MSc Alexandre Tronchot
email: alexandre.tronchot@etudiant.univ-rennes1.fr

Julien Berthelemy

Hervé Thomazeau

Arnaud Huaulmé

Paul Walbron

François Sirveaux

Pierre Jannin

) ; Paul

Validation of virtual reality arthroscopy simulator relevance in characterising experienced surgeons

Keywords:

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

Introduction

Surgical simulation aims to teach practical skills outside the operating room. It is necessary for safety [START_REF] Barger | Extended Work Shifts and the Risk of Motor Vehicle Crashes among Interns[END_REF] and organizational reasons related to the functioning of health facilities [START_REF] Erivan | Ten-year epidemiological study in an orthopaedic and trauma surgery centre: Are there risks involved in increasing scheduled arthroplasty volume without increasing resources?[END_REF] and the time constraints of teachers and students. In France, procedural simulation has been officially included in the medical and surgical training resident national curriculum since 2017 [START_REF] Bonnafous | Arrêté du 12 avril 2017 portant organisation du troisième cycle des études de médecine[END_REF]. Virtual reality (VR) simulation is a tool that complements traditional techniques on models [START_REF] Banaszek | Virtual Reality Compared with Bench-Top Simulation in the Acquisition of Arthroscopic Skill: A Randomized Controlled Trial[END_REF], or cadaveric subjects [START_REF] Henn | Shoulder Arthroscopy Simulator Training Improves Shoulder Arthroscopy Performance in a Cadaveric Model[END_REF]. It is particularly suitable for learning arthroscopy skills [START_REF] Walbron | Virtual reality simulator improves the acquisition of basic arthroscopy skills in first-year orthopedic surgery residents[END_REF][START_REF] Marcheix | Learning the skills needed to perform shoulder arthroscopy by simulation[END_REF][START_REF] Baumann | Lessons taught by a knee arthroscopy simulator about participants in a European arthroscopy training programme[END_REF]. Numerous researches have been dedicated to the development of this tool and its use. In two recent metaanalysis [START_REF] Frank | Utility of Modern Arthroscopic Simulator Training Models: A Meta-analysis and Updated Systematic Review[END_REF][START_REF] Aïm | Effectiveness of Virtual Reality Training in Orthopaedic Surgery[END_REF], one of the main outlined drawbacks was the difficulty in distinguishing performance levels (Construct Validity) with traditional metrics (such as duration and path length) in homogeneous experienced populations. Therefore, it seems adequate to search for new metrics or new methods of performance measurements.

In data mining, hierarchical clustering is a method of analysis which seeks to build a hierarchy of clusters with a linkage criterion. In time series analysis, dynamic time warping (DTW) is one of the algorithms for measuring similarity between two temporal sequences, which may vary in speed. Hierarchical clustering with DTW was used with success to distinguish different levels of expertise on studies about VR simulation in robotic surgery training [START_REF] Despinoy | Evaluation of contactless human-machine interface for robotic surgical training[END_REF] but have not yet been analysed in arthroscopic training.

It was hypothesized that a larger experience in surgical shoulder arthroscopy would be correlated with better performance on a virtual reality shoulder arthroscopy simulator and that experienced operators would share similar probe trajectories. The objective of this study was to analyse instrument trajectories and to determine the ability of the simulator to distinguish individual performance levels within cohorts of experienced surgeons of different expertise levels using innovative metrics and hierarchical clustering with DTW.

Methods

Population

The trajectories of two cohorts have been prospectively collected during international orthopaedic surgery congresses, French Society of Arthroscopy (SFA) held in Strasbourg-December 2018, and the International Shoulder Course, Val d'Isère-January 2019. Each participant answered a standardized questionnaire about his/her expertise in arthroscopy, his/her hobbies (Table 1). Informed consent was obtained for all the participants. Questions regarding the number of actual surgical procedures ever done before doing the exercise served to establish base-line experience levels for each cohort.

The first cohort called INTERMEDIATE is composed of 29 orthopaedic surgeons with an interest in arthroscopic surgery (surgeons at the end of the resident training phase and fellows), who have performed between one and 99 arthroscopic surgeries as principal operator possibly under supervision of a senior surgeon. The second cohort called EXPERT is composed of 28 orthopaedic surgeons with a subspecialisation in arthroscopic shoulder surgery who have performed more than 100 arthroscopic 4 surgeries as principal operator. This experience threshold was previously used in a similar study [START_REF] Fucentese | Evaluation of a virtual-reality-based simulator using passive haptic feedback for knee arthroscopy[END_REF]. Only actual surgical practice was assessed to determine the expertise level, therefore previous practice on any surgical simulator has not been take into account. Three INTERMEDIATE and two EXPERT trajectories data were incomplete.

The remaining 26 INTERMEDIATE and 26 EXPERTS participant trajectories data were analysed for the study. A two-sided power calculation was performed for the null hypothesis ("performance of experts is equal to that intermediate") with an acceptable type 1 error, α= 0.05 and acceptable type 2 error β = 0.20, power = 0.8. For the power calculation, data from previous shoulder arthroscopy validation study were used [START_REF] Rahm | Validation of a virtual reality-based simulator for shoulder arthroscopy[END_REF].

This resulted in a minimum number of 14 subjects in each group, which is in line with previously reported validation studies [START_REF] Fucentese | Evaluation of a virtual-reality-based simulator using passive haptic feedback for knee arthroscopy[END_REF][START_REF] Rahm | Validation of a virtual reality-based simulator for shoulder arthroscopy[END_REF][START_REF] Bayona | Assessment study of 11 insight ARTHRO VR ® arthroscopy virtual training simulator: face, content, and construct validities[END_REF][START_REF] Gomoll | Surgical Experience Correlates with Performance on a Virtual Reality Simulator for Shoulder Arthroscopy[END_REF].

Methods of assessment

The VirtaMed AG (Zurich) ArthroS™ arthroscopy simulator was used to collect trajectories. Face and construct validity of the simulator and its different modules have been assessed in previous studies for novice and experts population [START_REF] Fucentese | Evaluation of a virtual-reality-based simulator using passive haptic feedback for knee arthroscopy[END_REF][START_REF] Rahm | Validation of a virtual reality-based simulator for shoulder arthroscopy[END_REF][START_REF] Roberts | Validation of the updated ArthroS simulator: face and construct validity of a passive haptic virtual reality simulator with novel performance metrics[END_REF]. The FAST and SHOULDER modules were used for the study. The exercise corresponded to a basic procedure in shoulder surgery: the extraction of foreign bodies in a shoulder joint. "Catch the Stars Glenohumeral" is a semi-realistic immersion exercise with modelling of a pathology-free shoulder joint where the goal is to grasp and extract five foreign bodies (i.e., five stars).

The data collection of study participants was done according to a standardized model (Figure 1). First, the participant performed exercises on the FAST module using the arthroscopy simulator (1.1, 1.2 and 1.3 in figure 1). Second, the participant was introduced to the modelling of the right shoulder joint by performing a video-guided identification of the different structures of the joint in the SHOULDER module, procedure done in lateral decubitus position for all the participants. (2.1 in figure 1).

