Pierre Besset
email: pierre.besset@ensam.eu

Richard Béarée
email: richard.bearee@ensam.eu

Olivier Gibaru
email: olivier.gibaru@ensam.eu

FIR Filter-Based Online Jerk-Controlled Trajectory Generation

This paper presents a novel approach to generate online jerk-limited trajectories for multi-DOF robotic systems. Finite Impulse Response filters are used to efficiently turn low computational cost acceleration-limited profiles into jerk-limited profiles. Starting from a new setpoint, e.g. an event given by external sensors, and an arbitrary state of motion, i.e. with nonzero initial velocity and acceleration values, the proposed method can generate different shapes of jerk profile, including timeoptimal and fixed-time jerk-limited trajectories. A new definition of the velocity, acceleration and jerk kinematic limits can be instantaneously taken into account during the motion. Moreover, the very low calculation time (less than 1 microsecond) makes it possible to easily control a multi-DOF system during one control cycle (classically about 1 millisecond), while preserving time for other computer processing. The algorithm is experimentally tested on the new 7-DOF industrial robot KUKA LWR iiwa.

I. INTRODUCTION

Modern industrial robotic applications often require smooth trajectory generation. Robotic manipulation classically uses smooth profiles, such as jerk-limited trajectories, to avoid acceleration discontinuities, which can induce detrimental vibrations. A significant amount of work have been done to find efficient and elegant solutions to this problem of trajectory planning [START_REF] Erkorkmaz | Highspeed cnc system design. Part I: Jerk limited trajectory generation and quintic spline interpolation[END_REF], [START_REF] Jeong | Jerk limited velocity profile generation for high speed industrial robot trajectories[END_REF], [START_REF] Olabi | Feedrate planning for machining with industrial six-axis robots[END_REF]. As the previous robotic field, service robotics (collaborative) and mobile robotics have to solve the same problem but with online capability. Indeed, in this recent context, robots attempt to instantaneously react to unpredicted events originating from the environment and potentially from the physical human-robot interaction. Therefore, according to the context, the trajectory generation has to verify all or part of the following points:

• Trajectories should be modified during execution time (online capability) depending on external measurements or human commands;

• Trajectories should be time-optimal for productivity reasons;

• The smoothness of the trajectories, at least jerklimited, should depend on the expected behavior (e.g. human motion reproduction or end-point accuracy).

Existing solutions to the problem of time-optimal online trajectory generation can be found in the literature. In [START_REF] Béarée | Real-Time Planner in the Operational Space for the Automatic Handling of Kinematic Constraints[END_REF], authors propose an efficient real-time planner in the operational space, but only the acceleration-limited trajectory (trapezoidal velocity) is considered. In [START_REF] Kröger | Towards on-line trajectory computation[END_REF], [START_REF] Kröger | On-line trajectory generation: Basic concepts for instantaneous reactions to unforeseen events[END_REF], Kröger et al. introduce an elegant framework based on the current state of the system and the final desired state, that provide time-optimal accelerationlimited trajectories. The jerk-limited solution is presented, but not detailed, in their recent works [START_REF] Kröger | Opening the Door to New Sensor-Based Robot Applications -The Reflexxes Motion Libraries[END_REF]. One notes that this algorithm belonging to the well-known Reflexxes Motion Library [START_REF]Reflexxes Motion Library[END_REF] is no longer freely available for robotics research. Algorithms based on the analytical recalculation of the timeoptimal third order trajectory parameters can be found in [START_REF] Haschke | On-line planning of timeoptimal, jerk-limited trajectories[END_REF], [START_REF] Ramos | Time-optimal online trajectory generator for robotic manipulators[END_REF]. The computing time of such solutions is about 350 microseconds for one joint trajectory, which can be a significant limitation for multi-axis systems during one control cycle (typically 1 millisecond).

In this work, we propose a faster and more versatile methodology that exploits the property of FIR (Finite Impulse Response) filtering to generate on the fly jerk-limited trajectories, based on initial acceleration-limited trajectories. In addition to the real-time ability, the FIR-based method can be used to generate different jerk pattern (sine squared jerk, damped-jerk [START_REF] Béarée | New Damped-Jerk trajectory for vibration reduction[END_REF]) or smoother trajectories (snap-limited, crackle-limited,. . . ) by a simple modification of the filter coefficients or combination of filters.

The next Section presents the methodology adopted for the generation of jerk-limited trajectories. Section III details the procedure to generate online time-optimal jerk-limited trajectories. Section IV shows how the kinematic limits can be changed online. In section V, a fixed-time solution for multiaxis motion synchronization is proposed. Experimental results obtained with a collaborative 7 axis robot are presented in Section VI. Conclusion are drawn in section VII.

II. TRAJECTORY GENERATION WITH JERK LIMITATION

The methodology used in this paper is based on the convolution of acceleration-limited trajectories with a FIR filter inducing the final jerk pattern. For instance, convolving a step function of value e with a simple sliding average FIR filter produces a ramp function with ending value e and a slope value of e/T (with T the FIR filter time). Therefore, as illustrated in Figure 1, an acceleration-limited profile may be turned into a jerk-limited profile by FIR filtering. In [START_REF] Biagiotti | FIR filters for online trajectory planning with time and frequency-domain specifications[END_REF], Biagiotti et Melchiorri generalize this principle by combining FIR average filters in a cascade configuration to obtain smooth trajectories from simple rough step position commands. However, the resulting trajectories cannot be modified online (only zero initial velocity and acceleration conditions could be handled).

In this paper, the subscript f stands for "filtered", i.e. denotes the variables of the jerk-limited profile, which is the output of the algorithm. Moreover, for the online trajectory generation, it is assumed that the final velocity and acceleration are zeros, since if no new set-point is given, the goal is reached and the motion stops.

The inputs of the algorithm, presented in Figure 2 are the set-point q e , the initial state of the system (q 0f , v 0f , a 0f ), and the kinematics limits (v max , a max , d max , j max ). The algorithm is divided into two stages. The first one consists in generating an acceleration-limited (AL) trajectory, with is adapted for the following FIR filtering stage. The output of the filtering stage is a jerk-limited (JL) trajectory, which fulfills the requirements in terms of final position, initial state and kinematic limitations on the velocity, acceleration and jerk. As detailed in the remainder of this paper, the adaptation of the initial AL trajectory is based on simple analytical relations. One notes that for the time-optimal solution, this adaptation process can be repeated iteratively according to a decision tree.

