Alireza Rezaei
email: alireza.rezaei@u-psud.fr

Nicola Pellicanò
email: nicola.pellicano@u-psud.fr

Emanuel Aldea
email: emanuel.aldea@u-psud.fr

GPU-accelerated Height Map Estimation with Local Geometry Priors in Large Scenes

Detection and tracking of pedestrians in vast crowded areas is a complex problem addressed actively by the computer vision community. Proposed algorithms should ideally tackle issues of accuracy and speed at the same time. Lengthy computation times for high-quality optimizationbased algorithms relying on multiple sensors make them impractical to use on long and detailed sequences. Hence, an efficient acceleration scheme, which preserves the overall accuracy, is vital to be considered. In the current work, we iterate various steps taken to accelerate a multi-camera pedestrian detection algorithm formulated as an optimization of a height map with local scene geometry constraints. The work is performed using the NVIDIA CUDA framework which allows us to efficiently utilize GPU processors and optimize the various memory accesses. The final results show more than 1000x speedup on real data frames. With respect to preserving the output accuracy, we achieve an accelerated output which is more than 99.9% in agreement with the original results.

Introduction

Computer vision has made significant progress in the last decade toward more accurate, more robust and faster processing of the video data deluge that we create and use. Non-crowded scenes have represented for a long time the main area of interest for the computer vision community, and pedestrian detection algorithms evolved significantly in the last decade, addressing complex applications such as identification of people, grouping analysis, estimation of body parts, gesture based and trajectory based action analysis etc. However, as it has been already highlighted many times, all these methods are not appropriate when highdensity crowd analysis is performed, and new methods must be designed in order to cope with extreme clutter. While clutter is the main difficulty, practical considerations also raise difficult questions, i.e. the use and topology of a net-work of sensors, the transfer and the processing of the data, or the data fusion strategies. Technical difficulties widen the gap between proof-of-concept experiments aimed at highdensity crowded scenes and functional solutions.

Our work addresses a difficulty which is pervasive in pedestrian detection tasks relying on multiple camera networks with overlapping fields of view, namely the joint detection of pedestrians which project potentially in different sensors. During the last decade, a number of works [START_REF] Khan | Tracking multiple occluding people by localizing on multiple scene planes[END_REF][START_REF] Alahi | Sparsity driven people localization with a heterogeneous network of cameras[END_REF] addressed this matter under various simplificatory assumptions incompatible with strong clutter, which impede their applicability in high density areas. The underlying rationale is that at least an approximate detection at object level must be performed before relating uncertain detections from different views. On the other hand, a low-level fusion strategy, such as the one proposed in [START_REF] Eshel | Tracking in a dense crowd using multiple cameras[END_REF], is applied before the detection step and is expected to cope better with ambiguous scenes specific to crowded environments. Recently, the work of [START_REF] Pellicanò | Geometry-Based Multiple Camera Head Detection in Dense Crowds[END_REF] redefined the low-level fusion as a global height map estimation taking into account the local geometry of the scene, represented by the accurate ground plane location and the vertical vanishing line direction (Figure 1). The optimization was solved using a Loopy Belief Propagation algorithm, but the price for a high-quality solution compared to a heuristic based one is the computational cost (around 18 minutes per triplet of synchronized frames).

The objective of the current work is to reformulate the algorithm proposed in [START_REF] Pellicanò | Geometry-Based Multiple Camera Head Detection in Dense Crowds[END_REF] and to accelerate it by exploiting a massively parallel architecture, in order to render it compatible with practical applications. Section 2 offers a synthetic overview of the initial algorithm, then in Section 3 we highlight at different levels the various optimizations which were performed in order to adapt the algorithm to the considered architecture. Section 4 presents the results of the acceleration, and then we conclude in Section 5.

Algorithm overview

In [START_REF] Pellicanò | Geometry-Based Multiple Camera Head Detection in Dense Crowds[END_REF], the authors propose an unsupervised pedestrian detection method based on information derived from mul-1 tiple camera sources. They formulate the detector as a Markov Random Field (MRF) based stereo matcher, which has to minimize a global energy by assigning some labels to each pixel p of the reference image I:

E(l) = p∈I D p (l p ) + λ (p,q)∈N V p,q (l p , l q ) (1)
where: (i) given the label set L, l is a labeling assigning a value l p ∈ L to each p ∈ I; (ii) N is the set of edges of the image graph (4-connectivity is assumed); (iv) D p is the data cost function; (v) V p,q is the discontinuity cost function; (vi) λ is a regularization parameter. The detection task is performed by the use of a special unknown label implying that no pedestrian is present at that pixel.

The estimated scene geometry leads to the ability to project every pixel of one image in another one knowing its height above the ground plane (in meters). Thus, it is possible to bound the desired range of detection for a pedestrian to an interval of heights (from 1.4m to 2m in their experiments). As a consequence, they define the labels of the energy as height values (as opposed to depth values of classic stereo matching approaches). In terms of computational efficiency, such choice simplifies the problem of labels assignment, since each pixel, at a fixed range of heights, can live in a range of depths which is not fixed and which does not scale linearly.

The data cost is expressed as the combination of DAISY [START_REF] Tola | Daisy: An efficient dense descriptor applied to wide-baseline stereo[END_REF] dissimilarities between the reference pixel and the possible correspondences (lying along an epipolar segment) in each of the other views. The data cost is an input and can be precomputed, thus it has no impact when studying the acceleration of the optimization (except the initial access time).

The discontinuity cost derives from the observation that the direction of expected maximum height variation r p points towards the vertical vanishing point. Moreover, with the estimated geometry, the magnitude of such variation |∇ p | can be estimated. While such value depends also on the current estimated height of l p ,the authors suggest that pre-computing it for an average height leads to a negligible error, and thus this will be exploited as a fixed parameter in the acceleration. Being d(p, q ⊥ ) the projection of pixel q avr p , the discontinuity cost is estimated as [START_REF] Pellicanò | Geometry-Based Multiple Camera Head Detection in Dense Crowds[END_REF]:

D p,q (l p , l q ) = l p -l q -s p |∇ p |d(p, q ⊥ ) (2)
In computational terms, the pre-computation of such cost for any possible pairs of labels is not scalable with the domain cardinality, and thus should be avoided. However, the term s p |∇ p |d(p, q ⊥ ) could in theory be precomputed, since each node p relates to 4 possible values (one for each neighbor), and only 2 of them are distinct in absolute value (while the other two being their negation).

With the data cost, height variation magnitude, and a part of the discontinuity cost being possibly precomputed, the trade-off between computational speed and memory footprint becomes critical for the speedup of the algorithm.

To minimize the energy function (Equation 1), the Loopy Belief Propagation (LBP) algorithm has been used. LBP is an algorithm for computing approximate marginal statistics over graphs with cycle. LBP works over a graph by transmitting messages between nodes and computing beliefs at each node [START_REF] Tatikonda | Loopy belief propagation and Gibbs measures[END_REF]. Thus, each iteration of the algorithm consists in computing new messages between each neighboring nodes. An outgoing message from m to n is computed by considering all the incoming messages to m except the one coming from n itself. After convergence, the belief for a given node is computed by considering all the incoming messages to that node [START_REF] Ihler | Loopy belief propagation: Convergence and effects of message errors[END_REF].

GPU Implementation

In recent years, general purpose processing on GPU (Graphics Processing Unit) provided significant support for many scientific fields in which efficiency and speed is a vital factor. Specifically, NVIDIA CUDA framework has enabled us to rely on a GPU parallel environment with greater ease. [START_REF] Gray | Best practice guide-GPGPU[END_REF] describes the structure of NVIDIA GPUs in two levels: each GPU chip consists of streaming multiprocessors (SM), which have their own cores (Figure 2). CUDA uses the term 'grid of thread blocks', where multiple blocks map onto multiple SMs and multiple threads map onto the cores. Each thread block contains several threads. Each thread has its own local memory while it can also communicate with other threads inside the block via shared memory. Every thread has also access to a bigger and slower global memory. GPU groups threads together in execution warps. Threads inside a warp execute concurrently. The size of warps depends on the device being used (in our case it equals to 32). Generally, it is not necessary to follow the exact hardware specifications when utilizing CUDA; we can use more threads than cores and leave the scheduling to the hardware. Having these details in mind, the next section will describe our approach to the GPU implementation.

Basic optimizations

To optimize the pedestrian detection algorithm, we started with basic changes from a mostly serial CPU implementation to a highly parallel GPU code. The next three sections iterate over the main decisions taken in order to make the translation as efficient as possible.

Thread mapping

The first step to translate a serial algorithm to a parallel paradigm is to assign the responsibility of each processing unit (in our case GPU threads). In our problem, this means to specify how the grid of threads is related to our computation grids. As our data come from 2D images, if we decide on different dimensions for the thread blocks we need a proper translation between the two. In the remainder of this section N will denote the number of pixels, M the number of messages for each pixel and V number of labels. First we choose the responsibility of each thread block. One pixel per thread block: In this configuration, we will have N thread blocks either in 1D fashion or similar to the image in 2D with a (Width,Height) matrix of thread blocks. Each block tends to messages originating from only one pixel. So each thread depending on its block index tends to a different pixel. Multiple pixels per thread block: This way, we assign more than one pixel for each thread block in hope of doing more work in parallel. Each thread, depending on its block index and also its thread idx (for example third dimension of the thread index) knows which pixel to access.

In practice, putting computation of more than one pixel in a block will not give us any improvement. In our tested GPU we had the limit of 32 active blocks and 64 active warps per SM. This means with just 2 active warps per thread block we can theoretically achieve 100% occupancy for the GPU. In our case, for computing four messages for one pixel we need four active warps (assuming warp size is 32 and V < 32). Therefore, there is no actual need for more active warps per thread block.

We also have to arrange threads inside a block.

Each block containing M*V threads: Each thread is in charge of calculating the message for one neighbor regarding one particular label. Whether the formation is in one dimension or two will not affect the performance but structuring the block as M vectors of length V helps the programming process.

Each block containing M*32 threads: This means instead of using the number of labels as the width of the block we use the nearest power of two greater than V. This way we make sure each computation warp only deals with messages related two one neighbor. As mentioned in [START_REF] Han | Reducing branch divergence in GPU programs[END_REF], GPU architecture follows a Single Instruction Multiple Data (SIMD) execution model, which is not suited for kernels with divergent execution flow. Thus, as a general rule we need to make sure threads in the same warp follow the same (or similar) path. In our case, there is significant divergence between code execution paths for different messages. Therefore, in order to keep the divergence minimal in each warp, we chose the dimensions of the blocks to be M*32 (assuming the warp size is 32), limiting each warp to one message.

After deciding on how to map to the GPU threads, the conversion to parallel code is straightforward, as loop indexes are replaced by thread and block indexes. In the next two sections we describe in more details two major parts of the parallel code.