Third, the users performed the Catch the stars glenohumeral exercise in the SHOULDER module (2.2 in figure 1) in lateral decubitus which he/she performed 3 times. The first trial corresponding as the learning curve was removed from the study analysis.

Statistical Analysis 5 The Matlab-based program TRACAS (TRajectory Analysis for Computer

Assisted Surgery) was adapted to the arthroscopy simulator data and used for trajectory analysis. In the first step, the data extracted from the simulator were used to compute ten metrics (defined in Figure 2): duration, path length, velocity, acceleration, jerk, volume, economy of movement, motion smoothness, spectral arc length smoothness and depth. The analysis was done following previously published techniques [START_REF] Cotin | Metrics for Laparoscopic Skills Trainers: The Weakest Link![END_REF][START_REF] Hofstad | A study of psychomotor skills in minimally invasive surgery: what differentiates expert and nonexpert performance[END_REF][START_REF] Oropesa | Methods and Tools for Objective Assessment of Psychomotor Skills in Laparoscopic Surgery[END_REF][START_REF] Balasubramanian | A Robust and Sensitive Metric for Quantifying Movement Smoothness[END_REF].

Computation of all metrics and trajectory analysis were run on a dedicated computer (Xeon E5-1650V4 @3.60GHz with 32Go RAM) using Matlab_R2018b. Nonparametric test (two-sided Mann-Whitney U test) was used for these metrics between the different manipulators in order to observe which of these metrics showed statistical differences (p < 0.05). Box-plots representations were modelled. The second step of the study consisted of clustering the trajectories of participant using the ascending hierarchical classification with Dynamic Time Warping (DTW). DTW is relevant for the time series indexer because it has the ability to recognize shapes regardless of their time distortion or their phase shift [START_REF] Keogh | Exact indexing of dynamic time warping[END_REF].

Results

Metric Analysis

Analysis of the probe trajectory metrics revealed significant differences between the INTERMEDIATE and EXPERT groups (Table 2). Box-plots are represented in (Figure 3) and (Figure 4). The expert group significantly outperformed the intermediate group for the following metrics (Figure 3 There were no significant differences between the groups for the following metrics

Trajectory Analysis

The results of unsupervised clustering made from the DTW are given in (Figure 5). On this dendrogram, each datum is represented in abscissa and they all join one by one until forming a single cluster. The lower the groupings of clusters, the lower the DTW distance between the aggregated data and the more similar the trajectories. Since two categories are studied here, it was decided to compare the two largest clusters only.

The first cluster was composed of 48 similar trajectories, 100% of the data from the first category was derived from the EXPERT cohort (24/26 experts) while 93% of the second cluster was composed of data from the INTERMEDIATE cohort (26/26 intermediaries) and 7% from the EXPERT cohort (2 experts) for a total of 56 similar trajectories (Sensitivity of 92%, Specificity of 100%).

4.Discussion

About this loose body removal arthroscopic procedure in a virtual-reality simulated shoulder, the results confirmed that six among the ten metrics computed on the trajectories were able to distinguish between expert and non-expert operators: experts had more fluid (better result for spectral arc length smoothness), more economical (better results for economy and working volume), and more efficient (better results for time and speed) gesture. However, the acceleration and path length did not statistically differ between groups. In the same way, Rahm et al., [START_REF] Rahm | Validation of a virtual reality-based simulator for shoulder arthroscopy[END_REF] had noticed that the path length metric was not as discriminative as the others used in this simulator.

Although often used by VR simulators, path length does not seem to be correlated with expertise. The Motion Smoothness metric was calculated with different formulas in the past [START_REF] Cotin | Metrics for Laparoscopic Skills Trainers: The Weakest Link![END_REF][START_REF] Hofstad | A study of psychomotor skills in minimally invasive surgery: what differentiates expert and nonexpert performance[END_REF][START_REF] Balasubramanian | A Robust and Sensitive Metric for Quantifying Movement Smoothness[END_REF]. For this study, three formulas for motion smoothness were computed. The Hofstad computation [START_REF] Hofstad | A study of psychomotor skills in minimally invasive surgery: what differentiates expert and nonexpert performance[END_REF] was not significant and we choose to not represent it. The Cotin computation [START_REF] Cotin | Metrics for Laparoscopic Skills Trainers: The Weakest Link![END_REF] was just significant for this study. More interesting, our results of Spectral Arc Length Motion Smoothness were very significant and consistent with the results found by Balasubramanian et al., [START_REF] Balasubramanian | A Robust and Sensitive Metric for Quantifying Movement Smoothness[END_REF]. With these new types of metrics simply based on the simulator's raw trajectories, it was possible to significantly distinguish levels of expertise.

We demonstrated that clustering analysis with DTW was able to reliably discriminate between expert operators and intermediate operators in an unsupervised way. We suggest that this whole trajectory-based approach could be used to better 7 classify the performance level in a simulator. Cluster 2 is composed of the entire intermediate cohort and two experts. The declared ages of these two experts are near the median age of the intermediate cohort (31 years). Their presence in this cluster could be explained by a non-age-related experience. We did not find any impact of nationality on simulator performance. On the contrary, the intermediate cohort composed entirely of French participants is distinguished by the heterogeneity of the trajectories in the DTW classification.

Construct validity of the simulator had already been demonstrated for novices and experts [START_REF] Rahm | Validation of a virtual reality-based simulator for shoulder arthroscopy[END_REF] , [START_REF] Stunt | Validation of the ArthroS virtual reality simulator for arthroscopic skills[END_REF]. But, Stunt et al., [START_REF] Stunt | Validation of the ArthroS virtual reality simulator for arthroscopic skills[END_REF] faced difficulties to find significant differences in performance for the intermediate group (less than sixty arthroscopic surgeries as principal operator). Roberts et al., [START_REF] Roberts | Validation of the updated ArthroS simulator: face and construct validity of a passive haptic virtual reality simulator with novel performance metrics[END_REF] did not distinguish between surgical experience for a similar shoulder arthroscopic exercise with the performance metrics proposed by the simulator, questioning its interest for middle-grade trainees and consultant. Our study shown that it is possible for a VR arthroscopy simulator to distinguish between more levels of expertise by using new metrics or with clustering analysis with DTW.

The majority of the intermediate cohort had similar theoretical training in arthroscopy (84% had the French national diploma of arthroscopy). Nearly half of them had already received simulator training (46% vs. 27% for the experts). However, this group stands out from the experts because of the heterogeneity of their trajectories and their lower performance. It is therefore essential to combine practical simulator training with theoretical training. But, Manoharan et al., [START_REF] Manoharan | Competence in using the arthroscopy stack system: a national survey of orthopaedic trainees in the UK[END_REF] recently demonstrated the difficulty in the United Kingdom in creating a national training program for arthroscopic surgery. Walbron et al., [START_REF] Walbron | Virtual reality simulator improves the acquisition of basic arthroscopy skills in first-year orthopedic surgery residents[END_REF][START_REF] Walbron | Evaluation of arthroscopic skills with a virtual reality simulator in first-year orthopaedic residents[END_REF] reported on the efforts of the French College of Orthopaedic Surgery and Arthroscopic Society to propose a nationwide program of arthroscopic surgery learning with virtual reality simulator. First results were interesting but reached some limits due to the necessary but complicated organizational presence of a trainer with the student. Thus, creating a standardized nationwide program with regular monitoring of student performance remains complex in practice with only few successful examples in the world [START_REF] Strandbygaard | A structured four-step curriculum in basic laparoscopy: development and validation[END_REF]. Immediate and direct feedback on performance seems mandatory with ongoing guidance rather than having the student work autonomously with occasional feedback during spaced sessions with the trainer.

