Liang Ma
email: liangma@tsinghua.edu.cn

Damien Chablat

Fouad Bennis

Wei Zhang

Bo Hu

Franc ¸ois Guillaume

A novel approach for determining fatigue resistances of different muscle groups in static cases

Keywords: muscle fatigue, biomechanical muscle modeling, fatigue resistance, maximum endurance time, muscle groups

In ergonomics and biomechanics, muscle fatigue models based on maximum endurance time (MET) models are often used to integrate fatigue effect into ergonomic and biomechanical application. However, due to the empirical principle of those MET models, the disadvantages of this method are: 1) the MET models cannot reveal the muscle physiology background very well; 2) there is no general formation for those MET models to predict MET. In this paper, a theoretical MET model is extended from a simple muscle fatigue model with consideration of the external load and maximum voluntary contraction in passive static exertion cases. The universal availability of the extended MET model is analyzed in comparison to 24 existing empirical MET models. Using mathematical regression method, 21 of the 24 MET models have intraclass correlations over 0.9, which means the extended MET model could replace the existing MET models in a general and computationally efficient way. In addition, an important parameter, fatigability (or fatigue resistance) of different muscle groups, could be calculated via the mathematical regression approach. Its mean value and its standard deviation are useful for predicting MET values of a given population during static operations. The possible reasons influencing the fatigue resistance were classified and discussed, and it is still a very challenging work to find out the quantitative relationship between the fatigue resistance and the influencing factors.

Relevance to industry :

MSD risks can be reduced by correct evaluation of static muscular work. Different muscle groups have different properties, and a generalized MET model is useful to simplify the fatigue analysis and fatigue modeling, especially for digital human techniques and virtual human simulation tools.

Introduction

Muscle fatigue is defined as "any reduction in the ability to exert force in response to voluntary effort" [START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF], and it is believed that the muscle fatigue is one of potential reasons leading to musculoskeletal disorders (MSDs) in the literature [START_REF] Westgaard | Work-related musculoskeletal complaints: some ergonomics challenges upon start of a new century[END_REF][START_REF] Kumar | Musculoskeletal risk factors in cleaning occupation: a literature review[END_REF][START_REF] Kim | Knowledge and opinions of designers of industrialized wall panels regarding incorporating ergonomics in design[END_REF]. Great effort has been contributed to integrate fatigue into different biomechanical models, especially in virtual human simulation for ergonomic application, in order to analyze the fatigue in muscles and joints and further to decrease the MSD risks [START_REF] Badler | Simulating Humans: Computer Graphics Animation and Control[END_REF][START_REF] Vignes | Modeling muscle fatigue in digital humans[END_REF][START_REF] Hou | Research on virtual human in ergonomic simulation[END_REF].

In general, mainly two approaches have been adopted to represent muscle fatigue, either in theoretical methods or in empirical methods [START_REF] Xia | A theoretical approach for modeling peripheral muscle fatigue and recovery[END_REF]. One or more decay terms were introduced into existing muscle force models in theoretical fatigue models, and those decay terms were mainly based on physiological performance of muscles in fatigue contraction. For example, a fatigue model based on the intracellular pH [START_REF] Giat | A musculotendon model of the fatigue profiles of paralyzed quadriceps muscle under FES[END_REF] was incorporated into Hill's muscle mechanical model [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF]. This fatigue model was also applied by [START_REF] Komura | Calculation and visualization of the dynamic ability of the human body[END_REF] to demonstrate the fatigue of different individual muscles. Another muscle fatigue model [START_REF] Ding | Mathematical models for fatigue minimization during functional electrical stimulation[END_REF] based on physiological mechanism has been included into the Virtual Solider Research Program [START_REF] Vignes | Modeling muscle fatigue in digital humans[END_REF], and in this model, dozens of parameters have to be fit for model identification only for a single muscle. As stated in [START_REF] Xia | A theoretical approach for modeling peripheral muscle fatigue and recovery[END_REF], "these theoretical models are relatively complex but useful at the single muscle level. However, they do not readily handle task-related biomechanical factors such as joint angle and velocity." Meanwhile, several muscles around a joint are engaged in order to realize an action or a movement around the joint, and mathematically this results in an underdetermined equation while determining the force of each engaged muscle due to muscle redundancy and complex muscle force moment arm-joint angle relationships. Although different optimization methods have been used to face this load sharing problem [START_REF] Ren | Predictive modelling of human walking over a complete gait cycle[END_REF], it is still very difficult to val-idate the optimization result and further the fatigue effect, due to the complexity of anatomical structure and the physiological coordination mechanism of the muscles.

