Quentin Mercier

Amandine Aftalion
email: amandine.aftalion@ehess.fr

Optimal speed in Thoroughbred horse racing

The objective of this work is to provide a mathematical analysis on how a Thoroughbred horse should regulate its speed over the course of a race to optimize performance. Because Thoroughbred horses are not capable of running the whole race at top speed, determining what pace to set and when to unleash the burst of speed is essential. Our model relies on mechanics, energetics (both aerobic and anaerobic) and motor control. It is a system of coupled ordinary differential equations on the velocity, the propulsive force and the anaerobic energy, that leads to an optimal control problem that we solve. In order to identify the parameters meaningful for Thoroughbred horses, we use velocity data on races in Chantilly (France) provided by France Galop, the French governing body of flat horse racing in France. Our numerical simulations of performance optimization then provide the optimal speed along the race, the oxygen uptake evolution in a race, as well as the energy or the propulsive force. It also predicts how the horse has to change its effort and velocity according to the topography (altitude and bending) of the track.

Introduction

Very little is known about the optimal strategy for a Thoroughbred horse to run and win a race. Because the racing career of a Thoroughbred horse is not so long, and therefore the number of racing opportunities is limited, any information that can help to determine a horse ability according to the race distance or to optimize how to regulate its speed along the race can be crucial.

Due to limitations in the measurement of the mean oxygen uptake ( V O2) for a horse at high exercise, no information is available on the full V O2 profile in a race, depending on the distance. Up to now, for Thoroughbreds, only measurements on treadmills have been obtained using masks [START_REF] Eaton | Maximal accumulated oxygen deficit in Thoroughbred horses[END_REF][START_REF] Gauvreau | Comparison of aerobic capacity between racing standardbred horses[END_REF][START_REF] Rose | Maximum O2 uptake, O2 debt and deficit, and muscle metabolites in Thoroughbred horses[END_REF][START_REF] Mukai | Effects of three warm-up regimens of equal distance on V O 2 kinetics during supramaximal exercise in Thoroughbred horses[END_REF]. Nevertheless, a portable mask technology has been developed [START_REF] Art | Validation of a portable equine metabolic measurement system[END_REF], but to our knowledge no track tests have been yet performed on Thoroughbred racehorses. It is only for Standardbred and endurance horses that some study have been made, see [START_REF] Cottin | Oxygen consumption and gait variables of Arabian endurance horses measured during a field exercise test[END_REF] for instance. A review book on horse physiology is [START_REF] Hodgson | The athletic horse: principles and practice of equine sports medicine[END_REF] and some information can also be found in the report [START_REF] Evans | Training and Fitness in Athletic Horses A report for the Rural Industries Research and Development Corporation[END_REF]. What is known is that horses have a high aerobic capacity, about twice that of human beings, due to a high capacity for oxygen carriage and extractions, as well as a high stroke volume [START_REF] Hodgson | Principles and Practice of Equine Sports Medicine[END_REF][START_REF] Gerard | Energetic considerations of exercise. The Athletic Horse: Principles and Practice of Equine Sports Medicine (Second Edition)[END_REF]. They reach the maximal oxygen uptake ( V O2 max ) much quicker than humans, nevertheless no precise estimate of the time needed to reach a steady state V O2 is known. Similarly, no information is available about the distance or time at which the V O2 starts decreasing at the end of a race. Eventually, for long distances, it is not known whether the V O2 remains almost constant for the most part of the race January 6, 2021 1/17

or oscillates around a mid value, and then at which period and at which amplitude.

Estimates on the proportion of energy derived from aerobic and anaerobic pathways during competitive events have been made according to breed and length of race [START_REF] Gerard | Energetic considerations of exercise. The Athletic Horse: Principles and Practice of Equine Sports Medicine (Second Edition)[END_REF] but the relationship that exists between performance and anaerobic capacity remains to be determined. This paper will provide pieces of information for all these issues. Reference [START_REF] Spence | Speed, pacing strategy and aerodynamic drafting in Thoroughbred horse racing[END_REF] is the only one that we know of where pacing strategy for horses is analyzed, together with the effect of drafting. Several directions of study have been investigated to better understand the effort or mechanical work developed by horses. One of them is to measure the propulsive force either on a force plate [START_REF] Davies | External mechanical work in the galloping racehorse[END_REF] or with an instrumented horse shoe [START_REF] Robin | Ground reaction forces in the horse at the walk, trot and gallop measured with an instrumented shoe[END_REF][START_REF] Roland | Design and demonstration of a dynamometric horseshoe for measuring ground reaction loads of horses during racing conditions[END_REF]. More on the biomechanics of athletic horses can be found in [START_REF] Barrey | Biomechanics of locomotion in the athletic horse[END_REF] for instance.

The aim of this paper is to provide a mathematical model able to predict how a Thoroughbred horse should regulate its speed over the course of a race in order to optimize performance. We will see how this depends on the distance to run, but also on the shape and topography of the track. It is based on the model developed in [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF][START_REF] Aftalion | How to run 100 meters[END_REF][START_REF] Aftalion | Optimizing running a race on a curved track[END_REF] for human races and it is adapted here to horses to fit the data.

Based on data provided by France Galop and the Mc Lloyd tracking device in French horse races, we are able to model the optimal horse efforts and velocity for a fixed distance, depending on the curvature and change of slopes and ramps. Our model yields in particular information on the V O2 profile.

Materials and methods

Data

The data consist of two dimensional position and speed sampled at 10 Hz for horses racing in Chantilly (France), at the end of the 2019 Thoroughbred horse racing season. The races were all run on a PSF track in a standard surface condition. The data are provided by France Galop the French operational body for flat horse racing , and are from roughly ten races. The tracking system is developed by Mc Lloyd. It is a miniaturized device which does not bring any discomfort to the horse or the jockey. It weighs 90g and is put by the jockey under the saddle. Reliable data is obtained thanks to a patented positioning technology and robust mobile network data transmission, even during crowded events. The accuracy of the system was validated on horses by France Galop by comparing 1ms-accurate photo finish data on more than one hundred races with gap data on the finish line obtained after processing latitude and longitude data: the device mean accuracy was confirmed to be 25cm or 2 hundredth of a second. It is now used by France Galop on all races to provide live position information to the audience. The tracking system provides the latitude and longitude data sampled at 10Hz, as well as the velocity. The latitude and longitude data given by the tracker are projected over a reference track leading to the position of horses in the race which is therefore given as live information to the audience on all races.

Model

Once we have these raw data, we have to smooth the speed using a third order Savitzky-Golay filter. Therefore, the raw data provide two curves for each horse sampled at 10Hz:

• the curve of time vs distance from start, projected on the reference track, so that each horse runs the same distance,

• the curve of velocity vs distance from start, projected on the reference track.

January 6, 2021 2/17

These curves will allow to determine all the parameters specific to horses. The model of [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF][START_REF] Aftalion | How to run 100 meters[END_REF][START_REF] Aftalion | Optimizing running a race on a curved track[END_REF] yields an optimal control problem based on a system of coupled ordinary differential equations for the instantaneous velocity v(s), the propulsive force per unit of mass f (s), the anaerobic energy e(s), where s is the distance from start. The system relies on Newton's second law of motion and the energy balance which takes into account the physiology of horses. The energy balance is between the aerobic contribution V O2 or σ(e), the anaerobic contribution e(s) and the power developed by the propulsive force. The mechanical part takes into account the positive slope or negative ramp α(s), the bending of the track and the control on the variations of the propulsive force. A crucial piece of information to be taken care of is the centrifugal force in the bends. For this purpose, the horse is identified with a bending rod. The centrifugal force does not act as such in the equation of movement but limits the propulsive force through a constraint which yields a decrease in the effective propulsive force in the bends.

The physiology of the horse is taken into account through a number of parameters:

• the maximal propulsive force per unit of mass f M ,

• the global friction coefficient τ which encompasses all kinds of friction, both from joint and track. In total, f M τ is the maximal velocity,

• the maximal decrease rate and increase rate of the propulsive force which is related to the motor control of the horse: a horse, like a human being, cannot stop or start its effort instantaneously, but needs some time or distance to do it. This is what our control parameters u -and u + will provide,

• the total anaerobic energy or maximal accumulated oxygen deficit e 0 ,

• the V O2 profile as a function of distance, namely the distance d 1 at which the maximum of V O2 is reached, the distance d 2 at which V O2 decreases, and the relative decrease with respect to the maximum value. This is a curve σ(s) where s is the relative distance from start, but in fact, in the model, it is a curve σ(e(s)) where e(s) is the remaining anaerobic energy. The profile of σ is to be identified from the data.

These parameters are not measured or given but they are computed numerically for each horse and each race: they are the parameters that allow the optimal control problem to best fit the data, as we will explain below. Therefore, for a fixed distance, the model predicts the final time, the velocity curve and the effort developed by the horse to produce the optimal strategy. It depends on the geometry of the track, the ramps and slopes and the physiology of the horse.