A. AL Generator

The first stage of our methodology consists in generating the time-optimal AL trajectory, which respects the following kinematic constraints and boundary conditions :

|v(t)| ≤ v max if t < T a + T v |a(t)| ≤ a max else |a(t)| ≤ d max (1)
The acceleration-limited profile, also known as bang-bang profile, can be build in three phases: first an acceleration phase during which the acceleration is constant: |a(t)| = a max . During the second phase the acceleration is zero, i.e. the maximal velocity has been reached and is constant:

|v(t)| = v max .
Finally in the third phase comes the deceleration, constant as well: |a(t)| = d max . The durations of these three phases are respectively noted T a , T v and T d , as illustrated in Figure 3.

The displacement ∆q is given by relation 2:

∆q = q e -q 0 = T a (v r + v 0 ) 2 + T v v r + T d v r 2 , (2)
with:

T a = v r -v 0 a r ; T d = -v r d r . (3)
To ensure that the trajectory is time-optimal, we set the acceleration, deceleration, and cruising velocity values at the maximum values:

a r = s • a max ; d r = -s • d max ; v r = s • v max ,
where a r , d r and v r are the reached acceleration, deceleration and velocity values and s is the sign of the velocity during the cruising phase. In order to take into account a non-zero initial velocity, s can be determined by: s = sign(q e -q stop ),

where q stop represents the position reached after the fastest full stop [START_REF] Haschke | On-line planning of timeoptimal, jerk-limited trajectories[END_REF]. If the initial velocity v 0 is greater than the maximum velocity: s • v 0 > s • v r , the velocity has to be reduced to v r . In this case, a r = -s • a max and we have a double deceleration profile where a r and d r have the same sign [START_REF] Haschke | On-line planning of timeoptimal, jerk-limited trajectories[END_REF]. Finally, we determine suitable values for T a , T d and T v . Let us note ∆q min the minimum displacement for which v r is reached, when T v = 0. If s • ∆q ≥ s • ∆q min then the maximal velocity is reached during the motion. Consequently we have:

T a = v r -v 0 a r ; T v = ∆q -∆q min v r ; T d = -v r d r . ( 5
)
Otherwise, the maximal velocity is not reached and the velocity profile is wedge-shaped. The maximum reached velocity v r is then updated:

v r = s • - v 2 0 ar -2∆q 1 dr -1 ar , (6)
and

T a = max(0, vr-v0 ar ); T v = 0; T d = max(0, -vr dr ).
Finally, the triplet (T a , T v , T d ) defines the time-optimal acceleration-limited profile which respects the kinematic bounds.

B. Adaptation of the initial AL profile for FIR filtering

The parameters (q 0 , v 0 ) used to generate the initial AL profile have to be adjusted according to the initial state (q 0f , v 0f , a 0f ). The relationships between the parameters of the non-filtered and filtered profiles are given by equations ( 7) and [START_REF]Reflexxes Motion Library[END_REF].

v 0 = v 0f + T j a 0f 2 , ( 7
)
q 0 = q 0f + T j v 0f + T 2 j a 0f 3 + T j 2 (T a a r + T d d r ) , ( 8
)
with T j the filter time, as shown in Figure 4.

time Once the AL profile is generated, the test T v > T j should be checked. If T v is less than T j , an adaptation of the AL profile is required to keep the jerk of the JL profile from exceeding its limits, as shown on Figure 5. Such an adaptation consists in re-generating the AL profile with a lower value of v r which satisfies T v = T j and (5).

C. FIR Filtering and Jerk pattern

Starting from the previous AL profile, a JL trajectory can be obtained using a FIR moving average filter, as shown on Figure 4. The length of the filter T j is classically chosen to reach the maximum value of the jerk: T j = max(|a r -a 0f |, a max , d max )/j max . In order to respect the desired initial acceleration of the profile, the coefficients of the filter should be initialized at a 0f . One notes that the resulting JL trajectory is not time-optimal for all kinds of motion because the jerk time is fixed instead of the maximum jerk value. Other jerk patterns can be generated by changing the coefficients of the filters. For instance, jerk patterns can be used to reduce residual vibrations in the system as described in [START_REF] Béarée | New Damped-Jerk trajectory for vibration reduction[END_REF]. Figure 6 presents two examples of trajectories generated using Damped-Jerk and sine squared Jerk patterns.

III. TIME-OPTIMAL JL TRAJECTORY

Many industrial applications require time-optimal trajectories for productivity reasons. To ensure the time-optimality, the maximum jerk value should be fixed instead of the filter time. Therefore the filter time should be changed while filtering to satisfy the following relationships:

T ja = |a r -a 0f | j max ; T jv = |a r | j max ; T jd = |d r | j max , (9)
where T ja , T jv and T jd are the durations of the constant jerk stages, as shown in Figure 7. Before generating the AL profile, the sign s has to be determined, i.e. the fastest full-stop position q stop which can be reached under the kinematic bounds, as stated in (4). According to the state of the system, the acceleration profile leading to the fastest full-stop of the system can take different shapes, as Figure 8 shows. If q stop = q e then the full-stop profile should be kept as a result with no further calculation. If sign(a 0f ) = sign(a r ), the sign s should be reevaluated at t = |a 0f /j max |. In the case of a sign change, the trajectory is recalculated from the time T p where s changes. Figure 9 shows such a trajectory.

A. Adaptation of the AL profile

The relationships between (q 0 , v 0 ) and (q 0f , v 0f , a 0f ) given by ( 7) and (8) cannot be used for time-optimal generation since the length of the filter is not constant in the general case. Equations ( 10) and ( 11) are derived to take this difference into account.

v 0 = s • a r a 0f j max -s • a 2 0f 2j max + v 0f , (10)
q 0 = a r s • v 0f 2j max + a 2 0f 4j 2 max +q 0f - a 3 0f 6j 2 max -s(a r +d r ) v r 2j max .
(11) In addition to the adaptation previously described in II-B, which makes sure that the jerk bound is respected, the AL profile should also be checked to make sure that the solution given after filtering is time-optimal. If T ja > T a or T jd > T d , then the acceleration profile after filtering saturates at a value less than its bound, respectively a max or d max , as illustrated on Figure 10. In that case, the initial AL profile should be adapted in order to produce the desired time-optimal jerklimited trajectory. Figure 11 presents the decision table used to select the adaptation to be applied to the AL profile. As an example, if the initial AL profile exhibits T d < T jd and T v > T jv , a Type III-b adaptation is needed. In that case, the deceleration stage does not saturate, i.e. |d r | < d max . Therefore

T d = T jd ⇒ -v r d r = -s • d r j max , hence: d r = -s sj max v r . ( 12
)
Once the AL profile is adapted, it should be re-checked for a possible second adaptation, as shown in Figure 12.

