Thomas Gaspard

Antoine Bertout

Joël Goossens

Emmanuel Grolleau

Pascal Richard

WIP : Feasibility analysis of real-time periodic multi-phase tasks on unrelated multiprocessor platforms

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

I. INTRODUCTION

Heterogeneous processors have gained widespread use in mainstream computing platforms, particularly in System-on-Chip (SoC) designs found in personal mobile devices. These processors exhibit diverse architectures and incorporate specialized computing units such as GPU (Graphics Processing Unit), DSP (Digital signal processors), NPU (Neural Processing Unit), and others. As a result, applications can be decomposed into subtasks and executed on processors specifically suited to their characteristics, thereby optimizing performance. In certain scenarios, subtasks may be compiled for different architectures and executed conditionally based on the system context.

Emerging platforms, such as the Mediatek Helio X20® or ARM big.LITTLE®-equipped boards, consist of heterogeneous CPUs that share the same Instruction Set Architecture (ISA) and feature full cache coherence. In this configuration, task migration or preemption becomes even more feasible. These platforms typically comprise a cluster of fast but energyconsuming (big) cores and a cluster of slower but more energyefficient (LITTLE) cores. The performance of such systems naturally relies on the hardware implementation of identical instructions.

In the realm of real-time literature, heterogeneous platforms are classified as unrelated, where tasks can be executed at different speeds or rates across different processors. In this paper, we adopt the multi-phase model to represent applications, whereby each subtask is executed within its dedicated phase. Consequently, each phase of the same task can be executed on different processors (migrations are allowed) and at a different speed. We focus on the feasibility problem of optimally scheduling a set of multi-phase periodic real-time tasks on unrelated platforms. We first show that employing the seminal techniques for period (mono-phase) tasks is not optimal and propose a novel approach.

It has been shown that constructing a feasible schedule from a workload assignment can be done in polynomial time [START_REF] Lawler | On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming[END_REF][2] [START_REF] Bertout | Workload assignment for global real-time scheduling on unrelated multicore platforms[END_REF]. In this paper, we will not address the construction of the schedule itself, but rather, we will focus on the processing of the optimal workload assignment of multiphase periodic tasks upon unrelated platforms.

The structure of this work-in-progress paper is as follows. In Section II we present the multi-phase periodic task model and define the unrelated platforms. Section III is dedicated to the state of the art, where we review the works on unrelated platforms and the similar model of tasks. In Section IV, we apply a well-known result to our model and highlight its sub-optimality in this context. Section V introduces a less constrained technique, and Section VI is dedicated to future works and conclusion.

II. MODEL OF COMPUTATION AND ASSUMPTIONS

a) Unrelated multiprocessors: The hardware is modelled as a set of m unrelated processors Π . = {π 1 , . . . , π m } (the symbol . = means is equal by definition to). Each task τ i is also characterised by its processor affinities, i.e., with the execution rates r j i for the task τ i upon processor π j . When a job executes for a duration of t at a rate r, the job progresses by t • r.

b) Periodic task: The application is modelled as a set of n periodic implicit-deadline tasks Γ . = {τ i | i = 1, . . . , n}. Each task τ i is characterised by two parameters (C i , T i ) where C i is the worst-case execution time on a fictional processor, and T i is the release period. Each task releases a job every period T i . The first job is released at t = 0 (i.e., we consider synchronous tasks), the k th at t = k × T i and has to complete by (k + 1) × T i (tasks are said to have implicit deadlines). The utilisation of a task τ i is u i . = Ci Ti . c) Phases and processor rates: Each task is composed of sequence of ρ i phases: (τ i,0 , . . . , τ i,ρi-1 ), and computation requirement c j i for j ∈ 0, ρ i -1 , such as ρi-1 j=0 c j i = C i . The novelty of this model is to define processor rates at phase level. For each phase τ i,j , and for each processor

τ i,0 τ i,1 τ i,2
Fig. 1. A task with three phases visualised as an oriented graph

τ 0 i,0 τ 0 i,1 τ 1 i,0 τ 1 i,1
... Fig. 2. A task with two phases visualised as an oriented graph where the jobs are separated π k , r k i,j denotes the execution rate of τ i,j upon processor π k . To ease the presentation of the problem, we consider that the WCET is uniformly distributed among the phases (∀τ i ∈ Γ, ∀j ∈ 0, ρ i -1 , c j i . = Ci ρi ). However, as shown in Section IV, introducing a WCET per phase in the equations to reflect real-world applications is straightforward.

d) Phases activation: More formally, the sequence of phases is defined as follows. Without loss of generality, we can assume that κ i . = Ti ρi is an integer. κ i is the relative deadline of each phase of task τ i . The parameter κ i also determines the release dates of the phases. Relatively to the job release time, the phase τ i,j is defined to be released at time j • κ i and must be completed by time (j + 1) • κ i . Consequently, for each task there is at most one active (released but not completed) phase.

A task can be visualised as an oriented graph. In Figure 1 is shown such a graph with a task τ i made of three phases.

In the following, we will use the notation τ k i,j for the j th phase of the k th job of the task τ i . Such an example is given in Figure 2.

e) Timeline: At the system level, the sequence of phases forms a periodic pattern that we refer to as a timeline. Such a timeline is periodic, of period H . = lcm {T i | i ∈ 1, n }. An example of task set is given in Table I, and the corresponding timeline is drawn in Fig. 3. We emphasize that this visual representation differs from a schedule as no assumptions are made here regarding the order in which phases of different tasks will be executed and on which processors they will be executed.

f) Chunk: In this paper, a chunk will correspond to the 'granularity' of schedule, i.e., within a chunk, there will be no releases of phases. By definition the (largest) size of the chunk is κ min . = gcd {κ i | i ∈ 1, n }. We will divide the timeline into chunks of duration κ min . We will thus characterise a chunk by the set of phases that must be scheduled within. We label the different chunks as follows: t 0 , t 1 , t 2 , . . .. Hence, t a represents the time interval [a • κ min , (a + 1) • κ min ).

g) Preemptions and migrations: We assume that preemptions and migrations have zero cost. There is no restriction on the migration at the job or phase level, including inside a chunk.

τ i T i ρ i κ i τ 1 15 3 5 τ 2 3 1 3 τ 3 2 2 1 TABLE I A MULTI-PHASE PERIODIC TASK SET
As a timeline is periodic, it is sufficient to build a workload assignment on [0, H) for the chunks t 0 , t 1 , . . . , t H κ min -1 . Our workload assignment problem will be formulated as a linear program with x a i,j as variables. x a i,j is the workload of the phase of task τ i on π j during the chunk t a , see details in sections IV and V.

III. STATE OF THE ART

In this section, we provide a brief review of the state of the art on unrelated platforms and present related models that are formally and/or semantically different from our multi-phase model.

A. Optimal scheduling algorithms on unrelated Platforms

Optimal real-time scheduling algorithms for unrelated platforms all follow the same two-step methods. In the first step, the workload of tasks on processors is computed, either by solving a linear programming problem (LP) [START_REF] Lawler | On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming[END_REF], [START_REF] Bertout | Workload assignment for global real-time scheduling on unrelated multicore platforms[END_REF] or by using an ad hoc algorithm [START_REF] Chwa | Optimal real-time scheduling on two-type heterogeneous multicore platforms[END_REF]. In the second step, the (pattern) schedule is constructed offline [START_REF] Bertout | Template schedule construction for global real-time scheduling on unrelated multiprocessor platforms[END_REF], typically through bipartite matching, Birkhoff-Von Neumann decomposition [START_REF] Lawler | On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming[END_REF], or by using an ad hoc algorithm [START_REF] Chwa | Optimal real-time scheduling on two-type heterogeneous multicore platforms[END_REF], on a one-time unit and then rolled out into a concrete schedule within intervals delimited by successive absolute deadlines (deadline partitioning). The seminal work by Baruah [2] introduced this approach. For twotypes multi-core unrelated processors (clusters), the work of Chwa et al. [START_REF] Chwa | Optimal real-time scheduling on two-type heterogeneous multicore platforms[END_REF] propose tailored algorithms for the two-step. Bertout et al. [START_REF] Bertout | Template schedule construction for global real-time scheduling on unrelated multiprocessor platforms[END_REF] addressed the problem of n-types clustered platforms with the objective of reducing inter and intra-cluster migrations.

B. Related Model a) Multi-Phase Multi-Thread (MPMT) [START_REF] Courbin | Scheduling of hard real-time multi-phase multi-thread (MPMT) periodic tasks[END_REF]: The multiphase model adopted in this work corresponds to the singlethreaded version of the MPMT model. In the MPMT a task (referred to as a process in [START_REF] Courbin | Scheduling of hard real-time multi-phase multi-thread (MPMT) periodic tasks[END_REF]) is divided in consecutive phases. Each phase consists of multiple threads, and each thread requires a dedicated processor for execution. Notice that the authors consider the preemptive scheduling of periodic MPMT tasks on identical multiprocessor platforms. Therefore, their results do not directly apply to the unrelated platform model considered in our research.

b) Multi-frame [START_REF] Mok | A Multiframe Model for Real-Time Tasks[END_REF] (MF) task model: The multi-frame approach characterises a sporadic task with an implicit deadline and a cyclic pattern of executions, expressed by a vector of WCET. More precisely, each task job is executed with the WCET at the vector index corresponding to the job index modulo the vector size. Our multi-phase model is formally equivalent to the multi-frame model applied to periodic tasks,

τ 1 τ 2 τ 3 ρ 0 ρ 1 ρ 2 ρ 0 ρ 1 ρ 2 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 ρ 0 ρ 1 t 0 1 2 3 ... 29
Fig. 3. Timeline of the task set given in Table I as the execution of ℓ jobs of an MF task with a strict period of T /ℓ (with ℓ frames) is equivalent to the execution of one job of a multi-phase task with a period of T and ℓ phases. However, they are semantically different in the sense that we consider a single job as a task execution with different "parts", with different processor rates (and computing requirements).