Parallel reduction

Part of the message passing process used in this algorithm is to compute minimum and sum of messages. Both of these fall under the family of reduction vector operations (i.e. deriving one value from a K-sized vector). Therefore, to parallelize these operations we can follow the same procedure. To choose the best option, we considered two conventional approaches to a parallel reduction mentioned in [START_REF] Harris | Parallel prefix sum (scan) with CUDA[END_REF] and [START_REF] Luitjens | Faster parallel reductions on Kepler[END_REF]. The first approach uses each thread block shared memory to reduce at each step two elements of the vector. This means we will need a vector of size V in shared memory for each reduction. The reduction algorithm made possible on new GPUs [START_REF] Luitjens | Faster parallel reductions on Kepler[END_REF], uses a process called shuf-fling to communicate a variable between threads inside a block. This way there is no need for shared memory and synchronized access. After testing in our case, which we uses around 400k blocks and 25 labels, shuffling has been more effective as expected, although the difference is not considerable but significant.

Computing the discontinuity cost function

During each LBP iteration, considerable time is dedicated to the calculation of the discontinuity cost function. This part includes many summations and multiplications which, as mentioned before, are the same at each iteration. Therefore it is a good idea to pre-compute this part and exclude it from the run-time and turn it into a single memory look up. Each run of the function needs six arguments: two pairs of pixel locations and two labels. Since the locations are always neighbors, we can reduce the input to one pair for a pixel location and another argument indicating the direction of the neighbor. This means for completely pre-computing this function we will need a five dimensional matrix. The size of the discontinuity matrix S DM will be equal to:

S DM = H * W * 4 * V 2 (3)
where H and W are the input image height and width respectively. Complete pre-computation of the function led to speed up in runtime of each iteration; but at the same time cost more than 4GB of graphics memory, for 400k pixels and a label set of 25. This could prove problematic if we decide to increase the label-set or use larger images. Therefore another approach was used to divide the function in two parts: one was to be computed beforehand and one to be calculated at every iteration. The part of the function related to the geometry is precomputed and stored in a matrix with a size proportional to the size of the image. The computation depending on the labels proved to be very simple and manageable in each iteration. This way we decreased the amount of required memory to around 1.5GB which is a fairly reasonable usage. This new approach also decreased the number of global memory accesses. Before we had V 2 global memory access per message. Now we have only one per message. This gave us an overall improvement on iteration runtime.

Further optimization

In this part, we will briefly iterate over some further optimizations done in order to run the algorithm as efficiently as possible. Some of these points are general best practices which will work on any parallel GPU program, but in some cases the changes make sense only in the context of our problem.

Memory optimization

In the process of minimizing the mentioned energy function, there are three major group of stored data: precomputed data cost function, partially pre-computed discontinuity cost function and previously calculated messages. All these sources are needed for computing new messages, therefore in each iteration there are many memory reads from the GPU global memory. Also, in the final step of the computation we need to store the messages in the global memory again to be used in the next iteration. This makes a calculated memory access scheme vital for the algorithm to run as fast as possible. We will mention two general important points to achieve a better access time.

Benefiting from memory hierarchy: In some cases we need to use the same data more than once. The data cost function in our case is frequently needed. In these cases, it is not advisable to access the global memory each time. The best choice is to load the data once and store it in a faster memory, either shared memory or each threads local memory. The choice between the two depends how much of each memory we have available to use. Overuse of local memory can lead to using more registers which can reduce the overall occupancy of GPU. For example, in our tested GPU each block was limited to 65536 registers and 98KB of shared memory.

Coalesced access: [START_REF] Harris | Optimizing CUDA[END_REF] considers coalescing memory operations as one of the general optimization directives for a parallel program which can lead up to 10x speedup. In simple words, this means consecutive threads access consecutive memory addresses (among other conditions). When accessing incoming messages to each pixel it is not possible to maintain a coalesced access but in other cases maintaining such access gave us considerable improvement.

Instruction optimization

After optimizing the memory operations, the bottleneck of the kernel falls on too many mathematical instructions. Mostly in the discontinuity cost function, floating point division and multiplication caused a considerable delay. One possible solution to alleviate this problem is to sacrifice accuracy for more speed by using GPUs fast mathematical instructions. Actual usefulness of this approach obviously varies case by case. In our algorithm, considerable trial and error with these functions led to use of fdividef, fmul rd and fsub rd instead of regular division, multiplication and subtraction in limited cases. This gave us reasonable imprecision which did not affect the end result while saving significant time.

Algorithm optimization

The last step in our optimization process was to investigate whether or not any change in the core algorithm can be helpful. Following the advise of [START_REF] Felzenszwalb | Efficient belief propagation for early vision[END_REF], we decided to alternatively calculate only half of the messages in each iteration. Basically, if we divide the pixels into two subsets A and B following a checkerboard pattern, in each iteration we only compute the outgoing messages of pixels belonging to either A or B. This makes sense because the messages sent from nodes in A only depend on outgoing messages of nodes in B. The same can be said about calculation of messages from nodes in B. [START_REF] Felzenszwalb | Efficient belief propagation for early vision[END_REF] also describes the new message from node p to q at iteration t as: if t is odd (even) then

mt p→q = m t p→q if p ∈ A (if p ∈ B) m t-1 p→q otherwise (4)
This means the new messages are almost the same as the standard ones, and regarding convergence this method also converges to the same fixed point i.e. after convergence m t-1 p→q = m t p→q While being an approximation, this change practically did not affect the end result of the algorithm but it managed to decrease the iteration time by half.

Results

In this section, we will report the recorded execution time of the algorithm using real captured data on a selected platform. For testing, we have used a system with NVIDIA Geforce GTX1080 graphic card and 8GB of graphical memory. The system is also equipped with an Intel® Core™ i7-6900K CPU with 3.20GHz processing speed. The iteration time was measured using NVIDIA's own profiler nvprof. Figure 1 shows a sample input/output of the algorithm which is extracted from a sequence of captured images. The images cropped down to the size of 781*621 which will give us around 500K pixels to work with. As mentioned before, the algorithm is tested with a label set of size 25; from 1.4 meters to 1.975 with inclusion of a special label for pixels without heads. The algorithm consists of three main kernels -one for pre-computation, one for each iteration of loopy belief propagation, and one for final belief computation. The first and last kernel will run once for each image while the second one will run depending on the number of iteration needed for convergence, in this case 100. Figure 3, shows the general time-line of the program. As we can see, about 98% of the runtime is occupied by the iteration kernel, which makes it the most important kernel to optimize. Figure 1 shows the execution time of these kernels using ten consecutive frames.

Since more than 98% of execution time belongs to the iteration kernel, we will conduct our comparison with CPU code only for this kernel. We achieve over 3000x speed-up for each iteration. As said earlier, the process of getting to Step by step optimizations and their time in seconds this level of efficiency involved several general purpose optimizations as well as some strategies specific to our problem. Table 2 shows the progression of iteration time for a sample frame in different version of the algorithm, from initial serial code to the current most optimized state. It was crucial to make sure no part of the optimization introduces any intolerable deviation from the original output. Inherently, CPU and GPU codes give us slightly different floating point operations which can cause differences in final labels. On top of that, as covered in section 3.2, we have used two approximations namely the use of fast arithmetic operations and computing alternatively only half of the outgoing messages. Depending on each frame these changes cause some misidentification and misjudgment of height labels. A comparison of both versions of the algorithm was done on a set of 15 consecutive frames. Table 3 shows the result of this comparison. Three metrics are considered: the number of pixels in which both algorithms detect a head but disagree on the height (second column); the number of pixels in which the two implementations disagree on the presence of a head (third column), and finally average of absolute difference for the pixels which disagree on the height. As we can see, considering the total number of pixels, very few misidentifications happen and also the Figure 4 shows the overall execution time for each frame in both CPU and GPU code. These times include the reading of precomputed data cost function from disk. Overall, a solid 1100x speedup means we can process a long video sequence which used to take several days, in a matter of minutes.

Conclusions

In the current text, we iterated over several steps taken to optimize and accelerate the multiple camera pedestrian head detection algorithm [START_REF] Pellicanò | Geometry-Based Multiple Camera Head Detection in Dense Crowds[END_REF] and adapt it to run efficiently on GPU using NVIDIA CUDA framework. We presented the details of the adaptation for the GPU structure as well as problem specific approximations which helped reaching greater speed without losing precision. In the end, we achieved over 1000x speed up in overall execution time for each frame, while preserving over 99.9% of original outputs.

Figure 1 :Figure 2 :

 12 Figure 1: Regents Park Mosque dataset, Dense sequence [11]. Three camera views are used.

Figure 3 :

 3 Figure 3: Overall time-line of the execution of belief propagation for one frame with 100 iterations

Figure 4 :

 4 Figure 4: Comparison between the CPU and GPU code in total execution time for each frame

Table 1 :

 1 Time taken by the three main kernels in 10 consecutive frames. Times are in millisecond

	Frame	precomputation	Average iteration	Belief computation
	1	3.854	3.781	3.201
	2	4.268	4.267	5.415
	3	3.66	4.147	4.016
	4	3.788	3.799	3.054
	5	3.899	4.401	3.983
	6	3.936	4.032	3.203
	7	3.886	3.934	3.704
	8	3.79	3.8	3.235
	9	5.031	4.262	3.116
	10	4.284	4.101	3.094
	average	4.0396	4.0524	3.6021
	Version			Iteration time
	CPU code			∼11s
	Naive GPU implementation		∼1s
	Using initial pre-computation	∼0.25s
	Memory optimizations		∼0.05s
	Improved pre-computation		∼0.02s
	Using fast instruction approximation	∼0.01s
	Message passing approximation	∼0.004s

Table 2 :

 2

Table 3 :

 3 Result of comparing the output of optimized code with the original CPU code. Window size is 781*621 (485001 pixels)

	Frame	Number of pixels with wrong height	Number of pixels with wrong identification of head presence	average absolute difference of height (meter)
	1	18	5	0.025
	2	44	62	0.025
	3	41	12	0.025
	4	34	11	0.025
	5	31	5	0.025
	6	2	3	0.025
	7	47	112	0.025
	8	14	9	0.025
	9	15	3	0.025
	10	3	2	0.025
	11	41	32	0.032
	12	60	15	0.046
	13	43	10	0.04
	14	50	15	0.027
	15	18	1	0.038


Laurent Vivien

Johann Osmond

Jean Marc Fédéli

Delphine Marris-Morini

Paul Crozat

Jean François Damlencourt

Eric Cassan

Yves Lecunff

Suzanne Laval

42 GHz p.i.n germanium photodetector integrated in a silicon-on-insulator waveguide

come   L'archive ouverte pluridisciplinaire


P Guarnido-Lopez

Isabelle Ortigues

J David

S Polakof

Gonzalo Cantalapiedra-Hijar
email: gonzalo.cantalapiedra@inrae.fr

I Ortigues-Marty

Comparative analysis of signaling pathways in tissue protein metabolism in efficient and non-efficient beef cattle: acute response to an identical single meal size