8

We still need to demonstrate that the results for this method can be extrapolated to other exercises and simulators. Indeed, the Catch the Stars Glenohumeral exercise is identical for each attempt: the position as well as the order of appearance of the stars are the same; similarly, the anatomy and dimensions of the simulated shoulder are always the same. This invariance of the exercise makes the measurement of DTW very effective because each participant will have to use the same path. The experience level was only assessed by the number of arthroscopies performed as the principal operator, in line with previous studies' experience thresholds [START_REF] Rahm | Validation of a virtual reality-based simulator for shoulder arthroscopy[END_REF][START_REF] Stunt | Validation of the ArthroS virtual reality simulator for arthroscopic skills[END_REF]. Although the exercise was conducted under similar conditions for the two cohorts (at a congress and not in an OR environment), the results may have been distorted by inherent conference distractions. This may explain the presence of some outliers in the study. Also mentioned by Fucentese et al,, [START_REF] Fucentese | Evaluation of a virtual-reality-based simulator using passive haptic feedback for knee arthroscopy[END_REF] collecting data during congress might cause a selection bias by attracting participants who are highly interested in medical education and virtual reality training. Subgroup analysis for sports, video game and musical instrument practices were limited by the small sample size and the heterogeneity of the subgroups, we choose to not represent the results. Finally, this study is about technical skills of the surgeon only. Studying non-technical skills including cognitive and interpersonal skills is also of high importance [START_REF] Yule | Surgeons' Non-technical Skills in the Operating Room: Reliability Testing of the NOTSS Behavior Rating System[END_REF].

Conclusion

This study demonstrated the Construct Validity of the VR shoulder arthroscopy simulator within groups of experienced surgeons. With new types of metrics simply based on the simulator's raw trajectories, it was possible to significantly distinguish levels of expertise. We demonstrated that clustering analysis with Dynamic Time Warping was able to reliably discriminate between expert operators and intermediate operators.

Acknowledgements

We thank all participants of the study, and the VirtaMed team, Martina Vitz and Benjamin Grauer for the technical support.

METRICS DEFINITION Duration (DUR)

Execution time between the first time the hand moves the probe, until the probe has been released at the end of the task. It is measured in seconds (s).

Path Length (PL)

Sum of the total distances travelled by the probes (camera and grasper) during the execution of the task. It is measured in millimetres (mm).

Average Velocity (AV)

Average linear speed of the probe during the task. It is measured in millimetres per second (mm.s -1 ).

Average Acceleration (AA)

Average instantaneous acceleration of the probe during the task. It is measured in millimetres per second squared (mm.s -2 ).

Average Jerk (AJ)

Average jerk (derivative of the acceleration) during the task, also known as "smoothness". It is measured in millimetres per second cubed (mm.s -3 ).

Working volume (WV)

Volume of the convex hull for each trajectory. It is measured in millimetres cubed (mm 3 ).

Economy of motion (ECO)

Ratio between the working volume and the path length. It is measured in mm 2 .

Motion Smoothness (HMS)

Total change in acceleration of the tip of the instrument. It is measured in millimetres per second cubed (mm.s -3 ) and is normalized by the duration of the task.

Spectral Arc Length Smoothness (SPARC)

Smoothness of the linear movement speed profile using the spectral arc length method. Unitless measure.

Depth (D)

Total distance travelled by the tip of the instrument along the z-axis. It is measured in millimetres (mm).

  ): Duration (s), p<0.01 (Figure 3.1); Average velocity (mm.s -1 ), p<0.01 (Figure 3.2); Economy (mm 2 ), p<0.01 (Figure 3.3); Motion Smoothness (mm.s -3 ), p=0.05 (Figure 3.4); Spectral Arc Length Smoothness, p<0.01 (Figure 3.5) and Working Volume (mm 3 ), p<0.01 (Figure 3.6).

(Figure 4 ):

 4 Path length (mm), p=0.08 (Figure 4.1); Average Acceleration (mm.s -2 ), p=0.39 (Figure 4.2); Average Jerk (mm.s -3 ), p=0.41 (Figure 4.3) and Depth (mm), p=0.11 (Figure 4.4).

FigureFigure 1 .

 1 Figure

Figure 2 .Figure 3 .Figure 4 .Figure 5 .

 2345 Figure 2. Metric definitions

Table 1 .

 1 Baseline Characteristics of participants NOTE. Data are reported as median [interquartile range] or number (percentage).

	TOTAL	P-VALUE
	(n=52)

* Independent-samples Student's t-test † Pearson's chi-squared test METRICS Values (

Median [Interquartile range (IQR), 25% -75%] p † INTERMEDIATE EXPERT Path_Length PL 695mm

  [IQR, 467-1090] 671mm [504-846] 0.62 Duration DUR * 116.1s [82.8-154.2] 70.9s [56.4-86.3] <0.01 Average_Velocity AV * 6.9mm.s -1 [4.7-9.5] 10.0mm.s -1 [7.4-11.6] <0.01 Average_Acceleration AA 92.3mm.s -2 [61.0-141.4] 116.6mm.s -2 [79.2-144.2] 0.09 Average_Jerk AJ 3717mm.s -3 [2431-6087] 4211mm.s -3 [3017-5604] 0.31 Economy ECO * 33.8mm 2 [14.8-50.5] 19.3mm 2 [9.1-25.6] <0.01 Motion Smoothness HMS * 1.5.10 6 mm.s -3 [2.6E+5-3.5E+6] 4.6.10 6 mm.s -3 [3.1E+5-7.2E+5] 0.05 Spectral Arc Length Smoothness SPARC * -37.8 [-60.7 --26.5] -21.2 [-31.0 --16.8] <0.01

	Depth D	123.8mm [97.1-155.0]	116.6mm [92.1-138.8]	0.11
	Working Volume WV *	28236mm 3 [7193-52229]	12687mm 3 [5753-22309]	<0.01

Table 2 .

 2 Metric results for INTERMEDIATE and EXPERT categories, p-valuesSignificant p-values are highlighted in grey and with an * mark.

† Mann-Whitney U p-value

Conflicts of interest:

The author(s) do not declare any inherent conflicts of interest in the study.

F. Sirveaux is a consultant for Tornier Wright Medical outside the submitted work.

Sources of funding:

No funding was received to support of this work.

Author contribution:

Tronchot: Designed the methodology, collected, analysed, and visualized the data, wrote the manuscript. Acted as a project lead supervisor.