Muscle fatigue is often modeled and extended based on Maximum Endurance Time (MET) models at joint level in empirical methods. These models are often used in ergonomic applications to handle task-related external parameters, such as intensity of the external load, frequency of the external load, duration, and posture. In these models, the MET of a muscle group around a joint was often measured under static contraction conditions until exhaustion. Using this method can avoid complex modeling of individual muscles, and net joint strengths already exist in the literature for determining the relative load [START_REF] Anderson | Maximum voluntary joint torque as a function of joint angle and angular velocity: Model development and application to the lower limb[END_REF][START_REF] Xia | A theoretical approach for modeling peripheral muscle fatigue and recovery[END_REF]. The most famous one of these MET models is Rohmert's curve [START_REF] Rohmert | Ermittlung von Erholungspausen für statische Arbeit des Menschen[END_REF] which was usually used as guideline for designing the static contraction task. Besides Rohmert's MET model, there are several other empirical MET models in the literature [START_REF] El Ahrache | Percentile values for determining maximum endurance times for static muscular work[END_REF]. These MET models are very useful to evaluate physical fatigue in static operations and to determine work-rest allowances [START_REF] El Ahrache | Comparison of rest allowance models for static muscular work[END_REF], and they were often employed in biomechanical models in order to minimize fatigue as well. For example, [START_REF] Freund | A dynamic model of the forearm including fatigue[END_REF] proposed a dynamic model for forearm in which the fatigue component was modeled for each single muscle by fitting Rohmert's curve in [START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF]. [START_REF] Rodríguez | A joint-level model of fatigue for the postural control of virtual humans[END_REF] proposed a half-joint fatigue model, more exactly a fatigue index, based on mechanical properties of muscle groups. The holding time over maximum endurance time is used as an indicator to evaluate joint fatigue. In [START_REF] Niemi | A static shoulder model based on a time-dependent criterion for load sharing between syner-gistic muscles[END_REF], different fiber type composition was taken into account with endurance model to locate the muscle fatigue into single muscle level. However, in MET models, the main limitations are: 1) The physical relationship in these models cannot be interpreted directly by muscle physiology, and there is no universality among these models. 2) All the MET models were achieved by fitting experimental results using different formulation of equation. It has been found that muscle fatigability can vary across muscles and joints. However, there is no general formulation for those models. 3) Differences have been found among those MET models for different muscle groups, for different postures, and even for different models for the same muscle group. Due to the limitation from the empirical principle, the differences cannot be interpreted by those MET models. Thus, it is necessary to develop a general MET model which is able to replace all the experimental MET models and explain all the differences cross these models. [START_REF] Xia | A theoretical approach for modeling peripheral muscle fatigue and recovery[END_REF] proposed a new muscle fatigue model based on motor units (MU) recruitment [START_REF] Liu | A dynamical model of muscle activation, fatigue, and recovery[END_REF] to combine the theoretical models and the task-related muscle fatigue factors. In this model, properties of different muscle fiber types have been assumed to predict the muscle fatigue at joint level. However, in their research, the validation of their fatigue model was not provided. Furthermore, the different fatigability of different muscle groups has not been analyzed in details in this model. Fatigability (the reciprocal of endurance capacity or the reciprocal of fatigue resistance) can be defined by the endurance time or measured by the number of times of an operation until exhaustion. This measure is an important parameter to measure physical fatigue process during manual handling operations [START_REF] Lynch | Muscle quality. I. Age-associated differences between arm and leg muscle groups[END_REF][START_REF] Clark | Gender differences in skeletal muscle fatigability are related to contraction type and EMG spectral compression[END_REF][START_REF] Larivière | Gender influence on fatigability of back muscles during intermittent isometric contractions: A study of neuromuscular activation patterns[END_REF][START_REF] Hunter | Influence of aging on sex differences in muscle fatigability[END_REF].