Keywords:

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

Introduction

Residual feed intake (RFI) is one of the preferred criteria for improving feed efficiency through genetic selection. This is because RFI has moderate heritability [START_REF] Taussat | Relationship between feed efficiency and slaughter traits of French Charolais bulls[END_REF] and selecting for it can decrease feed-related costs [START_REF] Herd | Reducing the cost of beef production through genetic improvement in residual feed intake: Opportunity and challenges to application[END_REF] without negatively affecting other performance traits of the reproductive herd like the mature BW of breeding cows [START_REF] Johnson | The history of energetic efficiency research: Where have we been and where are we going?[END_REF]. However, selecting animals for RFI may have some drawbacks. First, RFI ranking of beef cattle is not repeatable when diet changes [START_REF] Asher | Consistency of feed efficiency ranking and mechanisms associated with inter-animal variation among growing calves[END_REF][START_REF] Coyle | Repeatability of feed efficiency in beef cattle offered grass silage and zero-grazed grass[END_REF][START_REF] Lahart | The repeatability of feed intake and feed efficiency in beef cattle offered high-concentrate, grass silage and pasture-based diets[END_REF]. This raises concerns about the reliability of RFI-based selection when animals are fed differently. Second, RFI selection may have unintended effects on other important animal traits, such as immunity or reproductive performances [START_REF] Wang | Impact of selection for residual feed intake on breeding soundness and reproductive performance of bulls on pasture-based multisire mating[END_REF][START_REF] Zerjal | Assessment of trade-offs between feed efficiency, growth-related traits, and immune activity in experimental lines of layer chickens[END_REF], as all these traits involve various energy-demanding mechanisms that can compete with each other when energy availability is limited. Balanced breeding programmes and avoidance of potential negative effects require a deeper understanding of the mechanisms behind RFI [START_REF] Cantalapiedra-Hijar | Biological determinants of between-animal variation in feed efficiency of growing beef cattle[END_REF].

Continuous protein synthesis and degradation rates (referred to as protein turnover) are associated with high energy expenditure and have been suggested as key determinants of RFI in beef cattle [START_REF] Richardson | Biological basis for variation in residual feed intake in beef cattle. 2. Synthesis of results following divergent selection[END_REF][START_REF] Cantalapiedra-Hijar | Biological determinants of between-animal variation in feed efficiency of growing beef cattle[END_REF]. Protein turnover could potentially be a key determinant of the tradeoffs between production efficiency and robustness traits in livestock [START_REF] Rauw | Immune response from a resource allocation perspective[END_REF]. We confirmed previously in beef cattle that lower protein turnover in skeletal muscle, as indicated by in vivo fractional protein degradation rate, was associated with the efficient RFI phenotype [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF]. However, this finding was observed when animals were fed high-starch diets, but not high-fibre diets, possibly due to differences in the amount and profile of absorbed nutrients (mainly glucogenic vs ketogenic) and the resulting insulin response.

The main challenge in evaluating potential biological mechanisms underlying RFI, such as protein turnover, is to determine whether the correlation is due to a covariation with feed (nutrients) intake level or a true cause-and-effect relationship [START_REF] Cantalapiedra-Hijar | Biological determinants of between-animal variation in feed efficiency of growing beef cattle[END_REF]. Due to the strong influence of nutrient intake level on metabolism [START_REF] Reeds | Nutrient intake and protein turnover[END_REF][START_REF] Boisclair | Nutrient Utilization and Protein Turnover in the Hindlimb of Cattle Treated with Bovine Somatotropin[END_REF][START_REF] Lobley | Nutritional and hormonal control of muscle and peripheral tissue metabolism in farm species[END_REF], it is challenging to determine whether the association between a specific mechanism and the RFI phenotype is due to differences in feeding level and appetite [START_REF] Lines | Selection for residual feed intake affects appetite and body composition rather than energetic efficiency[END_REF], or is related to host-related factors regardless of nutrient intake [START_REF] Nkrumah | Relationships of feedlot feed efficiency, performance, and feeding behavior with metabolic rate, methane production, and energy partitioning in beef cattle[END_REF]. The molecular pathways involved in tissue protein metabolism exhibit high sensitivity to short-term and acute effects of nutrient intake, particularly the rise in glucose and amino acid absorption, as well as the subsequent insulin response. This sensitivity holds true for both monogastric [START_REF] Norton | The leucine content of a complete meal directs peak activation but not duration of skeletal muscle protein synthesis and mammalian target of rapamycin signaling in rats[END_REF] and ruminants [START_REF] Mann | Insulin signaling and skeletal muscle atrophy and autophagy in transition dairy cows either overfed energy or fed a controlled energy diet prepartum[END_REF]. We propose that the association between RFI and tissue protein metabolism primarily arises from variations in the quantity of absorbed insulinogenic nutrients and amino acids, which serve as the primary drivers of protein synthesis signalling pathways [START_REF] Laplante | mTOR signaling at a glance[END_REF]. Our specific hypotheses are as follows: (i) the signalling pathways related to protein metabolism will exhibit similarities between efficient and non-efficient RFI cattle when they are fed the same diet and exposed to an identical nutritional stimulus (i.e. an identical and single meal size) and (ii) the differences observed in the correlation between RFI and protein metabolism across diets [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF] will diminish when employing a uniform and consistent meal size before assessment.

Hence, the objective of this study was to determine if specific proteins involved in the regulation of the protein metabolism in cattle tissues were activated differently in efficient and inefficient RFI cattle. To evaluate tissue metabolic responses in animals subjected to an identical nutritional stimulus, all animals (including both low and high RFI) were fed a single and identical meal size consisting exclusively of the concentrate part of their diet, 3 h before slaughter. The tissues under focus are the skeletal muscle, which is the main body protein pool, and the liver, which is the major site for amino acid catabolism. We targeted the signalling pathways of protein degradation (adenosine monophosphateactivated protein kinase [AMPK], autophagy and ubiquitin pathways) and synthesis (serine/threonine protein kinase [Akt], mammalian target of rapamycin [mTOR], ribosomal protein S6 kinase [S6K] cascade) and the main hepatic transaminases related to amino acid catabolism.

Material and methods

The experiment was conducted at INRAE, Centre Auvergne-Rhône-Alpes, France. The experimental protocol was approved by the Ethics Committee of the Auvergne-Rhône-Alpes region and the French Ministry of Higher Education, Research and Innovation (Authorisation number: APAFIS #16194-2016101016361277 v6 delivered on the 14th of January 2019).

Animals, dietary management and experimental design

The present work consisted of a postmortem study that followed an in vivo trial [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF], which used a total of 32 extreme RFI fattening young bulls (561 ± 49.1 kg of BW; 1.24 ± 0.21 kg of average daily gain [ADG]; 9.06 ± 0.93 kg/d of DM intake; with a difference of DM intake between extreme RFI individuals = 1.10 kg/d). Half of the bulls were fed a highstarch corn-silage plus cereal-based diet, while the other half received a high-fibre grass-silage plus beet pulp-based diet. Both diets had a forage-to-concentrate ratio close to 60:40. Following an initial 84-d feed efficiency test conducted over 100 Charolais bulls in two consecutive years (2019 [n = 49] and 2020 [n = 51]), we selected the eight most extreme RFI individuals (the four lowest and highest RFI cattle) within each diet and experimental year for the in vivo analysis of protein metabolism and N partitioning using indirect methods. All 32 extreme RFI animals were slaughtered at the INRAE experimental abattoir (Saint Genès de Champanelle, France) at the end of the in vivo study.

In the present postmortem study, we assessed protein metabolism pathways in the skeletal muscle and the liver from the 32 extreme RFI individuals immediately after their slaughter (16 animals from 2019 and 16 from 2020). Slaughter took place over the course of one month (June 2019 and June 2020), with one animal from each diet and RFI class (low and high RFI) being slaughtered daily. Altogether four animals were slaughtered per week on the same day. This approach ensured that the average experimental duration (from the start to the slaughter) remained similar across treatments. Animals were fed their respective total mixed ration for approximately 200 d prior to the day of slaughter (Guarnido- [START_REF] Lopez | Evaluation du métabolisme protéique comme déterminant des variations individuelles d'efficience alimentaire chez le bovin en engraissement[END_REF].

Animals underwent a fasting period beginning 15 hours prior to slaughter, followed by a single test meal containing an equivalent quantity of their respective concentrate (2.5 kg as fed) and fed 3 h prior to slaughter. Because of their nature, these concentrates presented a more contrasted nutritional composition (high-starch and high-protein concentrate or high-fibre and low-protein concentrate; Table 1) compared to their corresponding total mixed ration (Guarnido- [START_REF] Lopez | Evaluation du métabolisme protéique comme déterminant des variations individuelles d'efficience alimentaire chez le bovin en engraissement[END_REF]. We checked that all animals completely consumed their meal. The rationale is as follows. During the postprandial period, circulating insulin and subsequent amino acid uptake in the skeletal muscle tissue increase. The postprandial period represents the physiological window during which intramuscular anabolic signalling is stimulated [START_REF] Cuthbertson | Anabolic signaling deficits underlie amino acid resistance of wasting, aging muscle[END_REF]. Previous exploratory analyses of protein turnover rates in beef cattle (Cantalapiedra-Hijar et al., 2020) indicated that the intensity of protein synthesis signalling pathways was rather low when assessed in a non-stimulated condition, i.e. in a fasting state (Cantalapiedra-Hijar G., personal observation). In this regard, glucose intravenous infusion in dairy cows resulted in a rise in insulinemia, which was sufficient to activate within 1 h the mTOR pathway in the muscle of dairy cows [START_REF] Mann | Insulin signaling and skeletal muscle atrophy and autophagy in transition dairy cows either overfed energy or fed a controlled energy diet prepartum[END_REF]. Based on these findings, we decided to conduct the postmortem study with animals in the postprandial state, following a single meal size administered after a 12-hour food deprivation period. This methodology has been utilised in studies involving rats, piglets and horses [START_REF] Norton | The leucine content of a complete meal directs peak activation but not duration of skeletal muscle protein synthesis and mammalian target of rapamycin signaling in rats[END_REF][START_REF] Gazzaneo | Intermittent bolus feeding has a greater stimulatory effect on protein synthesis in skeletal muscle than continuous feeding in neonatal pigs[END_REF][START_REF] Loos | Pathways regulating equine skeletal muscle protein synthesis respond in a dose-dependent manner to graded levels of protein intake[END_REF]. Because the explored signalling pathways respond to a meal stimulus, it was important to set all individuals at the same postprandial time. The purpose of the short food deprivation period was to stimulate a prompt and complete consumption of the provided meal while minimising differences between individuals due to non-synchronised nutritional conditions (i.e. meal consumption at different times). By offering only the concentrate as the meal, we aimed to minimise any discrepancies in both the duration and quantity of feed consumed among the animals that would likely be observed if the total mixed ration had been used instead. This protocol was designed to guarantee that key proteins involved in the regulation of the protein metabolism in cattle tissues would be assessed in the same nutritional conditions for all animals fed the same diet. Finally, we chose a 3 hours interval between the last meal and slaughter because it corresponds to the postprandial absorption peak of glucose and insulin in cattle [START_REF] Jenny | Effects of high grain feeding and stage of lactation on serum insulin, glucose and milk fat percentage in lactating cows[END_REF][START_REF] Evans | Postprandial patterns of plasma glucose, insulin and volatile fatty acids in ruminants fed low-and highroughage diets[END_REF], both substrates acting as signals for protein synthesis and degradation [START_REF] Reeds | Nutrient intake and protein turnover[END_REF]. All animals were slaughtered strictly 3 hours after the distribution of their respective test meal.