For instance, one can imagine that the compiler decomposes the binary into different phases with different rates for CPUs, FPGAs, GPUs, etc. Nonetheless, to the best of our knowledge, there is no existing work on the execution of MF tasks on unrelated platforms. c) Graph-based models: Similarly to an MF task, a multi-phase task can be represented using a directed graph [START_REF] Stigge | The digraph real-time task model[END_REF] representation, where each phase is represented as a vertex and the edges symbolize the sequential execution between phases. Phases may be considered as multiple mono-phase tasks with equal periods linked by simple precedence constraints. Nevertheless, our model is more restrictive because a phase can not start as soon as the previous phase completes its execution but only at its set release time. Graph-based models also possess the expressiveness to represent conditional executions and parallel execution models of fork-join type. The constraint of executing each phase within its time window can be viewed as a horizontal and sequential version of the Gang model [START_REF] Goossens | Optimal Scheduling of Periodic Gang Tasks[END_REF], but we do not consider parallelism in this work.

d) Asynchronous task model: Precedence constraints can be encoded by modifying offsets and deadlines [START_REF] Bouchentouf | Dynamic scheduling of real-time tasks under precedence constraints[END_REF] to analyze tasks as independent. As we set phase release times inside a task, our model could be viewed a special case of the periodic mono-phase model with asynchronous releases and implicit deadlines by transforming each phase into a task with an offset equal to the relative deadline of the previous phase. However, explicitly considering the offsets would not simplify the problem resolution.

e) PREM and phased execution: the PRedictable Execution Model (PREM) [START_REF] Pellizzoni | A predictable execution model for cots-based embedded systems[END_REF], [START_REF] Alhammad | Schedulability analysis of global memory-predictable scheduling[END_REF] divides the execution of the runnables (executable entities) into three distinct and consecutive phases to avoid memory contention in a timetriggered system. Each task starts by executing a memory phase in which all the data and instructions needed by the task are read from main memory and stored in local private memory (e.g., cache or scratchpad). The subsequent computation phase can then perform the actual computation of the task using data and instructions that are readily available in the private memory, without any need to access the interconnect or shared main memory. Lastly, a last memory phase is performed at the end of each task to write back the updated data into the main memory. The phased execution of tasks is typically used in conjunction with a scheduling algorithm that exploits the fact that access to the shared main memory and bus are only going to occur during the memory phases of the tasks. Therefore, the scheduler can avoid contention in the main memory by not overlapping memory phases in different cores (i.e., such parallelism is forbidden). The memory phases (read & write) are assumed to be non-preemptive, while the execution phases can be preemptive. Our model is different in that the read and write phases are non-preemptive. Additionally, it is not equivalent to consider memories as specific types of processors. Lastly, we do not aim to avoid parallelism between phases of different tasks (memory phases in the PREM model). On the contrary, it is encouraged to achieve optimal scheduling. f) Distributed models: Sun and Liu [START_REF] Sun | Synchronization protocols in distributed realtime systems[END_REF] present a model in which tasks are decomposed into subtasks, similar to the way we decompose ours into phases. However, our approach differs in that the subtasks can only be executed on a single processor, whereas we consider that the phases can run on several and could migrate between them.

IV. CHUNK BY CHUNK APPROACH

In the following, without loss of generality, we will assume that κ min = 1 (by dividing, for all tasks, T i by κ min and multiplying all the rates by the same factor, we actually solve an equivalent problem where κ min = 1).

Baruah [START_REF] Baruah | Feasibility Analysis of Preemptive Real-Time Systems upon Heterogeneous Multiprocessor Platforms[END_REF] proposes an exact feasibility test for the scheduling of mono-phase tasks on unrelated multiprocessors. The feasibility problem is equivalent to solve a Linear Program (LP). A solution to the LP exists iff a feasible schedule exists. A first approach could be to apply the Baruah's method chunk by chunk. That is, considering H independent problems by evenly distributing the workload of each phase within each chunk. More formally, by solving independently the LP 1 for each t ∈ 0, H -1 .

LP 1. m j=1 x t i,j × r λi(t) i,j = C i κ i × ρ i ∀i ∈ 1, n (1)
m j=1 x t i,j ≤ 1 ∀i ∈ 1, n (2)
n i=1 x t i,j ≤ 1 ∀j ∈ 1, m (3)
where λ i (t) . = ⌊ t mod Ti κi ⌋ is the index of the current phase of task τ i at time t, and x t i,j ≥ 0 ∀i ∈ 1, n , ∀j ∈ 1, m is the fraction of the chunk t assigned to τ i on π j . Equation [START_REF] Lawler | On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming[END_REF] ensures that the phases will meet their deadlines; while equations ( 2)-( 3) ensure no intra-phase parallelism nor overloaded processor, respectively. As explained in Section II, the WCET of each task is here shared uniformly among the phases, but the LP can be readily extended by modifying Ci ρi to c λi(t) i

in [START_REF] Lawler | On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming[END_REF]. Notice that the same remark applies for LP 2 later in Section V. Such a solution forces some kind of fairness: each phase progresses by the same amount during each chunk. This solution is not optimal as, sometimes, a phase's execution should be delayed to ensure schedulability of other phases. For instance, let's take a look at the task set given in Table II. τ 1 is divided into two phases. The timeline of this task set is drawn in Fig. 4. The LP problem to solve for the first chunk is given by (4).

τ 1 τ 2 τ 3 ρ 0 ρ 1 ρ 0 ρ 0 T 2 2 2 C 2 1 1 π 1 1 0 1 1 π 2 0 1 0 0 TABLE II AN EXAMPLE TASK SET NOT FEASIBLE WITH A CHUNK BY CHUNK APPROACH τ 3 τ 2 τ 1 ρ 0 ρ 0 ρ 0 ρ 1
                                   x 0 1,1 = 1 x 0 2,1 = 1 2 x 0 3,1 = 1 2 x 0 1,1 + x 0 1,2 ≤ 1 x 0 2,1 + x 0 2,2 ≤ 1 x 0 3,1 + x 0 3,2 ≤ 1 x 0 1,1 + x 0 2,1 + x 0 3,1 ≤ 1 x 0 2,2 + x 0 2,2 + x 0 3,2 ≤ 1 (4)
This LP has no solution: x 0 1,1 + x 0 2,1 + x 0 3,1 = 2 > 1. However, the task set is trivially feasible: τ 1,0 can be executed on π 1 during the whole first chunk, and τ 1,1 on π 2 for the second chunk, while τ 2 and τ 3 share the second chunk on π 1 during the second chunk to meet their deadlines.

In the following section, we will present a less constrained approach for the workload assignment of multi-phase task sets which does not fail on such examples.

V. TOWARDS AN OPTIMAL MULTI-PHASE TECHNIQUE

A second approach is to consider the problem globally, relaxing the fairness constraint of the first approach. The constraint now applies to the boundaries of each phase (of each job within the hyperperiod). Indeed, each phase must be fully executed at these instants. More formally, ∀τ i ∈ Γ, ∀ρ ∈ 0, ρ i , ∀α ∈ 0, H Ti -1 :

αTi+(ρ+1)κi-1 t=αTi+ρκi   m j=1 x t i,j × r ρ i,j   = C i ρ i (5)
This finer grained approach to the multi-phase problem is introduced in the LP 2.

LP 2.                                αTi+(ρ+1)κi-1 t=αTi+ρκi   m j=1 x t i,j × r ρ i,j   = C i ρ i ∀i ∈ 1, n ∀α ∈ 0, H T i -1 ∀ρ ∈ 0, ρ i -1 m j=1 x t i,j ≤ 1 ∀i ∈ 1, n ∀t ∈ 0, H -1 n i=1 x t i,j ≤ 1 ∀j ∈ 1, m ∀t ∈ 0, H -1 with x t i,j ≥ 0 ∀i ∈ 1, n , ∀j ∈ 1, m , ∀t ∈ 0, H -1 . Proposition 1. LP 2, has a solution if and only if Γ is feasible.
The proof is left for future work.

VI. FUTURE WORKS AND CONCLUSION

Formal guarantees

We still need to demonstrate that LP 2 produces an optimal solution, so that we can verify the feasibility of the multiphase problem in an unrelated multiprocessor platform. It is quite clear that both approaches can be exponential, but this also needs to be demonstrated formally. The first approach is local and consists in replicating the solution of several LPs over the entire hyperperiod. The second approach is global and aims at computing the solution of one LP over the entire hyperperiod. It is also important to carefully quantify the number of variables and constraints generated as a function of the input size of the problem (distribution of the periods, number of tasks, number of phases per task, distribution of the WCET among phases, number of processors, etc.). We also plan to effectively release the constraint on the uniform distribution of periods and WCETs among phases.

Experiments

We intend to generate synthetic systems to compare both approaches in practice, for example on the success ratio metric, assuming that LP 2 is optimal. However, this requires to generate tasks and rates with variability whose total load can be controlled, which is not trivial, especially with a rate per phase. We do not rule out the use of linear programming to generate feasible sets. Next, it will be necessary to evaluate the runtime of the two approaches, in order to verify not only their practicality, but also whether under certain conditions there is any point in favouring the sub-optimal solution if the optimal solution is not practicable.

Conclusion

In this work-in-progress, we have presented the problem of scheduling multi-phase periodic tasks in an unrelated multiprocessor platforms. A first approach using a state-of-theart technique for mono-phase task proved to be sub-optimal. We are currently working on the assessment of the second approach optimality. Next, we will evaluate the runtime of the two approaches, in particular to verify under certain conditions (e.g. with non-harmonic periods) if it is reasonable to favour the sub-optimal solution if the optimal one is not practicable.

Fig. 4 .

 4 Fig. 4. Timeline of the task set given in TableII

ACKNOWLEDGEMENTS

The authors would like to express their gratitude to Quentin Bontemps for the valuable ideas and insightful discussions pertaining to this work.