Lorenza Pacini

Claire Lesieur
email: claire.lesieur@ens-lyon.fr

A computational methodology to diagnose sequence-variant dynamic perturbations by comparing atomic protein structures

may come

A computational methodology to diagnose sequence-variant dynamic perturbations by comparing atomic protein structures

Introduction

Proteins are constituted by amino acids chemically interacting via their atoms in a three-dimensional structure. The biological function of a protein relies on the protein dynamics, i.e. the motions of the atoms that constitute the protein structure [START_REF] Henzler-Wildman | Dynamic personalities of proteins[END_REF]. Thus, amino acid mutations that impact the protein structure and/or dynamics can impact the protein function [START_REF] Thusberg | Pathogenic or not? and if so, then how? studying the effects of missense mutations using bioinformatics methods[END_REF]. Computational and experimental approaches are available to investigate the impact of mutations on structure and dynamics [START_REF] Achoch | Protein structural robustness to mutations: an in silico investigation[END_REF][START_REF] Mclaughlin | The spatial architecture of protein function and adaptation[END_REF][START_REF] Naganathan | Modulation of allosteric coupling by mutations: from protein dynamics and packing to altered native ensembles and function[END_REF][START_REF] Vuillon | From local to global changes in proteins: a network view[END_REF][START_REF] Haliloglu | Adaptability of protein structures to enable functional interactions and evolutionary implications[END_REF][START_REF] Degiacomi | Molecular assembly of the aerolysin pore reveals a swirling membrane-insertion mechanism[END_REF][START_REF] Ramaswamy | Deep learning protein conformational space with convolutions and latent interpolations[END_REF][START_REF] Leitner | Mapping energy transport networks in proteins[END_REF][START_REF] Zhang | Prody 2.0: Increased scale and scope after 10 years of protein dynamics modelling with python[END_REF][START_REF] Kamada | Cancerassociated p53 tetramerization domain mutants: quantitative analysis reveals a low threshold for tumor suppressor inactivation[END_REF]. Nevertheless, investigating the impact of mutations on the slow dynamics underlying function and folding is still a challenge because it involves multiple time and length scales. There are large collective motions (e.g. interface, chain, large domain) with fluctuations from millisecond to second and small collective motions (e.g. 3D/2D structures) with fluctuations from microsecond to milliseconds [START_REF] Henzler-Wildman | Dynamic personalities of proteins[END_REF][START_REF] Munoz | When fast is better: protein folding fundamentals and mechanisms from ultrafast approaches[END_REF]. Such timescales are difficult to simulate at high resolution using molecular dynamics [START_REF] Lesieur | Editorial overview: Theory and simulation[END_REF][START_REF] Perilla | Molecular dynamics simulations of large macromolecular complexes[END_REF]. New ultra-fast spectroscopies combined with computational approaches are efficient but remain complicated techniques [START_REF] Herbst | Femtosecond infrared spectroscopy of bacteriorhodopsin chromophore isomerization[END_REF][START_REF] Kolano | Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy[END_REF][START_REF] Mizutani | Ultrafast dynamics of myoglobin probed by time-resolved resonance raman spectroscopy[END_REF][START_REF] Fang | Mapping gfp structure evolution during proton transfer with femtosecond raman spectroscopy[END_REF][START_REF] Schneider | Elucidating binding mechanisms and dynamics of intrinsically disordered protein complexes using nmr spectroscopy[END_REF][START_REF] Findsen | Picosecond time-resolved raman studies of photodissociated carboxymyoglobin[END_REF]. Another difficulty is to monitor how the dynamics propagate from the local scale (amino acid level) to larger scales in the protein in order to assess how mutations impact the dynamic propagation paths [START_REF] Buchenberg | Time-resolved observation of protein allosteric communication[END_REF].

Atomic motions and dynamics can be tracked by measuring changes in atomic interactions (link weights) inferred from network models of picture(0,0) (-35,0)(1,0)30 (0,35)(0,-1)30 picture picture(0,0)(35,0)(-1,0)30 (0,35)(0,-1)30 picture "main˙revision˙2" -2021/9/29 -page 2 -#2 picture(0,0) (-35,0)(1,0)30 (0,-35)(0,1)30 picture picture(0,0)(35,0)(-1,0)30 (0,-35)(0,1)30 picture 2 Pacini and Lesieur protein structures combined with MD simulations [START_REF] Gheeraert | Exploring allosteric pathways of a v-type enzyme with dynamical perturbation networks[END_REF][START_REF] Leitner | Mapping energy transport networks in proteins[END_REF]. More generally, dynamics of networked system is susceptible to link perturbations [START_REF] Unicomb | Threshold driven contagion on weighted networks[END_REF][START_REF] Battiston | Debtrank: Too central to fail? financial networks, the fed and systemic risk[END_REF]. Experimentally, broadband dielectric spectroscopy (BDS) monitors slow protein dynamics with a dielectric signal sensitive from the local dipole environments to the global organization of the protein structure such that sequence mutations local perturbations to large-scale thermal unfolding perturbations can be studied, showing the relation between interaction changes and dynamics changes [START_REF] Bourgeat | Experimental protein molecular dynamics: Broadband dielectric spectroscopy coupled with nanoconfinement[END_REF][START_REF] Bourgeat | Experimental diagnostic of sequence-variant dynamic perturbations revealed by broadband dielectric spectroscopy[END_REF]. The atomic interactions made by an amino acid with its first shell neighbors depict how the atoms occupy the space locally and in turn what is the remaining empty space where local side chain motions can take place. We have shown that in a protein structure every amino acid has a similar moderate average number of atomic interactions (link weights) with its first-shell neighbors, regardless its position and type [START_REF] Dorantes-Gilardi | In proteins, the structural responses of a position to mutation rely on the goldilocks principle: not too many links, not too few[END_REF]. This means there is always empty space available for atomic motions around amino acids. Moreover, the local atomic average is conserved for variants maintaining structural integrity. Keeping the average constant while modifying the number and position of the sidechain atoms at the mutation site, necessarily modifies the way atomic interactions are allocated among the neighbors. It follows that variants with structural integrity carve the local empty space differently by using different link solutions that lead to the same average number of atomic interactions and structural integrity but may impact the protein dynamics.

We therefore made the hypothesis that allocating atomic interactions to structural levels (1D, 2D, 3D and 4D) and measuring the difference in allocation between variants will reveal areas with structural-level dynamic perturbations associated with the mutations. This hypothesis is consistent with the scale of dynamics associated with structural levels and with experimental evidences showing the impact of mutations on individual structural levels or a combination of them [START_REF] Munoz | When fast is better: protein folding fundamentals and mechanisms from ultrafast approaches[END_REF][START_REF] Degiacomi | Molecular assembly of the aerolysin pore reveals a swirling membrane-insertion mechanism[END_REF][START_REF] Gersting | Loss of function in phenylketonuria is caused by impaired molecular motions and conformational instability[END_REF]. To verify the hypothesis, protein variants that maintain structural integrity but have different folding dynamics are analyzed. The structural integrity guarantees that differences in the atomic-contacts allocation are limited to dynamics perturbation and do not involve structural perturbations. If this condition was not fulfilled, there would be ambiguity on the consequences of the allocation changes. Such dynamic-centered perturbations are common in diseases related to misfolding and aggregation [START_REF] Bemporad | Protein misfolded oligomers: Experimental approaches, mechanism of formation, and structure-toxicity relationships[END_REF] and are also certainly at play in drug failure since local protein dynamics impact the thermodynamics and kinetics of drugs binding [START_REF] Amaral | Protein conformational flexibility modulates kinetics and thermodynamics of drug binding[END_REF].

