Noah Spahn
email: ncs@csu.fullerton.edu

When can we migrate? A model for Approaching Legacy System Migration

Migration of a legacy application is not purely a technical endeavor; there are organizational and business perspectives to consider as well. Drawing from both current process models and relevant standards for software maintenance, this paper presents a model for the planning of a successful Legacy Systems Migration. The novel feature of the model is the intentional and transparent reassessment of the software migration with stakeholder involvement.

I. INTRODUCTION

It is a known fact that maintenance is a necessary part of any E-type software system [START_REF] Lehman | On understanding laws, evolution and conservation in the large program life cycle[END_REF]. Lehman and Belady have shown that any useful software system that remains in use will undoubtedly become 'more legacy' over time [START_REF] Lehman | Program Evolution: Processes of Software Change[END_REF]. Systems that are in use must be maintained in order to keep from impeding the business that the system is meant to facilitate. Thus the discussion of legacy system migration must necessarily be preceded by a discussion of how maintenance fits into the software lifecycle.

This paper is intended for software engineers and managers who are considering the migration of a legacy system, and would like to navigate around potential migration hazards. The research was done by maintainers of a large monolithic legacy system, who are interested in a successful migration to a Service Oriented Architecture.

II. OPERATION, MAINTENANCE AND MIGRATION

The definitions of terms are in conformance with the International Standard Organization (ISO). In the document on Software life cycle processes (ISO/IEC 12207-2008 ) defines software operation as: supporting customers of the software product during its operation. According to the Standard, the purpose of software maintenance is to provide cost-effective change management for a delivered software product [START_REF]2002 information technology -software life cycle processes[END_REF].

ISO/IEC 14764 [START_REF]Standard 14764 on Software Engineering -Software Maintenance[END_REF] (Software Engineering standard on the Maintenance Life Cycle Processes) posits migration as a departure from the cyclical maintenance activities of categorizing modification requests (MR). The MRs can be classified as either a correction or enhancement (also known as modernization). Each classification also has two types of further categorization: corrections can be either corrective or preventative and enhancements are either adaptive or perfective. According to the standard such distinctions do not need to be made prior to making the modification.

For the context of this paper, maintenance and migration are viewed as two distinct activities which can have the common goal of extending the operational lifespan of a software product. The International Standards Organization (ISO) places migration as one of the outcomes of a software maintenance plan ( Section 6.4.10.2 [START_REF]2002 information technology -software life cycle processes[END_REF]). Modernization is the adaptive (or perfective) enhancement of an existing software product (according to section 3.7-3.8 [START_REF]Standard 14764 on Software Engineering -Software Maintenance[END_REF]).

III. STRUCTURED MIGRATION PROCESS

Four researchers from Utrecht University [START_REF] Khadka | A structured legacy to soa migration process and its evaluation in practice[END_REF] endeavored to define a process which consolidated the planning and execution stages of a legacy system migration. After conducting an orderly examination of the academic literature and analyzing seventeen case studies, they arrived at a holistic six-phase process. The first three phases pertain to legacy migration planning and the last three for the target system implementation and management.

Figure 1 shows that the planning begins with Legacy System Understanding (LSU) and proceeds in a clockwise direction to Target System Understanding (TSU) and Migration Feasibility Determination (MFD). The Implementation phase on the lower half of the graphic begins with Candidate Service Idenitification (CSI) to Implementation (Imp.) and ends with Deployment and Provisioning (D&P). We have concluded that the six phases of the structured process seek to answer four questions; two questions for each part.

In Migration Planning, the questions are: 1) How would we migrate? 2) Is migration feasible? Likewise, in the Implementation and Management phase, there are similar questions:

1) How will we migrate?

2) How will migration be managed? With a clearer sense of all the phases involved in a legacy system migration, the next question would naturally be one of measuring (or achieving) success within such a process. A continued examination of the existing literature would reveal a valuable study that was presented at the ninth joint conference on Computer Science and Engineering: Success Factors Model for migrating legacy systems to a service oriented architecture (SOA) [START_REF] Galinium | Success factors model: Case studies in the migration of legacy systems to service oriented architecture[END_REF].

Building upon the work work Thomas Erl, these researchers define a target system as the conglomeration of sub-systems Fig. 1: Structured Process for the migration of Legacy Systems(source: [START_REF] Khadka | A structured legacy to soa migration process and its evaluation in practice[END_REF]) which are modeled after the functionality of a company's business process [START_REF] Erl | Service-Oriented Architecture (SOA): Concepts, Technology, and Design[END_REF]. The principal premise of the Success Factors Model asserts that a successful legacy to SOA migration is strongly correlated with keeping both technical and business stakeholders involved at all phases: from planning through deployment [START_REF] Galinium | Success factors model: Case studies in the migration of legacy systems to service oriented architecture[END_REF], [START_REF] Cherbakov | Soa antipatterns: the obstacles to the adoption and successful realization of service-oriented architecture[END_REF], [START_REF] Galinium | Factors affecting success in migration of legacy systems to a service-oriented architecture (soa)[END_REF]. They visual representation of the model follows a linear progression from left to right starting with the Legacy System and ending at the goal of an SOA (figure 4). The model shows three routes to get from legacy to SOA: Technical, Business, Technical & Business. The study concluded that those factors which involved participation from both Technical and Business constituents had the strongest correlation with the success of the migration effort. The most significant factors, which were applied and found to be significant in all of the migration efforts studied were:

IV. SUCCESS FACTORS MODEL

1) Potential of the legacy system to be migrated (into SOA) 2) Strategy of migration 3) (SOA) Governance

According to the categorization of this paper, the 'Potential of the legacy system' was the only factor that was 'Technical only', since the scope of the legacy system's potential was limited to the ability to reuse existing components in the application. The remaining two significant success factors require the joint participation of of Business and Technical parties.

V. ANALYSIS

The six-phase migration process, clearly defines the phases in migration planning and execution. Such clear definitions allow legacy migration to be discussed with a more precise vocabulary.

Next, we looked at the success factors in legacy migration efforts to find that continued collaboration between technical and non-technical stakeholders is strongly correlated with a successful migration. A shared vocabulary enables both business and technical stakeholders to have clear communication throughout the migration planning and implementation.

The pinnacle of planning is the MFD phase, since it is the last stage before proceeding from planning to implementation. However, it was noted that eighty two percent of the case study migration efforts skipped the the MFD phase. We propose a model that addresses the challenge of performing the MFD phase within a legacy system migration.

VI. PROPOSED MODEL

Having examined several models for software maintenance and migration we propose a model for the maintenance of legacy systems that both anticipates and works towards a system migration. For the remainder of this paper we will call this model MALSM (Model for Approaching Legacy System Migration). The model began by coalescing the maintenance activities outlined in the IEEE Std 14764-2006 (Section 5.5 [START_REF]Standard 14764 on Software Engineering -Software Maintenance[END_REF]) into the holistic structured migration process [START_REF] Khadka | A structured legacy to soa migration process and its evaluation in practice[END_REF]. Maintenance is correctly depicted as a cyclical process (fig 3). However migration is a clear departure from the cyclical process and we see value in making migration planning (including component prototyping) as part of the maintenance cycle.

In fact, MALSM proposes to incorporate migration planning into the cyclical maintenance activities so that stakeholders can have regular and meaningful communication about the status of a pending migration. Consider the similarities and differences between the MALSM model (fig 4) and the maintenance process model in fig 3.

A. Tools, Methods and Process

The iterative enhancement model [START_REF] Grubb | Software Maintenance: Concepts and Practice[END_REF] is a common framework for approaching software maintenance. Practitioners of this model can consistently deliver small portions of functionality that have been developed in planned iterations. In the MALSM, we suggest the Agile approach to iterative enhancement. The foremost advantage of this approach is the value placed on a shortened feedback loop [START_REF] Highsmith | Agile Project Management -Creating Innovative Products[END_REF] as a communication channel between stakeholders. Within this model, a Scrum Master can help facilitate the collaboration of business and technical stakeholders. Technical managers, business analysts and end users will share a common vocabulary of the structured phases of migration to clearly categorize the outcomes of a unit of work.

A common practice among Agile adherents is the visual tool from the Kahnban method. Although not a strict requirement of the MALSM, it can provide a means to communicate by placing a high value on the transparency of the work being done. The Agile philosophy and the Kahnban method are often paired with the Scrum process and this is a suggestion of the MALSM. Within a Scrum process, a product backlog of 'proposed work items' is constantly being added to, regularly maintained and drawn from at the planning of each work iteration. The items from a prioritized product backlog are the input to the MALSM. Next, the work items are attended to within the context of a software maintenance process. Work items are discussed by all stakeholders present to gather input from both technical and non-technical stakeholders with the shared vocabulary and process understanding provided by the MALSM.