Maxime Naudé

Peter Faller

Vincent Lebrun
email: vlebrun@unistra.fr

A closer look at Type I left-handed β-helices provides a better understanding in their sequence-structure relationship: towards their rational design

Keywords: repeat protein, β-solenoid, sequence-structure relationship, protein design

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

Introduction

It is commonly accepted that protein's function arises from their structure, which itself is encoded in the sequence according to the Anfinsen's principle. [START_REF] Anfinsen | Principles That Govern the Folding of Protein Chains[END_REF] The sequence of proteins with tandem repeats is composed of a repetition of a short sequence, usually forming an elementary structural unit. [START_REF] Kajava | Β-Rolls, Β-Helices, and Other Β-Solenoid Proteins[END_REF][START_REF] Kajava | Tandem Repeats in Proteins: From Sequence to Structure[END_REF][START_REF] Chakrabarty | DbStRiPs: Database of Structural Repeats in Proteins[END_REF] They are particularly suitable for protein design, with their interactions it allow to conceive objects that can be tunable, leading to interesting properties regarding the design of repetitive units that are reminiscent of legos. [START_REF] Kajava | Tandem Repeats in Proteins: From Sequence to Structure[END_REF][START_REF] Tripp | The Tolerance of a Modular Protein to Duplication and Deletion of Internal Repeats[END_REF][START_REF] Javadi | Tandem-Repeat Proteins: Regularity plus Modularity Equals Design-Ability[END_REF][START_REF] Parmeggiani | Designing Repeat Proteins: A Modular Approach to Protein Design[END_REF] Some databases dedicated to Structural Repeats in Proteins have been developed to facilitate their identification and analysis. [START_REF] Chakrabarty | DbStRiPs: Database of Structural Repeats in Proteins[END_REF][START_REF] Paladin | Improved Data Extended Classification for Protein Tandem Repeat Structures[END_REF][START_REF] Pellegrini | Tandem Repeats in Proteins: Prediction Algorithms and Biological Role[END_REF][START_REF] Jorda | Identification of Tandem REpeats in Sequences with a K-MeanS Based Algorithm[END_REF] Noteworthy, such protein folds can be used as domains to mediate oligomerization of chosen protein in a predictable way, [START_REF] Voet | Computational Design of a Self-Assembling Symmetrical β-Propeller Protein[END_REF][START_REF] Norn | Computational Design of Protein Self-Assembly[END_REF][START_REF] Correnti | Engineering and Functionalization of Large Circular Tandem Repeat Protein Nanoparticles[END_REF][START_REF] Vanderstraeten | Synthetic Protein Scaffolds for the Colocalisation of Co-Acting Enzymes[END_REF] allowing the design of higher order assemblies for example. [START_REF] Leibly | A Suite of Engineered GFP Molecules for Oligomeric Scaffolding[END_REF][START_REF] Luo | Protein Assembly: Versatile Approaches to Construct Highly Ordered Nanostructures[END_REF][START_REF] Gwyther | Better Together: Building Protein Oligomers Naturally and by Design[END_REF][START_REF] Zhu | Protein Assembly by Design[END_REF] So far, most designed repeat proteins were based on α-helical units (eg AlphaRep, Ankyrin repeat). [START_REF] Parmeggiani | Designing Repeat Proteins: A Modular Approach to Protein Design[END_REF][START_REF] Plückthun | Designed Ankyrin Repeat Proteins (DARPins): Binding Proteins for Research, Diagnostics, and Therapy[END_REF][START_REF] Urvoas | Artificial Proteins from Combinatorial Approaches[END_REF] Besides, β-solenoids are particularly represented in the fiber-forming proteins, such as the cytoskeletal bactofilin BacA for examples, and hence have been used to design nano-wires for example. [START_REF] Kajava | Β-Rolls, Β-Helices, and Other Β-Solenoid Proteins[END_REF][START_REF] Deng | The Structure of Bactofilin Filaments Reveals Their Mode of Membrane Binding and Lack of Polarity[END_REF][START_REF] Scotter | Metal Ion-Dependent, Reversible, Protein Filament Formation by Designed Beta-Roll Polypeptides[END_REF][START_REF] Peralta | Engineering Amyloid Fibrils from β-Solenoid Proteins for Biomaterials Applications[END_REF][START_REF] Altamura | A Synthetic Redox Biofilm Made from Metalloprotein-Prion Domain Chimera Nanowires[END_REF] β-solenoids are defined as a single polypeptide backbone winding around an axis, forming β-sheets and β-turns. [START_REF] Kajava | Β-Rolls, Β-Helices, and Other Β-Solenoid Proteins[END_REF][START_REF] Kajava | Tandem Repeats in Proteins: From Sequence to Structure[END_REF][START_REF] Cheng | The Supramolecular Chemistry of β-Sheets[END_REF] The β-solenoids are appealing because of their mechanical properties and stability, as the amino acids oriented inwards form a hydrophobic core that stabilize the folding. [START_REF] Baarda | In Silico Stress-Strain Measurements on Self-Assembled Protein Lattices[END_REF][START_REF] Peng | Bottom-up Synthesis of Protein-Based Nanomaterials from Engineered β-Solenoid Proteins[END_REF] . This super structure was first discovered by Yoder et al. in 1993. [START_REF] Yoder | New Domain Motif: The Structure of Pectate Lyase C, a Secreted Plant Virulence Factor[END_REF] Different types exist, depending on the number of β-strands per rung, on the number of amino acids (a.a) per rung, etc. [START_REF] Kajava | Β-Rolls, Β-Helices, and Other Β-Solenoid Proteins[END_REF][START_REF] Cheng | The Supramolecular Chemistry of β-Sheets[END_REF] For example, the β-helix designates a β-solenoid with three β-sheets per rung (triangular section), and can be right-handed or left-handed. This latter, is subdivided in two types. The Type II left-handed β-helix (eg antifreeze protein CfAFP) is composed of 15 amino acids per rung versus carbonic anhydrase). Another difference is that the Type I is usually a trimerization domain (Figure 1a), while the Type II remains monomeric. [START_REF] Choi | Analysis of the Sequence and Structural Features of the Left-Handed β-Helical Fold[END_REF] The first structural study on Type I left-handed β-helix (hereafter abbreviated as LβH-I)domain, based on seven LβH-I protein structures, mostly focused on the torsion angles within the main chain. [START_REF] Iengar | Conformational and Sequence Signatures in β Helix Proteins[END_REF] They distinguished the β sheets of the helix in its trimeric form, and showed that the LβH-I sequence can be built from a repetition of a hexa-peptide motif. They highlighted a few important features in the sequence, such as the importance of glycine residues at the fourth position of the β strands, the need of small residue at the first position, and the composition of the hydrophobic core.

Later, Cohen and coworkers (i) confirmed that the sequence of a LβH-I is essentially the repeat of a hexad, (ii) introduced the nomenclature for it, ie two residues for the turn (called "loop" before) (T1 and T2) and four residues for the β-strand (B1 to B4), and (iii) reported the position-dependent residue propensities. Of note, the latter was performed based on the eight crystal structures available at that time (by sequence alignment), and excluding the rungs at the N-ter and C-ter extremities. [START_REF] Choi | Analysis of the Sequence and Structural Features of the Left-Handed β-Helical Fold[END_REF] This work was completed with an experimental study assessing the tolerance of the LβH-I fold for the different amino-acids at each position of the hexad. [START_REF] Choi | Site-Directed Mutagenesis Demonstrates the Plasticity of the β Helix: Implications for the Structure of the Misfolded Prion Protein[END_REF] In parallel, modelling studies also contributed, for example in evidencing the importance of proline and glycine residues in LβH-I for maintaining the integrity of the fold. [START_REF] Haspel | De Novo Tubular Nanostructure Design Based on Self-Assembly of β-Helical Protein Motifs[END_REF] While these previous studies established the first understanding of the sequence-structure relationship in LβH-I (compatibility and occurrence of residues in the hexad) they suffer from two limitations. Indeed, the reduction of the whole LβH-I sequence to only one hexad means considering (i) all three sides of the helix as equivalents, while they are not exposed to the same environment (due to the trimeric assembly, see Figure 1a) and (ii) that all rungs are equivalents, which is not true since the rungs at the extremities have particular roles. [START_REF] Choi | Analysis of the Sequence and Structural Features of the Left-Handed β-Helical Fold[END_REF][START_REF] Choi | Site-Directed Mutagenesis Demonstrates the Plasticity of the β Helix: Implications for the Structure of the Misfolded Prion Protein[END_REF] Indeed, the rungs at the N-ter and C-ter of the helix have key roles in nucleating and terminating the fold. As such, they belong to the so-called "capping motifs" at the extremities of the helix that are the structural elements, belonging to the β-helix or not, responsible for the solubility of the protein. [START_REF] Parmeggiani | Designing Repeat Proteins: A Modular Approach to Protein Design[END_REF][START_REF] Courtemanche | The Leucine-Rich Repeat Domain of Internalin B Folds along a Polarized N-Terminal Pathway[END_REF][START_REF] Richardson | Natural β-Sheet Proteins Use Negative Design to Avoid Edge-to-Edge Aggregation[END_REF][START_REF] Bryan | Structure-Based Prediction Reveals Capping Motifs That Inhibit β-Helix Aggregation[END_REF][START_REF] Prabha | Characterization of Left-Handed Beta Helix-Domains, and Identification and Functional Annotation of Proteins Containing Such Domains[END_REF] It has indeed been shown that removing capping motifs of β-solenoids, including left-handed β-helix, allows their self-assembly into fibers. [START_REF] Scotter | Metal Ion-Dependent, Reversible, Protein Filament Formation by Designed Beta-Roll Polypeptides[END_REF][START_REF] Peralta | Engineering Amyloid Fibrils from β-Solenoid Proteins for Biomaterials Applications[END_REF][START_REF] Peng | Bottom-up Synthesis of Protein-Based Nanomaterials from Engineered β-Solenoid Proteins[END_REF] While these aspects were considered by Balaram and coworkers first, and later by Prabha and Balaji, they did not provide a specific analysis of the sequence of these N-terminal and C-terminal rungs. [START_REF] Iengar | Conformational and Sequence Signatures in β Helix Proteins[END_REF][START_REF] Prabha | Characterization of Left-Handed Beta Helix-Domains, and Identification and Functional Annotation of Proteins Containing Such Domains[END_REF] In this present work, we propose a sequence analysis of LβH-I by taking into account simultaneously that (i) LβH-I have three different faces due to the trimeric arrangement and that (ii) LβH-I is composed of three rungs of different nature. It has been performed only with sequences extracted from proteins of which the crystal structure in available in the Protein Data Bank (PDB), hence leaving no doubt on the position occupied in the LβH-I fold by each considered amino-acid. This analysis also helps identifying and understanding the role of specific motifs in initiating, stabilizing or terminating the fold. Overall, this study aims at (i) improving our understanding of the LβH-I's sequence-structure relationship, and at (ii) providing guidelines for the rational design of protein sequences able to adopt a stable LβH-I fold.