For the ease of completeness, we provide below the full optimal control problem though the results of the paper do not require to understand it and the reader can skip this paragraph as a first reading. Instead of writing the equations of motion in the time variable, we write them using the distance from start s. This amounts to dividing by v the derivatives in time in order to get the derivatives in space. We also write the equations per unit of mass. Let d be the length of the race and g = 9.81 the gravity. Let c(s) denote the curvature at distance s from the start, which is provided for each track and α(s) be the slope coefficient. Let v 0 be the initial velocity. Let e 0 , f M , τ , u -, u + and the function σ be given. They are identified for a specific race and horse. The optimal control problem (where minimizing the final time is equivalent to minimizing the integral of the inverse of the velocity) coming from the Newton law of January 6, 2021 3/17 motion and the energy conservation is

min v,f,e,u d 0 1 v(s)
ds, where

v ′ (s) = 1 v(s) - v(s) τ + f (s) -gα(s) , v(0) = v 0 , e ′ (s) = σ(e(s)) -f (s)v(s) v(s) , e (0)
= e 0 , e(s) ≥ 0, e(d) = 0,

f ′ (s) = u(s) v(s) , u -≤ u(s) ≤ u + ,
and under the state constraint

f (s) 2 + v(s) 4 • c(s) 2 ≤ f 2 M for s ∈ [0, d].
The optimal control problem for horse performance is solved using Bocop, an open licence software developed by Inria-Saclay France [START_REF] Bonnans | BOCOP -A toolbox for optimal control problems[END_REF]. The discretisation used to solve the system of coupled ODE's is set to one point every two meters which ensure the same accuracy whatever the race. The solution of the optimal control problem yields, for each race and each horse, the optimal effort, regulation of velocity and the V O2 profile depending on the length and topography of the race. Our aim is therefore to identify the physiological parameters f M , τ , u + , u -, e 0 , σ(e) from the available data.

Identification process

The identification process is made through a bi-level optimization procedure looking for minimizing errors between the response of a Bocop simulation and the data through the following objective

min p |t simu (d) -t data (d)| + 1 N N i=1 (v simu (s i ) -v data (s i )) 2 ,
where the subscript simu (resp. data) refers to the variable extracted from the simulation (resp. data). The distance s 1 = 0 is the beginning of the race and s N = d is the length of the race, while s i are intermediate distances. The parameter p refers to the vector containing all the variables to identify:

p = (τ, e 0 , f M , d 1 , d 2 , σ M , σ f , u -, u + ),
where σ M is the maximal value of σ and σ f its final value. The objective is made up of two parts: first the difference in final time at the end of the race d, and then the mean square error over the speed measured at N points. For our identification process, N is taken equal to one thousand which ensures a good accuracy for the objective calculation (since it is one point every 2 meters for the longest race as the Bocop calculation) while keeping a relative low time cost. The points are evenly distributed between s 1 and s N .

The algorithm used here is a particle swarm optimization method [20] available in the pyswarm library in Python 3, part of the family of the heuristic optimization methods [START_REF] Wahab | A Comprehensive Review of Swarm Optimization Algorithms[END_REF]. The main advantage of such a method is its good ability to explore the design space and its ease of use and implementation. A swarm of designs {p i } is tested, that is the optimal control problem is solved for these parameters using Bocop, and its objective value is calculated. The performance influences the speed and the January 6, 2021 4/17 direction of the particles inside the design space for the next iteration. At the end of the process, the best score particle is kept as the result. The stopping criterion of the algorithm is set such that the algorithm is unable to find new particles for which the objective is at least 10 -7 better than the best score observed until then. For all the examples treated in this paper, the swarm size is set to 50 particles and a maximum of 150 iterations. Each identification process has reached the stopping criterion described. This method is well suited to our problem since the objective space has a lot of local minimizers so that gradient based methods can get stuck in local minimizers. It insures overall robustness in the results as the design space is always well explored before converging toward a particular area of the design space.

Topography of the tracks

We have studied three types of races: a 1300 meters, a 1900 meters and a 2100 meters in Chantilly. The GPS track is shown in Fig 1 . The 1300m starts with a straight, then the usual way to match smoothly a straight and an arc of a circle since the curvature increases linearly. It is used for train tracks and roads as well. It allows smooth variations of velocities which are more comfortable for horses. The specificity of the track is that there is a bend of 500 meters before the final straight, where the track is going down in the first quarter (about 1.5%) and going up in the second quarter (about 2%). In the 2100 meters, there is also a bend of 400 meters just after the start, which is first going up and then down. We will see that curvature and altitude have a strong effect on the optimal velocity.

Let us point out that the track is banked but, because data correspond to horses close to the inner part of the track, the banking is not meaningful for the data and will not be taken into account here.

Results

Among all the available data, we have watched the videos of the races and chosen three races and three horses with the following criteria: no interactions (or at least very little) with other horses that modified the speed, no specific strategy from the jockey to regulate the speed. Therefore, we have chosen horses which seemed to be close to have run a race which would have been similar if they were alone, and could be qualified as their optimal race. We are going to present three significant races of 1300m, 1900m and 2100m. They are all races on a PSF track in a standard surface condition. The 1300m was for 2-year-old horses, the 1900m for 3-year-old and the 2100m for 4-year-old.

We describe below the results of our simulations.

meters

The parameters identified for this race are in Table 1. The velocity data (raw and Table 1. Identified parameters for the 1300m. We measure σ in J/s/kg but we want to plot the results in terms of ml/mn/kg knowing that one liter of oxygen produces roughly 21 kJ (see [START_REF] Hanak | A field study of oxygen consumption and estimated energy expenditure in the exercising horse[END_REF] and also [START_REF] Peronnet | Table of nonprotein respiratory quotient: an update[END_REF]), so we have to multiply our data for σ by 60/21. For a maximal value of σ equal to 47, this yields a V O2 max of 133.6 ml/mn/kg. We see that the V O2 is increasing for about 400 meters, while the force is decreasing. Then when the V O2 decreases, the force and thus the velocity increase until 900 meters when the slope and end of race lead to a )

τ e 0 f M d 1 d 2 σ M σ f u - u + 3

Raw data Smoothed data Identification

decrease of force and velocity. We point out that the value of the propulsive force is higher than the ones found in [START_REF] Robin | Ground reaction forces in the horse at the walk, trot and gallop measured with an instrumented shoe[END_REF][START_REF] Roland | Design and demonstration of a dynamometric horseshoe for measuring ground reaction loads of horses during racing conditions[END_REF] but the velocity is also much higher. Distance (m)

F orce (N • m -1 ) (b) fM ftot f (s)

meters

The parameters identified for this race are in Table 2. The velocity data (raw and Table 2. Identification parameters for the 1900 meters. We observe that there is a strong start with the maximal velocity being reached in 300 meters. Then the velocity decreases. Between 900 and 1400m, we see the effect of the bend: at the beginning of the bend, the track is going down and the horse slightly speeds up; then the centrifugal force reduces the velocity but the velocity increases again at the end of the bend. The end of the race is with a strong January 6, 2021 7/17

τ e 0 f M d 1 σ M d 2 σ f u - u + 3
F orce (N • m -1 ) (b) fM ftot f (s)
acceleration before the final slight slow down. The V O2 curve vs distance and propulsive force vs distance are plotted in Fig 6 . We see that V O2 is increasing for about 400 meters, while the force starts at maximal value. Then, the V O2 is constant, the force and the velocity decrease to a mean value. At the end, the V O2 decreases when the residual anaerobic energy reaches a third of its initial value. The effect of the bend and centrifugal force are obvious: it leads to a decrease in propulsive force and velocity. We see on the force profile that there is a very strong acceleration in the end. It can only take place after the bend where the centrifugal force reduces the available propulsive force. In Fig 7, we have plotted a zoom on the velocity curve for the identified parameters, and then have removed the effect of the slope (flat track), of the curvature (straight track) and of both (flat, straight track). This allows to notice the specific effects: a bend reduces strongly the velocity (pink curve); on the real track (brown curve), because the first part of the bend is going down, the reduction in propulsive force and velocity is not so strong. The end of the track is uphill and one notices that the velocity curves corresponding to going up (brown, red) cannot provide a speeding up as high as the two others. The combination of slopes and ramps of this track (red curve) reduce the velocity and final time in total though the profile is very similar. We also point out that though these are local effects, they have a global influence on the strategy since they change the mean velocity.

meters

The parameters identified for this race are in Table 3. The velocity data and the Table 3. Identification parameters for the 2100 meters. Here, the V O2 is modified to match the behaviour observed in [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the VO2 response[END_REF] for humans where the V O2 first reaches a peak value, before the mean race value. The first bend going up requires a rise of V O2 at the beginning of the race. We observe that there is a strong start with the maximal velocity being reached in 200 meters. Then the velocity decreases and reaches a plateau. This plateau has been analyzed for human race in [START_REF] Aftalion | Pace and motor control optimization for a runner[END_REF] and is related to a turnpike phenomenon. It is very likely that the horse optimal velocity for long races can be analyzed with this mathematical tool as well.

τ e 0 f M d 1 σ 1 d 2 σ M d 3 σ f u - u + 3
F orce (N • m -1 ) (b) fM ftot f (s)
The first bend has a strong curvature and therefore reduces drastically the velocity as we can see in Fig 9b: the propulsive force is reduced in the first bend. In the last bend, as in the previous race, the velocity decreases and increases again at the end of the bend. The end of the race is similar to the 1300 meters, with a strong acceleration before the final slow down. The horse in this race is not as good in terms of performance as the one in the 1900m and he cannot maintain his velocity similarly at the end of the race.