B. Maintain, Modernize or Migrate?

Utilizing a categorization schema that is similar to the categorization of MRs outlined in IEEE Std 14764-2006, the MALSM extends the categorization by proposing to loosen the constraints imposed by the design structure of the existing system and allowing for an expanded Adaptive and Perfective categorizations (section 6.2 [START_REF]Standard 14764 on Software Engineering -Software Maintenance[END_REF]). For example: instead of considering Perfective modifications to the existing legacy system as improvements to extend it's lifespan, perfective modifications could be the trimming of extraneous features that are no longer utilized. In addition, adaptive modifications could be an opportunity to prototype a component of a Target System.

The prototype can serve as a proof of concept, which can later be loosely integrated into the Legacy System and reused in the Target System. These categorizations can be done either before or after the work has been completed.

The categorization serves to provide meaningful semantics for discourse.

At every meeting the software developer gets to hear the voice of the end user, which is the best way to ensure a successful requirements elicitation [START_REF] Wiegers | Software Requirements, Third Edition (Pro-Best Practices)[END_REF]. This can be achieved by regular stakeholder meetings with facilitated intentional conversations around the status of the software project. Allowing the non-technical stakeholders to be engaged at all stages in the maintenance and migration process.

Migration feasibility determination is a crucial point in any migration effort. Rather than simply acknowledge the need to determine feasibility and move on, the proposed model makes the MFD an explicit (and critical) stage in the software maintenance process. It is the deciding factor for the activities to be carried out in the next sprint.

Comparing the activities of the structured migration process to the proposed model, the disproportionate emphasis on migration planning should be evident. This corresponds to the perilous nature of a wholesale software migration effort and the common error of avoiding the MFD phase before leaving the planning activities and moving on to the enticingly green fields of developing a target system (migration implementation). It was noted that the Structured Migration Process was created to address a need for a consolidated model that provided a holistic view of the migration process (from legacy to target system). In developing the model the authors observed that the majority of software migration efforts sampled did not have an activity to explicitly determine the feasibility of the software migration effort before continuing onto implementation.

The negligence of failing to consider migration feasibility was illuminated by examining the factors of successful software migration efforts. Indeed, according to this observational study, there appeared to be a strong correlation between incorporation of technical and non-technical stakeholders in the oversight of an entire migration effort. The partnership between business and technology was determined to be the key to a successful migration.

In order to address the convenient omission of MFD, we sought to create a model that repeatedly visited the MFD phase with a heterogeneous group of stakeholders from both technical and non-technical groups. By having the stakeholders present at sprint planning and review meetings to discuss maintenance activities as a way of piloting a wholesale migration.

VIII. CAVEATS

The MALSM will have limited utility if there is not enthusiastic support from all levels of management. Indeed, unless upper management is aware of the paradigm and middle managers are actively supportive of the plans for understanding both legacy and target systems in order to determine migration feasibility, the migration efforts will likely flounder. A model built on communication between stakeholders cannot work without being communicated.

IX. FUTURE WORK

In order to properly evaluate the utility of the MALSM there would need to be several case studies evaluated to investigate correlation between adherence to the model and the stakeholder satisfaction with the outcome. Considerations for cost estimation and cost savings could be calculated.

It was noted during the development of this model that the ability to estimate efforts expended on migration would be particularly challenging since maintenance efforts are comingled with migration and modernization efforts. This categorical challenge even stymies a post-migration analysis of efforts expended. If this part of the software process could stand to be improved, it must first be measured [START_REF] Humphrey | Managing the Software Process[END_REF]. The challenge lies in what to measure.

Additionally, it may be helpful to have a specifically challenging case study software migration effort follow the MALSM and document the process along the way, noting the particular challenges and benefits of the model to suggest areas for future refinement of the model.

X. CONCLUSION

The two models were analyzed, decomposed and synthesized with the findings of several other research studies (and standards) to develop a revised structured model for migrating legacy systems without abandoning the current maintenance effort. A holistic view of the migration effort which includes both planning and execution brought to light the omission of a critical phase: that of determining migration feasibility based on a solid understanding of both the target and legacy systems. This glaring omission was shown to be of paramount importance when we considered that two of the three top factors which contribute to the success of migration efforts include both technical and non-technical (business) stakeholders. A complete holistic process was developed with the intent to address these two problems: the visiting of migration feasibility determination and the collaboration between technical and business stakeholders in all phases of migration. The collaboration was intended to be intentional throughout the process, but with emphasis during the migration feasibility determination.

Fig. 2 :

 2 Fig. 2: Success Factors model (source:[6])

Fig. 3 :

 3 Fig. 3: IEEE Std 14764-2006 Maintenance Process [4]

Fig. 4 :

 4 Fig. 4: Iterative Model for migration of Legacy Systems


Minh Khoa Nguyen

Léonard Jaillet
email: leonard.jaillet@inria.fr

Stephane Redon
email: stephane.redon@inria.fr

Minh Khoa Nguyen
email: minh-khoa.nguyen@inria.fr

Stéphane Redon

Stéphane Redon

As-Rigid-As-Possible molecular interpolation paths

Keywords: As-Rigid-As-Possible, morphing, interpolation path, molecular structures

published or not. The documents may come

Introduction

During the last few decades, important progresses in experimental approaches have allowed to generate massive collections of databases describing three dimensional (3D) structures of biomolecular systems, especially proteins [1,2]. While the quality and diversity of the obtained structures keep growing, each representation only corresponds to an instantaneous snapshot of one of the possible (meta)stable states of the system. However, in many cases, these systems may undergo large conformational changes corresponding to the transi-tions between the existing stable states. A good understanding of these transitions is essential since they are known to play important functional roles [3,4]. Moreover, transitions are rare events, i.e. the transition time between two (meta)stable states is typically very small compared to the time spent in the stable states. Therefore, they are extremely difficult to capture experimentally. To address this difficulty while taking advantage of the increase in computational resources, various simulation methods have attempted to reconstruct such representative conformational changes [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF].

Molecular Dynamics (MD) based on Newton's second law is a reference method to study the evolution of a given molecular system [START_REF] Haile | [END_REF]. However, such an unbiased approach is typically unable to capture large structural rearrangements. Particularly for proteins, these rare events require the systems to cross over high energy barriers, resulting in a prohibitive computational cost. Therefore, many methods have been designed to facilitate these crossings and/or to reconstruct the transition paths between given conformations [7].

Some methods directly extend the MD principle [8]. Hence, in this category, we find the Transition Path Sampling framework which attempts to model these rare events and characterizes free energy pathways by focusing on the most interesting part of the simulation [9]. Other methods neglect (at least at some stage) the dynamic aspects of the system. For example, Gaussian network models [10][11][12][13] use mass-and-spring coarse-grain representations to find the normal modes of collective motions. Low frequency motions, which typically participate in large conformational rearrangements, can then be extracted [14]. The Nudged Elastic Band [15] and String [16] methods search for the Minimum Energy Path (MEP) between two stable states from an initial path. Recently, motion planning methods from Robotics have also been applied to the exploration of conformational spaces in order to predict transition paths [17][18][19][20].

The approach presented in this paper is a molecular morphing method [21] which primarily relies on geometrical information. The simplest morphing method to generate a path between two given structures is linear interpolation. However, it tends to create intermediate conformations with artifacts such as artificial scalings, expansions, distortions or unrealistic bond lengths and angles. Hence, more sophisticated methods have been designed to address these problems. The Morph-Pro method corrects the intermediate conformations generated by linear interpolation such that the distances between any two consecutive alpha carbons (C α ) lie within a particular range [22]. The Climber method [23] interpolates the adjacent C α distances while imposing harmonic restraints among them. The well-known morph server [24] analyzes the difference between two given conformations by locating representative hinges. A path is then generated between these conformations by restrained interpolation using an adiabatic mapping technique. The generated paths are reasonably good in the sense that large chemical distortions are avoided. Their visual representations can help to better understand protein motions. With the FRODA method [25], rigidity analysis, geometric constraint and steric constraint are applied to simulate transition paths. A recent geometric target-ing method inspired by FRODA but using a different mathematical formalism has also been proposed [26].