Results

In order to better understand the sequence-structure relationship of LβH-containing proteins, we could identify 78 different crystallographic structures (complete list Table S1) querying UniProt IDs in the PDB. The analysis procedure that was followed is illustrated Figure 1. First, for each structure, all the parts of the protein not belonging to the LβH fold were omitted. As a consequence, all interactions between the canonical parts of LβH and other parts of the proteins have not been considered in this study. Second, the three different faces of the helix within the trimer were discriminated (Figure 1c). They were labelled according to their solvent exposition: buried (b), exposed (e) and partially exposed (p). [START_REF] Prabha | Characterization of Left-Handed Beta Helix-Domains, and Identification and Functional Annotation of Proteins Containing Such Domains[END_REF] They are composed of six residues, hence termed hexad, four for the β-strand (B1, B2, B3, B4), and 2 for the turn (T1, T2). [START_REF] Choi | Analysis of the Sequence and Structural Features of the Left-Handed β-Helical Fold[END_REF] Of note, in positions B1 and B3, the residues are oriented inwards, hence forming the hydrophobic core, whereas the others are oriented outwards of the helix (Figure 1d).. Third, the three different parts of the helix, the Nterminal capping rung (N), the Inner rungs (I), and the C-terminal capping rung (C) were distinguished (Figure 1d). As a consequence, the resulting analysis reports 9 different hexads repeats, which can be presented in a table of 3 lines (parts of the helix) and 3 columns (sides of the helix) (Figure 1e). The full table is given Figure S1, but for clarity sake, in the following main text, it will be presented line per line. The result of this analysis is the position-dependent residue propensity for each hexad. In other words, for the different positions, each amino acid is depicted as its letter, and the size of the latter is proportional to the occurrence of the amino acid (Fig. 2, 5 and 7).

Hereafter, each residue's position will be referred to as its position within its hexad: eg B3(C.e) refers to the residue in position B3 of the hexad located in the C-terminal Capping rung (C) of the helix, in the face of the helix that is the most exposed to the solvent (e).

Analysis of the inner rungs.

The inner rungs are the most represented in the analysis, with more than 72 % of residues, as they constitute the core of the helix. We observe that the three hexads of inner rungs are relatively close in composition (Figure 2), explaining their proximity with results reported by Cohen and coworkers. [START_REF] Choi | Analysis of the Sequence and Structural Features of the Left-Handed β-Helical Fold[END_REF] Yet, our analysis reveals some differences, notably at the T1 and T2 positions. The residues composing the hydrophobic core, at B1 and B3 positions, are very similar for the three faces (Figure 2). This is expected since, because they are buried, they should be the least impacted by the differences in solvent-exposition of the faces.

B3 position is occupied almost exclusively by Ile, Val and Leu (Figure 2). Even though Ile is the most frequent for all 3 faces, only 16 % of the inner rungs contain 3 Ile (at B3 (b), (e) and (p)). Instead, the most frequent configurations are (Ile)2(Val/Leu)1 and (Ile)1(Val)1(Leu)1 (Figure S3).

While Valine and Alanine are the most frequent at B1 position, residues with an alcohol or a thiol group (Ser, Thr, Cys) are also commonly encountered (Figure 2, S4 and S5). Indeed, these group make H-bonds with the main chain of residues i-3 (ie B4) and i+18 (ie same B1 position, in the following rung) (Figure 3a). Plus, in the case of Thr, the methyl group of its side chain also participates to the hydrophobic core. Interestingly, even if the residue B1 is usually part of the hydrophobic core, a non-negligible proportion of Asn is found at the B1(I.p), ca. 10%. Internal Asn ladders have been observed, also interacting with the main chain of B4 (e) (Figure 3b), and evidenced by tables analysing the interdependence of residue composition of adjacent rungs (Figure S8). This is reminiscent to the Asn ladders observed and described in the less regular βhelix of PelC and PelE. [START_REF] Yoder | Unusual Structural Features in the Parallel β-Helix in Pectate Lyases[END_REF] It is also interesting to observe Pro at B1(I.p) positions, as it is normally "forbidden" at this position. [START_REF] Choi | Site-Directed Mutagenesis Demonstrates the Plasticity of the β Helix: Implications for the Structure of the Misfolded Prion Protein[END_REF] Actually, it is observed only right after one of the loop that that are often observed at this corner, that are very variable in length and structure. In such case, the side chain of Pro is oriented towards the interior of the helix, contributing to the hydrophobic core while imposing a torsion on the main chain, bringing it back to the canonical fold (Figure 3c). Noteworthy, the presence of a loop prior the B1(I.p) position seems not compatible with the presence of Ser or Thr at this position, probably because of their hydrophilicity. It is then preferred to accommodate a Val, an Ile or a Pro. In the face (e) that positions where side chain point outward the helix (ie B2, B4, T1 and T2) are mainly occupied by polar and charged residues while they are less frequent for the other two faces.

) a) c) d) B 1 B 4 B 1 B 4 T 1 T 2
Focusing on position B4, the main observations made previously are confirmed: [START_REF] Choi | Analysis of the Sequence and Structural Features of the Left-Handed β-Helical Fold[END_REF][START_REF] Iengar | Conformational and Sequence Signatures in β Helix Proteins[END_REF] (i) Glycine is frequent in this position, and (ii) especially at the face (e) (74% in this study). As can be seen from the Figure 2, B4(I.b) can also be the start of the variable loops observed at the corners (b-e), although these loops mainly substitute the positions T1-2(I.b) (Figure 2, φ symbols).

In the canonical rungs (I), prolines are scarce and found almost exclusively in positions T1. It accounts for 1%, 8% and 14% of the residues for the faces (e), (b) and (p), respectively. Interestingly, the latter (ie T1(I.p)) is at the interface of the helices in the trimers, together with T2(I.p) and B2(I.p), explaining why residues with short side chains are most often observed (Ala, Pro, Gly, Ser) (Figure 2).

Globally looking at the composition of the full rung (I), it appears clearly that polar residues are mainly located at positions T1(I.e) and T2(I.e), which corresponds to the corner well-exposed to the solvent, but which do not contain variable loops. Indeed, the most frequent residues there are Asp for T1(I.e) and Asn for T2(I.e). They both participate to the global stabilization of the fold as their side chain are involved in H-bonds with one or two adjacent rungs: at T1(I.e), Asp and Asn accept a H-bond of the amide (nitrogen) of the residue i-18, and at T2(I.e), Asn can form Asn ladders, [START_REF] Haspel | De Novo Tubular Nanostructure Design Based on Self-Assembly of β-Helical Protein Motifs[END_REF] thus interacting with residues i-18 and i+18 (Figure 3d).

The position T2(I.p) is located at the interface between the three helices, at the centre of the trimer. While Gly residues are often observed, probably to avoid clashes, Asn and aromatic residues are able there to perform stabilizing interactions, reinforcing the cohesion of the trimeric assembly (Figure 4). Interestingly, the alternating of Gly and Asn from one rung to the next, is the most frequent pattern observed at T2(I.p) (Figure S13). Finally, at the B2 positions, the most frequent amino acids are β-branched, while one might have expected more polar residues as B2 is oriented outwards. The reason might be that β-branched residues are favourable for the formation of β-sheets. [START_REF] Pal | β-Sheet Propensity and Its Correlation Withparameters Based on Conformation[END_REF] Of note, of the B2 positions of inner rungs, the face (e) is where long and polar residues (Lys, Glu, Arg) are most observed, together representing 32% of the residues (9 % for (b) and 16 % for (p)).

Analysis of the rungs at the extremities of the LβH-I.

In order to avoid aggregation of the β-helix, by intermolecular interactions via its ends, or even the uncontrolled continuation of the fold, the helix is capped at both N-ter and C-ter. [START_REF] Richardson | Natural β-Sheet Proteins Use Negative Design to Avoid Edge-to-Edge Aggregation[END_REF][START_REF] Bryan | Structure-Based Prediction Reveals Capping Motifs That Inhibit β-Helix Aggregation[END_REF] Most of the times, the main capping element is a loop or non-repetitive structure, interacting with the end of the β-helix, covering and its main chain's amides groups and its hydrophobic core, at least partially. On top of that, it has been observed that the first (N-ter) and last (C-ter) rungs of the β-helix also display specific structural motifs participating to its termination (ie prevent further βinteraction). [START_REF] Richardson | Natural β-Sheet Proteins Use Negative Design to Avoid Edge-to-Edge Aggregation[END_REF][START_REF] Bryan | Structure-Based Prediction Reveals Capping Motifs That Inhibit β-Helix Aggregation[END_REF] Hereafter, they will be described into more details, and in particular, how they arise from the sequence. Of note, these first (N) and last (C) rungs, not only vary significantly compared to the rungs (I), but also from a β-helix to another.

The N-terminal rung -promoting the nucleation of the β-helix. The height of the residue's letter is proportional to its frequency. φ : residues involved in a loop or a non-canonical conformation ; ∅ : missing residues.

The N-terminal rung (N) already differs visually from the others, (I) and (C). Indeed, its turns are distorted, with a tendency to fold back towards the centre of the helix. Richardson & Richardson described them as upward protrusion, which help preventing continuation of β interactions towards N-terminus. [START_REF] Richardson | Natural β-Sheet Proteins Use Negative Design to Avoid Edge-to-Edge Aggregation[END_REF] In addition to this role, these turns might actually have a nucleation role, orienting the fold towards a LβH-I rather than more linear or less regular β-fold. In particular, we identified a specific triad, that both favours (i) the formation of a turn and (ii) the interaction with an adjacent rung (towards C-ter). The sequence of these nucleation motifs is X-Pro-Z (B4-T1-T2), where X = His, Asp or Glu, and Z = Ser, Thr, Asp, or Asn (Figures 5 and6). While Pro induces a structural constrain onto the main chain, hence forcing the formation of a turn, the side chains of X and Z form a hydrogen bond together, thus narrowing the angle of the turn. In addition, the side of Z forms a hydrogen bond with the main chain amide of the residue (i+18), which favours the formation of the helix. These triads are mostly found at (b) and (p), resulting in Pro being the most frequent residue at T1(N.b) and T1(N.p). Compared to the rungs (I), the residues compositing the hydrophobic core are similar, yet being more often composed of three Ile at B3 positions (Figure S3). Only in B1(N.e) one could note a higher proportion of Pro, which often occurs when the backbone returns to the canonical LβH-I fold, as shown before with the (I) rungs (vide supra).

On the other hand, the positions B2 differ from (I) to (N) as we observe here a higher proportion of non-β-branched or of polar residues. Another difference with rungs (I) is the presence of loops (visible as Phi symbols in Figure 5) at T1,2(N.p) in addition to T1,2(N.b).

Of note, there is no clear preference for a particular face for the start of the LβH-I fold. Indeed, for roughly half of the β-helices analysed here, the first (N) β-strand was of the face (p), and one fourth for faces (b) and (e).

The C-terminal rung -ending the fold.

Figure 7. Occurrence of residues for each one of the eighteen positions in the C-terminal rung (N). The height of the residue's letter is proportional to its frequency. φ : residues involved in a loop or a non-canonical conformation ; ∅ : missing residues.