Discussion

Results on V O2

From experiments on human races from 400m to 1500m (that is of similar duration of the races we analyze here) [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the VO2 response[END_REF][START_REF] Billat | Differential modeling of anaerobic and aerobic metabolism in the 800-m and 1,500-m run[END_REF], it is expected that the V O2 curve vs distance is

• increasing to a maximal value and then decreasing for short exercises,

• increasing and reaching the maximal value V O2 max , and then decreasing at the end of the race when the residual anaerobic energy is less than 30%,

• reaching a peak value which is higher than the value along the race for moderate length exercises.

Our simulations and identifications yields that the behaviour is the same for horses. The results of our simulations even provide precise information on the V O2 curve all along the race. We see in Fig 4a ,6a, and 9a, that

• the maximal value of O2 is reached in about 400m, that is about 20 to 30 seconds from start, which is indeed much quicker than humans, (this is consistent with [START_REF] Eaton | Maximal accumulated oxygen deficit in Thoroughbred horses[END_REF]),

• the 1300 meters is a short exercise where V O2 increases and decreases,

• for the 1900 and 2100 meters race, the V O2 reaches a mean value during the race and decreases about 500 meters from the finish line, when, as for humans, the residual anaerobic energy is about a third of the initial value. The launch of the sprint is optimal when the V O2 starts decreasing at the end of the race, and this follows from optimal control theory. Numerically, we can observe that the decrease in V O2 and increase in velocity at the end of the race take place at the same time, January 6, 2021 10/17

• in a 2100 meters race, the V O2 first reaches a peak value before decreasing to a plateau value.

The longer the horse can maintain its maximal value V O2 max , the better the performance is. Because the change of slope in V O2 is related not exactly to the distance but to the available stock in anaerobic capacity, a high anaerobic capacity is all the more important to maintain high velocities all along the race. A strong acceleration at the beginning of the race allows to reach the maximal value of V O2 quickly and is the best strategy. Jockeys often start slower than the optimal strategy being afraid that if the horse accelerates too strongly with respect to its capacity at the beginning of the race, then the drop in velocity at the end of the race will be bigger.

It is very likely that for horses, as for humans and explained in [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the VO2 response[END_REF], results on treadmills are very different from measurements during a race. Indeed, as pointed out in [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the VO2 response[END_REF], exercise at constant velocity yield contradictory results with exercises in a real race in terms of V O2 or pacing strategy. On a treadmill, the V O2 increases to reach a maximum value, whereas on a real race a decrease of V O2 is observed. Our numerical simulations obtained with varying parameters around the identified ones illustrate that in many cases, the best performance is achieved with a fast start, when the pace at the beginning of the race is higher than the pace at the end. While a fast start helps to speed up V O2 kinetics, and limits the participation of the anaerobic system in the intermediate part of the race, nevertheless, if it is too fast, it has the potential to cause fatigue and have an overall negative effect on the performance. Therefore, a departure which is too fast with respect to the horse's capacity increases the participation of the anaerobic system at the beginning of the race and can damage the final performance. Eventually, a fast start does not necessarily induce a good performance. But a high value of v V O2 max can allow a faster start velocity without increasing the O 2 deficit.

As soon as there is a slope or ramp, the V O2 is also impacted. Here, in Chantilly, the slope coefficients are strong enough to change the optimal velocity but not strong enough to modify the overall V O2 profile. The only effect is on the optimization of performace. For stronger slopes, one would need to take into account additionally that V O2 increases with slopes both downhill and uphill as explained in [START_REF] Self | Speed and incline during Thoroughbred horse racing: racehorse speed supports a metabolic power constraint to incline running but not to decline running[END_REF] and [START_REF] Rose | Kinetics of VO2 and VCO2 in the horse and comparison of five methods for determination of maximum oxygen uptake[END_REF].

Energy

Horses have two distinct types of energy supply, aerobic and anaerobic. In our model, it is e 0 which estimates the anaerobic energy supply. For human races, recent research suggest that energy is derived from each of the energy-producing pathways during almost all exercise activities [START_REF] Gastin | Energy System Interaction and Relative Contribution During Maximal Exercise[END_REF], which is what we also observe in our simulations.

In Table 4, we have computed the percentage of anaerobic energy to the total energy according to the length and duration of the race. The horse of the 1900m race has a very strong V O2 max and therefore uses a lower anaerobic energy. We point out that the values estimated in [START_REF] Eaton | Maximal accumulated oxygen deficit in Thoroughbred horses[END_REF] on a treadmill seem to be under estimated with respect to ours: for an exercise of duration 130 seconds, they find an anaerobic contribution of around 30%, which is smaller than our value. Indeed, in a race, for a similar duration of exercise, velocities and forces are much higher than on a treadmill, leading to a bigger contribution of the anaerobic supply [START_REF] Gastin | Energy System Interaction and Relative Contribution During Maximal Exercise[END_REF].

Fig 10 provides the evolution of the anaerobic energy of the three horses vs the distance to the finish line. We have highlighted the points where the V O2 changes slope. There is a strong anaerobic energy consumption both at the beginning and end of the race, that we can identify through strong slopes. The beginning of the race corresponds to the distance to reach V O2 max .

The horse of the 1300m is a young horse and he needs more time (or a bigger distance) to reach V O2 max (1214m vs 839 or 889 for the two others). The horse of the January 6, 2021 11/17 1900m has a higher V O2 max , therefore he needs less anaerobic energy when he is at V O2 max to maintain his speed. Indeed, once the maximal oxygen uptake is attained, the extra energy to increase speed further is provided by anaerobic pathways. So if a horse has a higher V O2 max , he will use less anaerobic energy to run at the same speed as a horse with a lower V O2 max . In total, this explains why the horse of the 1900m has a lower contribution of the anaerobic energy to the total energy. Green: horse of the 1300m, orange: horse of the 1900m, pink: horse of the 2100m. The distances are highlighted when the energy has a change of slope and the energy value at this point is indicated as well: it is at the beginning of the race when the V O2 max is reached, and at the end of the race, when the V O2 starts decreasing.

Effect of slopes, ramps and bends on a race

As evident from the data of [START_REF] Tan | Grip and limb force limits to turning performance in competition horses[END_REF], on a tight bend, horses slow down a lot. In Chantilly, the bends have a radius of at most 100m, which is not tight, but still has a strong impact on the optimal velocity.

To better illustrate this effect, we choose a race of 1900 meters and set an imaginary slope or ramp or bend for one third of the race at the beginning, middle or end of the race (that is roughly 630m) with the following configuration :

• either a positive slope of +3% for 630m,

• or a negative slope of -3% for 630m,

• or an arc of circle with a curvature of 1/100m For a slope going up, as illustrated in Fig 11, the best time is obtained when the slope is at the end of the race. Indeed, a good horse can still provide a strong effort at the end of the race, even if he is tired. If the slope is at the beginning, it has a strong effect on the velocity which cannot reach its maximum value. In the middle of the race, the slope reduces the mean velocity and therefore the final time.

For a ramp going down, as illustrated in Fig 12, it is the opposite: the best time is obtained when the ramp is at the beginning of the race. Indeed the horse speeds up more easily and more quickly. Effect of slope on the velocity profile for a 1900m race. Flat straight track (blue), +3% slope for 630m at the beginning of the race (orange), +3% slope for 630m at the middle of the race (green), +3% slope for 630m at the end of the race (red). Effect of ramp on the velocity profile for a 1900m race. Flat straight track (blue), +3% ramp down for 630m at the beginning of the race (orange), +3% ramp for 630m at the middle of the race (green), +3% slope for 630m at the end of the race (red). For the bend, as illustrated in Fig 13, the best time is obtained when the bend is at the middle of the race. This is where it has the smallest decrease on the velocity profile. At the beginning, it prevents the horse from reaching its maximal speed. Of course, here the effect is exaggerated with respect to a real race because the bend is long but it yields the general flavour. Similarly, at the end, it prevents the horse from sprinting.

Fig 13.

Effect of bend on the velocity profile for a 1900m race. Flat straight track (blue), bend for 630m at the beginning of the race (orange), bend for 630m at the middle of the race (green), bend for 630m at the end of the race (red). Therefore, we have seen that the optimal velocity has to be analyzed with respect to the changes of slopes, ramps or bends in order to optimize the horse effort. For a given track, it is very likely that the turnpike theory of [START_REF] Aftalion | Pace and motor control optimization for a runner[END_REF] should yield more precise and detailed analytical estimates of the increase of decrease of velocity.

Conclusion

Thanks to precise velocity data obtained on different races, we are able to set a mathematical model which provides information on how horses have to regulate their speed and effort on a given distance. It relies on both mechanical, energetic considerations and motor control. The process consists in identifying the physiological parameters of the horse from the data. Then the optimal control problem provides information on the velocity, the propulsive force, the V O2 and the anaerobic energy. We see that horses have to start strongly and reach a maximal velocity. The velocity decreases in the bends; when going out of the bend, the horse can speed up again and our model can quantify exactly how and when. The horse that slows down the least at the end of the race is the one that wins the race. We understand from the optimal control problem that this slow down is related to the anaerobic supply, the V O2 max and the ability to maintain maximal force at the end of the race. Therefore, horses that have a tendency to slow down too much at the end of the race should put less force at the beginning and slow down slightly through the whole race in order to have the ability to maintain velocity at the end.

From our simulations, we are also able to get information on the V O2 profile, such as when steady state V O2 is reached, when the decay of V O2 starts. The ability to maintain a high V O2 for a long time is related to the ability to maintain velocity. The V O2 starts to decrease when the residual anaerobic energy is too low, and this corresponds to the optimal time to launch the sprint for a long race.