B. Filtering

In order to keep the jerk at its maximum value, the length of the filter is dynamically switched according to relation [START_REF] Haschke | On-line planning of timeoptimal, jerk-limited trajectories[END_REF]. The trajectory starts at t = 0 with a filter initialized at a 0f with a length T ja . Then the filter is changed at t = T a to a length T jv , initialized at a r . Finally at t = T a + T v the length of the filter is set to T jd .

IV. ONLINE CONSTRAINTS CHANGE

Many recent robotic applications require online adaptation of the kinematic bounds of the trajectory. For instance, the detection of human presence in the workspace of the robot can enable a "smooth" mode in which the kinematic limits are lowered, or the jerk pattern modified. In the case where the velocity or acceleration limit is lowered during the motion, a transition stage may be required. If the initial acceleration and velocity are such that the new velocity limits are or will be exceeded, i.e. if:

s • v 0f + a 0f 2 • a 0f j max > s • v max , (13)
then the deceleration stage, noted a ds (t), should bring the velocity back into its news bounds as soon as possible. Let us note the duration of this stage T ds , as shown in Figure 13. We want:

v 0f + T ds t0 a ds (t)dt = s • v max .
For the fastest solution, we saturate the jerk:

v ds (t) = -s • j max t 2 2 + a 0f t + v 0f . We want v ds (T ds ) = s • v max , hence -s • j max 2 T 2 ds + a 0f T ds + v 0f -v max = 0, if T ds > -s•amax-a 0f jmax
then a ds should saturate at -s • a max .

V. MULTI-DIMENSIONAL CASE / AXIS SYNCHRONIZATION

In the general case, several Degrees-Of-Freedom (DOF) are controlled simultaneously to reach a targeted position. Classically, it is wished that the motion of each DOF ends at the same time. Consequently, for a motion involving n DOFs, with T i the duration of the time-optimal trajectory for the i-th DOF, n -1 trajectories have to be adjusted to the fixed time constraint T f ix given by the slowest DOF

T f ix = max i T i . (14)
This adjustment can be done in several, very different ways.

Here, we propose to take benefits from the filtering strategy to easily compute the fixed-time solution for the JL profile. Indeed, without consideration of time-optimality, if an AL profile is directly convolved with a FIR filter of time duration T j , the resulting profile is T j seconds longer than the initial AL profile.

Noting τ f ix the fixed time duration of the AL profile, we have the relation τ f ix = T f ix -T j . The AL profile used here allows arbitrary initial velocities and the fixed-time solution for such a profile can be simply calculated. Considering the rescaling of the whole time evolution with the parameter α = τ f ix /τ i :

τ a = αT a ; τ v = αT v ; τ d = αT d . (15)
The parameters to be determined are a r and d r , respectively the reached acceleration and deceleration. From ( 2) and (3) it is straightforward to derive the relationships ( 16) and ( 17):

a r = - q 0 + τ a • v 0 + τ d • v 0 /2 + τ v • v 0 -q e τ 2 a /2 + τ a • τ d /2 + τ a • τ v , (16)
d r = - v 0 + τ a • a r τ d , (17)
where q 0 and v 0 are the ones used to generate the initial trajectory, i.e. the ones given by the relationships ( 7) and [START_REF]Reflexxes Motion Library[END_REF]. Thus the axis synchronization problem for JL trajectories is solved with a set of simple equations. Since no iterative root-finding is required, these equations are almost instantaneous to evaluate. Equations ( 10) and ( 11) stand for trajectories with a jerk profile saturated at a given value j max . Since synchronized trajectories do not respect that condition anymore, another method has to be developed. In order to minimize the computation time, it is wished to avoid iterative resolutions of equations. By setting the jerk at its maximum value for the first acceleration phase and preserving the length of the filter for the deceleration phase T jd , as shown on Figure 14, the problem is simplified. T ja and T jv can be expressed as

T ja = s • a r -a 0f j max , T jv = s • a r j max . (18)
The unknown parameters are the reached accelerations and deceleration a r and d r , which are the solution of the system (19):

A • a 2 r + B • a r + C = 0 d r = - v 0f +(a 0f +ar) * Tja/2+ar * (τa-Tja)+ar * Tjv/2 τ d (19)
with

A = - a 0f +jmaxsτa 2j 2 max B = τa(T jd +τa+τ d +2τv) 2 + a 2 0f 2j 2 max + a 0f s(T jd +2τa+τ d +2τv) 2jmax C = q 0f -q ef + v 0f T jd 2 + τ a + τ d 2 + τ v - a 3 0f 6j 2 max -s • a 2 0f (T jd +2τa+τ d +2τv) 4jmax
This system gives an exact solution while s

• a 0f ≤ s • a r . Otherwise, T ja = -s• ar-a 0f
jmax , thus the expression of A, B and C changes. This approach is very fast, since it only requires to find the roots of a second order polynomial function. Figure 15 shows an example of synchronized jerk-limited trajectories.

VI. RESULTS

The algorithm has been tested for all types of configurations to validate its results. In any case the algorithm generates the solution in less than 1 microsecond on a regular laptop i7 / 2.7GHz running with Windows 7 as operating system. At the best of the authors knowledge this is the fastest time-optimal jerk-limited trajectory generator. A typical control cycle time being 1 ms, our algorithm can easily handle any multi-axis Fig. 14. The approach chosen to deal with joint synchronization is to constrain the value of the jerk for the first acceleration stage and the filter length for the second one. system and save a significant amount of time to the controller for other tasks, e.g. monitoring, obstacle avoidance or any data processing. Besides, such a small calculation time enable instantaneous reactions to unforeseen events, as introduced in [START_REF] Kröger | Towards on-line trajectory computation[END_REF].

VII. CONCLUSION

A method to generate online jerk-limited trajectories based on FIR filtering of adapted acceleration-limited profiles is presented in this paper. This generator handles non-zero velocity and acceleration as well as non-symmetric acceleration bounds. Time-optimal and time-fixed solutions for multi-axis synchronization are also proposed. The proposed method generates time-optimal trajectories in less than one microsecond on a regular laptop.

Fig. 1 .

 1 Fig. 1. FIR filtering principle for jerk-limited trajectory synthesis.

Fig. 2 .

 2 Fig. 2. Proposed online trajectory generation algorithm based on FIR filtering.

Fig. 3 .

 3 Fig. 3. A classic acceleration-limited profile. ar, dr and vr are the reached acceleration, deceleration and velocity values.

Fig. 4 .

 4 Fig. 4. The initial acceleration-limited trajectory (blue dashed line) is filtered to give a jerk-limited profile (solid red line).

