massive N. V., Kontich, Belgium). Inlet CO 2 was obtained from a gas cylinder (Alphagaz N48, Air Liquide, France) with a δ 13 C of -50.2±0.2‰. The outlet air of the chamber was regularly shunted and was sent to the loop to measure its 12 C/ 13 C isotope composition and thus the on-line carbon isotopic discrimination (∆ obs ). The gas inside the loop was introduced into the EA for GC as described above. ∆ obs during photosynthesis was measured following the method described by ref. (31). Air with 2% oxygen was from a cylinder (Crystal gas mixture, Air Liquide, France). When light was turned off, the leaf was immediately removed from the open system and one half was frozen in liquid nitrogen. The other half (still attached to the peduncle) was placed in the closed system for dark respiration measurements (see above).

Starch extraction

The protocol for starch extraction was similar to that described in (30). The frozen leaf material was lyophilised and powdered. 50 mg of leaf powder was suspended with 1 mL of distilled water in an Eppendorf tube (Eppendorf Scientific, Hamburg, Germany). After centrifugation, the pellet was washed four times with 95% ethanol at room temperature and starch was extracted by HCl solubilization and precipitated with cold methanol. After lyophilisation, starch was transferred to tin capsules (Courtage Analyze Service, Mont Saint-Aignan, France) for isotope analysis.

NMR analyses

Leaves used for NMR spectroscopy were fed for 2 h at 21°C and 400 µmol m -2 s -1 with either water (control), 12 C-substrates or 13 C-substrates in a large plexiglass chamber (surface area 450 cm²) connected in parallel to the sample air hose of the LI-6400 (Li-Cor Inc., Lincoln, NE), allowing CO 2 mole fraction monitoring.

NMR measurements were carried out as described in ( 9) and (32) from perchloric acid extracts prepared from 5 g of frozen leaf material. Spectra were obtained using a Bruker spectrometer (AMX 400) equipped with a 10-mm multinuclear probe tuned at 161.9 and 100.6 MHz for 31 P-and 13 C-NMR, respectively. The assignment of 13 C resonance peaks was carried out according to (33). Identified compounds were quantified from the area of their resonance peaks using fully relaxed conditions for spectra acquisition (pulses at 20 s intervals). Peak intensities were normalized to a known amount of the reference compound (maleate for 13 C and methyl-phosphonate for 31 P) that was added to the sample (internal standard).

C-enriched molecules

The positional 13 C-labeled molecules (99% 13 C in the considered position) were purchased from Eurisotop (Saclay, France). Pyruvate was dissolved in distilled water and the pH was adjusted to 6.7 with NaOH. To obtain non-fully labeled solutions (∆ obs experiments), the labeled compounds were mixed with industrial glucose (δ 13 C = -9‰) or pyruvate (δ 13 C = -21‰) from Sigma. The resulting overall composition of glucose and pyruvate solutions was checked to be 2500‰ and 1400‰, respectively. In each case, the final concentration was 0.015 mol L -1 . The solutions were fed to the leaves through the transpiration stream.

Calculations

The procedure used to calculate the decarboxylation rates of 13 C-enriched substrates in the light from apparent ∆ obs values has already been explained in detail (9). Briefly, the difference between apparent ∆ obs values obtained with and without substrate addition is considered to reflect the additional decarboxylation flux in the light. Using mass balance equations, it can be shown that the decarboxylation rate r day has the following form: (3), any isotopic dilution of the substrate is taken into account as the observed carbon isotope discrimination is always a net value that integrates the decarboxylation of natural (that is, not added) Pyr or Glc molecules, both before and after 13 C-substrate addition.

A similar procedure applies to dark-respired CO 2 measurements. In other words, CO 2 that is produced in darkness ( 13 C-percentage λ global ) after a light period with 13 C-enriched substrate feeding comes from respiratory oxidation of new photosynthates (the 13 C-percentage in the net fixed carbon is λ fixed ), photosynthates from the previous light period in the greenhouse ( 13 C-percentage λ previous ), and additional C coming from the 13 C-enriched substrate fed to the leaf ( 13 C percentage λ s ). The night-decarboxylation rate has the following form: (during which 400-800 mmol C m -2 have been fixed) (34). It should be noted that possible variations in these coefficients do only introduce very slight errors in the estimate of the 13 Cenriched substrate decarboxylation r night , because of the strong 13 C-enrichment of the substrate (that is, the λ p value is always very small compared to λ global or λ s and may be neglected). Again, that relationship is somewhat modified with positional enrichments to take into account the different origin of decarboxylated CO 2 (as described in 9).

Clustering analysis

The 13 C-NMR data were represented as an isotopomic array as described in (35). The positional isotopic abundances (in 13 C-percentage) relative to the natural 13 and 400 µmol m -2 s -1 PPFD. CO 2 mole fraction was 140 µL L -1 , 400 µL L -1 , and 1000 µL L -1 .

At t = 2 h, leaves were immediately frozen in liquid nitrogen for perchloric acid extraction.

Perchloric extracts were analysed for positional 13 or Glc-6-phosphate (closed disks) ratio. Phosphorylated compounds were measured by 31 P-NMR on the same samples used for 13 C-NMR after 13 C-labeling. Lines stand for linear regressions. The regression with DHAP/Glc-6-P is significant (F=61.7, P<0.08).