Measurements and sampling

Plasma and tissue sampling

Blood samples were collected at slaughter (i.e. 3 h after the test meal) by jugular venipuncture once the animal was hung. Blood samples were collected into a 9 ml EDTA tube (BD vacutainer, Ply-mouth, UK) and centrifuged at 2 500g for 10 min at room temperature to obtain the plasma, which was subsequently stored at À20 °C until the analysis of metabolites (glucose, urea and bhydroxybutyrate) and hormones (insulin and IGF-1) concentrations.

Approximately 30 min after the slaughter (i.e. 3 h and 30 min after the distribution of the test meal), both hepatic (caudal lobe) and muscle (longissimus dorsi) tissues were sampled in order to evaluate the postprandial protein synthesis and degradation intracellular pathways, as well as the activity level of key enzymes involved in amino acid metabolism. Tissue samples were quickly cut into small squares of 1 cm  1 cm, frozen immediately in liquid nitrogen and stored at À80 °C awaiting analyses.

Laboratory analyses

Concentrates were analysed for DM, organic matter (OM), CP, NDF, ADF and starch, and their feeding values were calculated as described in Guarnido- [START_REF] Lopez | Evaluation du métabolisme protéique comme déterminant des variations individuelles d'efficience alimentaire chez le bovin en engraissement[END_REF]. Chemical composition and feeding values are reported in Table 1.

Plasma metabolites and hormone analyses

For plasma glucose, b-hydroxybutyrate and urea concentrations, spectrophotometric analyses were conducted using an automated analyzer (Arene 20XT, Thermo Scientific, Vaanta, Finland). Urea analyses were conducted in duplicate, whereas glucose and b-hydroxybutyrate analyses were performed as single measurements. The accuracy profile (NF V03-110: 2010) of the automated analyser for the concentration of these three metabolites ranged between 0.05 and 0.90 g/L, yielded an average accuracy of 101% and a CV for replicates averaging 8%. Plasma insulin and IGF-1 concentrations were determined by radioimmunoassay (RIA) as detailed in [START_REF] Bonnet | Common and diet-specific metabolic pathways underlying residual feed intake in fattening Charolais yearling bulls[END_REF].

Molecular analyses

The relative abundance of proteins involved in protein synthesis and degradation were quantified in hepatic and muscle samples through western blot. Frozen samples were homogenised on ice in 2 ml lysis buffer (150 mM NaCl, 10 mM Tris, 1 mM ethylene glycol tetraacetic acid, 1 mM ethylenediaminetetraacetic acid [EDTA], pH 7.4, 100 mM Sodium Fluoride, 4 mM Sodium pyrophosphate, 2 mM Sodium Orthovanadate, 1% Triton X100, Protease inhibitor cocktail to 1% V/V) before using (Merck Sigma, P8340 Saint Quentin Fallavier). The homogenate obtained was centrifuged at 10 000g for 15 min at 4 °C. Protein concentration was determined using a Bicinchoninic acid assay protein assay kit (Thermo Pierce Biotechnology, IL, USA) according to the protocol of the manufacturer with bovine serum albumin (BSA) as a standard. Proteins were then denaturised in a 1X Laemmli solution (+b-mercaptoethanol) at 95 °C for 5 min, and separated onto polyacrylamide gels (Biorad, Mini-Protean TGX stain-Free gels, 4-20%, 4568095, Mitry-Mory).

Equal amounts of protein (20 lg per well) were loaded into gels set on a Mini-Protean tetra electrophoretic system (Bio-Rad) and run in a 1X buffer (Running buffer 10X, Tris-Base, Glycine and SDS) for 2 h at 80 V at ambient temperature and 70 V until the blue reached the bottom of the gel. After migration, separated proteins were transferred onto polyvinylidene fluoride membranes (Biorad, Trans-Blot Turbo, transfer kit, LF PVDF, 1704275). After washing in a phosphate-buffered saline-Tween 1X solution, membranes were blocked for 1 h at room temperature in blocking solution, BSA 5% PBS-Tween 0.1%. A total of eight samples were disposed by gel balanced for the experimental effects (RFI group, diet type and the experimental period [2019 and 2020]). In addition, two standard samples of each tested protein were included in every gel. For protein synthesis, we analysed the Akt, mTOR and S6K pathways. Antibodies utilised for Akt pathway were Akt (pan, rabbit ab #4691, Cell Signalling, Ozyme, France) and pAkt (Serine 473, rabbit Ab #4060, Cell Signalling, Ozyme, France). Antibodies utilised for mTOR pathway were mTOR (pan, rabbit ab #2972, Cell Signalling, Ozyme, France) and pmTOR (Serine 2448, rabbit Ab #39182, Cell Signalling, Ozyme, France). Antibodies utilised for S6K pathway were S6K (S6 ribosomal protein #2217 Cell Signalling, Ozyme, France) and pS6K (Serine 235/236, rabbit Ab #2211, Cell Signalling, Ozyme, France). For protein degradation (proteolysis), we analysed the AMPK, ubiquitin-like modifier activating enzyme 1 (UBA1) and the microtubule-associated protein 1A/1B-light chain 3 (LC3) pathways. Antibody utilised for UBA pathway was UBA1 (Anti-E1 Ubiquitin Activating Enzyme 1, ab228793, Abcam, Amsterdam). Antibody used for LC3 pathway was LC3B (Anti-LC3B antibody produced in rabbit, $1 mg/mL, L7543, Sigma Aldrich-Merck, France). For AMPk, we utilised AMPk (AMPKalpha #2532, Celle Signalling, Ozyme, France) and pAMPk (Phospho-AMPKalpha (Thr 172) #2531, Cell Signalling, Ozyme, France).

Membranes were incubated overnight at 4 °C, and protein bands were visualised by Infrared Fluorescence using the Odyssey Imaging System (LI-COR Inc. Biotechnology, Lincoln, NE, USA) and quantified by Odyssey infrared imaging system software (Application software, version 1.2). Finally, we normalised intracellular results of the activation of both protein synthesis and degradation according to the total quantity of proteins in these pools.

Hepatic enzyme activities

Enzyme activities of the two main hepatic transaminases, alanine amino-transferase (ALT) and aspartate amino-transferase (AST) were evaluated. Samples were homogenised with ice-cold buffer (10 w:v; 50 mM Tris, pH 7.4, 250 mM sucrose, 1 mM EDTA, 1 mM Sodium orthovanadate, 5 mM Sodium pyrophosphate, 50 mM NaF, 1 mM EDTA, 1% V/V Triton X100, 1% V/V Protease Inhibitor Cocktail). The homogenate was centrifuged at 10 000g for 15 min at 4 °C and the supernatant was used immediately for enzyme assays. Analyses were conducted through spectrophotometry using a microplate reader (Infinite Ò 200 PRO NanoQuant, Tecan, Grödig, Austria) as described in Polakof et al., (2008) and adapted for protein concentrations in cattle tissues through a previous set-up.

Statistical analysis

Statistical analyses were performed in R (RStudio Core Team, version 1.1.463, 2018). To test the effects of the diet, the RFI group and their interaction on all postmortem analysed variables, a general linear model was run that included the year, the nature of the diet, the RFI group, and their interactions as fixed effects (Experimental year, Diet, RFI, Year  Diet, Year  RFI and Diet  RFI);

Y ijkl = l + D i + A k + E j + D i  A k + E i  A k + D i  E j + ijkl .
where Y ijkl is the dependent variable for animal i, belonging to the RFI group l, receiving diet j, in year k, l is the overall mean; Dj is the fixed effect of diet used (j = high-starch and high-fibre concentrate), A k is the fixed effect of the year (k = 2019 and 2020), El is the fixed effect of the RFI group (l = efficient and non-efficient), Dj  Ak is the interaction between the effect of the diet and the year, El  A k is the interaction between the effect of the year and the RFI group, Dj  El is the interaction between the effect of the diet and the RFI group and the ℰ ijkl is the residual term. The triple interaction and the week in which the animals were slaughtered were also tested. However, since they did not show significance across all variables, they were not incorporated into the model. Finally, and just in the case of western-blot analyses, we added the effect of the agar gel number (n = 4 and 16 for protein degradation and synthesis proteins, respectively) as a fixed effect.

Mean values are reported with associated errors (SEM). When the Diet  RFI interaction was significant, the means for the different RFI groups were compared using Tukey's significant difference multiple comparison. Effects were declared significant when P 0.05, and a trend was considered when 0.05 < P < 0.10.

Results

All variables were assessed for normality and followed a normal distribution (P > 0.05). Interactions between year, RFI group and diet type were tested, and they were non-significant and are not presented in the tables (P > 0.05). Furthermore, the statistical effect of the gel number on western-blot results was not significant (P > 0.05). We have decided not to present the analysis of hepatic mTOR and AMPK proteins due to poor image quality and intensity of bands.

Plasma metabolites and hormone concentrations

Regardless of the composition of the test meal, efficient RFI animals had higher plasma b-hydroxybutyrate concentrations (+28.6%; P = 0.03) than non-efficient animals (Table 2). In contrast and only when fed a high-starch concentrate as the test meal, animals with efficient RFI showed higher insulinemia (+35.2%, Diet  RFI interaction; P = 0.03) and a tendency towards higher glycaemia (+16.7%, Diet  RFI interaction; P = 0.08) compared to animals with non-efficient RFI. There were no differences in plasma urea or IGF-1 levels across the tested concentrates or RFI groups (P ! 0.13). With regards to the diet effect, animals fed a highstarch concentrate had higher plasma insulin concentrations (+21%, P = 0.05) and a lower glucose-to-insulin ratio (À35.3%, P = 0.01) compared to animals fed a high-fibre concentrate.