In the field of Computer Graphics, powerful tools have been proposed to easily manipulate and animate meshes. This article extends the As-Rigid-As-Possible (ARAP) approaches that are well known to perform such operations. These methods generate new shapes while attempting to preserve the rigidity of the reference shape. The ARAP idea was initially introduced by Alexa et al. for interpolation in two dimensions (2D) [START_REF] Alexa | Proceedings of the 27th annual conference on Computer graphics and interactive techniques[END_REF]. Since then, it has received a lot of attention. Apart from its 2D extensions [START_REF] Choi | [END_REF][START_REF] Igarashi | [END_REF][30][START_REF] Baxter | Proceedings of the 6th international symposium on Non-photorealistic animation and rendering[END_REF], methods for 3D meshes have also been developed for interactive manipulation [START_REF] Sorkine | Symposium on Geometry processing[END_REF][33][START_REF] Borosán | Proc. of Eurographics[END_REF][START_REF] Zollhöfer | Eurographics[END_REF] and interpolation [START_REF] Sorkine | Symposium on Geometry processing[END_REF][START_REF] Chao | [END_REF][START_REF] Levi | Visualization and Computer Graphics[END_REF][38]. It is this latter application that inspired the present work. Some alternatives to ARAP are As-Isometric-As-Possible (AIAP) methods that preserve distances within the mesh [39,40], the As-Similar-As-Possible method (ASAP) which enforces the preservation of the mesh tetrahedrals [41], path deformation techniques based on physics [42] or possible dynamics [START_REF] Huang | Visualization and Computer Graphics[END_REF], and interpolation methods based on specific surface representations [START_REF] Sorkine | Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing[END_REF][START_REF] Lipman | [END_REF][46][START_REF] Botsch | Symposium on Geometry Processing[END_REF][START_REF] Paries | Computer Graphics and Applications[END_REF]. The reader is referred to [START_REF] Botsch | Visualization and Computer Graphics[END_REF] for a survey of mesh deformation techniques.

This article is organized as follows: Section 2 describes the ARAP methodology and its application to molecular structures. Section 3 shows the results of the ARAP interpolation method on several benchmarks. The pros and cons of the method are discussed in Section 4. Finally, Section 5 concludes this work and gives some suggestions for future improvements and applications.

Methodology

This section presents the ARAP methodology to generate molecular paths. It follows the global framework presented in Figure 1. The input is a pair of conformations (intial and target) described by a set of atoms embedded in the 3D cartesian space and connected by covalent bonds. The output is a morphing path comprising intermediate conformations.

As we will see, ARAP interpolations can be efficiently computed and lead to consistent paths which tend to preserve the local rigidity of the models.

Preprocessing

The ARAP method requires a one-to-one atom mapping (or matching) between the intial and target structures. To obtain such a mapping, a sequence alignment of the residues in both structures is first performed using the MUS-CLE method [50]. Atoms of the aligned residues are then mapped by their PDB names 1 . More robust mapping methods could be introduced, but this is outside the scope of this paper. Only matched atoms will be treated by the Fig. 1: Global framework to generate molecular paths from an initial to a target conformation using the ARAP interpolation method.

ARAP method and hence labeled as ARAP atoms, whereas the other atoms are labeled as non-ARAP atoms.

The ARAP topology is then built as follows: each ARAP atom is considered as a vertex and each bond between two ARAP atoms in the initial structure as an edge 2 . However, such a simple construction may produce disconnected components3 due to missing residues in the initial structure or a broken connectivity when discarding non-ARAP atoms. Since the ARAP method requires that all vertices belong to one connected component, two additional procedures are done to recover the connectivity. First, if the initial molecular system is itself disconnected due to missing residues, we reconnect the structure by creating an edge between the atom just before and the one just after each missing group of residues 4 . Second, to create connections among ARAP atoms, we use a recursive procedure called connect, which operates as follows. For each ARAP atom a, connect(a, m i ) is first called for each neighbor m i of a. The procedure creates an edge between a and m i if m i is also an ARAP atom. If m i is not an ARAP atom, then connect(a, n j ) is called on each neighbor n j of m i . To ensure the termination of the recursive procedure, we check that each atom is visited at most once. The next step forms groups of connected non-ARAP atoms. Each group is associated to a reference ARAP atom that shares a bond with one of the atom in the group. As we will see later, the non-ARAP atom positions are deduced from the ARAP atom positions.

ARAP interpolations may only provide deformations. Hence, we need additional constraints to define the global rotation and translation of the system. To determine these constraints, we perform a 3D structural alignment of the initial and the target structures. This cancels the need for any global rotation, while improving the robustness of the solution obtained, by removing large local rotations during the ARAP interpolation. We use the fast QCP variant developed by Liu et al. [51] of the quaternion superposition methods [52,53] to align both structures. The last step of the preprocessing phase settles the translational part by arbitrarily forcing one of the ARAP atoms to have a linear motion in the final ARAP interpolation.

In summary, the preprocessing phase provides the following inputs to the ARAP algorithm: the (aligned) vertex positions of the initial and target states, the edges that link these vertices together, and the index of the constrained vertex.

As-Rigid-as-Possible interpolation

As stated in the introduction, several implementations and extensions of the ARAP methodology exist. Here, the proposed formalism follows the original extension for 3D meshes [START_REF] Sorkine | Symposium on Geometry processing[END_REF].

ARAP formalism

Let us assume a molecular structure described by n vertices v 1 , . . . , v n connected by edges. Let p i ∈ R 3 be the vertex positions in the initial state S and p i their positions in the target state From the molecular topology, one can extract n sets N i , where each set consists of a central vertex v i and all the vertices whose topological distances to this vertex is lower than a given value k. The maximum distance used to define a set is typically k = 1, thus including only the one-ring neighbors which are the vertices directly connected to the central vertex v i .

S for i ∈ [1, n].
From each topological set, one can define a cell C i consisting of the vertex positions of the set N i for a given state. An example of cells obtained from a molecular system using the one-ring neighborhood is shown in Figure 3.

Let us now consider C i and C i , the cells centered on vertex i in the initial and target states, respectively. As stated in [START_REF] Sorkine | Symposium on Geometry processing[END_REF], if the transformation C i → C i is rigid, there exists a rotation R i such that:

p i -p j = R i (p i -p j ), ∀j ∈ N i (1)
Hence, Sorkine et al. define the approximate rigid transformation between the two cells as the rotation R i that minimizes the cell deformation energy E(C i , C i ), also called ARAP cell energy: where ω ij is the weight associated to edge e ij connecting v i with v j in N(i).

E(C i , C i ) = j∈Ni ω ij ||p i -p j -R i (p i -p j )|| 2 (2)
By default, ω ij = ω ji = 1. However, it is possible that ω ij = ω ji . Figure 4 shows an example of a rotation R i which minimizes the ARAP cell energy.

Considering now all the cells, an ARAP transformation minimizes the total ARAP energy E defined as:

E = i ω i E(C i , C i ) = i ω i j∈Ni ω ij ||p i -p j -R i (p i -p j )|| 2 (3)
where ω i is the cell weight. We will now explain how to compute an interpolation path from such a formulation.

Resolution method

The ARAP interpolation method is described in Algorithm 1. It takes as inputs the vertex positions in the initial and the target states, the desired number of states L in the interpolation path and the index c of the constrained vertex. First, ARAP cells are generated for the initial and target states (lines 2 and 3). Then, each ARAP rotation R i is computed (line 4) by minimizing the cell energy in Equation 2, using the fast QCP variant developed by Liu et al. [51].

Each intermediate state l along the path has a corresponding value t defined at line 6. Hence, t ∈ [0, 1] and the structure is at the initial and target states when t = 0 and t = 1, respectively. For each value of t, an interpolated rotation R i (t) is evaluated for each cell (line 8). An option to perform such an interpolation is presented in Section 2.2.3. Then, the position of the constrained vertex at t is computed by linearly interpolating its positions in the initial and target states (line 9). Finally, for each intermediate state, we compute the vertex positions p i (t) by minimizing the total ARAP energy in Equation 3 (line 10), i.e.

p 1 (t), . . . , p n (t) = arg min i ω i j∈Ni ω ij ||p i (t) -p j (t) -R i (t)(p i -p j )|| 2 (4)

Algorithm 1: ARAP molecular interpolation

Input : The initial vertex positions p i . The target vertex positions p i . The edges.

The number of states L in the interpolation path. The index c of the constrained vertex. Output: A smooth path consisting of L states. The method to solve such a system is presented in Section 2.2.4.