In [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF], we constructed a new muscle fatigue model in which the external task related parameters are taken into consideration to describe physical fatigue process, and this model has also been interpreted by the physiological mechanism of muscle. The model has been compared to 24 existing MET models, and great linear relationships have been found between our model and the other MET models. Meanwhile, this model has also been validated in comparison to three theoretical models. This model is a simpler, theoretical approach to describe the fatigue process, especially in static contraction cases.

In this paper, further analysis based on the fatigue model is carried out using mathematical regression method to determine the fatigability of different muscle groups. We are going to propose a mathematical parameter, defined as fatigability, describing the resistance to the decrease of the muscle capacity. The fatigue resistance for different muscle groups is going to be regressed from experimental MET models. The theoretical approach for calculating the fatigue resistance will be explained in Section 2. The muscle fatigue model in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF] is going to be presented briefly in Section 2.1. A general MET model is extended from this fatigue model in Section 2.2. The mathematical procedure for calculating the fatigability contributes to the main content of Section 2.3. The results and discussion are given in Section 3 and 4, respectively.

Method

A dynamic muscle fatigue model

A dynamic fatigue model based on muscle active motor principle was proposed in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF]. This model was able to integrate task parameters (load) and temporal parameters into manual handling operation in industry. The differential equation for describing the reduction of the capacity is Eq. (1). The descriptions of the parameters for Eq. ( 1) are listed in Table 1.

dF cem (t) dt = -k F cem (t) MVC F load (t) (1)
The fatigue model in Eq. ( 1) can be explained by the MUbased pattern of muscle [START_REF] Liu | A dynamical model of muscle activation, fatigue, and recovery[END_REF][START_REF] Vøllestad | Measurement of human muscle fatigue[END_REF]. According to muscle physiology, larger F load means more type II motor units are involved into the force generation. As a result, the muscle becomes fatigued more rapidly, as expressed in Eq. (1). F cem represents the non-fatigue motor units of the muscle. The decreased fatigability is expressed by term F cem (t)/MVC due to the composition change in the non-fatigue muscle fibers. k is a parameter indicating the fatigue ratio, and it has different constant values for different muscle groups. In a general case, it is assigned as 1. This dynamic model has been mathematical validated in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF] with static MET models and other existing dynamic theoretical models. Much more detailed explanation can be found in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF].

The integration result of Eq. ( 1) is Eq. ( 2), if F cem (0) = MVC. Equation (2) demonstrates the fatigue process under an external load F load (u). The external load can be either static external load or dynamic one.

F cem (t) = MVC e t 0 -k F load (u) MVC du (2)

The extended MET model

A general MET model can be extended by supposing that F load (t) is constant in static situation to predict the endurance time of a muscle contraction. MET is the duration in which F cem falls down to the current F load . Thus, MET can be determined by Eq. ( 3) and (4).

F cem (t) = MVC e t 0 -k F load (u) MVC du = F load (t)
(3)

t = MET = - ln F load (t) MVC k F load (t) MVC = - ln( f MVC ) k f MVC (4)
In comparison to the empirical MET models, the extended MET model is obtained from a theoretical muscle fatigue model based on motor units recruitment mechanism. In the extended MET model, only one parameter k remains variable. This parameter is defined as fatigability (fatigue ratio) in the theoretical model, and this parameter could enable us to explain the differences found in empirical MET models.

Regression for determining the fatigability

In [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF], two correlation coefficients were selected to analyze the relationship between the extended MET model (k = 1) and the empirical MET models. One is Pearson's correlation r in for evaluating the linear relationship and the other one is intraclass correlation (ICC) for evaluating the similarity between two models. The calculation results are shown in columns r and ICC 1 of Table 2. There are still large differences from 1 in ICC 1 column, while high linear correlations have also been found between the extended MET model and the empirical MET models. That means that the extended MET model can fit the empirical MET models for different muscle groups by adjusting the parameter k.