Unlike the (N) rung, of which structural features might prevent aggregation on the N-ter of the helix while helping the proper LβH-I fold to initiate, the particular features observed for rung (C) only explain its termination role. When looking at the structures and their sequence, three main features are prominent.

First, the absence (empty set symbol in Figure 7) of residues at two positions, T1-2(C.b), resulting in a shorter loop for c.a. 70 % of the helices analysed here (Figure 8a). This diminishes the number of H-bond accessible on the C-terminal end, and also constrains the hydrophobic core to be smaller than in the rest of the helix, resulting in higher proportion of Val and Ala at positions B3(C). Indeed, observing three Val at the B3 positions of rung (C) is significantly more common than in other rungs (Figure S3).

Second, the presence of a Pro in position B4(C.e) in c.a. 60 % of the structures analyzed, often accompanied with a Pro in the following T1 position (Figure 7). As evidenced in the past, a Pro in position B4 is not compatible with the LβH-I fold. [START_REF] Choi | Site-Directed Mutagenesis Demonstrates the Plasticity of the β Helix: Implications for the Structure of the Misfolded Prion Protein[END_REF] It is then used in this C-terminal rung as a "helix breaker" to prevent the formation of a new β sheet by steric interaction, as the side chain of Pro at this position is oriented in the same axis as the helix (Figure 8b) hence sterically forbidding further β interaction.

Third, on the contrary to the start of the LβH-I, its termination has a marked preference for a face, with ca 90 % of the analyzed structures ending in face (p) (Table S2). In such case the last occupied position of the canonical fold is B4(C.p) (Table 7). The following residues are the start of the capping loop, which often (in ca 75 % of the observed structures, Table S2) was forming a βhairpin with the final β-strand (C.p). Although being beyond the scope of this study as it does not belong to the LβH-I fold per se, the residue composition of this hairpin of the structures analyzed here is provided Figure S6. As a last remark, these C-ter capping motifs can further stabilize the trimer of β-helices, either by covering the hydrophobic core of another β-helix, or by further folding as an α-helix along the groove formed by two β-helices (Figure S40).

Discussion

The analysis presented herein is not a classical bioinformatics study performed on a large set of LβH-I. Instead, it was limited to those of which the crystal structure was available in the PDB. As a consequence, this analysis suffers from biases inherent to the narrow scope and redundancy of the sequences studied (majority of them belong to the family of UDP N-Acetyltransferase). This doesn't affect its relevance regarding the comprehension of the sequence-structure relationship, but limits its representativeness of the LβH-I-containing proteins. Its aim however was to provide a better understanding of the sequence-structure relationship for the LβH-I fold, based on structural features that are established with certainty by X-ray structure visualization. This allowed to identify interesting structural features, even if not present in a majority of sequences, such as the presence of Pro in B1(I.e) after a loop for example. In addition, we considered (i) the different nature of the different types of rung as well as (ii) the solvent exposure of the different face of the helix within the homo-trimer. This is the first time these two aspects are taken together into account during the analysis of LβH-I, thus distinguishing nine hexads. It first confirmed all previous observations, and second allowed the new description of specific motifs such as the Xxx-Pro-Zzz triad in the rung (N) for example, revealing the importance of this latter in initiating the LβH-I fold. While being clearly part of the β-helix, this work clearly shows that both rungs (N) and (C) should be considered as being part of the so-called "capping motif" as they possess structural features that are clearly involved in preventing the continuation of the LβH-I fold. This is in line with Richardson & Richardson viewpoint, but not with that of Bryan et al. as these rungs are ending or starting with an element of regular secondary structure. [START_REF] Richardson | Natural β-Sheet Proteins Use Negative Design to Avoid Edge-to-Edge Aggregation[END_REF][START_REF] Bryan | Structure-Based Prediction Reveals Capping Motifs That Inhibit β-Helix Aggregation[END_REF] Complementary information, especially concerning the succession of residues from a rung to another at one particular position, were not discussed in the main text but are given the supplementary information (Figures S7 to S40).

Finally, to people aiming at designing LβH-I proteins, we advise to base their sequence design on the structural features highlighted above while following these general principles: -In the most conserved positions, restrictions must be respected while optimizing the stability of the hydrophobic core and the number of H-bonds involving between adjacent rungs. -Use the positions where a higher diversity of residues is allowed to introduce stabilizing interactions between adjacent rungs (eg Asn ladders or electrostatic interaction Lys/Glu), and, to introduce charged residues in order to provide the entire protein with a net charge to ensure its solubility at the desired pH range. In other words, adjust its pI according to the final application of the protein.

-Optimise the sequences of (N) and (C) rungs to initiate and terminate the LβH-I folding, respectively.

Conclusion

This detailed analysis of the LβH-I fold is a significant step forward in our understanding of its sequence-structure relationship. Indeed, the analysis we provide reflects the trimeric nature of LβH-Is, as well as the nucleation and termination roles of their extremities, with a higher fidelity than before. It thus paves the way for the rational design of peptides and proteins folding into LβH-I, therefore particularly relevant for fields such as artificial protein design or bottom-up biomaterial synthesis.

Materials and methods

All the protein crystal structures analysed here (Table S1) have been downloaded from the Protein Data Bank (PDB). The Uniprot code was used to search for structures and a similarity search has been performed to find similar other structures. The Database of Structural Repeats in Proteins (DbStRiP) developed by Chakrabarty and Parekh has also been used as a source of information . [START_REF] Chakrabarty | DbStRiPs: Database of Structural Repeats in Proteins[END_REF] PyMOL software was used to visualize all 78 structures and to generate pictures. 40 For each LβH-I, the labelling of the faces and the rung types as well as the position annotation for each aminoacid were performed manually, following the methodology illustrated in Figure 1. Excel 2016 (Microsoft) was used to process data regarding sequences.

Figure 1 .

 1 Figure 1. Methodology following for this LβH-I analysis. a) Selection of trimeric LβH-I crystallographic structures from the PDB and identification of the canonical folding. b) Removal of structural elements that to not belong to the canonical LβH-I. c) Labelling of the faces of the LβH-I as a function of their exposition to the solvent (buried (b), exposed (e), and partial (p)) originating from the trimeric arrangement. d) Separation of the different types or rungs: N-terminal (N), inner (I) and C-terminal (C). e) The analysis of these nine different hexads finally yields a table reporting the occurrence of the amino-acids in the 54 different positions identified.

Figure 2 .

 2 Figure 2. Occurrence of residues for each one of the eighteen positions of the inner rungs. The height of the residue's letter is proportional to its frequency. φ : residues involved in a loop or a non-canonical conformation ; ∅ : missing residues.

Figure 3 .

 3 Figure 3. Illustration of stabilizing interactions observed in inner rungs. a) Thr, Ser and Cys in B1 can make H-bonding with the main chain of the amino acid, at the B4 position preceding it, the Thr methyl group oriented towards the hydrophobic core (PDB : 3IXC). b) An internal asparagine ladder (at the B1 position) interacting like the residues displayed in a) (PDB : 4EQY ). c) Overview of the hydrophobic core (B1 and B3 side chains) and highlighting the Pro residue participating to the hydrophobic core after a loop (in pink) (PDB: 1XHD). d) Illustration of the hydrogen bonds involving Asp and Asn positions T1 and T2 of rung (I), face (e) (PDB: 1OCX).

  b

Figure 4 .

 4 Figure 4. Examples of interactions located at the interface between the three helices, involving residues at position T2(I.p). a) Phe residues interactions (PDB: 1LXA), b) Asn ladders and interactions between helices (PDB: 4EQY)

Figure 5 .

 5 Figure 5. Occurrence of residues for each one of the eighteen positions in the N-terminal rung (N).The height of the residue's letter is proportional to its frequency. φ : residues involved in a loop or a non-canonical conformation ; ∅ : missing residues.

Figure 6 .

 6 Figure 6. Illustrations of nucleation motif Xxx-Pro-Zzz in a rung (N). a) His-Pro-Thr (PDB: 3HSQ) and c) Asp-Pro-Ser (PDB: 4E6U). The dashed square highlights the rung n+1, ie the first inner rung (I).

Figure 8 .

 8 Figure 8. Characteristic motifs observed in rungs (C) illustrated at the C-terminal end of the UDP-N-acetylglucosamine acyltransferase (PDB: ILXA). a) Deletion of residues at positions T1,2(C.b) results in a shorter corner (in red). b) Presence of the Pro-Pro motif at the B4-T1(C.p), with the sidechain of the first oriented along the axis of the helix.

Acknowledgments

Authors thank the Agence Nationale de la Recherche for funding (ANR-18-CE07-0022-01). They also thank O. Sénèque, M. Paternostre, W. Ghattas, V. Forge and C. Ménard-Moyon for helpful discussions.


Leandro C Tabares
email: leandro.tabares@cea.fr

Davis T Daniel

José Luis Vázquez-Ibar

Cyrille Kouklovsky

Valérie Alezra

Sun Un

Using the noncanonical metallo-amino Acid [Cu(II

Keywords: RIDME, High-field EPR, genetic code expansion, Cu(II) spin-label, structural biology