Signalling pathways related to muscle protein degradation and synthesis

Efficient RFI bulls had a lower potential for intracellular protein degradation in the muscle, as evidenced by their lower (À34.6%, P = 0.04) ubiquitin relative abundance (UBA1) compared to nonefficient bulls, regardless the composition of the test meal (Table 3). There were no differences in autophagy (LC3) or AMPK signalling pathways between the two RFI groups (P ! 0.30). With regards to protein synthesis, efficient RFI bulls had a lower abundance of total Akt (À24.4%, P = 0.05) and S6K (À26.8%, P = 0.07) proteins, but no changes were observed in their respective phosphorylated forms (P ! 0.11). In contrast, phosphorylated mTOR protein abundance was higher (+35%, P = 0.01) in non-efficient bulls regardless of the composition of the test meal, and a similar trend was observed for total mTOR abundance (+24.3%, P = 0.10). Overall, efficient RFI individuals presented a higher potential for protein synthesis in the muscle regardless of the composition of the test meal, as supported by their higher (+30.2%, P = 0.05) pS6K to S6K ratio. However, the pmTOR to mTOR ratio was higher in nonefficient RFI bulls when fed high-starch concentrate (+40.1%, Interaction RFI Â Diet; P = 0.09). There were no diet effects observed for the analysed protein signalling pathways in the muscle (P > 0.05).

Hepatic amino acid metabolism and signalling pathways related to protein degradation and synthesis

As shown in Table 4, efficient RFI bulls had a trend for lower hepatic ALT activity (À14.2%, P = 0.06) compared to non-efficient bulls. We did not find any significant differences in hepatic protein synthesis (Akt and S6K) or protein degradation (LC3 and UBA1) pathways between the two RFI groups (P ! 0.08). With regards to the effects of the composition of the test meal, animals fed a high-starch concentrate had higher protein degradation potential, as indicated by UBA1 abundance (+52%, P = 0.001). Furthermore, only efficient RFI bulls showed a higher relative abundance of pAkt (+19.1%, Diet  RFI interaction; P = 0.04) and a trend for a higher pAkt to Akt ratio (+23.3%, Diet  RFI interaction; P = 0.09) compared to inefficient bulls when fed the high-starch concentrate.

Discussion

We previously reported, using different approaches, greater nitrogen use efficiency in efficient RFI beef cattle compared with their inefficient counterparts whatever the diet (Cantalapiedra-Hijar et al., 2020; Guarnido-Lopez et al., 2021; Guarnido-Lopez [START_REF] Liu | Relationship of residual feed intake and protein efficiency in lactating cows fed high-or low-protein diets[END_REF]. We also found that efficient RFI beef cattle exhibit more protein deposition in the carcass at the expense of the organs only when animals were fed a high-starch (vs highfibre) diet [START_REF] Meale | Exploration of biological markers of feed efficiency in young bulls[END_REF][START_REF] Bonnet | Common and diet-specific metabolic pathways underlying residual feed intake in fattening Charolais yearling bulls[END_REF][START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF]. Protein turnover is considered to be a major determinant of RFI variations [START_REF] Cantalapiedra-Hijar | Biological determinants of between-animal variation in feed efficiency of growing beef cattle[END_REF] and could contribute to explain the relationship between nitrogen use efficiency, body composition and RFI values previously reported. A positive association between protein turnover and RFI was observed, but this association was evident only in animals fed a high-starch diet and not in those on a high-fibre diet, which is also associated with higher insulinemia [START_REF] Bonnet | Common and diet-specific metabolic pathways underlying residual feed intake in fattening Charolais yearling bulls[END_REF][START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF]. However, it remained uncertain whether this association was primarily attributed to differences in nutrient signalling among animals with extreme RFI.

In the present study, our main findings show that, in response to an identical meal size and independently of the composition of the test meal (concentrates), the efficient RFI cattle had lower protein degradation rates in muscle and liver, as well as lower hepatic transaminase activity than the inefficient RFI cattle whatever the diet. They also demonstrated a faster activation of protein synthesis in muscle. These results highlight the potential role of protein synthesis and degradation in contributing to differences in protein deposition that characterise individuals with extreme RFI phenotype.

Relationship between protein degradation, nitrogen use efficiency and residual feed intake

In addition to their related energy expenditures, lower protein degradation rates and amino acid catabolism could potentially contribute to an improved RFI by enhancing the nitrogen use efficiency [START_REF] Cantalapiedra-Hijar | Natural 15N abundance in specific amino acids indicates associations between transamination rates and residual feed intake in beef cattle[END_REF]. The lower activation of tissue protein degradation pathways and transaminase activity, as observed in our study following a similar nutritional stimulus, in efficient compared to non-efficient RFI individuals may explain the previously reported association between nitrogen use efficiency and RFI [START_REF] Cantalapiedra-Hijar | Natural 15N abundance in specific amino acids indicates associations between transamination rates and residual feed intake in beef cattle[END_REF][START_REF] Liu | Relationship of residual feed intake and protein efficiency in lactating cows fed high-or low-protein diets[END_REF][START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF].

Lower protein degradation rates have been reported in muscle of efficient RFI cattle [START_REF] Mcdonaugh | Meat quality and the calpain system of feedlot steers following a single generation of divergent selection for residual feed intake[END_REF] and pigs [START_REF] Cruzen | Evidence of decreased muscle protein turnover in gilts selected for low residual feed intake[END_REF] fed ad-libitum compared to their non-efficient RFI counterparts. We analysed in the present study, the lysosomal (LC3) and the ubiquitin (UBA1) proteins as the two major markers of the intracellular protein degradation pathways. The lysosomal pathway plays a role in autophagy of long-lived proteins [START_REF] Mizushima | Autophagy fights disease through cellular self-digestion[END_REF], while ubiquitin pathway degrades damaged, non-functional or misfolded proteins [START_REF] Goldberg | Protein degradation and protection against misfolded or damaged proteins[END_REF]. We only found a significant difference in the ubiquitin pathway, which showed a positive correlation with RFI values (r = 0.42, P = 0.02), suggesting non-efficient RFI individuals may have a higher rate of degradation for damaged, misfolded or non-functional proteins in muscle. Indeed, the higher tissue protein degradation rate in non-efficient animals may result in increased post-transcriptional protein modifications [START_REF] Bozaykut | Regulation of protein turnover by heat shock proteins[END_REF], such as N-terminal acetylation, which can in turn increase the creation of misfolded proteins and trigger the ubiquitin pathway, leading to further protein degradation [START_REF] Hwang | N-terminal acetylation of cellular proteins creates specific degradation signals[END_REF]. The autophagy pathway was not found to be different, because it is mainly triggered by reduced ATP levels [START_REF] Wang | Autophagydependent ATP release from dying cells via lysosomal exocytosis[END_REF] likely not occurring in our postprandial conditions. In contrast to our findings, other studies have reported that genes associated with the autophagy pathway were downregulated in efficient RFI cattle [START_REF] Rolf | Genome-wide association analysis for feed efficiency in Angus cattle[END_REF], but signalling information was lacking on those studies, limiting the functional conclusions to be drawn.

Results concerning the transaminase activity in the liver suggest lower amino acid catabolism in efficient RFI cattle. Both AST and ALT play a role in amino acid transamination, a necessary step for most amino acids to be catabolised [START_REF] Torchinsky | The discovery of enzymatic transamination and its biological significance[END_REF]. Differences in amino acid catabolism between extreme RFI cattle were also suggested by [START_REF] Cantalapiedra-Hijar | Natural 15N abundance in specific amino acids indicates associations between transamination rates and residual feed intake in beef cattle[END_REF], who observed lower transamination rates of individual plasma amino acids in efficient vs non-efficient RFI Charolais bulls using indirect isotopic methods. Based on the findings of [START_REF] Righetti | Reutilization of Amino Acids in Protein Synthesis in HeLa Cells[END_REF], which reported that on average 80% of amino acids released from protein degradation can be recycled for protein synthesis, it can be inferred that approximately 20% of amino acids released during each round of protein degradation will be catabolised. Therefore, higher rates of protein degradation may result in greater amino acid catabolism, as observed in uremic human patients [START_REF] Mitch | Mechanisms that cause protein and amino acid catabolism in uremia[END_REF] or in ruminants [START_REF] Bequette | Amino acid metabolism in animals: an overview[END_REF]. In the current study, we found a positive correlation (r = 0.52, P = 0.004) between the hepatic protein degradation potential Table 4 Relative abundances of main proteins involved in the liver protein degradation and synthesis pathways and hepatic transaminase activity from extreme residual feed intake (RFI) young Charolais bulls fed an identical single meal size of either a high-fibre and low-protein concentrate or high-starch and high-protein concentrate 3 h before slaughter.

Diet

High Abbreviations: LC3 = Microtubule-associated protein 1A/1B-light chain 3; UBA1 = Ubiquitin-like modifier activating enzyme 1; Akt = Serine/threonine protein kinase; pAkt = Phosphorylated serine/threonine protein kinase; S6K = Ribosomal protein S6 kinase; pS6K = Phosphorylated ribosomal protein S6 kinase; AST = Aspartate aminotransferase; ALT = Alanine amino-transferase; AMPK = Adenosine monophosphate-activated protein kinase; mTOR = Mammalian target of rapamycin. * The results of AMPK and mTOR were not exploitable due to a weak expression of proteins in gels. ** Enzymatic activity was measured over thawed samples of liver from extreme RFI bulls. Units are given in milliunit (mU). a,b,c Averages with different letters on the same row are significantly different (P < 0.05).

(UBA1 protein abundance) and the hepatic transaminase activity (ALT activity), irrespectively of the diet, but did not observe any correlations with targets involved in the protein synthesis pathway (P > 0.27).

Residual feed intake and tissue protein synthesis pathway

Although both groups were fed the same meal size 3 h before slaughter, efficient RFI cattle had higher postprandial plasma concentrations of b-hydroxybutyrate (both concentrates) and insulin (only for the high-starch-high-protein concentrate) compared to their inefficient counterparts. These metabolic features, along with the changes observed in the mTOR pathway, may suggest that the absorption of nutrients and subsequent activation of muscle protein synthesis occurred faster in efficient RFI animals in response to an identical single meal size. However, it is worth noting that no changes were observed in the hepatic protein synthesis pathways, likely because insulin does not stimulate protein synthesis in visceral tissues to the same extent as it does in skeletal muscle [START_REF] Davis | Differential effects of insulin on peripheral and visceral tissue protein synthesis in neonatal pigs[END_REF].