1 for i = 1, . . . , n do 2 C i ← ComputeCell(p i ); 3 C i ← ComputeCell(p i ); 4 R i ← ComputeRotation(C i , C i ); 5 for l = 0, . . . , L -1 do 6 t = l L-1 ; 7 for i = 1, . . . , n do 8 R i (t) ← ComputeInterpolatedRotation(R i ,

Interpolating Rotations

We use the Spherical linear interpolation (Slerp) method to interpolate rotations [54]. This method produces a rotation at a constant angular velocity along the shortest great arc (i.e. a geodesic) on a unit quaternion sphere. The formula to compute a Slerp between two normalized quaternions p and q with parameter t ∈ [0, 1] is:

Slerp(p, q, t) = p(p * q) t (5)
where p * is the conjugate of p. This expression can be rewritten without the exponentiation as:

Slerp(p, q, t) = sin(1 -t)θ sin θ p + sin tθ sin θ q (6)
where cos θ = p•q is the inner product of two quaternions (i.e. if p = [s, (x, y, z)] and q = [s , (x , y , z )], then p • q = ss + xx + yy + zz ).

Finding ARAP positions

To find p 1 (t), . . . , p n (t) in Equation 4, one sets to zero the derivative of the expression inside arg min with respect to each unknown

p k (t) with k ∈ [1, n] \ c,
where c is the constrained index. This produces a system of linear equations LP = b where P is a (n -1) × 3 matrix concatenating the unknown vertex positions p k (t). b has the same size as P. The (n -1) × (n -1) matrix L is the discrete Laplace-Beltrami operator applied to P. Thanks to the one-connected-component property and the constrained vertex, this system always has a unique solution.

The details on constructing these matrices can be found in [START_REF] Sorkine | Symposium on Geometry processing[END_REF]. However, we also briefly give their derivation here. For the sake of simplification, let us drop t in Equation 4 for the moment. Then, setting to zero the derivative of the expression inside arg min of the equation with respect to p k , one obtains

∂E ∂p k = ∂ ∂p k ω k j∈N k ω kj ||p k -p j -R k (p k -p j )|| 2 + j∈N k ω j ω jk ||p j -p k -R j (p j -p k )|| 2 = 0 2ω k j∈N k ω kj p k -p j -R k (p k -p j ) -2 j∈N k ω j ω jk p j -p k -R j (p j -p k ) = 0   j∈N k (ω k ω kj + ω j ω jk )   p k - j∈N k (ω k ω kj + ω j ω jk )p j = j∈N k (ω k ω kj R k + ω j ω jk R j ) (p k -p j )
We now bring back t into the last equation to obtain,

  j∈N k (ω k ω kj + ω j ω jk )   p k (t) - j∈N k (ω k ω kj + ω j ω jk )p j (t) = j∈N k (ω k ω kj R k (t) + ω j ω jk R j (t)) (p k -p j ) (7)
The last equation is used to construct matrices L and b when the constrained vertex c / ∈ N k . Hence, the diagonal and off-diagonal coefficients of L are those in front of p k (t) and p j (t) in the equation, respectively. The transpose of the right hand size in Equation 7 constitutes a row in b. When c ∈ N k , the following equation is used instead,

  j∈N k (ω k ω kj + ω j ω jk )   p k (t) - j∈N k \c (ω k ω kj + ω j ω jk )p j (t) =(ω k ω kc + ω c ω ck )p c (t) + j∈N k (ω k ω kj R k (t) + ω j ω jk R j (t)) (p k -p j ) (8)
As can be seen, L is a sparse, symmetric and positive definite matrix, for which we can thus compute a Cholesky decomposition. Because its coefficients are independent of t, this decomposition can be performed only once, and used to compute all intermediate states.

Implementation details

Cell definition We use the same cell definition as in [START_REF] Sorkine | Symposium on Geometry processing[END_REF], i.e. each cell is composed of one vertex and its one-ring neighbors. Also, we set the cell weights ω i = 1 for all cells and the weights associated to the cell edges ω ij = ω ji = 1.

Reaching the target Direct application of the ARAP method does not allow to reach the target perfectly, i.e. the computed conformation at t = 1 does not coincide with the target conformation. To overcome this limitation, we introduce for each edge e ij an extra rotation R ij and a stretching s ij . Hence, after solving for R i by minimizing Equation 2, we find for each set N i and each j ∈ N i the rotation R ij and stretching s ij such that:

s ij R ij R i (p i -p j ) = p i -p j (9)
The equation to find the atom positions for the intermediate conformations is adapted accordingly:

p 1 (t), . . . , p n (t) = arg min i ω i j∈Ni ω ij ||p i (t) -p j (t) -s ij (t)R ij (t)R i (t)(p i -p j )|| 2 (10)
where s ij (t) is linearly interpolated between 1 and s ij (i.e. s ij (t) = (1 -t) + ts ij ) and R ij (t) is computed as in Section 2.2.3.

Postprocessing

During the preprocessing phase, each non-ARAP atom n was associated to an ARAP atom a n . Hence, once an intermediate conformation is found at t, the position of the non-ARAP atom p n (t) can be deduced from the position of the ARAP atom p an (t) as:

p n (t) = p an (t) + R an (t) (p n -p an ) (11)
where R an (t) is the intermediate rotation of the cell whose central vertex is atom a n .

Results

We have implemented the proposed method in C++ as a module of the SAM-SON software platform [START_REF]Samson software for adaptive modeling and simulation of nanosystems[END_REF]. The Eigen library is used for solving the linear system of equations [START_REF] Guennebaud | [END_REF].

All benchmarks proposed here come from those proposed by the authors of the so-called geometric targeting method in [26]. Since the ARAP method requires connected components, only 13 benchmarks of single-chain molecules were considered. Among them, Heparin Cofactor II was removed because its target structure has too many missing residues (more than 20 consecutive residues). Therefore, only 12 benchmarks are presented in this article. For each case, the ARAP interpolation method is applied to generate a path composed of 20 conformations. The benchmark names, as well as the computational times, are shown in Table 1. The ARAP time is shown per conformation because it grows linearly with the number of conformations on the path.

As one can see, our methodology has globally a very low computational cost. The highest total time to obtain a path of 20 conformations is 1.082 s for DNA Polymerase which has 7313 ARAP atoms. The interpolation time per conformation per ARAP atom (not shown in the table) is about 0.006 ms for all benchmarks. This value is 13 to 82 times smaller for the same quantity shown in [26]. This is an advantage of the ARAP method, which solves a linear system LP = b where the square matrix L is sparse. Hence, the complexity for computing P is close to O(N ) where N , the diagonal size of L, is the number of ARAP atoms subtracted by 1 (the constrained vertex).

When visually inspecting the paths obtained by the ARAP method, the results appear reasonable except for Collagenase and Spindle Assembly Checkpoint Protein, where steric clashes occur. We provide in the following a more detailed analysis of the intermediate structures obtained from the ARAP path.

Bond lengths, bond angles and dihedral angles

To show the structure preservation property of the ARAP interpolation method, we look at how bond lengths, bond angles and dihedral angles in each intermediate conformation deviate from those in the initial conformation. Consider the first benchmark, 5'-Nucleotidase, which has a large rotation (about 90 • ) in one part of the structure (Figure 10a).

Figure 5a shows the absolute value of the change in bond length during interpolation with our approach. The horizontal axis shows the conformation sequence index along the path, where 1 corresponds to the first conformation and 20 corresponds to the last one. The vertical axis is the change in bond length in angstrom ( Å). The blue line plots the average change in bond length in each conformation. For each conformation, a blue bar shows the standard deviation from the mean value. The dotted and solid red lines plot the maximum and minimum values, respectively, of the change in bond length for each conformation. Therefore, the area between the maximum and minimum curves represents the range of the (absolute) change in bond length. The deviation area, i.e. the area between the blue bars, is where most values are observed.

The maximum curve in Figure 5a starts at 0 Å for the first conformation because here, it coincides exactly with the initial conformation. The maximum value increases until 0.616 Å (the 11 th conformation), then decreases and finally reaches 0.281 Å at the target conformation. Our method does indeed reach the target conformation. However, this final value is not 0 Å because the bond lengths of the target structures deviate from those of the initial structure to some degree. The deviation area is bounded in the range [-0.012, 0.025] Å, indicating a large amount of bond lengths generated by the ARAP interpolation do not deviate more than 0.025 Å from those in the initial conformation.