A mathmatical regression process is carried out to determine the parameter k as follows. From Table 2, it is observed that almost all the static MET models have high linear relationship with the extended MET model (for most models, r > 0.95), which means that each static model can be described mathematically by a linear equation (Eq. ( 5)). In Eq. ( 5), x is used to replace f MVC and p(x) represents the extended MET model in Eq. ( 4). m and n are constants describing the linear relationship between an existing MET model and the extended MET model, and they are needed to be determined in linear regression. It should be noticed that m = 1/k indicates the fatigue resistance of the static model, and k is fatigue ratio or fatigability of different static model.

f (x) = m p(x) + n (5)
Due to the asymptotic tendencies of the empirical MET models mentioned in El ahrache et al. ( 2006), when x → 1 (%MVC → 100), f (x) → 0 and p(x) → 0 (MET → 0), therefore, we can assume that n = 0. For this reason, only one parameter m needs to be determined. Since some empirical MET models are not available for %MVC under 15%, the regression was carried out in the interval from x = 0.16 to x = 0.99. With an space 0.01, N = 84 empirical MET values were calculated to determine the parameter m by minimizing the function in Eq. (6).

M(x) = N i=1 ( f (x i ) -m p(x i )) 2 = a m 2 + b m + c (6)
In Eq. ( 6), a = N i=1 p(x i ) 2 is always greater than 0, and

b = -2 N i=1 p(x i ) f (x i
) is always less than 0, therefore the fatigue resistance m can be calculated by Eq. ( 7).

m = -b 2a = N i=1 p(x i ) f (x i ) N i=1 p(x i ) 2 > 0 (7)
After regression for each empirical MET model, new ICC values were calculated by comparing f (x)/m and p(x), and they are listed in the column ICC 2 of Table 2. It is easy to prove that the regression does not change the r correlation. For this reason, only ICC is recalculated.

Results

Regression results

Both ICC correlations before regression and after regression are shown in Table 2. It should be noticed that the results ICC 1 before regression in Table 2 were slightly different from the results presented in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF], because the range of f MVC varies from 0.15 to 0.99 in this paper while it varied from 0.20 to 0.99 in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF] in order to validate the dynamic fatigue model. Some models were sensitive for such a change, e.g. Monod's model. However, the little change of the validation result does not change the conclusion in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF].

The ICC results for different muscle groups after regression are graphically presented in log-log diagram from Fig. 1 to 4 as well. The corresponding graphical results before regression can be found in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF]. The x-axis is the MET values predicted using the extended MET model at different relative force levels, while the y-axis is the MET values predicted using other MET models at corresponding relative force levels. The straight solid diagonal line is the reference line. For the other models, the one which approaches more closely to the straight line has a higher ICC. If a model coincides completely over the reference line, which means it is identical to the extended MET model.

Fatigue resistance results

The regression results (m) for each MET model are listed in Table 3. The mean value m and the standard deviation σ m were calculated for different muscle groups as well. The Monod's general model is eliminated from the calculation due to its poor ICC value. The intergroup differences are represented by the mean value of each muscle group. The Hip/back models have a higher mean value m = 1.9701, while the other human body segments and the general models have relative lower fatigue resistances ranging from 0.76 to 0.90, without big differences. The fluctuation in each muscle group, namely the intra muscle group difference, is presented by σ m . The stability in the general group is the best, and the hip/back model has the largest variation. There is no big difference between elbow and shoulder models.

Discussion

Result analysis

ICC: Almost all the ICC 2 are greater than 0.89, and only one is an exception (Monond and Scherrer, 0.4736). This exception is because of its relative worse linear correlation r with the extended MET model, while almost all the other ones have r over 0.96, and the Monod's model has only 0.6241. For the Monod's model, the linear error occurs mainly when the f MVC approaches to 0.15. This error is mainly caused by the way in which the Monod's model is formulated. This exception is eliminated in the following analysis and discussion.