In our previous in vivo study [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF], we did not directly measure the skeletal muscle protein synthesis rate. However, based on the data on carcass muscle proportion and protein degradation rates, we estimated slightly lower muscle protein synthesis rates in efficient RFI animals that were fed a corn-based diet. We attributed this finding to the lower circulating insulin and branched-chain amino acids in these animals compared to inefficient RFI bulls, as previously reported by [START_REF] Bonnet | Common and diet-specific metabolic pathways underlying residual feed intake in fattening Charolais yearling bulls[END_REF] and [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF]. In our present postmortem study, we found that the plasma insulin levels in animals fed the high-starch concentrate were greater in efficient RFI individuals than in inefficient RFI individuals, despite both groups being fed an identical meal size prior to slaughter. This finding could be interpreted as efficient RFI animals had a greater insulin responsiveness to nutrients than inefficient ones. This is also coherent with studies performed in chicken [START_REF] Gabarrou | Glucose-insulin relationships and thyroid status of cockerels selected for high or low residual food consumption[END_REF] and pigs [START_REF] Colpoys | Responsiveness of swine divergently selected for feed efficiency to exogenous adrenocorticotropic hormone and glucose challenges[END_REF], where RFI-efficient animals globally presented higher rises in plasma insulin levels following a glucose tolerance test. In contrast, the study by [START_REF] Fitzsimons | Effects of phenotypic residual feed intake on response to a glucose tolerance test and gene expression in the insulin signaling pathway in longissimus dorsi in beef cattle[END_REF] failed to evidence a significant difference in insulin responses and sensitivity across RFI groups following a glucose tolerance test. We did not observe the same finding when animals were fed the high-fibre concentrate, likely because the amount of glucose absorbed from the intestine with this kind of diet was not enough to elicit changes in insulin responsiveness [START_REF] Bonnet | Common and diet-specific metabolic pathways underlying residual feed intake in fattening Charolais yearling bulls[END_REF].

In the present study, in response to an identical single meal size, efficient RFI individuals had greater pS6K to S6K ratio in muscle tissues compared to their inefficient counterpart, irrespectively of the composition of the test meal. Phosphorylation of protein S6 kinase is one of the last steps leading to increases in protein synthesis [START_REF] Laplante | mTOR signaling at a glance[END_REF] by stimulating messenger RNA translation [START_REF] Kolitz | Eukaryotic initiator tRNA: finely tuned and ready for action[END_REF]. Thus, our results may be interpreted as a faster activation of protein synthesis for efficient RFI animals in response to an identical single meal size.

Considering the chronological sequence of events within the mTOR signalling cascade subsequent to its activation by insulin and amino acids [START_REF] Seiliez | An in vivo and in vitro assessment of TOR signaling cascade in rainbow trout (Oncorhynchus mykiss)[END_REF][START_REF] Wang | Fine-tuning of postprandial responses via feeding frequency and leucine supplementation affects dietary performance in turbot (Scophthalmus maximus L.)[END_REF], we can hypothesise that nutrient signalling on this pathway may have been faster in efficient RFI cattle. This hypothesis is compatible with the expected faster glucose and amino acid absorption rates when the same amount of concentrate should transit lower rumen sizes, anatomical feature found for these animals [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF]. These factors could have led to faster insulin secretion and protein synthesis activation in efficient RFI animals. Insulin secretion in response to meal intake can be as fast as 5 minutes in growing cattle [START_REF] Chase | Feeding behavior of steers fed a complete mixed ration[END_REF], leading to a peak in plasma insulin concentration between 1 and 4 hours after feeding, according to different experimental conditions in ruminants [START_REF] Jenny | Effects of high grain feeding and stage of lactation on serum insulin, glucose and milk fat percentage in lactating cows[END_REF][START_REF] Evans | Postprandial patterns of plasma glucose, insulin and volatile fatty acids in ruminants fed low-and highroughage diets[END_REF][START_REF] Vicari | Postprandial blood hormone and metabolite concentrations influenced by feeding frequency and feeding level in veal calves[END_REF]. In the same line, the greater b-hydroxybutyrate plasma concentrations in efficient RFI cattle, regardless of the composition of the concentrate, may also agree with a faster nutrient absorption rate in these animals. The portal and posthepatic absorption of b-hydroxybutyrate in animals with positive energy balance (as those used in this experiment) depends mostly on the amount of OM fermented in the rumen [START_REF] Loncke | Empirical prediction of net portal appearance of volatile fatty acids, glucose, and their secondary metabolites (b-hydroxybutyrate, lactate) from dietary characteristics in ruminants: A meta-analysis approach[END_REF][START_REF] Loncke | Empirical prediction of net splanchnic release of ketogenic nutrients, acetate, butyrate and b-hydroxybutyrate in ruminants: a metaanalysis[END_REF] and is primarily derived from the partial oxidation of butyrate in the rumen wall [START_REF] Noziere | Effect of composition of ruminally-infused short-chain fatty acids on net fluxes of nutrients across portal-drained viscera in underfed ewes[END_REF][START_REF] Loncke | Empirical prediction of net splanchnic release of ketogenic nutrients, acetate, butyrate and b-hydroxybutyrate in ruminants: a metaanalysis[END_REF]. In this regard, based on the abundance of proteins involved in mTOR pathway and gene expression in the rumen epithelium, [START_REF] Elolimy | Rapid communication: Residual feed intake in beef cattle is associated with differences in protein turnover and nutrient transporters in ruminal epithelium[END_REF] speculated that efficient RFI cattle could have an improved rumen absorption of nutrients such as butyrate.

The faster protein synthesis and lower protein degradation rates observed in our postmortem study in muscle of efficient RFI cattle, regardless of the nature of the concentrate, should logically have resulted in greater protein accretion rate. However, this finding was only observed in vivo in animals fed the corn-silage diet, and not in those fed the grass-silage diet [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF]. This difference is likely because the absorption rate and nature of nutrients varied more significantly between the two total mixed rations used in vivo than across the experimental concentrates used in this experiment.

Finally, as suggested by [START_REF] Berry | Cell biology symposium: genetics of feed efficiency in dairy and beef cattle[END_REF], there is evidence that RFI tends to favour the selection of later-maturing animals. All else being equal, animals of the same age that deposit proportionally more protein than fat for the same daily gain will, on average, be considered more efficient (low RFI). Considering the role of protein synthesis and degradation rates in protein retention, it is reasonable to anticipate that later-maturing animals (with low RFI) would display either lower protein degradation rates, higher protein synthesis rates, or a combination of both compared to earlier-maturing animals (with high RFI) of the same age.

Limitations of the study

The experimental design we employed in our study was inspired by previous research conducted in monogastric species, which focused on evaluating protein metabolism signalling pathways in response to an identical nutritional stimulus, namely a single and identical meal size, following a fasting period of 12 hours [START_REF] Norton | The leucine content of a complete meal directs peak activation but not duration of skeletal muscle protein synthesis and mammalian target of rapamycin signaling in rats[END_REF][START_REF] Murgas Torrazza | Leucine supplementation of a low-protein meal increases skeletal muscle and visceral tissue protein synthesis in neonatal pigs by stimulating mTOR-dependent translation initiation[END_REF][START_REF] Gazzaneo | Intermittent bolus feeding has a greater stimulatory effect on protein synthesis in skeletal muscle than continuous feeding in neonatal pigs[END_REF]. In these studies, the single meal given to the animals before the evaluation differed in both quality and quantity from the previous meals with the basal diet. The rationale behind this approach lies in the understanding that the activation state of the signalling pathways related to protein synthesis in muscle is mainly triggered by the immediate stimulus of absorbed nutrients (glucogenic nutrients and amino acids) and the subsequent insulin response [START_REF] O'connor | Regulation of translation initiation by insulin and amino acids in skeletal muscle of neonatal pigs[END_REF]. Similar mechanisms are expected to occur in ruminants, as evidenced by the activation of the mTOR pathway in the muscle of dairy cows triggered as early as 1 hour after glucose intravenous infusion [START_REF] Mann | Insulin signaling and skeletal muscle atrophy and autophagy in transition dairy cows either overfed energy or fed a controlled energy diet prepartum[END_REF]. Feeding all individuals in a similar manner before the assessment was intended to create a more comparable nutritional scenario for subsequent analysis.

Ruminants, however, differ from monogastric animals because of their particular digestive anatomy; their rate of feed digestion is much slower, and the profile and kinetics of nutrient absorption differ. As a result, the experimental model may present limitations and results should be interpreted with caution. Fasting for 12 hours in ruminants and distribution of a single and identical meal size to all animals do not entirely eliminate the influence of previous meals and historic intakes on the nutrient absorption and metabolism, including the studied signalling pathways. While we observed differential activation of the evaluated pathways across extreme RFI groups, this could lead to the interpretation that the differences are independent of feed intake levels since all animals received the same meal size before evaluation. However, we cannot disregard the possibility of a minor effect from the previously observed contrasting feed intake across both RFI groups [START_REF] Guarnido-Lopez | Protein metabolism, body composition and oxygen consumption in young bulls divergent in residual feed intake offered two contrasting forage-based diets[END_REF] on the outcomes we have obtained in the current study. Indeed, long-term or chronic effects of enteral oil infusion on insulin signalling pathways (Akt-mTOR-S6K) were found in the muscle of dairy cows in a fed steady-state [START_REF] Gingras | Long-chain omega-3 fatty acids regulate bovine whole-body protein metabolism by promoting muscle insulin signalling to the Akt-mTOR-S6K1 pathway and insulin sensitivity[END_REF]. Also, while the nutritional regulation of the protein synthesis can occur quickly (i.e. in the postprandial period), most mechanisms involved in protein degradation will be adjusted rather slowly (hours or days) [START_REF] Kimball | Control of protein synthesis by amino acid availability[END_REF]. Further controlled studies are needed to confirm the correlation between tissue protein turnover rates and RFI that is independent of the long-term effect of feeding level.

Finally, as previously justified, for the meal test, we only used the concentrate part of the two evaluated total mixed rations. These concentrates significantly differed in protein concentrations (118 vs 153 g metabolisable protein/kg DM) because they were designed to make the total mixed rations fed during the whole fattening period as iso-nitrogenous as possible (81.4 vs 85.6 g metabolisable protein/kg DM; Guarnido- [START_REF] Lopez | Evaluation du métabolisme protéique comme déterminant des variations individuelles d'efficience alimentaire chez le bovin en engraissement[END_REF]. The disparity in protein content may have led to a more intense activation of the protein synthesis pathway in animals fed the highstarch concentrate. Nevertheless, since no significant interactions were identified between the concentrate composition and RFI for any of the analysed variables, similar conclusions could be anticipated even when the diets exhibit less pronounced contrasts in terms of nutritional values.