Fig. 5 .

 5 Fig. 5. Example of an adaptation: Type II (see Fig. 11). To the left: nonadapted profile, Tv < T j . To the right, adapted profile, Tv = T j and |vr| < vmax.

Fig. 6 .

 6 Fig. 6. Example of trajectories with different jerk patterns: Damped-Jerk (dashed blue line) and sine squared Jerk (solid green line).

Fig. 7 .

 7 Fig. 7. Time-optimal jerk-limited profile with non-zero initial conditions, each derivative saturates at its bound.

Fig. 8 .

 8 Fig. 8. Examples of fullstops acceleration profiles.

Fig. 9 .

 9 Fig. 9. Example of an acceleration profile with a sign change at t = Tp.

Fig. 10 .

 10 Fig. 10. Example of an adaptation: Type III-b (see Fig. 11). To the left: non-adapted profile, T d < T jd . To the right, adapted profile, T d = T jd and |dr| < dmax.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Decision table for the selection of the adaptation of the AL profile. III-a III-c III-b

Fig. 13 .

 13 Fig.[START_REF] Béarée | Dissociated jerk-limited trajectory applied to time-varying vibration reduction[END_REF]. Example of a trajectory with a deceleration stage. Dashed blue line: acceleration. Full green line: velocity. The kinematic bounds are lowered at t = 5.3 s (vertical full line). A deceleration stage ensures that the velocity respects its new limit.

Fig. 15 .

 15 Fig. 15. Synchronization of time-optimal trajectories: initial trajectories to the left, synchronized trajectories to the right.

  The algorithm has been tested with a 7 DOF KUKA leightweigth LBR IIWA. The Fast Robot Interface (FRI) of the KUKA Sunrise.Connectivity collection of open interfaces was used for real time communication between the robot controller and an external computer, with a rate of 1 ms. Figures16 and 17show samples of experimental results with respectively synchronized and not synchronized joints. Dynamic new target references were randomly generated and the new trajectories updated online, with some changes in the kinematic limits.

Fig. 16 .

 16 Fig. 16. Experimental results for synchronized trajectories (7 joints). New targets are given on vertical lines. jmax is lowered at the full vertical line. Dashed lines: reference acceleration, full lines: measured acceleration. Top graph: joints 1 to 3; bottom graph: joints 4 to 7.


F Girard

F Barbault

C Gouyette

T Huynh-Dinh

J Paoletti

G Lancelot
email: lancelot@cnrs.orleans.fr

Dimer Initiation Sequence of HIV-lLai Genomic RNA: NMR Solution Structure of the Extended Duplex

The genome of all retrovirus consists of two copies of genomic RNA which are noncovalently linked near their 5' end. A sequence localized immediately upstream from the splice donor site inside the HIY-1 Ψ-RNA region was identified as the domain responsible for the dimerization initiation. It was shown that a kissing complex and a stable dimer are both involved in the HIV-lLai RNA dimerization process in vitro. The NCp7 protein activates the dimerization by converting a transient loop-loop complex into a more stable dimer. The structure of this transitory loop-loop complex was recently elucidated by Mujeeb et al. In work presented here, we use NMR spectroscopy to determine the stable extended dimer structure formed from a 23 nucleotides RNA fragment, part of the 35 nucleotides SLI sequence. By heating to 90°C, then slowly cooling this sequence, we were able to show that an extended dimer is formed. We present evidence for the three dimensional structure of this dimer. NMR data yields evidence for a zipper like motif A8A9.A16 existence. This motif enables the surrounding bases to be positioned more closely and permit the G7 and C17 bases to be paired. This is different to other related sequences where only the kissing complex is observed, we suggest that the zipper like motif AA.A could be an important stabilization factor of the extended duplex.

Introduction

Among all virus families, retroviruses are unique in having a diploid genome [START_REF] Coffin | RNA Tumor Viruses[END_REF]. Two copies of retroviral genomic RNA are joined within their 5' leader, at a 100 base long region called the dimer linkage structure (DLS), in order to form a noncovalent dimer as shown from the visualization of dimer structures by electron microscopy [START_REF] Bender | [END_REF](3)[START_REF] Chien | Electron microscopic analysis of the structure of RNA tumor virus nucleic acids[END_REF][START_REF] Murti | [END_REF]. The dimeric form of RNA has been proposed as being involved in recombination and reverse transcription (6)(7)(8)(9)(10)(11), GAG gene translation (12) and selective encapsidation of the genome in the viral particles (13)(14)(15). Even if the actual significance of such an arrangement is not absolutely clear, experiments have demonstrated that the dimeric structure is a determinant of the infection ability of the retrovirus (16)(17)(18). Extensive in vitro experiments showed that in vitro generated RNA fragments corresponding to the 5' end of the retroviral genome of HIV-1 (19)(20)(21), MoMuLV (22) or ASLV (23) can efficiently dimerize. Such experiments contributed to the knowledge of the nature of the dimer linkage structure. A sequence localized immediately upstream of the splice donor site (SD) inside the HIV-1 \f-RNA region has been identified as the domain responsible for the dimerization initiation (16,19,20,24). This 35 base long sequence, called SLI, is highly conserved and can adopt a stem-loop structure which presents a self complementary sequence in the loop. This autocomplementary sequence is usually GCGCGC, as in Lai isolate, or sometimes GUGCAC (Mal isolate) and is generally flanked by one 3' and two 5' adenosines. It has been shown that in vitro HIV-1 dimer formation was initiated by the annealing of the autocomplementary sequence via a loop-loop interaction between the two RNA molecules resulting in the formation of an unstable loop-loop complex at 37°C (21). This intermediary complex is converted at 55°C, or at lower temperatures in the presence of the NCp7 protein, into a more stable dimer involving all the stem-loop nucleotides in intermolecular interaction which is called the extended dimer (21 ,25-27). In vivo, the mechanism by which the dimer structure is formed has not yet been established even if it is currently thought to arise as mature virion assembles. The assembly of the genomic RNA molecules in the dimer is not a spontaneous process and involves active participation of some virion components during or shortly after release of the mature particles from the cell (28,29). In the same way, based on in vitro experiments, the conversion of the looploop complex, after recognition by the nucleocapsid protein, into the extended dimer has been proposed (30) to mimic the dimeric RNA maturation which occurs in vivo.