There are larger differences between the extended MET model and the empirical MET models, especially when the f MVC approaches to 0.15. Those differences can be explained by the interindividual difference in MET, and these differences are greater for the low %MVC [START_REF] El Ahrache | Percentile values for determining maximum endurance times for static muscular work[END_REF]. From the graphical representation, it can be noticed that the MET errors are mainly decreased in the range from 10 0 min to 10 1 min, which means the extended MET model after regression can predict MET with less error than using the extended MET model before regression(refer to Fig. 2, Fig. 4, Fig. 6 and Fig. 8 in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF]).

The greatest improvement of the fitness between the extended MET model and the empirical MET models is the Hip/Back model (Fig. 4). This approves that the extended MET model with a suitable fatigue ratio can adapt itself well to the most complex part of human body. The same improvement can be found for shoulder models and most of the elbow models. It should be noticed not all the MET models have been improved after the regression. Little fall can be found for the MET models (hand model) with ICC over 0.98 in the ICC 1 column. The possible reason is that it has already relative high ICC correlation, and the regression does not improve its fitness. However, those models after regression still have high ICC (> 0.95). As a summary, the regression approach achieves high ICC and improves the similarity between the extended MET model and the existing models. This proves that the extended MET model can be adapted to fit different body parts, and the extended MET model can predict the MET for static cases.

Fatigue resistance: The regression result of the fatigue resistance of different muscle groups were tested with normplot function in Matlab in order to graphically assess whether the fatigue resistances could come from a normal distribution. The test result shows fatigue resistances for general models and elbow models scatter near the diagonal line in the Fig. 5 and Fig. 6. Due to limitation of sample numbers in shoulder models and the large variance in hip/back models, the distribution test did not achieve satisfying result. Once there are enough sample models, it can be extrapolated that the fatigue resistances for different muscle groups for the overall population distributes in normal probability, therefore, the mean value locates in m ± σ could predict the fatigue property of 50% population.

Therefore, the mean value of m and its standard deviation σ m are used to redraw the relation between MET and f MVC , and they are presented from Fig. 7 to 10. The black bold solid line is the extended MET model adjusted by m and it locates in the range constrained by two slim solid lines adjusted by m ± σ m . After adjusting our fatigue model with m ± σ m , the extended MET fatigue model can cover most of the existing MET models from 15% MVC to 80% MVC. Although there is an exception in Hip/back model due to the relative large variability in hip muscle groups, it should also be admitted that the adjustment makes the extended MET model suitable for most of the static cases. In another word, the adjustment by mean and deviation makes the extended MET model suitable for evaluating the fatigue for the overall population.

The prediction by the extended MET model cannot cover the models for the %MVC over 80 as well as the interval under 15%. However in the industrial cases, it is very rare that the force demand can cross that limit 80% in static operations. Even if the physical demand beyond 80%MVC, the prediction difference in the extended MET model from the other MET

The extended MET model versus previous MET models

There are several MET models available in the literature, and they cover different body parts. These models are all experimental models regressed from experimental data, and each model is only suitable for predicting MET of a specific group of people, although the similar tendencies can be found among these models. Furthermore, those MET model cannot reveal individual differences in fatigue characteristic. However, it is admitted that different people might have different fatigue resistances for the same physical operation.

In comparison to conventional MET models, the general analytical MET model was extended from a simple dynamic fatigue model in a theoretical approach. The dynamic muscle fatigue model is based on muscle physiological mechanism. It takes account of task parameters (F load or relative load) and personal factors (MVC and fatigue ratio k), and it has been validated in comparison to other theoretical models in [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF]. Different from the other MET models, in this extended MET model, there is a parameter k representing individual fatigue characteristic.

After mathematical regression, great similarities (ICC > 0.90) have been found between the extended MET model and the previous MET models. This indicates that the new theoretical MET model might replace the other MET models by adjusting the parameter k. Therefore, the extended MET model generalizes the formation of MET models.