Fig 1 .Curvature (m - 1 )

 11 Fig 1. GPS track of the 1300m, 1900m and 2100m in Chantilly, France.

  .911 2731 5.150 421.5 559.9 47.0 40.71 -1.693e-03 1.504e-03 smoothed) and the velocity computed with our model are plotted in Fig3for the 1300m. We observe a very good match between the curves: there is a strong start with the maximal velocity being reached in 200 meters. Then the velocity decreases, and in particular in the bend, between 300 and 600 meters from start. Though the track is going down, the centrifugal force reduces the propulsive force as we see in Fig4b(the black curve shows the limitation due to the centrifugal force). It is only when reaching the clothoid, before the final straight, after 600 meters, that the horse can speed up again. The end of the race is uphill and the velocity decreases though the horse reaches the straight. Nevertheless, a decrease in velocity at the end of such a race takes place even on a flat track.The V O2 curve vs distance and propulsive force vs distance are plotted in Fig 4.

Fig 3 .V

 3 Fig 3. Velocity data for the 1300 meters race. Raw data, smoothed data, (t f = 76.544s) and computed velocity for the identified paramaters (t f = 76.544s).

Fig 4 . 2 (ml • mn - 1 • kg - 1 )

 4211 Fig 4. V O2 and propulsive force vs distance for the 1300 meters race. V O2 (left in green) and propulsive force (right): blue is the propulsive force f (s) in the direction of movement, black is the effective propulsive force f tot = f 2 + c 2 v 4 taking into account the centrifugal force, where c is the curvature.

  .73 2702 4.74 379 54.6 1392 40.3 -3.69e-03 3.94e-03 smoothed) and the velocity computed with the model are plotted in Fig 5 for the 1900 meters.

Fig 5 .Fig 6 . 2 (ml • mn - 1 • kg - 1 )

 56211 Fig 5. Velocity data for the 1900 meters race. Raw and smoothed data (t f = 116.460s) and computed velocity (t f = 116.460s) for the identified parameters.

Fig 7 . 5 V elocity (m • s - 1 )

 751 Fig 7. Effect of the slope and curvature on the velocity curve (zoom) for the 1900 meters race. Brown is the velocity curve of the race, red with the slope only (straight track), pink with curvature only (flat track) and green is a flat, straight track.

  .637 3567 5.50 304 51.29 524 46.71 1613 41.67 -1.928e-03 9.922e-04 computed velocity are plotted in Fig 8 for the 2100 meters. The V O2 and propulsive force are plotted in Fig 9.

Fig 8 .Fig 9 . 2 (ml • mn - 1 • kg - 1 )

 89211 Fig 8. Velocity data for the 2100 meters race. Raw and smoothed data t f = 130.933s) and computed velocity (t f = 130.933s) for the identified parameters.

Fig 10 .

 10 Fig 10. Evolution of the anaerobic energy vs the distance to the finish line.Green: horse of the 1300m, orange: horse of the 1900m, pink: horse of the 2100m. The distances are highlighted when the energy has a change of slope and the energy value at this point is indicated as well: it is at the beginning of the race when the V O2 max is reached, and at the end of the race, when the V O2 starts decreasing.

Fig 11 .

 11 Fig 11. Effect of slope on the velocity profile for a 1900m race. Flat straight track (blue), +3% slope for 630m at the beginning of the race (orange), +3% slope for 630m at the middle of the race (green), +3% slope for 630m at the end of the race (red).

19 V elocity (m • s - 1 )

 191 Fig 12. Effect of ramp on the velocity profile for a 1900m race. Flat straight track (blue), +3% ramp down for 630m at the beginning of the race (orange), +3% ramp for 630m at the middle of the race (green), +3% slope for 630m at the end of the race (red).

19 V elocity (m • s - 1 )

 191 notopo tf = 116.373s begin tf = 118.176s middle tf = 116.842s end tf = 117.762s

Table 4 .

 4 Percentage of anaerobic contribution in the total energy during the race.

	Race length (m) duration (s) V O2 deficit
	1300	76	47.32%
	1900	116	32.09%
	2100	131	38.01%

  -1 for 630m.Fig 11,[START_REF] Davies | External mechanical work in the galloping racehorse[END_REF] and 13 provide the optimal velocity vs distance for the 1900 meters parameters. The common feature is that a local change of elevation or bend does not only produce a local change in velocity but changes the whole velocity profile and mean value.

	January 6, 2021	12/17

January 6, 2021

Acknowledgments

The authors also wish to thank France Galop and Mc Lloyd for providing the data and for their interest in this work. Finally, they are very grateful to Pierre Martinon for his advice on Bocop.

We also understand better the effects of altitude and the bends and find that they are not local effects producing a local perturbation on the velocity strategy, but on the contrary have a global effect on the whole race. Therefore, a good knowledge of the track and training are crucial to adapt the global pacing strategy rather than slowing down because of bends or slopes.

Future works will be devoted to taking additionally into account drafting and the horse psychology [START_REF] Sih | Behavioral syndromes: an ecological and evolutionary overview[END_REF] since an alternative strategy can be to stay behind to save energy and overtake in the last straight [START_REF] Spence | Speed, pacing strategy and aerodynamic drafting in Thoroughbred horse racing[END_REF].

To maximize an individual horse's potential for winning, it should be entered in races appropriate for its racing ability. Therefore information on a horse speed, endurance or running economy coupled with simulations can help to predict how a horse profile is adapted to some distances to run.


Salomé Poyer

Eric Benoit

Petit

Sofia Telitel

Denise Karamessini

Jean-François Lutz
email: jflutz@unistra.fr.

Laurence Charles
email: laurence.charles@univ-amu.fr

Promoting carboxylate salts in the ESI source to simplify positive mode MS/MS sequencing of acid-terminated encoded polyurethanes

Synthetic polymers prepared by solid-phase processes are often released from the solid support by hydrolysis, leading to an acidic α chain-end which can usefully be employed to perform mass spectrometry experiments in the negative ion mode. This is the case for sequence-defined polyurethanes, which exhibit very simple MS/MS pattern as deprotonated species in great contrast to data obtained in the positive ion mode. Indeed, after deprotonation of their acidic end-group, collision induced dissociation (CID) of these polymers proceeds via competitive cleavages of all carbamate bonds. This leads to a unique series of anionic fragments spaced by the mass of one of the other coding co-monomer, enabling their sequence to be readily deciphered. However, as their size increases, polyurethanes are best ionized in the positive mode but their dissociation pattern is far more complicated, with four to five fragmentation routes depending on the adducted cation. This spectral complexity could however be highly reduced when selecting precursor ions that have experienced H/Na (or any other alkali) exchange in their acidic α end-group: in these conditions, only one reaction is observed and yields pairs of complementary products. Mechanisms could be proposed to explain how, although part of an end-group, the acidic proton had a key role in the dissociation processes of polyurethane chains. Exchange of this proton was then further optimized by proper selection of the X -anion in the NaX salt supplemented to the electrosprayed solution. By allowing simplification of MS/MS data while avoiding signal dilution over multiple ion series, CID of these [M -H + zNa] (z-1)+ permitted to envisage reliable decoding of sequence-defined polyurethanes storing large amount of information.

Introduction

Tandem mass spectrometry (MS/MS) has proved to be an unrivaled technique to decipher binary information encoded in the backbone of sequence-defined polymers [START_REF] Charles | Tandem mass spectrometry sequencing of sequence-controlled and sequencedefined synthetic polymers[END_REF]. Such macromolecules are typically composed of co-monomers of different mass arranged in a controlled manner throughout the chain [START_REF] Colquhoun | Information-containing macromolecules[END_REF][START_REF] Lutz | Coding Macromolecules: Inputting Information in Polymers Using Monomer-Based Alphabets[END_REF][START_REF] Rutten | Encoding information into polymers[END_REF]. For example, digital polymers can be prepared with two co-monomers arbitrarily defined as the 0-and 1-bit of the ASCII code. However, reliable decoding of such digital messages requires MS/MS data to allow full coverage of the binary sequence regardless of the chain length. This means that the same dissociation reactions should occur in all repeating units, independently of their coding moiety. As previously demonstrated in the field of DNA storage, long messages are preferentially composed of individual chains containing about 100 coded residues rather than written in very long chains [START_REF] Church | Next-generation digital information storage in DNA[END_REF]. To achieve de novo sequencing in this size range, the number of fragmentation pathways should be limited to avoid extensive dilution of total ion signal. Since the types of backbone bonds dictate the dissociation behavior of synthetic macromolecules [START_REF] Wesdemiotis | Fragmentation pathways of polymer ions[END_REF], the structure of sequence-defined polymers has to be optimized to control their dissociation extent [START_REF] Charles | MS/MS-Assisted Design of Sequence-Controlled Synthetic Polymers for Improved Reading of Encoded Information[END_REF].