We have selected case studies that are involved in human diseases and exhibit slow folding dynamics perturbations. Firstly, the B-pentamers of two AB 5 toxins, cholera toxin (CtxB 5 ) and heat labile (LTB 5 ) distinguished by 18 mutated amino-acids over 103 resulting in different folding/unfolding mechanisms [START_REF] Dorantes-Gilardi | In proteins, the structural responses of a position to mutation rely on the goldilocks principle: not too many links, not too few[END_REF][START_REF] Zrimi | Cholera toxin b subunits assemble into pentamers -proposition of a fly-casting mechanism[END_REF]. Secondly, single amino acid variants of Transthyretin (TTR), that have different propensity to misfolding and amyloidosis [START_REF] Saelices | Uncovering the mechanism of aggregation of human transthyretin[END_REF][START_REF] Saraiva | Transthyretin mutations in hyperthyroxinemia and amyloid diseases[END_REF]. Lastly, the allocation measure is applied to the main proteases (Mpro's) of the coronaviruses SARS-CoV and SARS-CoV-2 to diagnose dynamic perturbations that might be responsible for higher virulence or drug resistance.

Materials and Methods

Protein structures

The Protein Data Bank (PDB, http://www.rcsb.org/pdb/) codes of the protein structures used in this study are the following. CtxB 5 : 1eei, LTB 5 : 1ltr, WT TTR: 1f41, T119Y TTR: 4tne, L55P TTR: 3djz, V30M TTR: 3kgs, T119M TTR: 1bze, SARS-CoV Mpro: 2h2z, SARS-CoV-2 Mpro: 6y2e. Chains D to H have been studied for CtxB 5 and LTB 5 , from position 1 to position 102. Chains A and B have been studied for the TTR variants, from position 10 to position 124. Chain A has been studied for the Mpro of SARS-CoV and SARS-CoV-2, from position 1 to position 305.

Sequence and structure alignment

TopMatch [START_REF] Sippl | Detection of spatial correlations in protein structures and molecular complexes[END_REF] has been employed to calculate the percentage of sequence identity and the Root Mean Square Deviation (RMSD) between two protein variants.

Protein structure visualization

Figures of protein structures have been produced with VMD [START_REF] Humphrey | VMD: Visual molecular dynamics[END_REF]. The secondary structure information for the visualization of protein structures in the cartoon representation has been read from the secondary structure records in the corresponding PDB files though Andrew Dalke's VMD script that can be found at www.ks.uiuc.edu/ Research/vmd/script_library/scripts/vmd_use_pdb_ss.

Amino acid network (AAN)

The Amino Acid Networks (AANs) of protein structures are created as described in [START_REF] Dorantes-Gilardi | In proteins, the structural responses of a position to mutation rely on the goldilocks principle: not too many links, not too few[END_REF]. In a nutshell, the AAN of a protein structure is an undirected network where nodes represent amino acids and weighted links exist between amino acids that are at distance lower than 5 Å, where the link weight is given by the number of atomic interactions between the two amino acids (number of atomic couples at distance lower than 5 Å).

Links classification

A (i, j) link between amino acid i and amino acid j in the AAN of a protein is classified as: 1D if i and j belong to the same chain and are first neighbors in the protein amino-acids sequence; 2D if i and j belong to the same secondary structure element (α-helix, β-sheet or coil as retrieved from the PDB file of the protein structure) and i and j are distant by less than five positions in the protein amino-acids sequence; 3D if i and j belong to different secondary structure elements or are distant by at least five positions in the amino-acids sequence; 4D if i and j belong to two different chains.

Local structural-level allocation of the atomic interactions

For each amino acid i and for each link category x, with x = 1D, 2D, 3D or 4D, we define the fraction of atomic interactions allocated by i to the category x as fw x,i = w x,i /w i , where w x,i is the number of atomic interactions of category x made by i (i.e. the sum of the weights w(i, j) of links of category x in the AAN, for all neighbors j of node i) and w i is the total number of atomic interactions made by i (i.e., the sum of the weights w(i, j) in the AAN, for all neighbors j of node i). Using fractions of atomic interactions allocated to a structural level instead of the number of atomic interactions allocated to a structural level allows us to measure the contribution of each level compared to one another.

Protein variants comparison

The local structural-level allocation of atomic interactions in two protein variants var1 and var2 is compared through the differences ∆fw

x,i = f var2 w x,i -f var1 w x,i = w var2 x,i /w var2 i -w var1
x,i /w var1 i with x = 1D, 2D, 3D, 4D for each amino acid i of the protein structure. Because all amino acids picture(0,0)(-35,0)(1,0)30 (0,35)(0,-1)30 picture picture(0,0)(35,0)(-1,0)30 (0,35)(0,-1)30 picture "main˙revision˙2" -2021/9/29 -page 3 -#3 picture(0,0)(-35,0)(1,0)30 (0,-35)(0,1)30 picture picture(0,0)(35,0)(-1,0)30 (0,-35)(0,1)30 picture

Diagnosis of sequence-variant dynamic perturbations 3

in globular proteins are expected to allocate some interactions to the 1D, 2D and 3D structural levels, fw x,i is expected to be significantly smaller than 1 for each x = 1D, 2D, 3D, 4D. Thus, ∆fw x,i , which is also for each structural level is likewise expected to be significantly lower than 1 and ∆fw x,i = ±0.4 is considered a large change in allocation of atomic contacts to the x structural level.

Results and Discussion

Our goal is to generate a tool capable of identifying multiscale slow dynamics perturbations resulting from mutations using only the X-ray structures of two protein variants. To do so, we compared the allocation of atomic interactions to structural levels, in protein variants that have similar structure (RMSD < 1 Å) and different experimental dynamics. Precisely, the fraction of atomic interactions involved in the 1D, 2D, 3D and 4D structural levels fw 1D,i , fw 2D,i , fw 3D,i , fw 4D,i , is computed for each amino acid i of a protein variant and compared between the two variants through the measure of ∆fw 1D,i , ∆fw 2D,i , ∆fw 3D,i , ∆fw 4D,i . The atomic interactions and the atomic interaction allocations are inferred from modeling the PDB X-ray structure by a spatial network (Materials and Methods). We use three cases of study to show the match between non-zero ∆fw x,i regions and experimental dynamic differences.