Conclusions

In contrast to our initial hypothesis, the findings from our study reveal that efficient RFI cattle demonstrated reduced muscle protein degradation and a tendency towards lower hepatic protein degradation and transaminase activity in response to the nutritional stimulus triggered by an identical meal size. These findings were irrespective of the composition of the two tested concentrates. Under the condition of this experiment, a biological association between protein degradation and amino acid catabolism was identified and could explain the greater nitrogen use efficiency reported in efficient RFI cattle. Regarding protein synthesis, our results indicate that regardless of the composition of the concentrate, when subjected to an identical single meal size prior to the evaluation efficient RFI cattle displayed a faster nutrient absorption in comparison to inefficient one. This enhanced response appeared to lead to a faster activation of muscle protein synthesis in efficient RFI cattle. Lower protein degradation and faster protein synthesis rates may account for the leaner carcasses typically observed in efficient RFI cattle.

Ethics approval

The protocol for this study was approved by the Ethics Committee of the Auvergne-Rhône-Alpes region and the French Ministry of Higher Education, Research and Innovation (Authorisation number: APAFIS #16194-2016101016361277 v6 delivered on 14th January 2019). The experiment was conducted at INRAE, Centre Auvergne-Rhône-Alpes, France.

Table 1

 1 Dietary characteristics of experimental concentrates given to young Charolais bulls 3 h before the slaughter.

	Item	High-fibre and low-protein concentrate	High-starch and high-protein concentrate
	Ingredient composition, kg DM
	Wheat grain	0.50	1.48
	Beet pulp	1.62	-
	Soybean meal	0.37	0.99
	Bicarbonate	-	0.20
	Minerals and vitamin mix*	0.10	0.10
	Chemical composition (g/kg of DM)
	OM	91.7 ± 0.37	93.2 ± 0.22
	CP	167 ± 0.38	289 ± 1.41
	NDF	311 ± 1.31	130 ± 3.15
	ADF	160 ± 1.10	52.9 ± 0.21
	Starch	131 ± 0.69	374 ± 2.07
	Starch/NDF (g/g)	0.42 ± 0.00	2.96 ± 0.87
	In vitro enzymatic digestibility 1 (%)
	DM	92.1 ± 0.39	94.9 ± 2.12
	Organic matter	92.9 ± 0.33	95.2 ± 1.20
	Feed Values 2
	Net Energy (Mcal/kg)	2.00 ± 0.00	2.22 ± 0.01
	MP (g/kg of DM)	118 ± 2.71	153 ± 8.58
	MP/Net energy (g/Mcal)	58.9 ± 0.20	68.9 ± 1.14
	RPB (g/kg of DM) 3	À3.87 ± 1.80	84.1 ± 4.95

Abbreviations: MP = metabolisable protein; RPB = rumen protein balance.

* Minerals and vitamin mix: 5% P, 25% Ca, 8% Mg, 0Á2 % Na, vitamin A (210 mg/kg), vitamin D3 (3.75 mg/kg), vitamin E (2 730 mg/kg) and vitamin B1 (4.5 mg/kg).

1 Estimated from enzymatic digestion with pepsin-cellulase

[START_REF] Aufrère | Etude de la prévision de la digestibilité des fourrages par une méthode enzymatique (Study of the forecast of the digestibility of fodder by an enzymatic method)[END_REF]

.

2 Feed values were estimated from chemical composition and proportion of ingredients (INRA, 2018).

3 Rumen protein balance, representing nitrogen intake minus non-ammonia nitrogen at the duodenum (INRA, 2018).

Table 2

 2 Postprandial plasma metabolites and hormones concentrations from extreme residual feed intake (RFI) young Charolais bulls fed an identical single meal size of either a highfibre and low-protein concentrate or high-starch and high-protein concentrate 3 h before slaughter. Values with different letters on the same row are significantly different (P < 0.05). Relative abundances of main proteins involved in the skeletal muscle protein degradation and synthesis pathways from extreme residual feed intake (RFI) young Charolais bulls fed an identical single meal size of either a high-fibre and low-protein concentrate or high-starch and high-protein concentrate 3 h before slaughter.

	Diet	High-fibre and low-	High-starch and high-		P-value
		protein concentrate	protein concentrate
	RFI group	Low RFI	High RFI	Low RFI	High RFI	SEM	Diet	RFI	Diet  RFI
	Number of animals	8	8	8	8
	Metabolites
	Glucose, g/L	0.61	0.65	0.66	0.55	0.039	0.64	0.53	0.08
	BHB, mmol/L	0.31	0.25	0.32	0.26	0.022	0.52	0.03	0.76
	Urea, g/L	0.17	0.16	0.19	0.18	0.014	0.13	0.43	0.93
	Hormones
	IGF-1, ng/mL	334	301	363	354	31.9	0.18	0.57	0.77
	Insulin, lU/mL	11.0 b	17.5 b	25.6 a	16.6 b	3.09	0.05	0.58	0.03
	Glucose to insulin ratio, g/lU	0.55	0.48	0.32	0.35	0.013	0.01	0.75	0.43
	Abbreviations: BHB = b-hydroxybutyrate.
	a,b Table 3
	Diet	High-fibre and low-protein	High-starch and high-		P-value
		concentrate		protein concentrate
	RFI group	Low RFI	High RFI	Low RFI	High RFI	SEM	Diet	RFI	Diet  RFI
	Number of animals	8	8	8	8
	Protein degradation
	AMPK	1.00	0.94	1.04	1.00	0.051	0.33	0.30	0.88
	LC3	1.03	1.00	0.98	1.02	0.061	0.83	0.91	0.62
	UBA1	0.79	1.31	0.87	1.12	0.173	0.74	0.04	0.47
	Protein synthesis
	Akt	1.02	1.11	0.84	1.00	0.060	0.06	0.05	0.65
	p-Akt	0.95	1.20	0.87	0.96	0.095	0.12	0.11	0.44
	p-Akt:Akt	0.94	1.06	1.02	0.96	0.081	0.87	0.76	0.26
	mTOR	0.92	1.17	0.86	0.97	0.124	0.25	0.10	0.54
	p-mTOR	0.93	1.26	0.74	1.21	0.159	0.23	0.01	0.39
	p-mTOR:mTOR	1.13	1.15	0.76	1.28	0.198	0.46	0.10	0.09
	S6K	0.81	1.03	0.87	1.35	0.205	0.31	0.07	0.50
	p-S6K	0.79	0.89	1.14	1.25	0.233	0.14	0.67	0.98
	p-S6K:S6K	1.18	1.01	1.48	0.88	0.187	0.64	0.05	0.26

Abbreviations: AMPK = Adenosine monophosphate-activated protein kinase; LC3 = Microtubule-associated protein 1A/1B-light chain 3; UBA1 = Ubiquitin-like modifier activating enzyme 1; Akt = Serine/threonine protein kinase; p-Akt = Phosphorylated serine/threonine protein kinase; mTOR = Mammalian target of rapamycin; pmTOR = Phosphorylated mammalian target of rapamycin; S6K = Ribosomal protein S6 kinase; pS6K = Phosphorylated ribosomal protein S6 kinase. et al., 2022) in agreement with conclusions found in dairy cows

Acknowledgements

The authors gratefully acknowledge the staff of the Experimental Unit Herbipôle (INRAE, Saint-Genès-Champanelle, France) for their diligent care of the experimental animals. The authors are also grateful to Arnaud Delavaud for helping us with IGF-1 analysis. This study was part of the PhD thesis of the first author (Guarnido Lopez, 2022).

Data and model availability statement

The data were not deposited in an official repository. Data are confidential but available to reviewers upon request.

Declaration of Generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the corresponding author used Chat GPT in the writing process in order to improve English grammar. After using this tool/service, the author reviewed and edited the content as needed, and takes full responsibility for the content of the publication.

Author ORCIDs

Guarnido-Lopez, P: https://orcid.org/0000-0002-5013-0888. Ortigues-Marty, I: https://orcid.org/0000-0002-0399-013X. David, J: https://orcid.org/0000-0002-5992-4190. Polakof, S: https://orcid.org/0000-0002-0976-8732. Cantalapiedra-Hijar, G: https://orcid.org/0000-0001-9486-8238.

Author contributions

Guarnido-Lopez, P: Conceptualisation, Formal analysis, Writing -Original Draft, Writing -Review & Editing.

Ortigues-Marty, I: Conceptualisation, Formal analysis, Writing -Review & Editing, Supervision.

David, J: Conceptualisation, Methodology.

Polakof, S: Conceptualisation, Formal analysis, Writing -Review & Editing.

Cantalapiedra-Hijar, G: Funding acquisition, Conceptualisation, Formal analysis, Writing -Review & Editing, Supervision.

Declaration of interest

None.

Financial support statement

The experimental trial was funded by APIS-GENE (Paris, France) as part of the EffiScience project. The first author received a scholarship from INRAE Phase division and APISGENE to conduct his PhD thesis.


Astrid Klipfel
email: astrid.klipfel@univ-artois.fr

Yaël Frégier
email: yael.fregier@univ-artois.fr

Adlane Sayede
email: adlane.sayede@univ-artois.fr

Zied Bouraoui
email: zied.bouraoui@univ-artois.fr

Optimized Crystallographic Graph Generation for Material Science

Graph neural networks are widely used in machine learning applied to chemistry, and in particular for material science discovery. For crystalline materials, however, generating graph-based representation from geometrical information for neural networks is not a trivial task. The periodicity of crystalline needs efficient implementations to be processed in real-time under a massively parallel environment. With the aim of training graph-based generative models of new material discovery, we propose an efficient tool to generate cutoff graphs and k-nearest-neighbours graphs of periodic structures within GPU optimization. We provide pyMat-Graph a Pytorch-compatible framework to generate graphs in real-time during the training of neural network architecture. Our tool can update a graph of a structure, making generative models able to update the geometry and process the updated graph during the forward propagation on the GPU side.

Introdution

New materials discovery is a fundamental challenge in material sciences where high-throughput screening based on machine learning models is largely employed to obtain materials with desired properties. Crystalline (crystal) material generation has recently received considerable attention, e.g. [START_REF] Xie | [END_REF][START_REF] Gibson | [END_REF][START_REF] Klipfel | [END_REF]. In our setting, we are interested in generating new crystal materials for developing new solar panels with a band gap enabling hydrolyse. This helps to solve problems related to clean energy production and storage, which is one of the major challenges facing our society. It can also be used to produce hydrocarbons from CO2, helping to reduce the carbon footprint of human activities.