In addition, different fatigue resistances have been found while fitting to different MET models, even for the same muscle group. Therefore, it is interesting to find the influencing factors on the parameter k and to analyze its statistical distribution for ergonomic application. In this paper, we tried to use the mean value and standard deviation of the regressed fatigue re-

Influencing factors on fatigue resistance

Although the MET models fitted from experiment data were formulated in different forms, the m can still provide some useful information for the fatigue resistance, especially for different muscle groups. The differences in fatigue resistance result is possible to be concluded by the mean value and the deviation, but it is still interesting to know why and how the fatigue resistance is different in different muscle groups, in the same muscle group, and even in the same person at different period. There is no doubt that there are several factors influencing on the fatigue resistance of a muscle group, and it should be very useful if the fatigue resistance of different muscle groups can be mathematically modeled. In this section, the fatigue resistance and its variability are going to be discussed in details based on the fatigue resistance results from Table 3 and the previous literature about fatigability. Different influencing factors are going to be discussed and classified in this section.

All the differences inter muscle groups and intra muscle groups in MET models can be classified into four types: 1) Systematic bias, 2) Fatigue resistance inter individual for constructing a MET model, 3) Fatigue resistance intra muscle group: fatigue resistance differences for the same muscle group, and 4) Fatigue resistance inter muscle groups: fatigue resistance differences for different muscle groups. Those differences can be attributed to different physiological mechanisms involved in different tasks, and influencing variables are subject motivation, central command, intensity and duration of the activity, speed and type of contraction, and intermittent or sustained activities [START_REF] Enoka | Mechanisms of muscle fatigue: central factors and task dependency[END_REF][START_REF] Elfving | Task dependency in back muscle fatigue-Correlations between two test methods[END_REF]. In those MET models, all the contractions were exerted under static conditions until exhaustion of muscle groups, therefore, several task related influencing factors can be neglected in the discussion, e.g., speed and duration of contraction. The other influencing factors might contribute to the fatigue resistance difference in MET models.

Systematic bias : all the MET models were regressed or reanalyzed based on experiment results. Due to the experimental background, there were several sources for systematic error. One possible source of the systematic bias comes from experimental methods and model construction [START_REF] El Ahrache | Percentile values for determining maximum endurance times for static muscular work[END_REF], especially for the methods with subjective scales to measure MET. The subjective feelings significantly influenced the result. Furthermore, the construction of the MET model might cause system differences for MET model, even in the models which were constructed from the same experiment data (e.g. Huijgens' model and Sjogaard's model in General models). The estimation error was different while using different mathematic models, and it generates systematic bias in the result analysis.

Fatigue resistance inter individual : besides the systematic error, another possible source for the endurance difference is from individual characteristic. However, the individual characteristic is too complex to be analyzed, and furthermore, the individual characteristic is impossible to be separated from existing MET models, since the MET models already represent the overall performance of the sample participants. In addition, in ergonomic application, the overall performance of a population is often concerned. Therefore, individual fatigue resistance is not discussed in this part separately, but the differences in population in fatigue resistance are going to be discussed and presented in the following part.

Fatigue resistance intra muscle group : the inter individual variability contributes to the errors in constructing MET models and the errors between MET models for the same mus- As mentioned in Section 1, the influences on fatigability from gender and age were observed in the literature. In the research for gender influence, women were found with more fatigue resistance than men. Based on muscle physiological principle, four families of factors were adopted to explain the fatigability difference in gender in [START_REF] Hicks | Sex differences in human skeletal muscle fatigue[END_REF]. They are: 1) muscle strength (muscle mass) and associated vascular occlusion, 2) substrate utilization, 3) muscle composition and 4) neuromuscular activation patterns. It concluded that although the muscle composition differences between men and women is relatively small [START_REF] Staron | Fiber type composition of the vastus lateralis muscle of young men and women[END_REF], the muscle fiber type area is probably one reason for fatigability difference in gender, since the muscle fiber type I occupied significantly larger area in women than in men [START_REF] Larivière | Gender influence on fatigability of back muscles during intermittent isometric contractions: A study of neuromuscular activation patterns[END_REF]. In spite of muscle fiber composition, the motor unit recruitment pattern acts influences on the fatigability as well. The gender difference in neuromuscular activation pattern was found and discussed in [START_REF] Larivière | Gender influence on fatigability of back muscles during intermittent isometric contractions: A study of neuromuscular activation patterns[END_REF], and it was observed significantly that females showed more alternating activity between homolateral and contralateral muscles than males.