The first case study is the comparison of the B-pentamers of two AB 5 toxins, Cholera toxin (CtxB 5 ) and human heat labile (LTB 5 ), which have 82% of sequence identity, structural and functional similarity but different folding/unfolding mechanisms: the formation of assembly intermediates is the rate-limiting step for CtxB 5 (fly-casting folding mechanism) while the folding of the monomers is the rate-limiting step for LTB 5 (inducedfit folding mechanism) [START_REF] Ruddock | Assembly of the b subunit pentamer of escherichia coli heat-labile enterotoxin: Kinetics and molecular basis of rate-limiting steps in vitro[END_REF][START_REF] Lesieur | A kinetic model of intermediate formation during assembly of cholera toxin b-subunit pentamers[END_REF]. Since the differences in folding dynamics of CtxB 5 and LTB 5 are known, they represent a good case-study to test the hypothesis that differences in protein dynamics are encoded by different local structural-level allocation in the protein structure.

The second case study is Transthyretin (TTR) with non-pathological and pathological variants that are involved in neurological and cardiac genetic diseases due to amyloid fiber formation [START_REF] Cendron | Amyloidogenic potential of transthyretin variants: INSIGHTS FROM STRUCTURAL AND COMPUTATIONAL ANALYSES[END_REF][START_REF] Ruberg | Transthyretin (TTR) cardiac amyloidosis[END_REF][START_REF] Saraiva | Transthyretin mutations in hyperthyroxinemia and amyloid diseases[END_REF]. The exact mechanism of amyloid formation of TTR is unknown and probably depends on the particular TTR variant and on the experimental conditions [START_REF] Dasari | Structural characterization of cardiac ex vivo transthyretin amyloid: Insight into the transthyretin misfolding pathway in vivo[END_REF][START_REF] Dasari | Two distinct aggregation pathways in transthyretin misfolding and amyloid formation[END_REF][START_REF] Hamilton | Transthyretin: a review from a structural perspective[END_REF][START_REF] Schmidt | Cryo-EM structure of a transthyretinderived amyloid fibril from a patient with hereditary ATTR amyloidosis[END_REF][START_REF] Yee | A molecular mechanism for transthyretin amyloidogenesis[END_REF]. The study of TTR variants challenges the sensitivity of the allocation measure to detect the impact of single amino acid mutations.

The third case study is the comparison of the Mpro of two coronaviruses: SARS-CoV and SARS-CoV-2. SARS-CoV is responsible of the Severe Acute Respiratory Syndrome (SARS) pandemic that caused 774 deaths in 2003 (www.who.int/csr/sars/country/ table2004_04_21/en/, last visited on 8th June 2020) and SARS-CoV-2 is responsible of the actual COVID-19 global pandemic that has caused more than three million deaths since December 2019 (covid19. who.int/, last visited on 20th May 2021). The inhibition of Mpro is one of the targets for the anti-viral treatment of COVID-19 thanks to its functional importance in virus replication [START_REF] Dai | Structure-based design of antiviral drug candiyears targeting the SARS-CoV-2 main protease[END_REF][START_REF] Yang | The crystal structures of severe acute respiratory syndrome virus main protease and its complex with an inhibitor[END_REF]. Moreover, the structural similarity between the SARS-CoV Mpro and the SARS-CoV-2 Mpro has motivated the research for common drugs that could inhibit the Mpro activity of a broad spectrum of coronaviruses [START_REF] Burley | How to help the free market fight coronavirus[END_REF][START_REF] Jin | Structure of mpro from SARS-CoV-2 and discovery of its inhibitors[END_REF][START_REF] Yang | Design of wide-spectrum inhibitors targeting coronavirus main proteases[END_REF]. We apply our methodology, validated through the first two case studies, to determine if a different dynamical behavior can be expected for the two Mpro, questioning the common drug strategy.

B-pentamers of AB5 toxins

CtxB 5 and LTB 5 have 18 mutated residues over 103 (82% of sequence identity and very similar structure (RMSD = 0.59 Å, Figure 1A), but different folding/unfolding mechanisms. We have compared the local structural-level allocation of the atomic interactions in LTB 5 with respect to CtxB 5 , for all amino acid positions, through the measure of ∆fw 1D,i = f LT B 5 w 1D,i -f CtxB 5 w 1D,i (and similarly for ∆fw 2D,i , ∆fw 3D,i and ∆fw 4D,i ). The pentameric unit has been employed for the calculations, whereas the results (Figure 1) are plotted only for chain E, for simplicity. The plots for all five chains are reported in Figure S1. The results that were not consistent in at least four chains over five have been considered as potential X-ray artifacts and not taken into consideration.

Differences in fw 2D ,fw 3D and fw 4D are found at the N-terminal, the C-terminal and at the central α-helix of the pentamers (Figure 1C, boxes), all involved in the interfaces of the pentamers (I 1 , I 2 and I 3 , respectively, Figure 1A and1B). LTB 5 allocates more atomic interactions to the 4D structural level at the I 2 and I 3 interfaces compared to CtxB 5 (∆fw 4D > 0), while it allocates less atomic interactions to the 2D structural level and more to the 3D structural level at the I 1 and I 3 interfaces compared to CtxB 5 (∆fw 2D < 0 and ∆fw 3D > 0). The fact that more atomic interactions are allocated to the maintenance of the interface compared to the secondary-structure fold means that the secondary structure will be more susceptible to unfolding compared to the tertiary and quaternary structure: when the interfaces are disrupted, the monomers are already unfolded. This is consistent with the experimental evidence that assembly intermediates are not detected by BDS during the thermal unfolding of LTB 5 , contrary to CtxB 5 [START_REF] Bourgeat | Experimental protein molecular dynamics: Broadband dielectric spectroscopy coupled with nanoconfinement[END_REF][START_REF] Bourgeat | Experimental diagnostic of sequence-variant dynamic perturbations revealed by broadband dielectric spectroscopy[END_REF]. The BDS results also corroborate the allocation measures because it detects dynamics perturbations in the N-terminal, C-terminal and central α-helix [START_REF] Bourgeat | Experimental diagnostic of sequence-variant dynamic perturbations revealed by broadband dielectric spectroscopy[END_REF]. More precisely, the allocation measure detects perturbations in the 2D, 3D and 4D structural levels known to cover fast microsecond to millisecond collective motions of small length scale (2D-3D structures) to slow millisecond to second collective motions of larger length scale (large domain, interfaces/chain). The same is observed by BDS, fast and slow collective motions for the N-and C-terminal domains and slow collective motions for the central α-helix which presents an allocation perturbation over a larger sequence domain (residue 58 to 81, Figure 1C) than the Cand N-terminal domains.

In addition, the central α-helix, where most of the changes in the allocation measure are observed, has only one mutation (ALA75THR), showing that the allocation measure tracks perturbations from the local scale, i.e. the site of mutations, to larger scales. The lack of mutations in the central α-helix makes it difficult to predict the helix dynamic perturbation from sequence and/or structural analysis. It emphasizes the specificity of the allocation measure in detecting multiscale dynamics perturbations.

The comparison of LTB 5 and CtxB 5 validates our hypothesis that differences in protein dynamics can be retrieved by differences in the local structural-level allocation of atomic interactions into the 1D, 2D, 3D and 4D classes. The next case studies challenge the sensitivity of the measure to detect the impact of milder sequence changes on protein folding dynamics.