The same type of plot for the linear interpolation method5 is shown in Figure 5d. The maximum values are significantly larger (the peak is around 1.118 Å at the 10 th conformation) compared with those of the ARAP interpolation method. The deviation area also has higher mean and standard deviation values, which suggests that the ARAP interpolation method preserves bond lengths much better than the linear interpolation method.

Similarly, Figure 5b and 5e show that ARAP interpolation preserves bond angles better than linear interpolation. On the other hand, the results on the dihedral angle do not differ greatly in both methods as shown in Figure 5c and 5f. This is because the chosen one-ring topology of a cell contains no dihedral angles, and hence dihedral angles are not constrained by the proposed method. Therefore, the method naturally favors global deformation through changes in dihedral angles, which is typically expected in conformational changes of molecular systems. Further discussion on the dihedral angle can be found in Section 4.1.

We have detailed above the results for 5'-Nucleotidase, but the same behavior is observed for all benchmarks. Table 2 shows the maximum mean value across the path regarding the change in bond lengths, bond angles and dihedral angles for each benchmark. For bond lengths and bond angles, the values in the ARAP columns are always lower than those in the Linear columns, i.e. the ARAP method preserves these quantities better. However, the results for dihedral angles do not follow this pattern. In fact, maximal changes in dihedral angles occur at the target conformation, which is why the values are the same for both methods in the table.

C α distance

Most coarse-grain methods use distances among C α atoms for morphing. In [10], these distances are interpolated between the initial and target structures, whereas in [22] those between consecutive C α atoms in intermediate structures are constrained to the range [3.7, 3.9] Å. Figure 6a shows how consecutive C α distances evolve along the path for 5'-Nucleotidase using the proposed method. Although the value range area is within [3.556, 3.822] Å, the deviation area indicates that most values lie in the range [3.775, 3.86] Å. This shows that the ARAP interpolation method tends to preserve consecutive C α distances as well. The linear interpolation method does not preserve well this quantity as shown inFigure 6b. As Table 2 indicates, this behavior is also observed for all benchmarks. Similar to bond lengths and bond angles, the changes in consecutive C α distances are always lower in the ARAP method than those in the linear method.

Motion of structures along the path

The closing motions of Adenylate Kinase and Calmodulin are captured by the ARAP method in Figure 7.

Figure 8a and 8b show the shear motions of one part of Alcohol Dehydrogenase and Dihydrofolate Reductase, respectively. This type of motion is also observed in Pyrophosphokinase (Figure 8c). A slight change in the loops (green and light blue) on the left is caused by a shear motion of the green helix and light-blue loop.

In Dengue 2 Virus Envelope Glycoprotein (Figure 10d), a closing motion of a hinged domain (red, orange and yellow) and a rotation of about 180 • of the small light blue loop are captured.

Our proposed method agrees with [26] for the motion of Diphtheria Toxin, i.e. a rotation of about 180 • of a large domain (see Figure 10b). Note that this motion was not found by the Yale Morph Server [24].

In Figure 10a, our method shows a rotation of about 90 • of one part of 5'-Nucleotidase. Figure 10c (DNA Polymerase) shows the closing motion of a group of green helices and slight rotations of other domains (red and yellow).

For Pyruvate Phosphate Dikinase (Figure 10e), one observes an opening motion carried out by two domains. One domain rotates by 90 • (yellow and orange) while another rotates less than 90 • (in green and purple).

Some weaknesses of the proposed method are shown in the case of Collagenase and Spindle Assembly Checkpoint Protein. Steric clashes are observed during the formation of the dark blue helix loops for both cases (Figure 9). This is due to the chosen Slerp method for interpolating rotations, which restricts the rotation angle to the range [0, π] while the helix formation requires more than this. In the case of Spindle Assembly Checkpoint Protein, one also observes clashes between the red strand and other strands of the beta sheet. This is because the ARAP method does not guarantee the absence of steric clashes between distant (non-bonded) parts of the structure.

Discussion

ARAP energy and dihedral angles

One classical way to represent molecular systems is through the use of dihedral angles (see e.g. [START_REF] Zhang | [END_REF]). Hence, one can represent the backbone of a protein in terms of φ, ψ, ω angles and the mobility of the side chains in terms of dihedral angles χ. This internal coordinate representation is commonly used, since it is known that bond lengths and bond angles vary only slightly at room temperature whereas major conformational rearrangements are often due to dihedral angle variations [START_REF] Schlick | Molecular modeling and simulation: an interdisciplinary guide: an interdisciplinary guide[END_REF].

There is an interesting connection between the internal coordinates representation and the ARAP energy definition. If a one-ring topology is used for ARAP cell construction, any move in internal coordinates is just a combination of rigid rotations of cells, which results in zero ARAP energy. On the other hand, conformations produced by ARAP interpolation may not result from changes in internal coordinates only. This is because our solution minimizes the ARAP energy, but may not zero it along the path. Therefore, conformations produced by ARAP interpolation may be close to conformations produced by changes in internal coordinates, but may also include changes in bond lengths and bond angles. This helps ARAP interpolation overcome two obstacles faced by internal coordinates moves: the impossibility to reach the target conformation due to fixed bond lengths and bond angles, as well as the loop closure problem [START_REF] Shehu | [END_REF].

Mesh quality

In our methodology, ARAP edges are "derived" from covalent bonds in molecular systems (through the recursive connect procedure). Hence, the ARAP topology constructed during the preprocessing phase does not form a mesh composed of triangles or tetrahedrons, as is typically the case in Computer Graphics. Despite the simple graph structure that we use, the result section has shown that the method works efficiently and produces paths with adequate geometric properties.

Large rotations

One limitation of the proposed methodology is that the amplitudes of local rotations are smaller than π. In our case, the initial structural alignment performed during the preprocessing phase alleviates this problem to some degree. However, this may not be sufficient to represent helix formation, as shown for Collagenase and the Spindle Assembly Checkpoint Protein. In 2D Computer Graphics, this issue can be resolved by correcting rotation angles so that the absolute angle differences between adjacent triangles are lower than π [START_REF] Baxter | Proceedings of the 6th international symposium on Non-photorealistic animation and rendering[END_REF]. Unfortunately, this cannot be directly applied in 3D because the rotation axes of adjacent triangles do not align. In the future, we would like to implement recent approaches to address this problem [START_REF] Kaji | Mathematical Progress in Expressive Image Synthesis III[END_REF].

Symmetry

The method presented so far is not symmetric, i.e. the result of an interpolation from the target to the initial structure would differ from that of the reverse direction. However, this property may be interesting for some applications. An easy way to tackle this limitation is to perform ARAP interpolation from both directions and take the average of the results. However, this solution would double the computational cost while in many cases, the result from one direction may be close to the average produced by a bi-directional solution. Hence, such a symmetric version of ARAP has not been implemented yet.

Conclusion

We have presented a new morphing method able to generate interpolation paths between two given molecular structures. This method relies on the ARAP paradigm used in Computer Graphics to edit complex meshes while preserving local characteristics of the structure. Despite being a purely geometrical approach, the proposed method generates interpolation paths at a very low computational cost, while preserving bond lengths and bond angles very well. Although the presented method has some limitations discussed above, we believe ARAP interpolation may find numerous uses, and we would like to investigate several extensions.

Obviously, ARAP interpolations may be complemented by an energy minimization step to locally improve the quality of the intermediate conformations (although this might significantly increase the cost of producing a path). In problems requiring more complex paths, one could imagine more sophisticated methods to locally improve the quality of ARAP interpolations. For example, the ARAP method could be combined with Steered MD [START_REF] Phillips | [END_REF] in order to find in which directions steered MD forces would be applied. ARAP interpolations could also be used to start off umbrella sampling for free energy calculations [62].

There are also many directions we would like to explore in order to extend the ARAP methodology to the field of structural biology. One of them is to improve ARAP interpolations by introducing more edges in the ARAP topology, based on for e.g. other pairwise interactions such as hydrogen bonds or dipolar interactions. Another possibility is to reflect chemical properties of molecules in the ARAP edges, by adjusting their weights according to, for e.g. known equilibrium bond lengths. We would also like to allow for large internal rotations thanks to recent ARAP improvements such as [START_REF] Kaji | Mathematical Progress in Expressive Image Synthesis III[END_REF]. Furthermore, the ARAP interpolation computation is highly parallelizable, since intermediate conformations can be computed independently from one another, and we plan to take advantage of this property to improve the efficiency of our implementation.