Meanwhile, in [START_REF] Mademli | Effect of voluntary activation on agerelated muscle fatigue resistance[END_REF] older men were found with more endurance time then young men in certain fatigue test tasks charging with the same relative load. One of the most common explanations is changes in muscle fiber composition for fatigability change while aging. The shift towards a higher proportion of muscle fiber type I leads old adults having a higher fatigue resistance but smaller MVC. Gender and age were also already taken into a regression model to predict shoulder flexion endurance [START_REF] Mathiassen | Prediction of shoulder flexion endurance from personal factors[END_REF]. In [START_REF] Nussbaum | Effects of age, gender, and task parameters on fatigue development during intermittent isokinetic torso extensions[END_REF], the effects of age, gender, and task parameters on muscle fatigue during prolonged isokinetic torso exercises were studied. It constated that older men had less initial strength. It was also found that effects of age and gender on fatigue were marginal, while significant interactive effects of age and gender with effort level were found at the same time.

Besides those two reasons, the muscle fiber composition of muscle varies individually in the population, even in a same age range and in the same gender [START_REF] Staron | Fiber type composition of the vastus lateralis muscle of young men and women[END_REF], and this could cause different performances in endurance tasks. Different physical work history might change the endurance performance. For example, it appeared that athletes with different fiber composition had different advantages in different sports: more type I muscle fiber, better in prolonged endurance events [START_REF] Wilmore | Physiology of Sport and Exercise[END_REF]. Meanwhile, the physical training could also cause shift between different muscle fibers [START_REF] Costill | Adaptations in skeletal muscle following strength training[END_REF]. As a result, individual fatigue is very difficult to be determined using MET measurement [START_REF] Vøllestad | Measurement of human muscle fatigue[END_REF], and the individual variability might contribute to the differences among MET models for the same muscle group due to selection of subjects.

Back to the existing MET models, the sample population was composed of either a single gender or mixed. At the same time, the number of the subjects was sometimes relative small. For example, only 5 female students (age range 21-33) were measured [START_REF] Garg | The effect of maximum voluntary contraction on endurance times for the shoulder girdle[END_REF], while 40 (20 males, age range 22-48 and 20 females, age range 20-55) were tested in shoulder MET model [START_REF] Mathiassen | Prediction of shoulder flexion endurance from personal factors[END_REF]. Meanwhile, the characteristics of population (e.g., students, experiences workers) could cause some differences in MET studies. Due to different population selection method, different gender composition, and different sample number of participant, fatigue resistance for the same muscle group exists in different experiment results and finally caused different MET models under the similar postures.

In Hip/back models, even with the same sample participants, difference existed also in MET models for different postures. The variation is possible caused by the different MU recruitment strategies and load sharing mechanism under different postures. [START_REF] Kasprisin | Joint angle-dependence of elbow flexor activation levels during isometric and isokinetic maximum voluntary contractions[END_REF] observed that the activation of biceps brachi was significantly affected by joint angle, and furthermore confirmed that joint angle and contraction type contributed to the distinction between the activation of synergistic elbow flexor muscles. The lever of each individual muscle changes along different postures which results different intensity of load for each muscle and then causes different fatigue process for different posture. Meanwhile, the contraction type of each individual muscle might be changed under different posture. Both contraction type change and lever differences contribute to generate different fatigue resistance globally. In addition, the activation difference was also found in antagonist and agonist [START_REF] Karst | Antagonist muscle activity during human forearm movements under varying kinematic and loading conditions[END_REF][START_REF] Mottram | Motor-unit activity differs with load type during a fatiguing contraction[END_REF] muscles as well, and it is implied that in different posture, the engagement of muscles in the action causes different muscle activation strategy, and as a result the same muscle group could have different performances. With these reasons, it is much difficult to indicate the contribution of posture in fatigue resistance because it refers to the sensory-motor mechanism of human, and how the human coordinates the muscles remains not clear enough until yet.