From organic chemistry to material science, Graph Neural Networks (GNN) have received increasing attention in a variety of tasks such as classification [START_REF] Schütt | [END_REF]Jørgensen et al., 2018;Gasteiger et al., 2020b;Gasteiger et al., 2020a;Chen et al., 2019;Choudhary and DeCost, 2021;[START_REF] Klicpera | [END_REF] and generation [Satorras et al., 2021;[START_REF] Xie | [END_REF][START_REF] Long | [END_REF][START_REF] Kelvinius | Graph-based machine learning beyond stable materials and relaxed crystal structures[END_REF][START_REF] Gibson | [END_REF]. Notice that organic molecules are composed of wide carbon chains with a limited variety of atoms, while crystal materials are three-dimensional periodic structures composed of a wide variety of chemical bonds and atoms. The periodic structure of crystals is often represented as a parallelepiped tiling, a.k.a crystal lattice or unit cell. While generating graph-based representations of organic molecules is straightforward, the periodic structure of crystals makes difficult graph processing when training a generative model, and in particular when a massively parallel environment is required. More precisely, generative models may update the geometry of a chemical structure during forward propagation. However, since the graph associated with a given structure is built from the local environment of atoms, a modification of the geometry leads to the modifications of the graph associated with the structure. Consequently, building a generative model with a dynamic graph is hard to achieve on a periodic structure compared to organic molecules.

When training graph-based generative models for material discovery, cutoff distance is a commonly used technique [START_REF] Schütt | [END_REF]Gasteiger et al., 2020b;Jørgensen et al., 2018]. It designates a relative distance threshold value above which no interaction between nodes is considered. In the same vein, [Jørgensen et al., 2018;Chen et al., 2019] suggests that k-nearest-neighbours (KNN) graphs can also be a good choice for GNN models. KNN-graph is a type of graph where all the nodes are connected to the k-nearest nodes. When processing small molecules, any naive strategy of computing the interatomic distances is feasible, allowing to compute KNN or cutoff graph in a short amount of time and reasonable memory. However, for periodic structures which are infinite, the search area should be carefully selected to avoid unnecessary calculation and memory saturation. In fact, the volume of the search space expands with the cube of the search radius. As such, possible graphs should be generated in milliseconds to be usable in practice during the training process. Moreover, a periodic structure is represented with a multi-graph where a given node can share multiple edges with another and with itself which brings more complexity to the graph generation process. Finally, for big structures, a processing strategy suitable for massively parallel environments should be used in order to deal with a batch of multiple structures at the same time.

Figure 1: Example of a crystal composed of two atoms, atom 0 is in the centre of the cell and atom 1 is in the bottom left corner. The edges of the graph Γ1 associated with a material are composed of the index of a source node, the index of the target node and the coordinate of the cell of the targeted node.

To address the aforementioned issues, we propose an efficient tool that solves KNN and cutoff graph generation for crystalline materials. We provide a compatible implementation with PyTorch that performs on GPUs 1 . We used an approach inspired by the KD-tree search algorithm adapted for periodic structures and propose a data structure adapted to massively parallel environments (GPUs) that effectively keeps track of the KNN of each atom. We empirically show the benefits of using our tool.

Crystallographic Graph Generation

A crystalline structure can be defined with a cloud of atoms and a repetition pattern that represent periodicity. The repetition pattern is often described as a parallelepiped called a lattice or a cell. The periodic structure is obtained with the tiling of the space by the crystal cell. Consequently, a given atom inside of the cell has multiple positions because of the tiling in space and the local environment of an atom which can overlap with adjacent repetition.

Crystallographic Graph

We follow [START_REF] Klipfel | [END_REF] to define the graph associated with a crystal material as an oriented graph where each edge is represented by triplets containing the index of the source node, the index of the destination node and the relative cell coordinate of the destination node. Figure 1 illustrates this representation. Notice that the definition of graph provided in [START_REF] Klipfel | [END_REF] generalizes to most of the graph definitions proposed in previous works [Jørgensen et al., 2018;Chen et al., 2019;Satorras et al., 2021]. We now recall the formal definition of crystalline structure [START_REF] Klipfel | [END_REF].

Definition 1. The representation space of featured materials M F is the disjoint union n∈N M F n where:

M F n = (ρ, x, z) | ρ ∈ GL d (R), x ∈ [0, 1[ n×d , z ∈ F n
Chemical materials are represented in M = M N , with atomic numbers as feature sequence z.

M F n is an infinite set of triplet ρ, x, z that represents all possible materials with n atoms. The atomic number has a chemistry reference, e.g. 1 for hydrogen or 6 for carbon.

1 Code available at https://github.com/aklipf/mat-graph Figure 2: The searching procedure continues while the evaluated area (in the parallelogram with continuous lines) doesn't fully overlap with the search area (represented as a red circle). The next evaluated area (in dotted lines) expends in the direction where the search area is not yet evaluated.

Definition 2. We call directed 2-graph Γ = (Γ 0 , Γ 1 , Γ 2 ) a triplet of sets together with applications:

• π 1 : Γ 1 → Γ 0 × Γ 0 , written π 1 (γ) = (src(γ), tgt(γ)) • π 2 : Γ 2 → Γ 0 × Γ 0 × Γ 0 We call Γ a directed 1-graph when Γ 2 = ∅.
The aforementioned graphs are often called multi-graphs or hyper-graphs because they generalise 1-graphs to dimensions ≥ 1. They are directed because we do not assume any symmetry on Γ w.r.t vertice permutations. Recall that π 1 and π 2 may not be injective.

Definition 3. Let M = (ρ, x, z) in M F
n be a material and c i > 0 for 1 ≤ i ≤ n denotes cutoff distances. We define a directed 2-graph Γ = Γ M,c by the graded components:

• Γ 0 = {1, . . . , n} • Γ 1 = (i, j, τ ) ∈ Γ 0 ×Γ 0 ×Z d ||ρ(x j -x i +τ )|| < c i • Γ 2 = (γ, γ ′ ) ∈ Γ 1 × Γ 1 src(γ) = src(γ ′ )
This graph construction includes many definitions of material graphs, making it versatile and usable in most contexts. This definition includes a graph built from a constant cutoff distance (i.e. c i is constant), a graph built from k nearest neighbour or built from chemical properties such as the covalent radii. For more detail, we refer to [START_REF] Klipfel | [END_REF].

Generation Process

To handle the periodic nature of crystalline, we adapt our graph generation process to work in a torus space. To this end, graph generation is performed by exploring the direct repetition of a cell where we start by evaluating the adjacent cell and extend the search area until we find all the edges. Our graph generation method is built upon two main parts: a searching algorithm and an ordered stack. Combined, the generation process follows an iterative process limiting the RAM usage by splitting the search area. Our generation process remains fast since only a few iterations are required, avoiding useless search areas.

Searching procedure Our search procedure is based on a classic KD-tree search strategy. As shown in Figure 2, a search radius is used to represent the area where connected nodes can exist. On the other side, we expand the explored area up to a search radius. As the search radius is defined with the KNN in the case of a KNN-graph, the search radius decreases over time when a new area is explored. The search procedure pseudo-code is given by Algorithm 1.

Algorithm 1 KNN graph generation algorithm Input: k: the k nearest connected atoms ρ: the shape of the lattice of the crystal x: the position of the nodes inside the lattice Output: a set of edges 1:

d max i ← ∞ 2: border ← (0, 0, 0) 3: Γ 1 ← {∅} 4: while any(d max i > closest distance(border, ρ)) do 5: extension ← next evaluated area(border, d max i ) 6: γ ← evaluate area(ρ, x, extension) 7: Γ 1 ← push nearest(Γ 1 , γ, k) 8:
border ← border + extension 9:

d max i ← max (i ′ ,j,τ )∈Γ1|i ′ =i d i ′ jτ 10: end while 11: return Γ 1
Ordered stack To keep track of the k closest points already discovered by our search procedure, we proposed an efficient data structure to store points. Our ordered stack first concatenates new data and then sorts them by distance. After that, the edges are filtered to keep only the KNN in the case of a KNN-graph or the edges under a given cutoff distance in the case of a cutoff-graph.

Algorithm 2 Push edges in and ordered stack Input: k: the k shortest edges Γ 1 : a list of edges γ: the list of edges to merge Output:

the list of the k shortest edges 1: Γ ′ 1 ← Γ 1 ∥ γ 2: Γ ′ 1 ← sort by distance(Γ ′ 1 ) 3: Γ ′ 1 ← stable sort by source index(Γ ′ 1 ) 4: d k i ← k nearest distance i (Γ ′ 1 ) 5: Γ ′ 1 ← {(i, j, τ ) ∈ Γ ′ 1 |d (i,j,τ ) ≤ d k i } 6: return Γ ′ 1
In addition to graph generation, our tool provides additional functionalities such as:

• Symmetric graph: as some GNN require symmetric directed graphs to perform specific message-passing schema, our tool includes a procedure that makes a given graph symmetric by adding missing edges while guaranteeing the uniqueness of the edges. during inference [START_REF] Klipfel | [END_REF][START_REF] Xie | [END_REF][START_REF] Klicpera | [END_REF].

Performance Evaluation

To evaluate the performance of our tool, we conducted experiments on Materials project [Jain et al., 2013] which is a dataset composed of 133420 crystalline materials studied with ab inito calculation. We considered the same setting as [START_REF] Xie | [END_REF] where structures composed of more than 64 atoms are removed since they are in general considered outliers. The experiments are performed on an Nvidia quadro RTX 8000 GPU.

CPU vs GPU We compared the time required to process all the structures for our tool with and without GPU optimization. We used a fixed batch size of 256 structures and generated the structures for the 16, the 32 and the 64 nearest neighbours of atoms. As shown in table 1, the KNN-graph generated on GPU is up to 40 times faster than an equivalent CPU library.

Complexity, inference time and RAM usage

To check the time complexity of our method, we compare the generation time of one batch with various KNN settings in Table 1 and batch size in Table 2). Experiments on batch size have been performed for a KNN-graph with 32 neighbours.

Conclusion

We propose an efficient tool to convert crystalline materials into graphs. Our library allows for reducing the time spent during prepossessing. More importantly, the graph conversion is quick enough to be used during the training process without the prepossessing step and updates the graph while updating the geometry of a given structure. Our tool opens new possibilities in generative networks for material science.

Table 1 :

 1 Processing time of 119701 filtered structures in seconds for CPU, GPU and Batch configurations. KNN denotes the number of neighbours in the graph while the cutoff distance defines a radius (in Angstrom) inside which all atoms are connected. Batch corresponds to the generation time for a single batch of 256 structures.

	• Triplets generation: We provide an implementation to
	generate triplets composed of two edges sharing the
	same source nodes during the run-time. This task is
	important because recent works use triplets information

Table 2 :

 2 Batch time corresponds to the building time of the graph for a single batch. Total time refers to the time required to process the entire dataset. RAM to the average GPU RAM used by our graph generation tool.

Acknowledgments

This work has been supported by ANR-22-CE23-0002 ERI-ANA, ANR-20-THIA-0004 and by HPC resources from GENCI-IDRIS (Grant 2022-[AD011013338]).