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

The technology for genetically encoding noncanonical amino-acids (ncAA) using exogenous engineered tRNA/aminoacyl-tRNA synthetase (tRNA/aaRS) pairs can site-specifically endow a protein with a new physical or chemical property in ways that conventional site-directed mutagenesis using the 22 canonical amino-acids cannot. [START_REF] Liu | Adding New Chemistries to the Genetic Code[END_REF] This genetic code expansion technology has the potential for greatly diversifying methods for paramagnetic spin-labeling of proteins for magnetic resonance applications. Such spin-labels can then be coupled with Electron Paramagnetic Resonance (EPR) techniques to study the tertiary and quaternary structures of proteins 2,3 by exploiting the magnetic dipolar interaction between two such labels. Two commonly used techniques are DEER [START_REF] Milov | Use of the Double Resonance in Electron Spin Echo Method for the Study of Paramagnetic Center Spatial Distribution in Solids[END_REF][START_REF] Larsen | Double Electron-Electron Measurement of Electron Disordered Solids Resonance Spin-Echo Spin Pair Separations Modulation : Spectroscopic in Orientationally Disordered Solids[END_REF][START_REF] Martin | Determination of End-to-End Distances in a Series of TEMPO Diradicals of up to 2.8 Nm Length with a New Four-Pulse Double Electron Electron Resonance Experiment[END_REF] (Double Electron-Electron Resonance) and RIDME [START_REF] Kulik | Electron Dipole-Dipole Interaction in ESEEM of Nitroxide Biradicals[END_REF][START_REF] Milikisyants | A Pulsed EPR Method to Determine Distances between Paramagnetic Centers with Strong Spectral Anisotropy and Radicals: The Dead-Time Free RIDME Sequence[END_REF] (Relaxation Induced Dipolar Modulation Enhancement). Conventional site-directed spin labeling (SDSL) sitespecifically introduces cysteine residues that are subsequently linked to nitroxide radicals using sulfur chemistry. This works best when the cysteines are unique and do not physically or functionally affect the protein. Using genetic-code expansion technology, the ncAA azidophenylalanine has been used, instead of cysteine, to attach a spin-label using azido-alkyne clickchemistry. [START_REF] Widder | Double Nitroxide Labeling by Copper-Catalyzed Azide-Alkyne Cycloadditions with Noncanonical Amino Acids for Electron Paramagnetic Resonance Spectroscopy[END_REF][START_REF] Abdelkader | Protein Conformation by EPR Spectroscopy Using Gadolinium Tags Clicked to Genetically Encoded P-Azido-L-Phenylalanine[END_REF] Since it is bio-orthogonal, azido-phenylalanine provides a biologically and chemically unique labeling site. A conceptually simpler and more elegant approach has been to express the targeted protein directly with a paramagnetic ncAA. The first demonstration of this approach was the genetic incorporation of the nitroxide radical 2,2,5,5-tetramethyl-pyrrolin-1-oxyl amino acid into E. coli thioredoxin. [START_REF] Schmidt | A Genetically Encoded Spin Label for Electron Paramagnetic Resonance Distance Measurements[END_REF] More recently, 3-pyrazolyltyrosine (PyTyr) has been introduced into the cyan fluorescent protein and the histidine kinase response-regulator CheY.

Cu(II) chelation to the ncAA resulted in the PyTyr:Cu(II) spin-label. [START_REF] Merz | Site-Specific Incorporation of a Cu 2+ Spin Label into Proteins for Measuring Distances by Pulsed Dipolar Electron Spin Resonance Spectroscopy[END_REF] DEER was used to measure the distance between a PyTyr:Cu(II) center and a nitroxide label attached to a cysteine residue. [START_REF] Merz | Site-Specific Incorporation of a Cu 2+ Spin Label into Proteins for Measuring Distances by Pulsed Dipolar Electron Spin Resonance Spectroscopy[END_REF] An alternative approach has been to use classical site direct mutagenesis to encode the HXXXH (dHis) motif optimally placed in helical structures that can form a high affinity rigid Cu(II) binding site. [START_REF] Cunningham | The Double-Histidine Cu 2+ -Binding Motif: A Highly Rigid, Site-Specific Spin Probe for Electron Spin Resonance Distance Measurements[END_REF][START_REF] Ghosh | The Cu2+-Nitrilotriacetic Acid Complex Improves Loading of α-Helical Double Histidine Site for Precise Distance Measurements by Pulsed ESR[END_REF][START_REF] Ackermann | Pulse Dipolar EPR Spectroscopy in Proteins: CuII-CuII and Nitroxide-Nitroxide Cases[END_REF] dHis:Cu(II) spin labels have been demonstrated to yield RIDME traces with narrow distance distributions and good sensitivity.

The genetic incorporation of (2,2'-bipyridin-5-yl)alanine (BpyA) was the first demonstration of the use of genetic-code expansion to introduce a metal-binding ncAA into a protein. [START_REF] Xie | A Genetically Encoded Bidentate, Metal-Binding Amino Acid[END_REF] Bipyridine forms stable metal complexes and binds Cu(II) more strongly than any of the canonical aminoacids. [START_REF] Irving | The Stability of Metal Complexes of 1,10-Phenanthroline and Its Analogues. Part I. 1,10-Phenanthroline and 2,2′-Bipyridyl[END_REF][START_REF] Williams | Thermodynamic Considerations in Co-Ordination. Part VII. Solubility of the Histidine-H + System and Stability Constants, Free Energies, Enthalpies, and Entropies of Protonation of Histidine and Tryptophan and of Formation of Their Manganese(II), Iron(II), Cobalt(II), Nickel(II), Copper(II), and Zinc(II) Complexes[END_REF] BpyA can be viewed as a compact bio-orthogonal single amino-acid version of the dHis motif. Using the pEVOL-BipyA plasmid developed by Schultz and coworkers, [START_REF] Xie | A Genetically Encoded Bidentate, Metal-Binding Amino Acid[END_REF][START_REF] Young | An Enhanced System for Unnatural Amino Acid Mutagenesis in E. Coli[END_REF] we have for the first time encoded BypAs into proteins as spectroscopic labels. In particular, BypAs were introduced into the monomeric ultra-stable variant of the green fluorescent protein (muGFP) [START_REF] Scott | A Novel Ultra-Stable, Monomeric Green Fluorescent Protein for Direct Volumetric Imaging of Whole Organs Using CLARITY[END_REF] . Cu(II) were bound to form the BypA:Cu(II) spin-labels. Although GFPs are typically dimeric, muGFP was designed to be monomeric up to concentrations of 10 mg mL -1 or about 360 μM. [START_REF] Scott | A Novel Ultra-Stable, Monomeric Green Fluorescent Protein for Direct Volumetric Imaging of Whole Organs Using CLARITY[END_REF] By leveraging the sensitivity benefits of high microwave frequencies, we have been able to obtain high quality 94 GHz RIDME traces from BypA:Cu(II) labeled muGFP samples that were only 1 μL in volume having typical protein concentrations of less than 200 μM. Equally important, these measurements shed light on the existence and structure of muGFP dimers even at low concentrations. We demonstrate that the use of BypA:Cu(II) and high-frequency RIDME is an effective and sensitive approach for examining tertiary and quaternary structures of proteins. Expression plasmids for H25B, Y151B and H25B/Y151B (where B is denotes BpyA) mutants of muGFP were prepared and used to co-transform E. coli BL21-DE3 cells together with the pEVOL-BipyA plasmid to singly and doubly incorporate BpyA. As expected, the green fluorescence of the cells carrying the mutant plasmids was only observed in media supplemented with BpyA amino acid (Figure 1). The detection of the fluorescence was not only indicative of the BpyA incorporation but also of the correct folding of the muGFP which is required for the chromophore formation. The fluorescence of the H25B and Y151B variants were lower than that of the muGFP. H25B/Y151B had the lowest. Purification studies showed that this was a consequence of the protein expression level rather than misfolding. The yields of Y151B, H25B and H25B/Y151B were 50%, 25% and 10% with respect to the unmodified muGFP. The dependence of the expression level with the position [START_REF] Albayrak | Cell-Free Co-Production of an Orthogonal Transfer RNA Activates Efficient Site-Specific Non-Natural Amino Acid Incorporation[END_REF][START_REF] Schinn | Rapid in Vitro Screening for the Location-Dependent Effects of Unnatural Amino Acids on Protein Expression and Activity[END_REF] and the number of incorporated ncAA is a common occurrence that is not well understood [START_REF] Amiram | Evolution of Translation Machinery in Recoded Bacteria Enables Multi-Site Incorporation of Nonstandard Amino Acids[END_REF] The UV-visible spectra of the purified protein confirmed the BpyA incorporation (Figure 2). The calculated extinction coefficient for muGFP at 280 nm is 20.4 mM -1 cm -1 and the measured for 2,2'-bipyridine 24 , 13 mM -1 cm -1 . The ratios of the absorbance at 280 and 488 nm were: 0.41 for muGFP, 0.68 for H25B and Y151B, and 0.95 for H25B/Y151B, consistent with the incorporation of one BpyA residue in the single mutants and two in the double mutant. An important advantage of using 2,2'-bipyridine is that its complexation to copper, as well as other divalent metals such Co(II) and Ni(II), can be simply and accurately quantified by following the changes in UV-visible absorption at 280, 305 and 317 nm (Figure 2). [START_REF] Mason | The Electronic Spectra and Optical Activity of Phenanthroline and Dipyridyl Metal Complexes[END_REF][START_REF] Nakamoto | Ultraviolet Spectra and Structures of 2,2′-Bipyridine and 2,2′,2″-Terpyridine in Aqueous Solution[END_REF] By contrast, binding to the dHis and PyTyr are more difficult to follow. Quantification of Cu(II) loading is important given its propensity to promiscuously binding to proteins. Titrations of the BypA labeled variants with Cu(II) were conducted following the change in absorption at 317 nm (Figure 2, inset). To reach a plateau, 0.8 equivalents of Cu(II) per monomer was required for the H25B variant and 0.9 for the Y151B and 1.7 for the H25B/Y151B. The lower occupancy of H25B was likely due to the presence of Fe(II) that already occupied some of the BpyA centers. This was evident from the significant absorbance at 305 and 317 nm observed in the freshly purified H25B (Figure 2, black trace). Fe(II) occupancy was estimated to be 20%. However, the occupancy depended on the protein expression conditions (see Supporting Information for details). Spurious Fe(II) binding to BpyA has been previously observed. [START_REF] Mills | Computational Design of an Unnatural Amino Acid Dependent Metalloprotein with Atomic Level Accuracy[END_REF][START_REF] Bersellini | Multidrug Resistance Regulators (MDRs) as Scaffolds for the Design of Artificial Metalloenzymes[END_REF] Figure 3 shows 94 GHz field-swept echo-detected EPR spectra of the H25B:Cu(II), Y151B:Cu(II) and H25B:Cu(II)/Y151B:Cu(II) proteins. The three turning points in each spectra correspond to the three Cu(II) principal g-values (g x,y,z = hν/(γB x,y,z ) and g x ≥ g y ≥ g z ). The hyperfine structure seen in the conventional 9 GHz EPR spectra (Figure S1) was not resolved. The effects of g-straining or distribution in g-values appeared to be particularly strong in the g x region. The spectrum of H25B:Cu(II)/Y151B:Cu(II) was reproduced by the sum of those of H25B:Cu(II) and Y151B:Cu(II) (Figure 3A andS1A). The H25B:Cu(II) spectrum was axial, with principal g-values of 2.270, 2.062 and 2.062, indicative of a square-planar Cu(II) centers. [START_REF] Bennett | EPR Methods for Biological Cu(II): L-Band CW and NARS[END_REF][START_REF] Solomon | Copper Active Sites in Biology[END_REF] Y151B:Cu(II) could be decomposed into an axial H25B:Cu(II)-like (30% of the total intensity) component and another (70%) that was anisotropic with principal g-values of 2.242, 2.091 and 2.036. The Y151B:Cu(II)

shown in Figures 3A andB were representative. The actual compositions varied from sample to sample. The g z value of 2.24 with an effective 63/65 Cu hyperfine coupling (|A eff |) of 440 MHz (1470×10 -4 cm -1 ) (Figure 3B and S1B), compared to 540 MHz for H25B:Cu(II)) appeared to be an unusual combination, not corresponding to any of the previously values measured for Cu(II) complexes of bipyridine. [START_REF] Garribba | The Cu(II)-2,2′-Bipyridine System Revisited[END_REF] This suggested that the metal ligand sphere of the BypA:Cu(II) label was modified by its interactions with the protein. The sample-to-sample variation in the two components indicated that these interactions were likely to be weak.