To prevent sequence-dependent reactions, chemical groups of low MS/MS reactivity have to be selected to implement the code, such as the simple H/CH3 variation in alkyl segments used in most digital polymers developed in our group [START_REF] Roy | Design and synthesis of digitally encoded polymers that can be decoded and erased[END_REF][START_REF] Gunay | Chemoselective Synthesis of Uniform Sequence-Coded Polyurethanes and Their Use as Molecular Tags[END_REF][START_REF] Cavallo | Orthogonal Synthesis of "Easy-to-Read" Information-Containing Polymers Using Phosphoramidite and Radical Coupling Steps[END_REF]. Beside the nature of coding moiety, two main strategies can be envisaged to design the polymeric skeleton. One can first focus on developing synthesis protocols for production of highly monodisperse samples of long polymeric chains, which molecular structure can be further optimized to ensure their MS/MS readability [START_REF] Amalian | Controlling the structure of sequence-defined poly(phosphodiester)s for optimal MS/MS reading of digital information[END_REF]. For example, this approach was successfully implemented for sequence-defined poly(phosphodiester)s. These species were first synthesized as defect-free chains with degrees of polymerization (DP) above 100 using an automated synthesis protocol to implement phosphoramidite chemistry [START_REF] Al Ouahabi | Synthesis of Monodisperse Sequence-Coded Polymers with Chain Lengths above DP100[END_REF]. However, while readily ionized in the negative mode electrospray ionization (ESI), collisional activation of poly(phosphodiester)s led to very complex MS/MS data due to efficient cleavage of all phosphate bonds. Abundance of the resulting eight fragment series, also dispersed in terms of charge state, was too weak to allow full sequence coverage of chains with DP above 50 [START_REF] Amalian | Controlling the structure of sequence-defined poly(phosphodiester)s for optimal MS/MS reading of digital information[END_REF]. Specific phosphoramidite reagents were then prepared to introduce, between each group of 8 coded bits (i.e., one byte), an alkoxyamine linkage which cleavage required much lower energy compared to phosphate bonds. As a result, successful decoding of long byte-truncated poly(alkoxyamine phosphodiester)s could be achieved by performing a first activation stage to specifically induce homolysis of these C-ON bonds, followed by a second activation stage to sequence so-formed (small) single byte containing fragments [START_REF] Al Ouahabi | Mass spectrometry sequencing of long digital polymers facilitated by programmed inter-byte fragmentation[END_REF]. The alternative bottom-up strategy consists of (i) selecting candidates based on high MS/MS readability demonstrated for short oligomers and (ii) dedicating further efforts to maintain their simple dissociation pattern while increasing their chain length. This approach is explored here for sequenced-defined polyurethanes (PUs). When subjected to collision-induced dissociation (CID), PU oligomers with DP=4-12 exhibited very simple MS/MS pattern which revealed that only one reaction occurred in all repeating units, giving rise to a single fragment series due to a fixed negative charge in the oligomer α end-group [START_REF] Amalian | Negative mode MS/MS to read digital information encoded in sequence-defined oligo(urethane)s: A mechanistic study[END_REF]. As a result, coded sequences could be literally read from peak-to-peak distance in MS/MS spectra and a simple algorithm was created for automated extraction of digital data [START_REF] Burel | MS-DECODER: Milliseconds Sequencing of Coded Polymers[END_REF]. Alternatively, the code could be graphically read using Kendrick mass defect plots [START_REF] Poyer | Convenient graphical visualization of messages encoded in sequence-controlled synthetic polymers using Kendrick mass defect analysis of their MS/MS data[END_REF]. Although optimal for straightforward MS/MS decoding in the negative ion mode, sequence-defined PUs exhibited decreasing ionization yield as their chain length increased. This issue could be solved by performing positive mode ESI, but CID of cationized PUs led to more complex MS/MS data, as previously reported for related species [START_REF] Mass | Sequence analysis of an isocyanate oligomer by MALDI-TOF mass spectrometry using collision induced dissociation[END_REF][START_REF] Gies | MALDI-TOF/TOF CID study of polycarbodiimide branching reactions[END_REF][START_REF] Gies | Collision induced dissociation study of ester-based polyurethane fragmentation reactions[END_REF]. Because the nature of activated precursors was previously reported to highly influence CID outputs for some synthetic polymers [START_REF] Lattimer | Tandem mass spectrometry of lithium-attachment ions from polyglycols[END_REF][START_REF] Lattimer | Tandem mass spectrometry of poly(ethylene glycol) lithium-attachment ions[END_REF][START_REF] Tintaru | Conformational sensitivity of conjugated poly(ethylene oxide)poly(amidoamine) molecules to cations adducted upon electropsray ionization. A mass spectrometry, ion mobility and molecular modelling study[END_REF], different cationic adducts were investigated for their influence on the MS/MS behavior of PUs.

Experimental section

Chemicals

HPLC-grade solvents used to prepare sequence-defined PU samples (methanol, water, tetrahydrofuran) were purchased from Carlo Erba (Peypin, France). Salts used to promote formation of PU cationic adducts were ammonium acetate, sodium chloride, lithium fluoride, potassium iodide, rubidium bromide, cesium chloride, all obtained from Sigma (Saint Louis, MO) and used without further purification. Sequence-defined PUs used in this study (Supporting Information Table S1) were prepared according to a synthesis protocol published elsewhere [START_REF] Gunay | Chemoselective Synthesis of Uniform Sequence-Coded Polyurethanes and Their Use as Molecular Tags[END_REF]. Each PU sample (a few mg) was dissolved in THF and further diluted in methanolic solutions of different salts (see text).

Mass spectrometry

MS and MS/MS experiments were carried out a hybrid Q-TOF mass spectrometer Synapt G2-HDMS instrument (Waters, Manchester, UK). The ESI source was operated at 50°C, with a desolvation gas temperature of 100°C, using the following settings: i) negative ion mode: capillary voltage, -2.3 kV; sampling cone, -35-120 V depending on PU chain length; extraction cone, -4.0 V; ii) positive ion mode: capillary voltage, +2.8 kV; sampling cone, +30-80 V depending on the species (see text); extraction cone, +4.0 V. All ions were accurately mass measured using the orthogonal acceleration (oa) time of flight (TOF) mass analyzer, with external calibration based on sodium acetate clusters in the MS mode or using the precursor ion as an internal standard in the MS/MS mode. MS/MS experiments were performed by subjecting selected precursor ions upon collisions with argon in a collision cell.

Instrument control, data acquisition and data processing of all experiments were achieved with the MassLynx 4.1 programs provided by Waters. MS n experiments were performed with a QTrap3200 mass spectrometer (Applied Biosystems Sciex, Concord, ON, Canada) equipped with an ESI source operated in positive ion mode, using a capillary voltage of +5.5 kV and a cone voltage of +75 V. Zero-grade air was used as the nebulizing gas (at a pressure of 10 psi) whereas nitrogen was used as the curtain gas (at a pressure of 20 psi) as well as the collision gas. In this instrument, precursor ions selected by a quadrupole and activated in a collision cell led to fragments that were further selectively sampled in a linear ion trap for a second activation stage. Instrument control, data acquisition and data processing of all experiments were achieved using Analyst software (version 1.4.1) provided by Applied Biosystems.

Results and discussion

CID of PUs: positive vs negative ion mode

Sequence-defined PUs were prepared using a chemoselective multistep-growth process implemented on a hydroxyfunctionalized crosslinked polystyrene resin, and their cleavage from the solid support led to an acid moiety at their α chain-end [START_REF] Gunay | Chemoselective Synthesis of Uniform Sequence-Coded Polyurethanes and Their Use as Molecular Tags[END_REF]. This enabled their deprotonation in the negative mode ESI, which happened to be extremely useful for their MS/MS sequencing. Indeed, as detailed in a previous study [START_REF] Amalian | Negative mode MS/MS to read digital information encoded in sequence-defined oligo(urethane)s: A mechanistic study[END_REF], deprotonated PUs experienced one bond cleavage per unit upon CID, leading to very simple MS/MS data composed of a single series of fragments spaced by the mass of one of the other co-monomers (115.0 Da for bit-0 vs 129.1 Da for bit-1). These ai -product ions were formed upon cleavage of the O-CO bond in each carbamate group and all contained the deprotonated α termination, as exemplified in Supporting Information Fig. S1 with the 4-mer named P1. Accordingly, the 0001 sequence of P1 could be readily re-constructed from the peak-to-peak m/z distance equal to the mass of one or the other coding co-monomer. Although this simple MS/MS pattern was kept unchanged for longer PUs, deprotonation yield strongly decreased as the chain length increased. For the P2 16-mer shown in Supporting Information Fig. S2, acquisition time had to be increased by a factor of 50 in the MS/MS mode to ensure detectability of all fragments. This was no longer compatible with high throughput reading of low amounts of coded polymers and did not offer interesting perspectives for sequencing of much longer chains.