Transthyretin variants

The allocation measure is applied on T119Y and T119M non-pathogenic TTR variants [START_REF] Saelices | Uncovering the mechanism of aggregation of human transthyretin[END_REF][START_REF] Saraiva | Transthyretin mutations in hyperthyroxinemia and amyloid diseases[END_REF] and on L55P and V30M pathogenic TTR variants that have higher propensity to misfolding and amyloidosis than WT TTR [START_REF] Saraiva | Transthyretin mutations in hyperthyroxinemia and amyloid diseases[END_REF]. Again, the four TTR variants verify the condition of structural integrity with RMSD values: 0.58 Å, 0.58 Å, 0.54 Åand 0.34 Åfor L55P TTR, V30M TTR, T119Y TTR and T119M TTR, respectively, Figure 2A). The dimeric unit has picture(0,0)(-35,0)(1,0)30 (0,35)(0,-1)30 picture picture(0,0)(35,0)(-1,0)30 (0,35)(0,-1)30 picture "main˙revision˙2" -2021/9/29 -page 4 -#4 picture(0,0)(-35,0)(1,0)30 (0,-35)(0,1)30 picture picture(0,0)(35,0)(-1,0)30 (0,-35)(0,1)30 picture (and similarly for ∆fw 2D,i , ∆fw 3D,i and ∆fw 4D,i ). Positive (negative) values of ∆fw x,i mean that residue i allocates a higher (lower) fraction of contacts to the x structural level in LTB 5 compared to CtxB 5 .

been employed for the calculations, whereas the results (Figure 2C) are plotted only for chain A, for simplicity. The plots for both chains are reported in Figure S2 andS3. The results that were not consistent for both chains have been considered as potential X-ray artifacts and not taken into further consideration. The T119Y TTR variant shows very little changes in the allocation measure compared to the WT TTR, apart from the 98-103 segment (Figure 2B and2C). Changes in this segment, which is part of a long coil, are observed for the four variants, with changes in chain A different from the changes in chain B (Figure S2 andS3), suggesting a perturbation related to an X-ray bias rather than a dynamic fault. The absence of changes in the allocation measure of T119Y TTR is consistent with the fact that T119Y does not have dynamic perturbations underlying a propensity to amyloid formation [START_REF] Saelices | Uncovering the mechanism of aggregation of human transthyretin[END_REF].

In contrast, the pathogenic L55P TTR variant shows large differences compared to the WT due to the reallocation of the atomic interactions of many residues within the segment 50-58 (Figure 2C). The residues in the mutant devote more resources to 2D interactions and less to 3D interactions. This may increase the 3D-mobility of the segment 50-58 by having less atomic space occupancy in the mutant and hence more empty space for atomic motions, and in turn make the segment 50-58 available for intermolecular interactions with another monomer, pre-requisite for amyloid fiber formation. Intra-molecular perturbations, suggesting a folding perturbation, agree with the L55P fiber diameter similar to the TTR tetramer size [START_REF] Lashuel | The most pathogenic transthyretin variant, l55p, forms amyloid fibrils under acidic conditions and protofilaments under physiological conditions[END_REF], meaning that dissociation can happen but is not necessary for fiber formation and is consistent with L55P TTR amyloid fiber models [START_REF] Pacini | Induced perturbation network and tiling for modeling the l55p transthyretin amyloid fiber[END_REF][START_REF] Pacini | The crystal structure of amyloidogenic leu55 pro transthyretin variant reveals a possible pathway for transthyretin polymerization into amyloid fibrils[END_REF].

The changes in allocation of the other pathogenic variant, V30M TTR, are distinct from the ones of L55P TTR: lower absolute value and 2D and 3D perturbations in a segment (residues 86-92) at the dimer interface (Figure 2). This is consistent with the fact that ex-vivo studies have shown that V30M TTR unfolds prior to fiber formation, contrary to WT TTR [START_REF] Dasari | Structural characterization of cardiac ex vivo transthyretin amyloid: Insight into the transthyretin misfolding pathway in vivo[END_REF][START_REF] Schmidt | Cryo-EM structure of a transthyretinderived amyloid fibril from a patient with hereditary ATTR amyloidosis[END_REF]. Moreover, the V30M interface perturbations is also compatible with the mutation leading to dissociation prior fiber formation, in agreement with the diameter of the V30M fiber higher than the size of the TTR tetramer [START_REF] Serpell | Examination of the structure of the transthyretin amyloid fibril by image reconstruction from electron micrographs[END_REF].

We find little perturbation for the non-pathogenic T119Y variant and substantial perturbation for the pathogenic L55P and V30M variants pointing out a quantitative relation between allocation perturbations and pathology as previously shown by the localized frustration measure [START_REF] Ferreiro | Localizing frustration in native proteins and protein assemblies[END_REF][START_REF] Kumar | Localized structural frustration for evaluating the impact of sequence variants[END_REF]. Moreover, the L55P and V30M variants exhibit different allocation perturbations at different areas, adding a qualitative relation between the perturbations and pathology. The consistency between the different allocation measures for T119Y, L55P and V30M and their different experimental dynamics supports the reliability of the allocation measure in detecting dynamics perturbations even for single-mutation variants.

T119M exhibits the most perturbed allocation measure with three segments having changes in the allocations of 2D and 3D interactions. This result reveals that perturbations do not necessarily imply pathology, meaning that a prognostic of pathology cannot rely only on a diagnostic of perturbation quantification. Here prognostic is defined as 'knowledge beforehand of how a situation is likely to turn out' (www.merriam-webster.com/dictionary/ prognosis#note-1, last visited on 17th May 2021). For a prognostic, the perturbations diagnosed with the allocation measure must be somehow picture(0,0)(-35,0)(1,0)30 (0,35)(0,-1)30 picture picture(0,0)(35,0)(-1,0)30 (0,35)(0,-1)30 picture "main˙revision˙2" -2021/9/29 -page 5 -#5 picture(0,0)(-35,0)(1,0)30 (0,-35)(0,1)30 picture picture(0,0)(35,0)(-1,0)30 (0,-35)(0,1)30 picture classified as functionally (folding) tolerated or functionally (folding) harmful. Three perturbed segments introduce the possibility that changes on one segment compensate changes in others, resulting in functionally (folding) tolerated perturbations. The fact that L55P and T119M present perturbations in the 50-58 segment, but only L55P promotes fiber formation, supports the presence of a compensatory mechanism in T119M [START_REF] Dorantes-Gilardi | In proteins, the structural responses of a position to mutation rely on the goldilocks principle: not too many links, not too few[END_REF]. We calculated that T119M has a higher number of atomic interactions between these two segments than the WT TTR, hence the mobility of the segment 50-58 in L55P might be prevented in T119M by tightening it to the 21-26 segment. Yet, the higher atomic interactions could also provide more backup links contributing to the higher stability of T119M reported experimentally, and have thus no relation with the 50-58 segment allocation perturbations in L55P [START_REF] Mccutchen | Comparison of lethal and nonlethal transthyretin variants and their relationship to amyloid disease[END_REF][START_REF] Quintas | Tetramer dissociation and monomer partial unfolding precedes protofibril formation in amyloidogenic transthyretin variants[END_REF][START_REF] Dorantes-Gilardi | In proteins, the structural responses of a position to mutation rely on the goldilocks principle: not too many links, not too few[END_REF]. T119M is also known to prevent amyloidogenesis when present together with the V30M mutation in heterozygous individuals [START_REF] Alves | Compound heterozygoty in patients with ttr-related amyloidosis[END_REF] and to protect the V30M TTR tetramer from dissociation in vitro when some V30M chains are replaced by T119M chains [START_REF] Hammarstrom | Trans-suppression of misfolding in an amyloid disease[END_REF]. This is another type of compensatory mechanism where the dynamics perturbation caused by the V30M mutation on one chain is corrected by a dynamics perturbation caused by the T119M mutation on another chain. Again having three perturbed segments in T119M is consistent with a T119M-V30M compensatory mechanism but the allocation measure is not enough to determine the exact mechanism. In fact, the TTR cases show the complexity of establishing a prognostic that will link a perturbation diagnostic to pathology.