13 C- 1 -

 131 where d is the flow, S leaf surface area, V M the molar volume at air temperature, c e and c o are the CO 2 mole fractions in inlet and outlet air, respectively. λ values are13 C percentages (using delta-values is not possible because of large13 C-enrichments) in inlet CO 2 (subscript e), outlet CO 2 (subscript o), net fixed CO 2 (subscript fixed), and13 C-enriched added substrate (subscript s). This equation holds for homogeneously labeled substrates; it is somewhat changed for positional enrichments to take into account the different origin of decarboxylated CO 2 . This occurs when glucose or pyruvate are added: the C-1 atom of pyruvate is decarboxylated by PDH while the C-2 and C-3 positions are decarboxylated by the Krebs cycle. Similarly, the C-3 and C-4 atom positions of glucose are decarboxylated by the PDH reaction, the other being decarboxylated by the Krebs cycle. Taking advantage of positionally13 C-enriched substrates: pyruvate would specifically enrich the CO 2 produced by PDH while13 C-2-pyruvate would specifically enrich the CO 2 that comes from the Krebs cycle. The same applies to positional13 C-enrichment in glucose. It should be noted that, in contrast to the argument of ref.

  where λ p is a linear combination of λ previous and λ fixed . It is equal to 0.6 λ previous + 0.4 λ fixed after 2-3 h in the light under ordinary CO 2 /O 2 conditions (during which 200-400 mmol C m -2 have been fixed), and 0.5 λ previous + 0.5 λ fixed after 2-3 h in the light under high CO 2 conditions

Figure 2 .

 2 Figure 2. Decarboxylation rates and inhibition of decarboxylation by light calculated from

Figure 3 .

 3 Figure 3. Isotopomics array representation of 13 C-abundance in the carbon atom positions of major metabolites in detached leaves incubated with 13 C-substrates for 2 h at 21°C, 21% O 2

Figure 4 .

 4 Figure 4. Main panel. 13 C-abundance in glutamate (Glu, black bars), succinate (Succ, light-

Figure 5 .Fig. 2 )

 52 Figure 5. Relationship between the inhibition of the TCA cycle in the light (in %, data from

  Figure 2

  C-abundance (1.1%) are indicated by colors so that black cells indicate near-natural abundance, green and red cells indicate lower-and larger-than-natural 13 C-abundance. The clustering analysis was carried out with the Cluster software and the array is drawn using the TreeView software (both are from M. Eisen, Standford University).Figure 1. A, carbon isotope discrimination (∆ obs ) associated with photosynthesis of detached leaves (at 21°C and 400 µmol m -2 s -1 PPFD) fed with either 13 C-1-or 13 C-2-enriched Pyr under four CO 2 /O 2 conditions: 140, 400 or 1000 µL L -1 CO 2 in 21% O 2 , and 400 µL L -1 CO 2

	in 2% O 2 . B, carbon isotope composition (δ 13 C) of respired CO 2 in darkness after the
	corresponding light periods. When leaves experienced a light period under 2% O 2 , the carbon
	isotope composition was measured in either 21% (indicated as '400 µL L -1 , 2-21%') or 2% O 2
	(indicated as '400 µL L -1 , 2-2%'). Each value is the mean±SE of three measurements. The
	control δ 13 C value of respired CO 2 was -22.1±0.5‰.

  fructosyl and glucosyl moieties of sucrose, respectively. Red and green cells indicate13 Cabundances above and below the natural abundance (which is 1.1%). Below-natural abundance cells appear dark-green because the 13 C-abundance is still very close to 1.1%.

C-abundances by NMR. Each column is a separate set of experimental conditions. Cit, citrate; Fum, fumarate; Ido/Gal, uncertain Dhexofuranose belonging to the idose-galactose group; Mal, malate; Obt, oxobutyrate; SF and SG,

Acknowledgements. We thank the Institut Fédératif de Recherche 87 for its support through a Transversal Project grant. Dr G. Tcherkez wishes to thank Dr. Jean Vidal for valuable discussions on the manuscript.

Supporting information (for on-line publication) Table S1. Apparent photosynthetic carbon isotope discrimination (∆ obs ) and carbon isotope composition of dark-respired CO 2 (δ 13 C) of detached leaves fed with 13 C-1-or 13 C-3-Glc. Conditions during the light period are 140 and 400 µL L -1 CO 2 in 21% O 2 under 400 µmol m -2 s -1 PPFD. In water (no feeding conditions), the ∆ obs value is 21.6±1.2‰. The δ 13 C value of respired CO 2 in darkness in water (no feeding) is -34.5±0.8‰ and -42.0±0.7‰ after a light period under 140 µL L -1 and 400 µL L -1 CO 2 , respectively. Mean and SE of three measurements. Table S2. Glucose decarboxylation rates and inhibition of decarboxylation by light, using the method of ref. (9) and data from Table S1. The decarboxylation rates by PDH and the TCA cycle are given in the light (denoted as r light just below) and in the dark (denoted as r night just below).

Inhibition by light (calculated as 1 -r light /r night , in %) is indicated on the right. Table S3. Carbon isotope composition of starch (in ‰) after feeding leaves with 13 C-enriched Glc (δ 13 C=2500‰) or Pyr (δ 13 C=1400‰) in the light under different CO 2 /O 2 conditions (experiment described in Figure 1) and associated contributions of 13 C-enriched substrates to starch synthesis. The δ 13 C value of CO 2 used in the open system is -50.2‰. Samples were immediately frozen in liquid nitrogen after 2 hours feeding, for starch extraction. The contribution values assume refixation of the 13 C-enriched decarboxylated CO 2 and do not take into account the global δ 13 C of added substrate. Contributions given in µmol m -2 s -1 are average values over the 2 h feeding period. Mean and SD of three measurements.