In contrast, high ion current was recorded in the positive ion mode for PU oligomers. For example, a nearly 20-fold enhancement was measured for the doubly charged 16-mer P2 compared to the deprotonated molecule (Supporting Information Fig. S3). However, CID spectra recorded in the positive mode exhibited dissociation patterns far more complex than that observed for deprotonated PUs. As shown in Fig. 1(a), dissociation of protonated P1 was observed to proceed by cleavage of all bonds in the carbamate moieties, conducted to four product ion series (Supporting Information Table S2). Restricting the nomenclature proposed by Wesdemiotis et al. [START_REF] Wesdemiotis | Fragmentation pathways of polymer ions[END_REF] to the sole carbamate moiety, fragments containing the left-hand side group were named αi + when formed after cleavage of the CH2-O bond and ai + when resulting from the cleavage of the O-(CO) bond, while xj + and yj + ions both containing the  termination arose from cleavage at the O-(CO) and the (CO)-NH bonds, respectively (Fig. 1, top scheme). Although coverage of the 0001 sequence of P1 was achieved with each ion series observed in Fig. 1(a), this made these MS/MS data highly redundant. Moreover, product ions were also dispersed in terms of charge states when formed from multiply protonated species generated upon ESI of long PUs. In addition, at the energy levels requested to produce all fragments for full sequence coverage of increasing length PUs, extensive secondary dissociation occurred and further increased CID data complexity (Supporting Information Fig. S4 and Table S3). Alkali adducts could also be easily formed in the positive mode ESI since PUs contain carbonyl and ether oxygen atoms. However, their activation led to even more complex MS/MS spectra compared to protonated species, with an additional series of wj + ions (containing the  end-group and formed after cleavage of the CH2-O bond) as illustrated in S5), as commonly reported for polyethers where the strength of alkali adduction was shown to decrease for increasing size cations [START_REF] Lattimer | Tandem mass spectrometry of lithium-attachment ions from polyglycols[END_REF][START_REF] Lattimer | Tandem mass spectrometry of poly(ethylene glycol) lithium-attachment ions[END_REF]. In great contrast, highly simplified CID data were recorded for [P1 -H + 2Na] + ions, i.e., sodium adducts of P1 in which H/Na exchange has occurred in the carboxylic acid group of the α termination (Fig.

1(c)).

A major complete distribution of ai + fragments was observed, together with a few (less abundant) members of the xj + series. Surprisingly, the same MS/MS pattern was obtained for precursors containing alternative alkali, regardless of their size (Supporting Information Fig. S6). These last results suggested that the adducted cation was more tightly bound to the alkali carboxylate form of P1 compared to its acidic form. Interestingly, the same simple dissociation behavior still applied as the chain size was increased (vide infra), permitting to anticipate high benefits for full sequence coverage of much larger PUs. Overall, MS/MS data recorded for the salt form of P1 (Fig. 1(c)) strongly suggested that the acidic proton of the α end-group played a key role in the dissociation of [P1+H] + (Fig.

1(a))

and of [P1+Na] + (Fig. 1(b)). This is similar to reported works on unsaturated fatty acids, where formation of [M -H + 2X] + ions (with X = alkali) by Hsu and Turk [START_REF] Hsu | Distinction among isomeric unsaturated fatty acids as lithiated adducts by electrospray ionization mass spectrometry using low energy collisionally activated dissociation on a triple stage quadrupole instrument[END_REF] or of [M -H

+ Ba] + ions by Volmer et al. [START_REF] Zehethofer | Plasma free fatty acid profiling in a fish oil human intervention study using ultra-performance liquid chromatography/electrospray[END_REF] was shown to inhibit fragmentation mechanisms driven by the carboxylate while promoting charge-remote fragmentation along the hydrocarbon backbone to yield useful information about double bond location. These reactions were further investigated in order to understand the influence of the carboxylate α termination on CID of PUs, using the example of the small P1 4-mer for the sake of clarity.

Mechanistic study of PU dissociation

In order to understand how the nature of the dissociating precursor influenced its

Table S4).

The lack of αi + and yj + fragments in the CID spectrum of [P1 -H + 2Na] + (Fig. 1(c))

indicated that the α acidic proton also had a key role in the formation of these product ions.

Indeed, when generated from [P1 + Na] + (Fig. 1(b)), yj + ions were detected as sodiated species, strongly suggesting that the dissociating m/z 629.3 precursor was a sodium adduct of P1 rather that the protonated form of P1 having a sodium carboxylate as the α termination.

HDX experiments indicated that αi + species lacked one exchangeable proton while yj + ions had gained one (Supporting Information Table S2 and Table S4). Accordingly, proposed mechanisms involved a structure with intra-molecular hydrogen bonds which would favor i) experiments showed that they were formed according to a consecutive process (Supporting Information Fig. S8(a)). In contrast, when formed from [P1 + Na] + , abundance of yj + fragments strongly increased as their size decreased (Fig. 1(b)). As shown in MS 3 , monomer elimination was indeed a minor dissociation pathway for yj + formed from [P1+Na] + (Supporting Information Fig. S8(b)). Alternatively, these yj + ions were readily generated as secondary fragments after loss of CO2 from wj + ions, as supported by MS 3 data (Supporting Information Fig. S7).

transfer
Finally, CID data of Fig. 1 showed that ai + /xj + product ions were detected regardless of the type of precursor ion. However, relative ion abundance measured in these two ion series slightly varied as a function of the dissociating species. As previously shown in the case of deprotonated PUs [START_REF] Amalian | Negative mode MS/MS to read digital information encoded in sequence-defined oligo(urethane)s: A mechanistic study[END_REF], multiple reactions could be envisaged for the formation of these fragments. Exploring all these processes is beyond the scope of this study which, instead, In contrast, to account for the decreasing abundance of xj + ions as their size decreased, competitive reactions involving a 1,3-proton transfer were considered, as proposed by Gies and Hercules for related species [START_REF] Gies | Collision induced dissociation study of ester-based polyurethane fragmentation reactions[END_REF]. Consistent with HDX experiments (Table 1), a proton would be transferred either from the NH group onto the ether oxygen of the same carbamate moiety (Scheme 3, pathway 3 in blue). Alternatively, one could also consider transfer of the and m/z 177.1 in pathway 3. This result is consistent with the proposed strong solvation of the adducted cation by multiple oxygen atoms of P1 (Scheme 3), which would also prevent release of the naked alkali, regardless of its size (Supporting Information Fig. S6). In contrast, presence of the acidic α termination in [P1 + Na] + would favor the formation of a hydrogen bond network (as proposed in Scheme 2), preventing strong interaction with the adducted alkali and its facile release as its size increased, as observed when activating P1 cationized with K + , Rb + or Cs + (Supporting Information Fig. S5).

focused

Promoting formation of PU salts

Although Absolute abundance of these two species is indicated into parentheses. These experiments were performed with the Synapt G2-HDMS mass spectrometer.

Sequencing of long PU oligomers

These optimal conditions were then used for positive mode sequencing of the P2 16-mer.

When prepared in a methanolic solution of NaOH (1 mM), this oligomer was readily electrosprayed as the doubly charged [P2 -H + 3Na] 2+ species and was observed to dissociate into the two main ai z+ and xj z+ ion series (Fig. 3(a) and Supporting Information Table S5).

Using low activation energy level (e.g., 40 eV laboratory frame for data shown in Fig. 3(a masses of ai and xj species annotated in blue and red, respectively. Unlabelled minor peaks correspond to secondary fragments. In both spectra, the y-axis has been expanded by a factor of 3 for the sake of clarity. These experiments were performed with the Synapt G2-HDMS mass spectrometer.

Conclusion

The dissociation pattern of sequence-defined PUs in the positive ion mode could be strongly modified when changing their acidic α termination into a carboxylate moiety. This H/Na (or other alkali) exchange was easily performed in solution and permitted to efficiently suppress all reactions involving the carboxylic acid α group of these polymers. Collisional activation of carboxylate-terminated PUs mainly induced cleavage of the O-(CO) bond of carbamate group present in each repeating unit, which led to the formation of two fragment series only and hence fulfilled conditions requested for full sequence coverage by MS/MS. This is another example of the "MS/MS design" approach which aims at optimizing the structure of encoded polymers to control their dissociation behavior and hence improve their sequencing. However, in contrast to previously reported cases where synthesis works were requested, changing a carboxylic acid into its carboxylate form was readily achieved in the electrosprayed solution supplemented with the appropriate alkali salts. Thank to multiple charging during ESI, PUs of increasing length were efficiently ionized in the positive ion mode and exhibited the same simple MS/MS pattern regardless of their charge state. With this new methodology in hand, production of sequence-defined polymers to store multiple bytes of information can now be envisaged. S3. Activation energy is given in the center-of-mass frame. This experiment was performed with the Synapt G2-HDMS mass spectrometer. S4) and used to recover the 1101001011011000 sequence, using the precursor ion (C94H174N16O35 2+ , m/z 1043.6158) as an internal standard.

Fig. 1 (

 1 Fig. 1(b) for [P1 + Na] + . Lithiated P1 exhibited the same dissociation behavior whereas CID of [P1 + K] + , [P1 + Rb] + or [P1 + Cs] + conducted to the release of the naked alkali (Supporting Information Fig. S5), as commonly reported for polyethers where the strength of

Fig. 1 .

 1 Fig. 1. ESI-MS/MS spectra of (a) [P1 + H] + at m/z 607.4 (X=H), (b) [P1 + Na] + at m/z 629.3 (X=H) and (c) [P1 -H + 2Na] + at m/z 651.3 (X=Na). Top: dissociation scheme with X = H or Na. Peaks annotated in grey correspond to secondary fragments. # designates dehydration products while the peak annotated with a black triangle in part b) correspond to a sodiated monomer (see text). For the sake of clarity, most minor peaks were not annotated but a complete list of fragments is reported in Supporting InformationTable S2 for [P1 + H] + ,

Scheme 1 .

 1 Scheme 1. Mechanism proposed to account for the formation of wj + fragments upon CID of [P1 + Na] + at m/z 629.3.

Scheme 2 .

 2 Scheme 2. Mechanism proposed to explain the formation of αi + or yj + fragments upon CID of precursor ions holding the acidic α termination such as [P1 + H] + at m/z 607.4 and [P1 + Na] + at m/z 629.3.