6 Supplementary Material Section 3.1 and Section 3.2 presented the results on bond length, bond angle, dihedral angle and consecutive alpha carbon distances for only the first case in the benchmark list. The same results for all the cases and video clips of their motions can be found in the supplementary material.

Fig. 2 :

 2 Fig. 2: Example of ARAP connectivity construction starting from the ARAP atom marked with a star. ARAP atoms are grey, while non-ARAP atoms are white. (a) The initial molecular topology. (b) ARAP edges created from the marked ARAP atom and represented by bold lines. When the connect procedure stops for the marked atom, the top-left ARAP atom is not connected, since its connection to the marked atom passes through another ARAP atom.

Fig. 3 :

 3 Fig. 3: Some part of a protein (top-left) and examples of cell decomposition for three atoms: nitrogen (A/upper-right), carbon (B/lower-left), and oxygen (C/lower-right).

Fig. 4 :

 4 Fig. 4: The local ARAP rotation that maximizes rigidity. a. The initial cell. b. The target cell. c. The optimal rotation R i minimizes the sum of the rotated ||p j -p j || 2 distances (arrow bars), after aligning the cells such that p i = p i .

Fig. 5 :

 5 Fig. 5: Statistics of absolute changes in bond length, bond angle and dihedral angle for 5'-Nucleotidase. Results from ARAP interpolation for (a) bond length, (b) bond angle, (c) dihedral angle. Results from linear interpolation for (d) bond length, (e) bond angle, (f) dihedral angle.

Fig. 6 :

 6 Fig. 6: Statistics of Consecutive C α distances for 5'-Nucleotidase from (a) ARAP interpolation, (b) Linear interpolation.

Fig. 7 :

 7 Fig. 7: Open-to-close motions in the colored parts of (a) Adenylate Kinase. (b) Calmodulin.

Fig. 8 :

 8 Fig. 8: Shear motions. (a) Alcohol Dehydrogenase: shear motion of the colored part compared to the static grey part. (b) Dihydrofolate Reductase: shear motion of the blue strand of the beta sheet. (c) Pyrophosphokinase: shear motion of the green helix and the light blue loops.

Fig. 9 :

 9 Fig. 9: (a) Collagenase: Steric clashes occur in the formation of the dark blue helix loop. (b) Spindle Assembly Checkpoint Protein: steric clashes occur in the formation of the dark blue helix and the motion of the red loop.

Fig. 10 :

 10 Fig. 10: Motion of (a) 5'-Nucleotidase: Rotation of about 90 o of the colored part can be detected by paying attention to the red helix. (b) Diphtheria Toxin: Rotation of about 180 o of the colored part can be detected by observing the beta sheets, i.e. the red sheet goes behind the green one in the final picture compared with the first. (c) DNA Polymerase: Closing motion of the green helices. (d) Dengue 2 Virus Envelope Glycoprotein: The closing motion of the red and yellow domains by rotation. The small light blue loop also rotates by 180 o . (e) Pyruvate Phosphate Dikinase: The opening motion accomplished by the green-purple domain and the yellow-orange domain. Rotation of 90 o of the yellow domain can be seen clearly by the orientation of the orange arrows. Rotation of the green-purple domain is smaller.

Table 1 :

 1 Benchmark details and time of execution. A chain id is specified after each pdb code.

				No. of		CPU Time (ms)
	#	Protein Name	Initial/Target Structure	non-ARAP and ARAP atoms	Pre-processing	ARAP time per conformation	Post-processing	Total	Figure
	1	5'-Nucleotidase	1HP1(A)/1HPU(C)	2/4027	130	22	0	570	10a
	2	Adenylate Kinase	4AKE(A)/1AKE(A)	0/1656	47	9	0	227	7a
	3	Alcohol Dehydrogenase	8ADH(A)/6ADH(A)	4/2784	87	16	0	407	8a
	4	Calmodulin	1CFD(A)/1CFC(A)	1072/1166	49	6	33	202	7b
	5	Collagenase	1NQD(A)/1NQJ(B)	0/901	26	5	0	126	9a
	6	Dengue 2 Virus Envelope Glycoprotein	1OAN(A)/1OK8(A)	163/2962	95	17	6	441	10d
	7	Dihydrofolate Reductase	1RX2(A)/1RX6(A)	1/1268	36	7	0	176	8b
	8	Diphtheria Toxin	1DDT(A)/1MDT(A)	0/4021	130	23	0	590	10b
	9	DNA Polymerase	1IH7(A)/1IG9(A)	26/7313	261	41	1	1082	10c
	10 Pyrophosphokinase	1HKA(A)/1Q0N(A)	0/1267	34	7	0	174	8c
	11 Pyruvate Phosphate Dikinase	1KBL(A)/2R82(A)	7/6745	234	38	0	994	10e
	12 Spindle Assembly Checkpoint Protein	1DUJ(A)/1KLQ(A)	1479/1509	64	8	46	270	9b

Table 2 :

 2 Comparison of maximum mean values of changes in bond lengths, bond angles, dihedral angles and consecutive C α distances

	#	Protein Name	Bond change (×10 -3Å )	Angle change ( • )	Dihedral angle change ( • )	Consecutive Cα distance change (×10 -3Å )
			ARAP Linear ARAP	Linear ARAP Linear	ARAP	Linear
	1	5'-Nucleotidase	9	124	1.4	4.7	6.4	6.4	19	274
	2	Adenylate Kinase	16	92	2.7	5.0	14.4	14.4	46	97
	3	Alcohol Dehydrogenase	23	81	3.6	5.8	19.8	19.8	53	53
	4	Calmodulin	4	71	1.3	2.9	7.2	7.2	7	82
	5	Collagenase	18	118	3.5	6.0	14.5	14.5	57	188
	6	Dengue 2 Virus Envelope Glycoprotein	11	141	2.9	6.9	18.0	18.0	27	208
	7	Dihydrofolate Reductase	23	64	2.4	4.7	9.7	9.7	57	72
	8	Diphtheria Toxin	19	216	3.0	10.5	11.1	11.1	40	483
	9	DNA Polymerase	6	57	1.6	3.7	10.8	10.8	14	30
	10 Pyrophosphokinase	12	80	2.5	5.0	10.0	10.0	22	81
	11 Pyruvate Phosphate Dikinase	15	93	1.4	3.6	7.8	7.8	32	227
	12 Spindle Assembly Checkpoint Protein	23	200	4.8	10.5	22.8	22.8	46	352

More specifically, two atoms are mapped if they share the same name or have equivalent names such as 1HG2 and HG21.

Optionally, to build edges, one could consider only the bonds which are present in both the initial and the target structures.

A connected component is a set of vertices and edges such that any vertex can be reached from another vertex by traversing through a series of edges in the same component.

More specifically, the chosen atoms to create the edge are alpha carbon atoms

The path by the linear interpolation method is also generated after the initial and target conformations are 3D structurally aligned.

funding from the European Research Council through the ERC Starting Grant No. 307629.


Elda Miramontes
email: elda.miramontes.garcia@ifremer.fr

Antonio Cattaneo

Gwenael Jouet

Sebastien Garziglia

Gwenael

Implications of Sediment Dynamics in Mass Transport along the Pianosa Ridge (Northern Tyrrhenian Sea)

Keywords: Mass transport deposit, Contourite, Pockmark, Tyrrhenian Sea

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Corsica Trough is located between Corsica and the Tuscan Shelf in the Northern Tyrrhenian Sea. It has a maximum depth of 900 m and connects the Tyrrhenian and Ligurian Seas. The eastern flank of the Corsica Trough is formed by the Pianosa Ridge, a tectonic structure with typical slopes between 3° and 10°, reaching 20° locally (Fig. 1). This area is characterised by several submarine landslides and contourite deposits [START_REF] Cattaneo | Submarine Landslides and Contourite Drifts Along the Pianosa Ridge (Corsica Trough, Mediterranean Sea). In: Submarine Mass Movements and Their Consequences: 6th International Symposium[END_REF]Fig. 1). Contourite drifts are formed by the Levantine Intermediate Water (LIW), flowing northwards [START_REF] Artale | Simultaneous temperature and velocity measurements of the internal wave field in the Corsican Channel (Eastern Ligurian Sea)[END_REF] and generating stronger bottom currents on the Pianosa Ridge due to the Coriolis force. The sediment input from the Tuscan Shelf is low since sediment from Italian rivers is mostly trapped in the shelf and cannot reach the Corsica Trough [START_REF] Roveri | Deep-Water Contourite Systems: Modern Drifts and Ancient Series, Seismic and Sedimentary Characteristics[END_REF]. Most of the sediment deposited in the Corsica Trough is transported from the Corsica shelf by the Golo turbidite network [START_REF] Deptuck | Dimensions and architecture of late Pleistocene submarine lobes off the northern margin of East Corsica[END_REF]Fig. 1).