The structure of a loop-loop complex, corresponding to a model of the dimerization initiation complex of HIV-1 genomic RNA has been recently proposed from two dimensional nuclear magnetic resonance (NMR) spectroscopy (31). This model provides information to help in the understanding of the mechanical distortions that partially melt both of the stems within this metastable loop-loop initiation complex and may facilitate the spontaneous refolding of the extended dimer. Using this model, two structural features are especially important to explain the relative thermostability of the loop-loop complex compared to the extended dimer. Mujeeb et al. (31) point out that the kissing-loop is characterized by a fully underwound loop helix together with a triple base stack in which an adenine is stacked between an adenine and a cytosine of the opposite strand. Such a feature induces mechanical strain which leads to breakage of the G-C base pair at the junction of the stem and of the loop.

It was therefore of interest to study the evolution of this unusual structure as the extended dimer is formed. We present here a NMR study of the stable extended dimer which has been formed from a chemically synthesized 23 nucleotides RNA fragment, a part of the 35 nucleotide SLI sequence. This study details a complete structure determination of the extended dimer which is then compared to that which has been proposed for the loop-loop initiation complex (31), in order for us to assess the role of the adenosine residues flanking the autocomplementary GCGCGC sequence in the stabilization of the dimer, together with the structure of the central GCGCGC/GCGCGC double stranded motif.

Materials and Methods

Sample Preparation

Automated Synthesis (Applied Biosystem 392) of the 23-mer (Figure 1) was carried out using phosphoramidite chemistry (Perseptive). Two l51J.mol columns gave a total yield of 39.9% (average coupling yield 95.7%) based on a trityl assay. Deprotection was carried out using i) NH3 saturated ethanol at 55°C for 24 hours followed by 24 hours at room temperature ii) tetrabutylammonium-fluoride/tetrahydrofuran for 40 hours at room temperature. The crude product was desalted twice on a Sephadex G-10 column, eluting with O.IM then 0.05M triethylammonium acetate (TEAA). The eluted product, detected by UV absorbance at 260nm, was lyophilized then purified using preparative HPLC (Nucleosil 300A 5-C18 with O.OIM TEAA-CH3CN). The DMT-on fraction was detritylated with AcOH 80% and fractionated using HPLC with an ion-exchange column (Waters DEAE-8HR Protein Pak with IM NaCI-CH3CN gradient in Tris O.IM) then a C-18 reverse phase column. The purified oligonucleotide was exchanged with a Dowex 50W-X8 Na+ column then lyophilized to yield 7.5 mg (5% total yield purified product). The purity (100%) was verified with capillary electrophoresis, analytical HPLC and mass spectroscopy (ES): 7400mu.

For NMR experiments, in order to remove the paramagnetic impurities and the Mg 2+ ions, the sample was passed through a Chelex 100 column. It was dialyzed against a mixture of phosphate buffer (10mM, pH 5.7) and EDTA (2mM), then against EDTA (2mM) and finally against sterile water. The sample was concentrated by lyophilization and then dissolved in 90% H 2 O/10% D 2 0 or 100% D 2 0 argon-degassed solutions at a concentration of 2mM. The RNA was heated to 90°C and then cooled down slowly to 4°C. Phosphate buffer (IOmM, pH 5.7) and NaCI (100 mM) were finally added.

NMR Experiments

NMR experiments were carried out on a AMX-500 spectrometer operating at 11.74 Tesla and processed on a X32 computer. Spectra were recorded with 800 and 1024 complex data points in the tl and t2 dimensions respectively. All D 2 0 experiments were carried out with the carrier frequency set on the HDO resonance, and a sweep width of 4505 Hz and 4002 Hz in the F1 and F2 dimensions respectively. The residual H 2 O resonance was irradiated by a weak presaturation during 1s. For experiments in H20, a sweep width of 11112 Hz and 11003 Hz was used in the F1 and F2 dimensions respectively and the last /2 jump was replaced by a jump and return selective excitation (32) in order to remove the signal of water. Two-dimensional data sets for DQF-COSY, E-COSY, TOCSY and NOESY were recorded in the States mode. For 2D-spectra recorded in water, base line was corrected following the procedure described by Marion et al. (33). In D20, six NOESY spectra corresponding to 0, 50, 90, 120, 300 and 600ms mixing time were collected at 25°C with 32 scans for each t1 value, without removing the sample or changing the frequency or gain. The non-selective Tl was measured in the range of 1.8s for the H1', 2.0s for the H8/6 and 2.2s for the H2. This is due to the fact that in RNA the base and adenine H2 resonances relax at similar rates as already reported (34). Consequently, a delay of 4s was used in order to optimize the signal to noise ratio without noticeable difference in the spin-state magnetization of the different protons. The t1 data was zero-filled to 2048 points and processed with a sine 90 phase shifted function. This light apodization function does not distort the signal intensities, and the resulting peaks have no truncation effects. Correct phasing of the data in both dimensions was achieved with great care in order to quench the zero-quantum contribution in the cross-peak integration. The NOE values were obtained by dividing the volume of the cross-peak by the volume of the corresponding diagonal peak obtained at Oms mixing time. The cross-peak volumes were calculated with integration routines of the UXNMR software package.

Results

Spectral Assignment in the Stem Region of SL1

In general, the observed NOE patterns for the stem region of SLI were typical for a right-handed A-type helix. Figure 2 shows the NOESY spectra of the 23-mer recorded in H20:D20 (90: 10). The correlation between the two sharp resonance peaks at 7.40 and 6.62 ppm was an interesting starting point (data showed at 25°C in D20 on Figure 4). They are strongly correlated to the imino resonance peaks at 14.53 and 13.17 ppm and were characteristic of H2(A)-N3H(U) cross-peak in Watson-Crick base pairs. They were assigned to H2 of adenines A21 and A22, which are the only adjacent adenines in the stem and the corresponding imino protons to the U2 and U3 resonances. Hence, the imino-imino connectivities allow assignment of the imino resonances of the seven base pairs of the stem. End-fraying effect precludes the observation of the terminal base pair exchangeable resonance CI-023 at l5°C, but this resonance peak as well as its connectivities are easily observed at SOC (Figure 2). It should be pointed out that even thought the 07 imino resonance was broader than the central 0 imino resonance, it can still be observed. By using these resonances, the corresponding amino and H5 resonances of the cytosines are then assigned.