Scheme 3 .

 3 Scheme 3. Proposed mechanisms for the dissociation of [P1 -H + 2Na] + species: pathway 1 (in red) could rationalize consecutive production of ai + ions, whereas pathway 2 (in pink) and pathway 3 (in blue) would account for the competitive formation of xj + ions.

Scheme 3 ). 1 (

 31 Scheme 3). The latter reaction was also proposed since the enol form of carbamate moieties was evidenced in the formation of internal fragments at m/z 160.0, m/z 174.1 and m/z 275.1 (annotated in grey in Fig.1(c)), as described in Supporting Information Scheme S1. None of the reactions depicted in Scheme 3 produced the complementary fragments expected at m/z 152.1 (sodiated unit 1) and m/z 138.1 (sodiated unit 0) in pathway 1, m/z 292.1 in pathway 2

  [P1 -H + 2Na] + ions exhibited a highly promising CID behavior for MS/MS sequencing of larger PUs, it should be acknowledged that abundance of these species remained minor (about 30%) compared to [P1 + Na] + , when employing a 1 mM NaCl solution. Due to the low volatility of sodium salts, the amount of NaCl could only be increased up to 3 mM but with no significant improvement of the absolute (or relative) intensity of the [P1 -H + 2Na] + signal. Optimizing the declustering potential (DP) that governed ion sampling from the ESI atmospheric pressure into the vacuum side of the mass spectrometer allowed an overall signal enhancement although it was not specific to the sodium salt form of PUs: as observed for the P3 8-mer of sequence 01000010, increasing DP from 20 to 60 V allowed ion abundance to be roughly increased by a factor of about 3 for both [P3 + 2Na] 2+ at m/z 563.3 and [P3 -H +3Na] 2+ at m/z 574.3 (Supporting Information Fig.S9).Different NaX salts were then tested in order to optimize production of [P -H +zNa] (z-1)+ species, using the P4 8-mer of sequence 01000100 as a model. The best salt to use was found to be NaOH because it is both basic and highly dissociated in polar solvents, and hence allowed the [P4 -H + 3Na] 2+ ion at m/z 574.3 to be obtained with maximal absolute abundance (Fig.2(a)). Addition of 1 mM NaOH in the electrosprayed solution did not induce major loss of sensitivity, as shown in Fig.2(b) by similar abundances measured for [P4 + 3Na] 2+ when using NaI and [P4 -H + 3Na] 2+ when using NaOH.

Fig. 2 .

 2 Fig. 2. (a) Absolute abundance of the [P4 -H + 3Na] 2+ ion at m/z 574.3 (grey bars, with average values on top) as a function of the NaX salt supplemented to the electrosprayed solution. Error bars as standard deviation of 3 replicates. (b) ESI(+) mass spectra (m/z 550-580) showing relative abundance of [P4 + 2Na] 2+ (m/z 563.3) and [P4 -H + 3Na] 2+ (m/z 574.3) when using NaI, NaCH3COO and NaOH as the cationization agent (each at 1 mM).

Fig. 1 ,

 1 )) permitted to minimize secondary fragments (annotated in grey) while achieving full coverage of P2 sequence with the complete series of ai z+ ions (annotated in blue), detected with z = 1 from i = 1-12 and z = 2 from i = 10-16. The 1101001011011000 code written in the backbone of P2 was best read after charge state deconvolution of these MS/MS data (Fig.3(b)). Starting from a1 always expected with a 154.0 Da mass since it does not contain any coding moiety (top scheme), the sequencing task simply consisted of searching for the next ai+1 signal by adding either 115.1 Da (mass of the 0-bit) or 129.1 Da (mass of the 1-bit). As a result, following blue arrows in Fig.3(b) allowed the binary message to be readily deciphered. The partial sequence reconstructed when using xj species had, instead, to be read from the right-to the left-hand side (as indicated by red arrows below the x-axis of Fig.3(b)) and was used as a validation tool.

Fig. 3 .

 3 Fig. 3. (a) ESI(+)-MS/MS of [P2 -H + 3Na] 2+ at m/z 1076.6, using a 1.45 eV activation energy (center-of-mass frame). In the associated dissociation scheme (inset), singly and doubled charged fragments are respectively designated with single and double pattern, in blue for ai z+ and in red for xj z+ . Grey # symbols designate secondary fragments, including the two internal m/z 160.0 and m/z 174.0 ions. (b) Charge-deconvoluted MS/MS spectrum with

Fig. S1 .

 S1 Fig. S1. ESI(-)-MS/MS of the deprotonated P1 oligomer of sequence 0001 (top structure) detected at m/z 605.3. Activation energy is given in the center-of-mass frame. Starting from the a1 -ion always detected at m/z 131.1 since this ion does not contain any coding moiety, the 0001 sequence of P1 is reconstructed from the peak-to-peak distance equal to the mass of the 0-bit (115.0 Da) or of the 1-bit (129.0 Da): Δm/z (a2 -/a1 -) = 115.0, Δm/z (a3 -/a2 -) = 115.1, Δm/z (a4 -/a3 -) = 115.1 and Δm/z ([PU-1 -H] -/a4 -) = 129.0. This experiment was performed with the Synapt G2-HDMS mass spectrometer.

Fig. S2 .

 S2 Fig. S2. ESI(-)-MS/MS of the deprotonated P2 oligomer of sequence 1101001011011000 (top structure) detected at m/z 2084.2. Activation energy is given in the center-of-mass frame. This experiment was performed with the Synapt G2-HDMS mass spectrometer.

Fig. S3 .

 S3 Fig. S3. Relative intensity of some PUs when adducted to one (black dots) or two (white dots) sodium cations compared to deprotonated molecules obtained in the negative ion mode, as a function of their polymerization degree, n. These experiments were performed with the Synapt

Fig. S4 .

 S4 Fig. S4. ESI(+)-MS/MS of the doubly protonated P2 detected at m/z 1043.6, allowing full coverage of the 1101001011011000 sequence, as depicted by the inset scheme. However, the sequencing task was made difficult by numerous secondary fragments (designated by asterisks) of particularly high abundance in the low m/z range of the MS/MS spectrum. All peaks were not annotated for the sake of clarity but fragments useful for sequencing are all reported in TableS3. Activation energy is given in the center-of-mass frame. This experiment was performed with the

Fig. S5 .

 S5 Fig. S5. ESI(+)-MS/MS of P1 when adducted with (a) Li + at m/z 613.4, (b) K + at m/z 645.3, (c) Rb + at m/z 691.3 and (d) Cs + at m/z 739.3. Activation energies are given in the center-ofmass frame. Peaks annotated in grey correspond to secondary fragments. These experiments were performed with the Synapt G2-HDMS mass spectrometer.

Fig S6 .

 S6 Fig S6. ESI(+)-MS/MS of (a) [P1 -H +2Li] + at m/z 619.4, (b) [P1 -H +2K] + at m/z 683.3, (c) [P1 -H +2Rb] + at m/z 775.2 and (d) [P1 -H +2Cs] + at m/z 871.2. Activation energies are given in the center-of-mass frame. Peaks annotated in grey correspond to secondary fragments. These experiments were performed with the Synapt G2-HDMS mass spectrometer.

Fig. S7 .

 S7 Fig. S7. MS 3 experiments performed for [P1 + Na] + at m/z 629.3, showing that consecutive dissociation of wj + fragments (in pink) at (a) m/z 515.3 (w4 + ), (b) m/z 400.2 (w3 + ) and (c) m/z 285.1 (w2 + ) led to wj-1 + lower congeners as well as yj + ions (in orange) and xj-1 + ions (in red). These experiments were performed with the QTrap 3200 mass spectrometer.

Fig. S8 .

 S8 Fig. S8. MS 3 experiments showing that (a) dissociation of yj + produced from for [P1 + H] + at m/z 607.4 mainly led to yj-1 + , whereas (b) yj + produced from [P1 + Na] + at m/z 629.3 required highly energetic activation to generate wj-1 + (in pink) and, to a lower extent, xj-1 + (in red) and yj-1 + (in orange) ions. These experiments were performed with the QTrap 3200 mass spectrometer.

Fig. S9 .

 S9 Fig. S9. Influence of the declustering potential (DP) on the abundance of [P3 + 2Na] 2+ at m/z 563.3 (in blue) and of [P3 -H +3Na] 2+ at m/z 574.3 (in green). Error bars as standard deviation of 3 replicates. These experiments were performed with the Synapt G2-HDMS mass spectrometer.

Table 1 .

 1 Table S2 for [P1 + H] + , Table S4 for [P1 + Na] + and Table1for [P1 -H + 2Na] + . These experiments were performed with the Synapt G2-HDMS mass spectrometer. Activation energies are given in the center-of-Accurate mass measurement of fragments formed in CID of sodiated P1-Na (Fig.

	mass frame.

1(c)) using the precursor ion (C27H49N4O11Na2 + , m/z 651.3188 before H/D exchanges, left; C27H44D5N4O11Na2 + , m/z 656.3502 after H/D exchanges, right) as an internal standard. n.d.: not detected

  on the case of [P1 -H + 2Na] + . Because ai + and xj + ions obtained upon cleavage of the same O-CO bond in this m/z 651.3 precursor were measured with very different abundances (see a1 + vs x4 + , a2 + vs x3 + , a3 + vs x2 + , a4 + vs x1

+

in Fig.