The aims of this study are to present the size and the stratigraphic distribution of Mass Transport Deposits (MTDs) along the Pianosa Ridge, and to analyse their relationship with the pre-existing seabed morphology.

Material and Methods

In 2013 two cruises, PRISME2 and PRISME3 onboard the R/V Atalante and R/V Pourquoi pas?, respectively, took place along the Pianosa Ridge. Multibeam bathymetry, 72-channel high-resolution mini GI gun (50-250 Hz) and CHIRP (1800-5300 Hz) seismic reflection profiles, and Calypso piston cores were collected along the Pianosa Ridge. P-wave velocity was measured on whole core sections using a Geotek Multi-Sensor Core Logger (MSCL). The bulk sediment semiquantitative geochemical composition was measured with an Avaatech X-ray fluorescence (XRF) core scanner on split cores. Radiocarbon dating was obtained from bulk planktonic foraminifera at the Poznan radiocarbon laboratory. Radiocarbon ages were corrected for a marine reservoir effect of 400 years and calibrated using Calib 7.0.2 radiocarbon calibration software and the IntCal13 calibration curve [START_REF] Reimer | IntCal13 and Marine13 radiocarbon age calibration curves 0-50,000 years cal BP[END_REF]. MTDs were identified by seismic facies validated by the analysis of equivalent coring samples. MTD volume was calculated using the Kingdom Suite ® software by quantifying the time interval between the top and the bottom of the MTD from the seismic profiles. The velocity applied for time-depth conversion (1550-1560 m/s) was obtained from the velocity analysis calculated with the ProMAX ® Seismic Processing Software.

Results

Along the Pianosa Ridge 11 MTDs were identified in the Quaternary sediment record. Differences in location, volume and morphology were used to distinguish three areas of MTDs: A, B and C (Table 1). Seismic correlation with the borehole GDEC4-2 (Fig. 1; Angue Minto'o 2014) provided information about the stratigraphy in the region.

Area A shows a small submarine landslide associated to the destabilisation of the northern flank of the Elba Canyon (west of the Elba Island), where the slope is 8° to 20°. The headwall scarp and the deformation features of the MTD are clear in the bathymetry (Fig. 1). MTD A is of Holocene age, since it affects a regional reflector dated at 12 kyr.

Six small to intermediate size MTDs (B1 to B6) are observed in area B (Table 1). They are located west of the Pianosa Island, where the Pianosa Ridge forms an edge, generating in this zone high slope gradients (8-10°). The MTDs have an oval shape in plan view. B1, B2, B3 and B4 have a mean thickness of about 10 m; B5 and B6 mean thickness is 5 m. The continental slope in area B is starved during Pliocene and Quaternary. Pre-Messinian deposits (age deduced by correlation with the well Martina 1; Figure 1; [START_REF] Pascucci | Seismic Stratigraphy of the Miocene-Pleistocene Sedimentary Basins of the Northern Tyrrhenian Sea and Western Tuscany (Italy)[END_REF]) crop out in patches on the upper slope. Despite the reduced size of these MTDs, their thickness corresponds to about 30% of the Pliocene-Quaternary deposits in the lower continental slope. B6 and B5 MTDs are older than 542 kyr, while the sediment covering B4 was deposited at 452 kyr, based on correlation to the bottom of borehole GDEC4-2. B3 is close in time to B4 but separated by 300 kyr from B2 (Table 1). B1 is the most recent MTD (42-59 kyr) in area B, it took place about 80 kyr after B2. The frequency of MTDs in area B is irregular and it was higher during the Middle Pleistocene than during the Upper Pleistocene.

Four MTDs (C1 to C4) are located in area C, in the southern zone of the study area. Their volume ranges from 0.67 to 2.62 km 3 and they are characterised by a high width/length ratio, except for C2 that has a ratio lower than 1 (Table 1). Between 265 and 380 kyr BP C3 and C4 were formed, followed by C2 at 125-160 kyr (Table 1). The Pianosa Slump (C1) is the most recent MTD of area C and it was analysed in more detail than the other MTDs since it was sampled by the Calypso piston core PSM3-CS022 (Figs. 2,3). The top of C1 (i.e. the bottom of the post-slide hemipelagic sediment; Fig. 3) is dated at 42,326 cal yr BP and [START_REF] Cattaneo | Submarine Landslides and Contourite Drifts Along the Pianosa Ridge (Corsica Trough, Mediterranean Sea). In: Submarine Mass Movements and Their Consequences: 6th International Symposium[END_REF] report that the MTD disrupted sediments as old as 50,000 cal yr BP. C1 is overlain by about 17 to 20 m of sediment (Fig. 3). However, various morphological features related to MTD C1 can still be appreciated at the present seafloor (Fig. 2). The headwall domain of C1 has the same lateral extent as the Pianosa Slump deposit. However, the headwall morphology is rather irregular. In the northern part, the headwall is continuous, parallel to a normal fault (Fig. 2b). It is connected to the MTD by a sidewall that is 30 m high at the present seafloor. South of this zone, the slope is cut by multiple discontinuous headwalls and scarps. C1 is composed of three main parts (Figs. 2b,2c) deduced from seismic profiles, bathymetry and Calypso piston core PSM3-CS022 (Fig. 3): 1) strongly deformed sediment, 2) fold and thrust system and 3) tilted blocks. The strongly deformed sediment corre-sponds to chaotic seismic facies that are present in the central northern part and in the southern part of C1, where the headwall is discontinuous. In this area the surface morphology shows lobate shapes and crests that indicate a movement towards the west that turned slightly to the southwest, in the direction of increasing depth. The MTD thickness of the southern part shows that there is a deposit at the foot of the slope up to 50 m thick and a thicker zone (50-70 m) in the central distal part. In the northern part of C1 the internal structure of the deposit is more complex than in the southern part. In this area the toe domain of the MTD is formed by a fold and thrust system. The frontal part of C1 is composed of hemipelagites and turbidites that are deformed and compressed forming a system of thrusts, folds and pressure ridges, with individual pop-up blocks apparent in the frontal part of the MTD (Fig. 2). The greatest thickness of C1 deposits (up to 100 m) is found in the central zone of C1 where the pressure ridges are higher (up to 10 m high) and larger (up to 2 km long) (Fig. 2a). The proximal part consists of tilted blocks (Fig. 2b) that become more deformed towards the south. Tilted blocks are identified in seismic profiles where reflectors dip in different directions. Moreover, tilted blocks form crests that are apparent in the seafloor. Groundtruthing evidence of the presence of tilted blocks comes from the bottom of Calypso piston core PSM3-CS022 (Fig. 3). This core shows that a 4.7-m-thick debris flow deposit composed of mud with clay pebbles lies on top of the tilted blocks. These distinct seismic and lithological facies correspond to different degrees of deformation, which can be identified in the Calcium curve. When the degree of deformation is very high (debris flow deposit), the Calcium signal is very noisy (Fig. 3). Between the compressed frontal zone and the zone of tilted blocks, the sediment is strongly deformed, characterised by chaotic seismic facies and by a relatively smooth upper surface (Fig. 2). Since the pressure ridges and the crests are perpendicular to the mass movement, they provide information on the direction of the transport. The northern part of the Pianosa Slump comes from the area of the straight headwall parallel to the fault. The northernmost part moved slightly towards the northwest, while most of the sediment was transported in a southwest direction, towards the deepest part of the basin.

Multibeam bathymetry reveals the presence of many pockmarks located in the contourite drifts between areas B and C and close to MTDs in zone C (Fig. 1). The fluid escape paths over the Pianosa Slump are mainly found in the pressure ridges, in the crests and in the sidewall (Fig. 2b).

Discussion and Conclusions

Slope instability has been a recurrent process along the Pianosa Ridge, although the origin and the control factors could vary according to the location of the submarine landslides. The differences in size and volume between MTDs in area B and C are linked to an uneven sedimentation along the Pianosa Ridge related to the basement morphology and to bottom currents. Sediment accumulation on the slope in area C is much higher compared to area B. Therefore, more sediment could be potentially mobilised by mass transport during a single event, generating larger submarine landslides. In area B bottom currents are accelerated due to the higher slope gradient, preventing sediment deposition.