We calculated the principal g-values of (Cu[(2,2'-bipyridine)(H 2 O) 2 ]) 2+ (see Supporting Information). These qualitatively agreed with the measured values for 2,2'-bipyridine:Cu(II) and H25B:Cu(II) (Figure 3C). The g-values of a Cu(II) ion bound to the N 149 V 150 B 151 region as shown in Figure 3C exhibited an anisotropy similar to that of the anisotropic Y151B:Cu(II) component.

Molecular dynamics (MD) calculations indicated that asparagine-198 could also bind to the Cu(II) ion. The calculated g-values of this arrangement were only negligibly different (see Supporting Information). These calculations provided support that Y151B:Cu(II) had a square pyramid structure involving protein interaction. Further studies are in progress to ascertain experimental evidence.

The unlabeled muGFP also possessed a copper-binding site. In the presence of 1 equivalent of Cu(II), it had an axial EPR spectrum (orange trace, Figure 3A) that was distinct from those of the BypA:Cu(II). [START_REF] Ghosh | The Cu2+-Nitrilotriacetic Acid Complex Improves Loading of α-Helical Double Histidine Site for Precise Distance Measurements by Pulsed ESR[END_REF] N ELDOR-NMR spectroscopy showed that this binding site had at least one histidine (see Figure S3 and Supporting Information). In brief, we observed a distinct 2 MHz split doublet in 14 N ELDOR-NMR spectrum of the this native Cu(II) site that is consistent with the hyperfine interaction of Cu(II) with the distal ring nitrogen of a histidine ligand that is not observed for the BypA:Cu(II) centers. This along with the optical data indicated that the BypA sites outcompeted this native site for Cu(II) and the latter did not interfere with the BypA:Cu(II)-based measurements.

Figure 4A shows the 94 GHz Cu(II) RIDME traces from 1 μL samples of single and doubly BypA:Cu(II) labeled muGFP variants as well as muGFP:Cu(II) itself. The time trace of the 50 μM H25B:Cu(II)/Y151B:Cu(II) doubly labeled variant exhibit strong dipolar modulation with a 50% modulation-depth and was obtained in less than 4 h. Even after a 10-fold dilution, analyzable dipolar modulations were readily detectable (< 8 h). These RIDME dipolar modulations build-up during the mixing time interval in the pulse sequence during which the detected spins experience a change in their local magnetic field caused by the longitudinal relaxation of the spins to which they are coupled. For the simple case where the relaxation is exponential with time constant T 1 , the modulation-depth is given by

∆= ! !"# " #1 -exp ) #$ $%& % ' *+
Eq. 1

where Q exp has been include to account for the possibility that not all detected Cu(II) experience a dipolar interaction with another Cu(II). [START_REF] Wort | A General Model to Optimise CuII Labelling Efficiency of Double-Histidine Motifs for Pulse Dipolar EPR Applications[END_REF] For the H25B:Cu(II)/Y151B:Cu(II) variant, Q exp is unity.

The T 1 were essentially the same for all cases (Figure S2). The RIDME traces shown in Figure 4 were obtained with a 32 μs mixing time or approximately1.5 T 1 . This mixing time was a reasonable balance between good modulation depths, which increases with the mixing time, and significant loss in signal-to-noise which invariably occurs with increasing mixing time. [START_REF] Breitgoff | UWB DEER and RIDME Distance Measurements in Cu(II)-Cu(II) Spin Pairs[END_REF] The 30 K temperature was a compromise between short T 1 allowing fast shot-repetition times (153 μs) for efficient signal averaging and sufficiently long phase memory times to insure the 3 μs RIDME measurement window. Evidently, these compromises did not significantly affect the quality of the measurement. The modified-Tikhonov distance distribution of the RIDME traces. Fits (thin black lines) using the same background fits are superimposed in panel A and the horizontal gray traces are the differences between the modified-Tikhonov fits and measurements scaled by a factor of 5, except for 5 µM H25B:Cu(II)/Y151B:Cu(II) which has been scaled by 2.5.

Intrinsic to these RIDME measurements, where only a small fraction of the Cu(II) spins are excited and detected, is the possibility of orientation selection. This is because each position in the Cu(II) EPR spectra arises from a unique set of orientations of each BpyA:Cu(II) with respect to the magnetic field. In such cases, the orientation dependence of the dipolar interaction may not be sufficiently sampled to reliably extract the distance between the Cu(II) centers. The use of high magnetic-fields potentially exacerbates this problem since a smaller set of Cu(II) are excited and detected. However as further discussed in the Supporting Information, this did not seem to be a significant problem presumably due to the rotational flexibility afforded by the two Cβ bonds that link the BypA:Cu(II) label to protein backbone. MD calculation based on the muGFP crystal structure were used to ascertain the conformational space spanned by a BpyA:Cu(II) label. In these MD simulations, the label was modeled as [Cu(BpyA)(H2O)2] 2+ and the less bulky [Cu(BpyA)] 2+

(see Supporting Information for details). As expected, the simulations showed that the [Cu(BpyA)] 2+ was less sterically constrained, with 𝑟 ⃗ & ( #&'())) (the vector connecting the Cu(II) to Cα of the same residue) spanning a wider angular space than the hydrated complex (Figure 5A).

Both forms of the BpyA:Cu(II) spin-label were spatially very compact. The conformational space spanned by the Cu(II) atom was similar in size to that of a phenoxyl oxygen atom of a tyrosine amino-acid (Figure 5A). In spite of this compactness, the interatomic 𝑟 ⃗ +#&'())) vector subtended a large portion of an octant of a sphere (Figure S5). It is likely that due to the distributed nature of the Cu(II) Zeeman interaction this coverage was even greater than predicted by the MD simulations. This meant that the RIDME measurements likely sampled a large range of the orientation dependence of dipolar interactions resulting in the lack of any significant orientation selection effects in the measurements (see Supporting Information).

The dipolar spectra of the BypA:Cu(II) labeled variants were obtained by Fourier transforming (FT) their RIDME traces (Figure 4B). Superimposed on the FT of the H25B:Cu(II)/Y151B:Cu(II)

variant RIDME time trace are three calculated dipolar spectra (Pake patterns) corresponding to Cu(II)-Cu(II) distances of 4.2, 3.0, and 2.4 nm. The dipolar spectra of all three BypA:Cu(II) labeled variants appeared to have similar components. The FT do not allow for complete quantitative analyses of the dipolar interactions. The most common approach is based on Tikhonov regularization. Rather than the conventional approach where the background is fitted and then subtracted or, more properly, divided from the measured traces, we implemented and use the approach of Fábregas Ibáñez and Jeschke [START_REF] Fábregas Ibáñez | Optimal Background Treatment in Dipolar Spectroscopy[END_REF] which incorporates the background directly into the Tikhonov regularization, specifically the modified 𝐾 √-kernel, the details for which are found in their publication and which hereon will be referred to as the modified-Tikhonov calculation. A single stretch exponential model of the background of the RIDME time traces (Table S2) yielded the better fits and more stable solutions than polynominals. [START_REF] Abdullin | Pulsed EPR Dipolar Spectroscopy on Spin Pairs with One Highly Anisotropic Spin Center: The Low-Spin FeIII Case[END_REF][START_REF] Keller | Intermolecular Background Decay in RIDME Experiments[END_REF] The modified-Tikhonov fit of these traces are shown in Figure 4 and conventional Tikhonov regularization in Figure S6. For simplicity, a brute-force grid search that scanned the zero time-point, the region used to fit the stretch background and regularization parameter was used. For each zero time-point and fitting region, a stretch exponential was fit to the data which along with the chosen regularization parameter were then used for the modified-Tikhonov calculation. Shown in Figures 4C are the fits with the smallest RMS deviation from the measured data. The regularization parameters were typically 10 and corresponded to the corner of the l-curve. The numerical "noise" can be significant when the background is simply divided adversely affecting the analysis. This was much less evident with the modified-Tikhonov approach. Otherwise, as seen on Figure 4A the fits were free of systematic deviations.

The distance distributions returned by the modified-Tikhonov regularization (Figure 4C) were consistent with what we had qualitatively surmised from the Fourier transforms. The doubly and singly labeled variants had Cu-to-Cu distances corresponding approximately 2.5, 3.5 and 4 nm. In nearly all cases, there were low probability distances above 6 nm. However, these were not stable with respect to the scanned parameters. The presence of this feature likely arose from the inadequacies of modeling the background as a single stretch exponential. [START_REF] Keller | Averaging of Nuclear Modulation Artefacts in RIDME Experiments[END_REF] Moreover, the 3 µs dipolar evolution time was insufficiently long to make this feature reliable. The H25B:Cu(II)/Y151B:Cu(II) 3.9 nm distance was the expected distance based on the separation of H25 and Y151 in the muGFP crystal structure (Figure 5B). Since the hyperfine and optical measurements precluded the presence of Cu(II) bond to sites other than a BpyA, the simplest explanation for the 2.5 and 3.5 nm H25B:Cu(II)/Y151B:Cu(II) features, as well as the RIDME modulations of the singly label variants, was the presence of significant concentrations of dimers.