From the H2 resonances of A21 and A22, and taking into account that in A-type RNA the H2 proton of an adenine is at about 4-4.5A. from the HI' of its 3' flanking nucleotide paired uracil, the HI' resonance of the G23 terminal residue was directly assigned (Figure 3). Consequently, all H8/H6, HI' and H5 resonances were sequentially assigned. These assignments were completed by running a DQFCOSY and a NOESY spectra at 50ms. The COSY spectrum showed a lack of H1'H2' connectivities except for G7, A8, A9 and G23, indicating a North-conformation for all the others residues. In the 50ms NOESY spectrum, the threshold was adjusted in order to observe only the connectivities corresponding to the protons whose distance was less than 3.0A. i.e. Hl'-H2' (2.7A.), H8/6 with the H2' of the adjacent 5' residue (-2.5A. in A-type helix) and H2' -H3' belonging to the same residue in nucleotide with Northconformation (2.2A.). Then, the assignment of the H8/6 and H1' resonances was verified (Figure 3 and4) and completed by the assignment of the H2'. Moreover, the aromatic-aromatic connectivities between adjacent residues in the sequence were all observed. Other familiar connectivities such as H8/6(i)-H5(i-1) and H5(i)-H1'(i) were also observed, additionally supporting the assignment. Several special features, characteristic of the structure of this oligonucleotide, were found. All the nucleotides (C1 to G7 and C17 to G23) were Watson-Crick paired in the stem. The wobble U6-G 18 base pair showed strong correlation between the resonance NIH(Gl8) at 10.23 ppm and the resonance N3H(U6) at 11.62 ppm. This strong correlation are characteristic of the wobble U -G base pair whose the two NH are generally 2.5 Å distant.

Usual and Unusual NOEs in the A8-A16 Sequence

Using similar considerations than for the stem region, an unequivocal assignment of the A8-A 16 sequence was obtained. Some ambiguities due to overlapped crosspeaks were clarified by running NOESY spectra at 10, 20 and 30°C. The NOEs indicated a right-type helix for the sequence Gl0-C15. Initially using the H5 resonance peaks, the assignment of all the cytosine NH2 resonance signals were obtained and the paired guanine then assigned.

Several special features were characteristic of the structure of this sequence. The observation of continuous base-ribose connectivities (Figure 3) simplified the assignment process and provided strong evidence that all this macromolecule was structured. The NOESY spectra obtained in water showed that the G10, G12 and G14 imino protons were connected with the C15, C13 and C11 amino protons, indicating a Watson-Crick type base-pairing. We did not observe evidence for pairing of the amino protons of the A8, A9 and A16 adenines. The H2 resonances of A8, A9 and A16 were assigned by using their proximities (4-5A.) with their H1' and the HI' of their 3' neighbor nucleotide. The strong H2(A9)-H2(A16) and the medium H2(A8)-H2(AI6) as well as the weak H2(A8)-H5(C17) and H2(A9)-H5(C17) connectivities demonstrated the proximities of the adenines A8 and A9 with A16 (Figures 3 and4). Despite the fact that A16'is neighbored by two pyrimidines and A8 and A9 are neighbored by two purines, in the sequence, the H2(A16) is 0.15-0.31 ppm more upfield than H2(A8) or H2(A9). Moreover, we observed a very weak H2(A8)-H2(A9) NOE even at 450ms mixing time. Usually the induced ring-current effect is more important for purines than for pyrimidines (35); we have evidence here of NOE effect between H2 protons of A8 or A9 and A16 but not between A8 andA9, consequently we conclude that A 16 is stacked between A8 and A9. The N1H(G10) showed NOE with the H1'(A16). Moreover, the global correlation time computed by using the building up of H6-H5 was 6.0ns, in agreement with a duplex form of two times 23 nucleotides at 25°C. The H1'-H8/6 and H2'H8/6 connectivities indicate that the duplex is under a right Ahelix form, even for the sequence G 10-CIS. Nevertheless, these NOEs are described as qualitative data, therefore no conclusions can be drawn on the nature of the duplex.

Except for residues G7, A8, A9 and the 3' terminal G23, all nucleotides in SL1 adopt canonical C3'-endo sugar pucker. This conclusion is based on the absence of measurable J(H I' -H2') on the high-resolution DQF-COSY spectra. The measured values for the J(Hl'-H2) coupling constant for G7, A8, A9 and G23 were 7.6, 7.5, 7.6 and 6.4 Hz respectively. Since typical values for C2'-endo conformation are greater than 7 Hz, this can reflect a dynamic equilibrium between various conformers (36).

A-type Helical Conformation of the Duplex

A folding topology for SL 1 can be predicted from the observed NOEs and the DQF-COSY data. Watson-Crick base-pairing interactions, weak intensities of H8/6(i)-Hl'(i), strong intensities ofH8/6(i)-H2'(i-l), presence of H2(A)-Hl' crosspeaks and predominantly C3' -endo sugar conformations all indicate that the duplex, except G7, A8, A9 and G23, has an A-form helix typical of RNA. By contrast, no simplifying structural assumption, except for a right-helical type, can be made for the adenine A8, A9 and A 16 due to their missing pairing. These mismatches resulted in a modified rise angle of some nucleotides in order to stabilize the duplex by pairing of the central guanines and cytosines.

Structure Determination Experimental Constraints

Figure 5 shows a diagram summarizing the intemucleotide network observed in the NOESY spectra of the SLI dimer. Since two separate resonance peaks were never observed for the same proton in the two strands of the duplex in the NMR spectra, the same constraints were introduced for the corresponding nucleotides of the two strands but the annealing process was never constrained to give a symmetrical structure. A total of 482 NOE distance restraints, 266 dihedral restraints and 108 hydrogen bonding distance restraints were used for the structure calculations (Table I). This corresponds to 15 experimental constraints per residue. Two steps were used to translate NOEs into distance. In the first step, NOEs obtained at 50, 90 and 120ms were measured by volume integration and translated into distances by using the H5-H6 NOE as a reference (2.46A). The distances were used with an ±0.5 error bar. Then NOE cross-peaks that were seen only at 300ms or longer mixing times were put into the group of very weak NOE restrains with an assigned inter-proton distance of 4.0 to 7.0A. The intensities of cross-peaks involving exchangeable protons depend both on the interproton distance and on the rate of exchange with the solvent. Following this, the estimation of the intensities of the NOE were made by taking into account the line width of the resonance peaks. The 150ms and 250ms H2O NOESY data were used to estimate the interproton distances involving the exchangeable protons, using amino-H2(A) cross-peaks as a reference. For these NOEs, conservative upper limits were used corresponding to 3.5A (strong), 5A (medium) and 6A (weak). In the second step, a back-calculation using the total relaxation matrix was used in order to verify the structure by comparison with all existing and absent experimental NOEs.