Group B and C present some similarities in the time intervals of slope failure, since B1-C1 and B2-C2 are in similar ranges of age. In contrast, the other MTDs Fig. 3 From left to right: CHIRP seismic facies, core log, P-wave velocity and Calcium curves of Calypso piston core PSM3-CS022. Photo of the core and zoom with inverted colour. The red area shows a turbidite (peak in P-wave velocity), the orange area a debris flow deposit and the yellow area tilted blocks. Three calibrated radiocarbon ages are represented with red stars are not synchronous. When C3 and C4 were formed, area B was in stable conditions (Table 1).

The main predisposing factor in areas A and B seems to be the steep slopes (8-20° and 8-10°, respectively) (Fig. 1). Within the Elba Canyon, due to the Coriolis effect, bottom currents could produce a stronger erosion on the northern flank of the canyon, generating the observed canyon asymmetry with a steeper northern flank (Fig. 1). Incisions and oversteepening on the northern flank might have contributed to the formation of MTD A. In area B, six MTDs are stacked, and represent a relevant percentage (30%) of the Pliocene-Quaternary deposit in the lower slope, suggesting that an important proportion of the sediment accumulated on the middle and upper slope was affected by mass transport. The formation of submarine landslides in area C (Fig. 1) seems to be much more complex. Multiple factors may predispose slope instability in this area. Bottom currents may have an important role since they provide sediment and generate incisions with higher slopes, corridors of non deposition or slight erosion (moat with slope of 15°; Fig. 1). In addition, layers of coarser sediment in contourite drifts seem to influence the location of failure planes [START_REF] Miramontes | The Pianosa Contourite Depositional System (Corsica Trough, North Tyrrhenian Sea): stratigraphic evolution and possible role in slope instability[END_REF]) and could potentially stock more gas than the overlying mud. Numerous pockmarks affect the present day seafloor on top of the MTDs and contourite drifts (Figs. 1,2). Some of these pockmarks are clearly rooted on buried MTDs and turbidite lobes (Fig. 1). Such a configuration suggests that the emplacement of these sedimentary bodies might have influenced subsequent instability.

In the Pianosa Ridge the present-day seismicity is considered to be very low [START_REF] Scisciani | Active intraplate deformation within Adria: Examples from the Adriatic region[END_REF], while the area entered in a post-rift phase from the late Messinian [START_REF] Pascucci | Seismic Stratigraphy of the Miocene-Pleistocene Sedimentary Basins of the Northern Tyrrhenian Sea and Western Tuscany (Italy)[END_REF]. However, the location of the headwall of the Pianosa Slump next to a normal fault that affects the Pliocene-Quaternary record (Fig. 3) raises questions about the possible relationship between faulting and slope instability.

Fig. 1

 1 Fig. 1 Seafloor gradient map of the Corsica Trough with superposed location of morphosedimentary features: Mass Transport Deposits (MTDs), contourite drifts and turbidite lobes (see legend). Three bathymetric cross-sections across areas A, B and C show the location of MTDs

Fig. 2 a

 2 Fig. 2 a Isopach map of the Pianosa Slump (C1) overlaying the present seafloor gradient map with the location of Calypso piston core PSM3-CS022 b Morphological interpretation of C1 c Multichannel high-resolution mini GI gun seismic reflection profile across MTDs C1, C3 and C4 and line drawing with the interpretation

Table 1

 1 Characterisation of Mass Transport Deposits in areas A, B and C

	Area Name	(km 3 ) Volume	runout (km) Maximum	length (km) Maximum	(km) Width	ratio Width/Length	Age (kyr BP)
	A	A	0.10	1.5	1.2	3.6	3.0	<12
	B	B1	0.19	7.0	5.7	7.6	1.3	42-59
		B2	0.09	8.0	5.6	4.6	0.8	125-137
		B3	0.26	7.5	5.9	7.2	1.2	420-437
		B4	0.15	4.2	2.5	7.4	3.0	452->542
		B5	0.05	6.0	3.4	4.4	1.3	>542
		B6	0.04	6.2	3.6	4.4	1.2	>542
	C	C1	2.62	9.0	6.0	13.5	2.3	42-50
		C2	0.47	16.0	10.0	7.8	0.8	125-160
		C3	0.67	6.0	4.0	8.7	2.2	265-321
		C4	1.86	10.4	5.3	18.6	3.5	321-380

Acknowledgments

We thank the Captain and the crew of the PRISME2 and PRISME3 cruises (2013) onboard R/V Atalante and R/V Pourquoi pas?, respectively. We are grateful to reviewers M. Rovere and C. Campbell for their valuable suggestions that improved the manuscript. The thesis of Elda Miramontes is co-funded by TOTAL and Ifremer as part of the scientific project TOTAL-Ifremer PAMELA. We thank projects LabexMER ANR-10-LABX-19-01 Axis 4 for lab support.


Introduction

In supersymmetric (SUSY) theories the properties of bosons are related to those of fermions so that each particle has a partner with spin differing by 1/2. Supersymmetric extensions of the Standard Model [1,2] are very appealing since they offer a solution to the hierarchy problem. Indeed, the quadratic divergence in the Higgs mass radiative correction appearing in the Standard Model is exactly canceled by the supersymmetric particles contribution, leaving only a logarithmic term:

Δm 2 h ∝ m 2 susy ln (Λ UV /m susy ), (1)
where m susy is the mass splitting between the Standard Model particles and their supersymmetric partners and Λ UV is an ultraviolet cutoff. This equation tells us that the masses of the new SUSY particles should not be much bigger then the TeV scale, otherwise the correction to the Higgs mass would be unnaturally large compared to the electroweak breaking scale (∼ 100 GeV) and the hierarchy problem would appear again. So, these new particles can be produced at the LHC and generate by successive decays a cascade of SUSY particles, ending with the lightest supersymmetric particle, which is often the stable and weakly interacting lightest neutralino, implying a missing transverse energy signature.

During the cascade process, the lightest neutral Higgs boson (h) may also be emitted. In this case, the missing transverse energy produced in association with the Higgs can be exploited to reduce the background, making it possible to study the dominant decay channel h → b b, otherwise covered by the enormous QCD continuum.

Higgs searches in the h → b b channel

Exploiting the capabilities of the ATLAS detector [3,4] in missing transverse energy measurement and b-tagging, the passage of weakly interacting particles may be revealed and a bb pair with invariant mass peaking around the Higgs mass can be reconstructed. The following selection cuts were applied:

• E miss T > 300 GeV;
• exactly 2 b-jets with p T > 50 GeV;

• two light flavoured jets with p T > 100 GeV;

• no leptons with p T > 10 GeV.

The main sources of background are SUSY events themselves, as they contain lot of b-jets, both true and mistagged, and Standard Model top production. Moreover, the minimization of fake high transverse missing energy signals produced by instrumental effects is mandatory to control the QCD background.

For the Minimal Supergravity (mSUGRA) [5] point ( m 0 = 300 GeV, m 1/2 = 425 GeV, A = 1

LAPP-EXP-2007-04

Michele Consonni 200 GeV, tan β = 20, μ > 0 ), which is compatible with the last searches at LEP [6], a signal significance (S/ √ B) of order 10 can be achieved after 10 fb -1 . Figure 1 shows the invariant mass distribution of the bb pair for this scenario.

SUSY parameter determination

Once beyond the Standard Model phenomena are discovered, it is important to determine the masses and couplings of the new observed particles. At the LHC, mass informations are provided by thresholds and edges in the invariant mass plots of particles produced in SUSY cascades.

In the case under study, the Higgs is mainly produced by the cascade:

qL → χ0 2 q → χ0 1 hq. ( 2
)
As a consequence of two-body kinematics, the invariant mass of the Higgs-quark pair shows both a minimum and maximum value, related to different combinations of the masses of the involved SUSY particles.

Figure 2 shows the bb+jet invariant mass distribution, where the jet is the one minimizing the invariant mass, selected between the two with hardest p T in the event. The mass edge is clearly visible and its value can be obtained by a Gaussian convoluted triangular shape fit. The mass threshold can be measured in the same way by replacing the jet with the one maximizing the invariant mass.

The statistical error on the two values after 100 fb -1 is estimated to be of order 5 GeV and a systematic error of about 1% is expected on the jet energy scale.

Figure 1 .

 1 Figure 1. Invariant mass distributions of the bb pair.

Figure 2 .

 2 Figure 2. Invariant mass distributions of the bb+jet system.