The analysis of the B-pentamers of AB 5 toxins and of TTR variants has proven that the comparison of local structural-level allocation of atomic interactions is a good measure of differences in protein slow folding dynamics and of the propagation of perturbations from local to larger scales. To explore the issue of common drug design for protein variants sharing structural integrity, we compared the allocation measure of the main proteases (Mpro's) of two coronaviruses, SARS-CoV and SARS-CoV-2.

Main proteases of SARS coronaviruses

The Mpro's of SARS-CoV and SARS-CoV-2 have very similar sequence (12 mutations over 305 residues, corresponding to 96% identity, Figure 3B) and similar crystalline structure (RMSD = 0.71 Å for one Mpro chain, Figure 3B). The percentage of sequence identity between the two Mpro's is intermediate with respect to the previous case studies and the RMSD is as low (¡ 1 Å) so we assume that differences in the allocation measure will measure differences in protein dynamics. The main differences in atomic interactions allocation are located at the C-terminal domain (Figure 3A), where a higher fraction of atomic interactions is devolved to 2D and a lower fraction to 3D (∆fw 2D > 0, ∆fw 3D < 0 and longer αhelix, Figure 3C) in SARS-CoV-2 Mpro compared to SARS-CoV Mpro. Moreover, the C-terminal coil points towards the active domain in SARS-CoV-2 Mpro and towards the rest of the extra domain in SARS-CoV Mpro (Figure 3C). This is reflected by ∆fw 3D < 0 at residues residues 213 to 215 and GLN256 of the extra domain and ∆fw 3D > 0 at residue TYR154 of the active domain (Figure 3A and 3C). Much effort is devolved to the design of drugs to inhibit the Mpro's of a broad spectrum of coronaviruses [START_REF] Burley | How to help the free market fight coronavirus[END_REF][START_REF] Jin | Structure of mpro from SARS-CoV-2 and discovery of its inhibitors[END_REF][START_REF] Yang | Design of wide-spectrum inhibitors targeting coronavirus main proteases[END_REF][START_REF] Carli | Candidate binding sites for allosteric inhibition of the sars-cov-2 main protease from the analysis of large-scale molecular dynamics simulations[END_REF], but our results suggest differences in the enzyme dynamics, particularly of the C-terminal dynamics, a region whose role in the enzymatic activity has been shown experimentally for SARS-CoV (Shi and Song, 2006). This result supported by the other cases of study advises against a broad-spectrum drug strategy due to the large-scale dynamics differences that are likely to prevent same-drug recognition. It therefore counsels on more strain-customized drugs to inhibit the function of the Mpro of coronaviruses.

Conclusion

Our hypothesis that the difference in the allocation of atomic interactions to structural levels (1D, 2D, 3D and 4D) between variants will reveal areas with structural-level dynamic perturbations associated with the mutations is verified for the four cases of study. The methodology is fast and computationally-light and diagnoses robust and functionallysensitive mutations in terms of their impact on the multi-scale protein dynamics. As an example, the calculation for the 305 amino acids of the two SARS-CoV Mpro structures takes less than 2 seconds on a laptop with an Intel Core i7 processor. From the results, we can deduce that the allocation of the atomic interactions into structural levels contains information on the protein dynamics. Further application to a large database of protein variants will establish the ranges of structural allocation changes that associate with different dynamic perturbations. This could open-up novel directions towards the decoding of a protein dynamics from the static information contained in its crystalline structure. The allocation measure provides information which will allow digging into the molecular mechanisms underlying pathogenic mutations and compensatory mechanisms involving error-correcting mutations, and thus assist in assigning drugs to variants for personalized therapy.

Fig. 1 .

 1 Fig. 1. B subunits of AB 5 toxins. A: Superposition of the X-ray structures of the CtxB 5 pentamer (PDB: 1eei) and the LTB 5 pentamer (PDB: 1ltr). The 18 mutated positions are highlighted in one chain (cyan amino acids for CtxB 5 and green amino acids for LTB 5 ). B: Details of the X-ray structures of CtxB 5 and the LTB 5 . C: Changes in the local structural-level allocation of atomic interactions in LTB 5 compared to CtxB 5 (chain E), where ∆fw 1D,i = f LT B 5 w 1D,i -f CtxB 5 w 1D,i

Fig. 2 .

 2 Fig. 2. Transthyretin (TTR) variants. A: Superposition of the X-ray structures of the TTR dimer variants: WT (PDB: 1f41), T119Y (PDB: 4tne), L55P (PDB: 3djz), V30M (PDB: 3kgs) and T119M (PDB: 1bze). B: X-ray structure of the WT TTR dimer. The segments involved in changes in local structural-level allocation in the variants with respect to the WT are highlighted. C: Changes in the local structural-level allocation of atomic interactions in the TTR variants compared to WT TTR (chain A), where ∆fw 1D,i = f variant w 1D,i -f W T w 1D,i(and similarly for ∆fw 2D,i , ∆fw 3D,i and ∆fw 4D,i ). Positive (negative) values of ∆fw x,i mean that residue i allocates a higher (lower) fraction of contacts to the x structural level in the variant compared to the WT. Next to each plot, the segment involved in changes in local structural-level allocation of atomic interactions in the variants compared to the WT TTR are highlighted in the variant's X-ray structure, with the same color code as the boxes in the plots and panel B.The mutated position is highlighted in green.

Fig. 3 .

 3 Fig. 3. Main proteases (Mpro) of coronaviruses SARS-CoV and SARS-CoV-2. A: changes of local structural-level allocation of atomic interactions along the Mpro monomer, where ∆fw 1D,i = f SARS-CoV -2 w 1D,i

Acknowledgements

We thank the CNRS MITI for funding the work.

Funding

This work has been supported by the CNRS MITI (Grant OASIC project Go-Pro2018).

The Python code used for the production of the results is available at github.