Figure 3: Expansion of the aromatic to HI' proton region of the NOESY spectrum of the 23-mer dimer in DP at 2s•c obtained after a mixing time of 300ms. The green lines show the H2 resonance positions. The medium H2(A9)H5(C17) and the weak H2(A16)-H8(A9) connectivities are clearly seen. Figure 4: Expansion of the aromatic-aromatic region of the NOESY spectrum of the 23-mer dimer in 0 20 at 25°C obtained after a mixing time of 300ms. The green lines indicate the H2 resonance positions. The H2(A8)-H2(A16) and the H2(A9)-H2(Al6) are clearly seen while the H2(A8)-H2(A9) connectivity is showed very weak.

A 3'-endo conformation was assigned to all the ribose rings except G7, A8, A9 and G23 based on the absence of HI'-H2' COSY crosspeaks for these residues, and was achieved by restraining their endocyclic torsion angles. The glycosidic torsion angle X was set anti for all the residues, because of the lack of strong H8/H6-H1' NOE intensities. For the other five backbone dihedral angles, a set of loose A-form helix restraints (A-form backbone ±15°) were used to constrain the stem residues backbone (C1 to G7 and C17 to G23) and the G10-C15 sequence to adopt a right helix conformation (37,38).

Figure 5: Summary of the distance restraints for the sequence C5-GI2/CI3'-GI9'. The observed inter-strand restraints can be seen between the protons of the residues G7, A8 and A9 with C17' or A16'. Table 1: NMR restraints and statistics for the extended dimer. *R.m.s.d. Root-mean-square deviation± standard deviation in position of heavy atoms compared to the global average structure.

Structure

It has been shown that the SLI dimer can exist in principle in two different conformations: the kissing complex and the extended dimer (21,27). It is not easy, only on the basis of qualitative NOE data, to discriminate between the two conformations. Consequently, molecular dynamics calculations were done by two methods which discriminated the NOE constraints in order to evaluate the global conformation of the duplex. The first method fixed a linear duplex form i.e. pairing of C1 with G23 of the other strand, pairing of G10 with C15 of the other strand and so on. The second way fixed a kissing complex conformation, i.e. the pairing of C1 with G23 of the same strand, the pairing of G10 with C15 of the other strand and so on. The choice between the two conformations was made on the basis of the NOE energy and more importantly on the number of distance violations implicating the residues A8, A9, Al6 and their next residues G7, G10, C15 and C17 (Figure 5).

In both ways, one hundred random structures were generated using DYANA 1.5 (39). Then these basic starting structures were subjected to a simulated annealing-restrained molecular dynamic protocol using the X-PLOR program [START_REF] Brünger | X-PLOR: a system for X-ray crystallography and NMR[END_REF] with a force-field consisting of bond lengths, bond angles, improper angles, repulsive van der Waals potential (single quartic repulsion), experimental distances and dihedral angles constraints from the NMR data. The force constants were imposed to be 50 Kcal.mol-t.A-2 for NOE distances and 25 Kcal.moJ Initial velocities are assigned according to a Mawxell distribution at 2000K and all masses are set to uniform values (10 a.u.). Then an equilibration period consisted in 1000 steps of 10ps dynamics with a time step of 1 fs at 2000K. The first phase of the calculation consisted of 1000 steps of dynamical simulated annealing with a time step of 1 fs at 2000K. At the beginning of each cycle, the various force field terms were increased gradually to their maximum values. During the stage 2 of the calculation which consisted of 1000 steps of dynamics with the same time step, the temperature was cooled down to 300K. This protocol was finally followed by 1000 steps of Powell minimization. At this point, the converged structures were selected on the basis of their low NOE violation energies and their total energies. Finally the structure convergence was examined graphically.

The SL1 Dimer of HIV-1Lai Exists in an Extended Form in Our Sample

The dynamic molecular calculations for the extended duplex form yielded approximately 20 per cent of structures which converged to low energies with no significant violations. Whereas all runs trying to obtain the kissing complex form yielded numerous violations for the sequence involving the A8, A9 and A16 residues as well as their neighboring residues. We conclude that in our sample, the SLI dimer is in an extended configuration. It is therefore indicated that the observed NOE between H2-A8/H5-C17, H2-A9/H5-C17, H2-A16/H5-C17 were fundamental for the structure determination. Although in the NOE spectra (Figure 3), the signals of H5-C 17 and H 11 -C 17 are overlapping, however for the dynamic calculation it is impossible to restrain H 11 -C 17 to be close to adenines A8, A9 and A 16 in both of the possible structures. On the other hand, we confirmed from the molecular dynamic calculation, using data obtained by NMR, that the H5-C17 is closely adjacent to the H2 adenines of the zipper like motif. The 19 lowest energy structures were subjected to a simulated annealing-restrained molecular dynamics at 1000 K,in which the full van der Waals term was introduced.

The global average of the 19 structures is a symmetric homodimer (Figure 6) in which both SLI whole stem (C1 to G7 and C17 to G23) as well as the G10 to C15 sequence form canonical A-type helix. The part containing the unpaired adenines of A8, A9 and Al6 is stabilized by a partial stacking of the three bases into a zipper-like arrangement (Figure 7). This motif has the effect of unwinding the helix, leading to the elongation of the duplex.

The wobble U6-G18 base-pair locally distorts the stem conformation. The pairing of NIH with N3H of uracil in the place of N4H2 of cytosine in classic WatsonCrick base-pair induces an unusual spatial location of the bases and distorts the conformation of the phosphodiester backbone.

Discussion

We studied the structure of the extended dimer formed from the SL1 sequence part of HIV-1Lai; RNA sequence. Below we shall discuss the formation of this extended dimer and of the resulting structure family. It has already been proposed that the HIV-1 Lai dimerization process involves a twostep mechanism (27) operating via a looploop interaction through the autocomplementary nucleotides G257CGCGCG262 situated in the loop of the SLI sequence. This transient complex is then converted through enhancement of the temperature (27) or through the NCp7 nucleocapsid protein (30) to a more stable extended dimer. A determination of the kissing complex structure has been elucidated (31) and the loop-loop interaction confirmed for a slightly different RNA sequence.

The data presented here suggest that, by slowly cooling a chemically synthesized short RNA fragments (23nt) corresponding to the SLI sequence from HIV-lLai RNA, only an extended dimer is formed. Under the same conditions, only a kissing complex has been observed with the DIS sequence from HIV-IMal RNA (42).