1(c

)), they were not envisaged as complementary fragments formed during a charge-remote reaction but, instead, as ions produced via two distinct mechanisms. Since all ai + ions contained two Na (Table

1

Table S1 .

 S1 Structural description of PU oligomers used in this study.

		ionization tandem mass spectrometry. Rapid Commun. Mass Spectrom. 22 (2008) 2125-
		2133.
		n	sequence	composition	mass (Da)
	P1	4	α-0001-	C27H50N4O11	606.3476
	P2	16	α-1101001011011000-	C94H172N16O35	2085.2171
	P3	8	α-01000010-	C48H88N8O19	1080.6166
	P4	8	α-01000100-	C48H88N8O19	1080.6166

Table S2 .

 S2 These experiments were performed with the Synapt G2-HDMS mass spectrometer. Accurate mass measurement of fragments formed in CID of protonated P1 using the precursor ion (C27H51N4O11 + , m/z 607.3549 before H/D exchanges, left; C27H44D7N4O11 + , m/z 614.3988 after H/D exchanges, right) as an internal standard.

					before HDX			after HDX
				composition	m/zth	m/zexp	composition	m/zth	m/zexp
			a1 +	C6H13O3 +	133.0859	133.0860	C6H10D3O3 +	136.1048	136.1051
	α-containing	fragments	a2 + a3 + a4 + α1 + α2 + α3 +	C11H22NO5 + C16H31N2O7 + C21H40N3O9 + C6H11O2 + C11H20NO4 + C16H29N2O6 +	248.1492 363.2126 478.2759 115.0754 230.1387 345.2020	248.1489 363.2124 478.2756 115.0755 230.1386 345.2020	C11H18D4NO5 + C16H26D5N2O7 + C21H34D6N3O9 + C6H10D1O2 + C11H18D2NO4 + C16H26D3N2O6 +	252.1744 368.2440 484.3136 116.0816 232.1512 348.2208	252.1749 368.2439 484.3138 116.0827 232.1515 348.2204
			α4 +	C21H38N3O8 +	460.2653	460.2651	C21H34D4N3O8 +	464.2904	464.2906
			x1 +	C6H12NO2 +	130.0863	130.0852	C6H10D2NO2 +	132.0988	132.1008
	-containing	fragments	x2 + x3 + x4 + y1 + y2 + y3 +	C11H21N2O4 + C16H30N3O6 + C21H39N4O8 + C5H14NO + C10H23N2O3 + C15H32N3O5 +	245.1496 360.2129 475.2762 104.1070 219.1703 334.2336	245.1495 360.2123 475.2762 104.1068 219.1700 334.2331	C11H18D3N2O4 + C16H26D4N3O6 + C21H34D5N4O8 + C5H10D4NO + C10H18D5N2O3 + C15H26D6N3O5 +	248.1684 364.2380 480.3076 108.1321 224.2017 340.2713	248.1691 364.2381 480.3080 108.1315 224.2014 340.2716
			y4 +	C20H41N4O7 +	449.2970	449.2966	C20H34D7N4O7 +	456.3409	456.3432
						S4

Table S3 .

 S3 Accurate mass measurement of fragments formed in CID of [P2 + 2H] 2+ (Figure

		α-containing product ions	-containing product ions
			ai z+ ions			xi z+ ions
	i composition	m/zth	m/zexp	composition	m/zth	m/zexp
	1 C6H13O3 +	133.0859	133.0850 C5H10NO2 +	116.0706	116.0710
	2 C12H24NO5 +	262.1649	262.1658 C10H19N2O4 +	231.1339	231.1334
	3 C18H35N2O7 +	391.2439	391.2432 C15H28N3O6 +	346.1973	346.1968
	4 C23H44N3O9 +	506.3072	506.3080 C21H39N4O8 +	475.2762	475.2773
	5 C29H55N4O11 +	635.3862	635.3874 C27H50N5O10 +	604.3552	604.3557
	6 C34H64N5O13 +	750.4495	750.4488 C32H59N6O12 +	719.4185	719.4221
	7 C39H73N6O15 +	865.5128	865.5103 C38H70N7O14 +	848.4975	848.4976
	8 C45H84N7O17 +	994.5918	994.5902 C44H81N8O16 +	977.5765	977.5791
	9	C50H93N8O19 + C50H94N8O19 2+	1109.6552 555.3312	1109.6582 C49H90N9O18 + 555.3334 C49H91N9O18 2+	1092.6398 546.8236	1092.6423 546.8214
	10	C56H104N9O21 + C56H105N9O21 2+	1238.7341 619.8707	1238.7306 C55H101N10O20 + 619.8729 C55H102N10O20 2+	1221.7188 611.3630	1221.7178 611.3624
	11	C62H115N10O23 + C62H116N10O23 2+	1367.8131 684.4102	1367.8116 C60H110N11O22 + 684.4101 C60H111N11O22 2+	1336.7821 668.8947	1336.7800 668.8962
	12	C67H124N11O25 + C67H125N11O25 2+	1482.8764 741.9419	1482.8787 C65H119N12O24 + 741.9420 C65H120N12O24 2+	1451.8455 726.4264	1451.8480 726.4264
	13	C73H135N12O27 + C73H136N12O27 2+	1611.9554 806.4813	1611.9526 C71H130N13O26 + 806.4844 C71H131N13O26 2+	1580.9244 790.9659	1580.9332 790.9698
	14	C79H146N13O29 + C79H147N13O29 2+	1741.0344 871.0208	1741.0328 C76H139N14O28 + 871.0197 C76H140N14O28 2+	1695.9878 848.4975	1695.9958 848.4976
	15 C84H156N14O31 2+	928.5525	928.5551 C82H151N15O30 2+	913.0370	913.0364
	16 C89H165N15O33 2+	986.0842	986.0862 C88H162N16O32 2+	977.5765	977.5791
			αi z+ ions			yi z+ ions
	1 C6H11O2 +	115.0754	115.0763 C4H12NO +	90.0913	90.0913
	2 C12H22NO4 +	244.1543	244.1539 C9H21N2O3 +	205.1547	205.1541
	3 C18H33N2O6 +	373.2333	373.2327 C14H30N3O5 +	320.218	320.2183
	4 C23H42N3O8 +	488.2966	488.2977 C20H41N4O7 +	449.297	449.2979
	5 C29H53N4O10 +	617.3756	617.3787 C26H52N5O9 +	578.376	578.3771
	6 C34H62N5O12 +	732.4389	732.4427 C31H61N6O11 +	693.4393	693.4413
	7 C39H71N6O14 +	847.5023	847.5001 C37H72N7O13 +	822.5183	822.5180
	8 C45H82N7O16 +	976.5813	976.5745 C43H83N8O15 +	951.5972	951.5962
	9	C50H91N8O18 + C50H92N8O18 2+	1091.6446 546.3259	1091.6497 C48H92N9O17 + 546.3278 C48H93N9O17 2+	1066.6606 533.8339	1066.6587 533.8307
	10	C56H102N9O20 + C56H103N9O20 2+	1220.7236 610.8654	1220.7241 C54H103N10O19 + 610.8681 C54H104N10O19 2+	1195.7395 598.3734	1195.7408 598.3763
	11	C62H113N10O22 + C62H114N10O22 2+	1349.8025 675.4049	1349.8071 C59H112N11O21 + 675.4178 C59H113N11O21 2+	1310.8029 655.9051	1310.8035 655.8975
	12	C67H122N11O24 + C67H123N11O24 2+	1464.8659 732.9366	1464.8668 C64H121N12O23 + 732.9335 C64H122N12O23 2+	1425.8662 713.4367	1425.8635 713.4415
	13	C73H133N12O26 + C73H134N12O26 2+	1593.9449 797.4761	1593.9575 C70H132N13O25 + 797.4739 C70H133N13O25 2+	1554.9452 777.9762	1554.9419 777.9720
	14	C79H144N13O28 + C79H145N13O28 2+	1723.0238 862.0156	1723.0527 C75H141N14O27 + 862.0173 C75H142N14O27 2+	1670.0085 835.5079	1670.0164 835.5067
	15 C84H154N14O30 2+	919.5472	919.5516 C81H153N15O29 2+	900.0474	900.0472
	16 C89H163N15O32 2+	977.0789	977.0735 C87H164N16O31 2+	964.5869	964.5881

Table S4 .

 S4 Accurate mass measurement of fragments formed in CID of sodiated P1 using the precursor ion (C27H50N4O11Na + , m/z 629.3368 before H/D exchanges, left; C27H44D6N4O11Na + , m/z 635.3745 after H/D exchanges, right) as an internal standard. n.d.: not detected.

Acknowledgments L.C. and J.-F.L. thank the French National Research Agency (ANR project 00111001, grant numbers ANR-16-CE29-0004-01 and ANR-16-CE29-0004-02) for financial support. L.C. also acknowledges support from Spectropole, the Analytical Facility of Aix-Marseille University, by granting a special access to the instruments purchased with European Funding (FEDER OBJ2142-3341).

Promoting carboxylate salts in the ESI source to simplify positive mode MS/MS sequencing of acid-terminated encoded polyurethanes

Salomé Poyer, 1 Benoit Eric Petit, 2 Sofia Telitel, 2 Denise Karamessini, 2 Jean-François Lutz, 2* and Laurence Charles