The Y151B:Cu(II) variant exhibited a drop in modulation depth from 20% to 8% with a decrease in protein concentration (Figure 4A), consistent with a shift in the monomer-dimer equilibrium favoring monomers. Nonetheless, dimers were present even at 50 μM, in contrast to previous observations that muGFP is predominately monomeric below 360 μM [START_REF] Scott | A Novel Ultra-Stable, Monomeric Green Fluorescent Protein for Direct Volumetric Imaging of Whole Organs Using CLARITY[END_REF] . One would have also expected the background of the more concentrated sample to decay significantly faster since it results from a greater number of non-specific dipolar interactions between Cu(II) ions. [START_REF] Keller | Intermolecular Background Decay in RIDME Experiments[END_REF] However, only a small increase was observed (Table S2). This may have in part been due to compensating changes in the mono-dimer equilibrium. Further study is required. The 2.6 and 3.8 nm distances found for the Y151B variant suggested the presence of two different quaternary structures. The MD profile for rH25B:Cu(II)-Y151B:Cu(II) for the H25B:Cu(II)/Y151B:Cu(II) variant matched the prominent 3.9 nm feature in measured distance distribution independently of the quaternary model (Figure 5C). The smaller 2.4 nm feature of this variant had similar counterpart in the distance distribution of the singly labeled Y151B:Cu(II) variant (grey shaded region in Figure 5C). These coincided with the MD profile of rY151B:Cu(II)-Y151Bʹ:Cu(II) based on the usGFP. The 1 to 3.5 nm portion of the H77-H77ʹ distance profile of the unlabeled muGFP is also consistent with the usGFP quaternary structure. In contrast the 3.7 nm rY151B:Cu(II)-Y151Bʹ:Cu(II) profile matched the antiparallel structure of muGFP. The MD rH25B:Cu-H25B´ profile for the of H25B:Cu(II) variant based on usGFP and muGFP structures did not agree with the experimental data. The AlphaFold2 model-based distance profile could account for the longer half of the experimentally observed bimodal distribution that had maxima at 2.5 and 4.7 nm (Figure 5C). This suggested the possibility that in this variant some of the dimers had monomers in a parallel configuration.

The comparison of RIDME measurements and MD simulations indicated that different quaternary conformations are taken by the muGFP variants. This is not surprising since a single point-mutation can alter the GFP monomer/dimer equilibrium and modify the dimeric structure. [START_REF] Palm | The Structural Basis for Spectral Variations in Green Fluorescent Protein[END_REF][START_REF] Leibly | A Suite of Engineered GFP Molecules for Oligomeric Scaffolding[END_REF] Given this sensitivity, we cannot exclude the possibility that the BypA mutation or the Cu(II) influences the monomer/dimer equilibrium. However, we do know that for the BypA variants, no BypA:Cu(II):BypA centers that bridge the muGFP monomers were present. [START_REF] Yang | Diverse Protein Assembly Driven by Metal and Chelating Amino Acids with Selectivity and Tunability[END_REF] The titration experiments showed a nearly stochiometric Cu(II) to BpyA ratio excluding the presence of significant quantities of [BpyA2Cu] 2+ and [BpyA3Cu] 2+ complexes and any Cu(II) bond to other places in the protein. The crystal structures and MD calculations did not reveal any other sufficiently close candidate ligands, in particular glutamates and aspartates, that could form [Cu(BpyA)X] 1+ and bridged the two monomers. In the unlabeled variant, one of the histidines around H77 does come from the neighboring monomer in the usGFP model; hence, Cu(II) may have promoted dimer formation in this case. [START_REF] Leibly | A Suite of Engineered GFP Molecules for Oligomeric Scaffolding[END_REF] The binding of Cu(II) to unintended sites in the protein is a common issue of Cu(II)-based spin labels. [START_REF] Merz | Site-Specific Incorporation of a Cu 2+ Spin Label into Proteins for Measuring Distances by Pulsed Dipolar Electron Spin Resonance Spectroscopy[END_REF][START_REF] Cunningham | The Double-Histidine Cu 2+ -Binding Motif: A Highly Rigid, Site-Specific Spin Probe for Electron Spin Resonance Distance Measurements[END_REF] Based on our results, it is likely that BpyA is capable of outcompeting most forms of unintended Cu(II) binding site(s) such as those formed by multiple histidines. Moreover, overloading of the BypA sites can be quantitatively avoided using simple optical spectroscopy.

The genetic incorporation of metal-chelating amino acids appears to provide excellent opportunities for novel spin-labels with ideal properties. Here we demonstrated this using BpyA:Cu(II), but other metals, notably Co(II), will also work as spin-labels. BpyA:Cu(II) is chemical stable and structurally compact. MD simulation showed that it does not occupy much larger space than a tyrosine. Yet, it is sufficiently rotationally flexible that orientation selection does not appear to affect distance measurement. The compactness of the BpyA:Cu(II) label also insures that its contribution to the uncertainty in the distance measurements is minimal. The use of high microwave frequencies to perform Pulse Dipolar Spectroscopy measurements on these centers is not an obvious choice since the width of the Cu(II) EPR spectrum range increase linearly with frequency. In spite of this, we have shown that using 94 GHz RIDME it is possible to measure with good accuracy the dipolar interaction between two labels even for protein concentrations as low as 5 µM using only a 1 µL of sample. Here we demonstrated that the combination of the paramagnetic non-conventional amino acid BpyA:Cu(II) with high-frequency, high-magneticfield, EPR, resulted in a sensitive methodology for the study of the structure of proteins.

ASSOCIATED CONTENT

Supporting

Figure 1 .

 1 Figure 1. Fluorescence image of E. coli cells co-transformed with pEVOL-BipyA and the expression plasmids for muGFP and the H25B, Y151B and H25B/Y151B variants. The empty pET24a plasmid was used as negative control. The cells were grown on LB-agar media complemented with 0.1 mM IPTG, 0.025% arabinose and 0 (left) and 500 mM (right) BpyA.

Figure 2 .

 2 Figure 2. UV-visible absorption spectra of muGFP and BpyA labelled variants. The spectra were taken in the presence of 0 (black), 1 (red), 2 (green) and 3 (violet) equivalents of Cu(II) per protein monomer. Spectra have been normalized to the intensity of the 488 nm GFP chromophore absorption. The inset shows the variation in absorbance at 317 nm as a function of the added equivalents of Cu(II) per protein monomer at pH 6 normalized to unity.

Figure 3 .

 3 Figure 3. (A) The 30 K 94 GHz field-swept spin echo detected EPR spectra of 50 μM H25B:Cu(II)/Y151B:Cu(II) (black); 50 μM H25B:Cu(II) (red); 50 μM Y151B:Cu(II) (purple) and 200 μM muGFP:Cu(II) (orange). The cyan trace is the sum of the Y151B:Cu(II) and H25B:Cu(II) spectra. The six line Mn(II) spectrum centered at 3.36 T arises from the microwave cavity and provided a convenient magnet-field and amplitude standard. (B) The difference (blue) of the Y151B:Cu(II) spectrum (purple) and a scaled +10 G shifted H25B:Cu(II) (red) spectrum that revealed a second component of the Y151B:Cu(II) spectrum that was 0.7 of the integrated intensity. A spectrum of 500 μM 2,2'-bipyridine:Cu(II) (green) is also shown for comparison. The principal g-values of the BypA:Cu(II) centers are shown in the same colors as their spectra. The g x values were obtained from the 9 GHz spectra shown in the Figure S1. (C) The models of the muGFP BypA:Cu(II) centers and their DFT calculated Cu(II) g-tensors: the simple (Cu[(2,2'bipyridine)(H 2 O) 2 ]) 2 (top), NGB:Cu(II) model based on the protein sequence (middle) and a model based on molecular dynamics calculations (bottom) in which asparagine-198 donates an axial ligand.

Figure 4 .

 4 Figure 4. (A) 94 GHz Cu(II) RIDME traces of 50 (orange) and 5 μM (blue) H25B:Cu(II)/Y151B:Cu(II); 250 (green) and 50 μM (brown) Y151B:Cu(II); 200μM H25B:Cu(II) (red); and 200 μM Cu(II) added to 200 μM muGFP (cyan). The traces were obtained at the maximum of the corresponding field-swept echo-detected spectrum. (B) The Fourier transforms of each RIDME trace after subtraction of stretch exponential fits of their backgrounds. The black traces are the calculated dipolar spectrum of two ½-spin separated by the distances indicated. (C)

Figure 5 .

 5 Figure 5. MD simulations: (A) The positions spanned by the phenoxyl oxygen (pink spheres) of Y151 (outlined in light blue) and the Cu(II) ion of [Cu(II)BpyA(H 2 O) 2 ] 2+ (red spheres) and [Cu(II)BpyA] 2+ (orange spheres) models of the sidechain of Y151B:Cu(II) (outlined in green and blue). (B) Structural model of H25B/Y151B with the relevant distances indicated based on the dimeric usGFP structure. (C) The MD distance profiles for monomeric r H25B:Cu-Y151B:Cu and dimeric r Y151B:Cu-Y151B´:Cu , r H25B:Cu-H25B´:Cu and r H77:Cu-H77´:Cu based on the usGFP (red shaded) and muGFP (green shaded) crystal structures and an AlphaFold2 structure (blue shaded). The black traces show the corresponding RIDME results. All distance profiles have been normalized to their total probabilities. The grey shaded regions indicate distances arising consistent with the usGFP quaternary structure.

Information Available: 9

 9 GHz cw-EPR spectra of BpyA labeled variants of muGFP, characterization of the native Cu(II) binding site in muGFP, DFT calculations on a potential muGFP:Cu(II) site, orientation selection, T 1 measurements, background fits parameters of the RIDME traces, conventional Tikhonov regularization using DeerAnalysis, modeling of the quaternary structures muGFP and experimental and computational methods. Davis T. Daniel current address: Institute for Energy and Climate Research (IEK-9), Forschungzentrum Jülich GmbH, Jülich, 52424, Germany Corresponding Author: Leandro C. Tabares, Bat 532 Pce 215, F-91191 Gif-sur-Yvette, France, phone: +33-169087579, email: leandro.tabares@cea.fr. ACKNOWLEDGMENT We are grateful to Dr. Peter Schultz (Scripps Research) for his generous gift of the pEVOL-BipyA plasmid. We acknowledge Tania Tibiletti (I2BC) for her assistance with 9 GHz measurements. This work was financially supported by the ANR InVivoNanoSpin (ANR-20-CE11-0002) and the Ile-de-France DIM-ELICIT program. We benefited from the biophysics platform of the Institute for Integrative Biology of the Cell supported by French Infrastructure for Integrated Structural Biology (FRISBI, ANR-10-INBS-0005) and the Infrastructure for Biology, Health and Agronomy (IBISA).

  . To best of our knowledge, this is the first time two BpyA have been successfully incorporated into a single protein. Even with diminished

	No BpyA expression levels sufficient amounts of the ncAA labeled proteins were obtainable for 500 µM BpyA
	muGFP	H25B	muGFP	H25B
	spectroscopic applications.	Y151B		Y151B
	Control	H25B/Y151B	Control	H25B/Y151B

AUTHOR INFORMATION

Notes

The authors declare no competing financial interests.