A comparison of the sequence involved in the formation of the HIV-1 DISMal dimer with the sequence involved in the formation of the HIV TARTAR* complex or in the formation of the RNA I:RNA II complex of the ColE1 plasmid, Dardel and coworkers (42) suggests that the size of the loop, which is longer in DIS (9nt) than in TAR (6nt) or ColE1 (7nt), could explain why the symmetrical kissing-complex of DISMal is highly favored over the extended dimer. This however does not explain our finding which indicates that, in the case of the SLI sequence from HIV-lLai RNA which is 9 nt long like the DIS loop sequence of HIV-1Mal RNA, the extended dimer is favored. However, if both SLI sequences have the same loop size and the same stem sequence but a 0-U wobble base pair in HIV-!Mal RNA, the flanking nucleotides of the autocomplementary sequence in the loop are different: A8, A9 and A16 in SL1 HIV-1 Lai or G8, A9 and A 16 in SL1 HIV-1Mal. Consequently, it is interesting to examine the influence of the substitution of A8 by G8 on the relative stability of the extended dimer and of the kissing complex.

In the extended dimer, the three nucleotides A8, A9 and A16 are unpaired and should be regarded as a destabilizing component. But, we showed here that the conformation of the A8-A9-A16 motif corresponds to a zipper like structure which brings the surrounding bases closer together allowing a pairing of nucleotide G7 with nucleotide C17 in the extended duplex; this base pairing is not present in the kissing complex (31). This is in agreement with the results of Muriaux et al. (27) which show that the reactivity of the kissing-complex and of the extended dimer to Rnase V1 are different. Rnase V1, an enzymatic probe of helical structures, induces cuts between residues U253 and G254 (U6 and G7 in the present work) in the extended dimer but not in the kissing-complex, indicating a more stable structure of the duplex around the U253G254-G265C264 helix (U6G7/G18C17 here). The zipper like motif implies the partial stacking of the mispaired A8, A9 and A16 nucleotides. This stacking built around three adenines, that are not present in the DIS HIV-1 Mal sequence, should be an important stabilization factor in the SL 1 HIV1-Lai RNA extended dimer as compared to the loop-loop complex.

Mujeeb et al. (31) showed that the three adenines formed a pair of interloop triplebase arrays (A8-A9-C17) and suggested that these triple-base stacks are critical for stabilizing the kissing complex. We showed here that in the extended duplex, the three adenines A8, A9 and A16 form another stabilizing motif. These two conformational factors bring the importance of the three adenines to light, both in the initiation dimer and the extended dimer. It can be elucidated that the three adenines are well conserved among HIV-1 isolated (18), and replacing the three adenosines with uridines mean that dimerization is abolished in the in vitro situation (17). Kissing complex formation distorted the loop helix and ruptured the distal G7-C17 base-pair of end stem (31). Moreover, pairing of G and C in the loops occurred between two anti-parallel strands without a twist of the helix, while the same sequence forms a A-right helix in the dimer. Consequently, the transition between the kissing complex and the extended dimer should involve an increase of the stacking of G7-C17 and of the bases G10 to C15 in the G10CGCGC15 sequence. This conclusion is in agreement with the increase of hypochromicity observed by Theilleux-Delalande [START_REF] Theilleux-Delalande | Interaction ARN/proteines de nucleocapside chez VIH-1: leurs implications dans les relations structuresfonctions in vitro[END_REF] during the change of conformation of the 23-mer dimer of SL 1 HIV-1 Lai between 25 and 45°C. Taking into account these structural data, we suggest that an increase in the degrees of freedom, brought about by an increase in the temperature, or the interaction with NCp7, enables the energy potential barrier to be passed between the two conformations and releases the loop-loop complex of its mechanical energy, which has been stored to generate the linear duplex. The structural characteristics of the extended dimer presented here and their comparison with those of the kissing complex have taken a great step forward in the understanding of why linearization is a critical step in assembling the mature HIV-1 Lai genomic dimer.

On the basis of mutation effects, chemical probe results and modelisation, it has been proposed (44) that the guanine residue G273 (G8 here) forms a base triple in the deep groove with the purine of the kissing base-pair G277-C276' (G7-C'17) with H-bonds linking the Nl and N2 sites ofG273(G8) and the acceptor Hoogsteen sites of G276 (G8) in SLI HIV-1Mal• Although no evidence of this triple interaction has been observed in an NMR study of the complex ( 42), the lack of resolution and the low sensitivity for the NH2 'H resonance peaks region of adenines, that are close to those of water prevents this. We proposed that there are two stabilizing factors which favor the extended duplex in SLI HIV-lLai over the kissing complex in SL1 HIV-1Mal• Firstly, we suggest that the steric hindrance of 06 and N2H2 in guanine as compared to N6H2 and H2 in adenine respectively, hampers its stacking with the adenines in SLI HIV-lMal and lead to a favored loop-loop interaction in the equilibrium "kissing complex H extended duplex" in SLI HIV-1Mal. Secondly, the absence ofG7-C17 hydrogen bond in SLI HIV-lLai precludes the formation of the base-triple interaction and leads to the formation of a kissing complex with another structure which distorts the helix between the stem and the loop, leading to a less stable structure for the kissing complex than for the extended dimer. The mechanism driving to the change of conformational state induced by temperature effect or by interaction with NCp7 is now under investigation in our laboratory.

Figure 1 :

 1 Figure 1: Two possible conformations of the self-associated SLI HIV-Lai 23-mer: the kissing complex (top) and the extended dimer (below).

Figure 2 :

 2 Figure 2: Portions of the 500-MHz NOESY spectrum for the 23-mer dimer in H2O/D2O (90:10) solution at 5°C obtained after a I50ms mixing-time. a/ Connectivities between imino protons; b/ Connectivities between the imino protons and the amino or the aromatic protons.

  -1 .rad -2 for dihedral terms. The electrostatic terms were not present during the simulated annealing. Very close non-bonded contacts are first removed by 1000 steps of Powell minimization (41).

Figure 6 :

 6 Figure 6: Stereoview of the SL1 HIV-1Lai; extended dimer. The residues A8, A9 and A16 of the two strands are shown in red.

Figure 7 :

 7 Figure 7: Superposition of 9 structures for the G7-A8-A9-G10/C15'-A16'-C17' sequences of the extended dimer. The three adenines form a zipper like motif. The G7 and C17' at the end of the stem as well as the G10 and the C5' bases at the beginning of the central stem are paired. The high propeller-twist value (23°) of the pair G7-Cl7' is due to the proximity of the unpaired bases of A8 and A16 which are not coplanar.

Acknowledgements

This work was supported by a grant to J. Paoletti from Agence Nationale de Ia Recherche sur le Sida (ANRS) and to G. Lancelot from Physique et Chimie du Vivant (PCV). F. Girard was supported by Conseil Regional de la Region Centre and F. Barbault by the Ministere de l'Education Nationale, de Ia Recherche et de Ia Technologie (MENRT).