Fatigue resistance inter muscle groups: As stated before, the three different muscle fiber types have different fatigue re-sistances, and different muscle is composed of types of muscles with composition determining the function of each muscle [START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF]. The different fatigue resistance can be explained by the muscle fiber composition in different human muscle groups.

In the literature, muscle fiber composition was used measured by two terms: muscle fiber type percentages and percentage fiber type area (CSA: cross section area). Both terms contribute to the fatigue resistance of the muscle groups. Type I fibers occupied 74% of muscle fibers in the thoracic muscles, and they amounted 63% in the deep muscles in lumbar region [START_REF] Sirca | The fibre type composition of thoracic and lumbar paravertebral muscles in man[END_REF]. On average type I muscle fibers ranged from 23 to 56% for the muscles crossing the human shoulder and 12 of the 14 muscles had average SO proportions ranging from 35 to 50% [START_REF] Dahmane | Spatial fiber type distribution in normal human muscle histochemical and tensiomyographical evaluation[END_REF]. In paper [START_REF] Staron | Fiber type composition of the vastus lateralis muscle of young men and women[END_REF][START_REF] Shepstone | Short-term high-vs. low-velocity isokinetic lengthening training results in greater hypertrophy of the elbow flexors in young men[END_REF], the muscle fiber composition shows the similar composition for the muscle around elbow and vastus lateralis muscle and the type I muscle fibers have a proportion from 35 -50% in average. Although we cannot determine the relationship between the muscle type composition and the fatigue resistance directly and theoretically, the composition distribution among different muscle groups can interpret the MET differences between general, elbow models and back truck models. In addition, the fatigue resistance of older adults is greater than young ones could also be explained by a shift towards a higher proportion of type I fiber composition with aging. These evidences meet the physiological principle of the dynamic muscle fatigue model.

Another possible reason is the loading sharing mechanism of muscles. Hip and back muscle group has the maximum joint moment strength [START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF] among the important muscle groups. For example, the back extensors are composed of numerous muscle slips having different moment arms and show a particularly high resistance to fatigue relative to other muscle groups [START_REF] Jorgensen | Human trunk extensor muscles physiology and ergonomics[END_REF]. This is partly attributed to favorable muscle composition, and the variable loading sharing within back muscle synergists might also contribute significantly to delay muscle fatigue.

In summary, individual characteristics, population characteristics, and posture are external appearance of influencing factors for the fatigue resistance. Muscle fiber composition, muscle fiber area, and sensory motor coordination mechanism are the determinant factors inside the human body deciding the fatigue resistance of muscle group. Therefore, how to construct a bridge to connect the external factors and internal factors is the most important way for modeling the fatigue resistance for different muscle groups. How to combine those factors to model the fatigue resistance remains a challenging work. Despite the difficulty of modeling the fatigue resistance, it is still applicable to find the fatigue resistance for a specified population by MET experiments in regression with the extended MET model due to its simplicity and universal availability.

Limitations

In the previous discussion, the fatigue resistance of the existing MET models were quantified using m from regression. The possible reasons for the different fatigue resistance were analyzed and discussed. However, how to quantify the influence from different factors on the fatigue resistance remains unknown due to the complexity of muscle physiology and the correlation among different factors.

The availability of the extended MET model in the interval under 15% MVC is not validated. The fatigue resistance is only accounted from the 15% to 99% MVC due to the unavailability of some MET models under 15% MVC. For the relative low load, the individual variability under 15% could be much larger than that over 15%. The recovery effect might play a much more significant role within such a range.

Conclusions and perspectives

In this paper, fatigue resistance of different muscle groups were calculated by linear regression from the new fatigue model and the existing MET static models. High ICC has been obtained by regression which proves that our fatigue model can be generalized to predict MET for different muscle groups. Mean and standard deviation in fatigue resistance for different muscle groups were calculated, and it is possible to use both of them together to predict the MET for the overall population. The possible reasons responsible for the variability of fatigue resistance were discussed based on the muscle physiology.

Our fatigue model is relative simple and computation efficient. With the extended MET model it is possible to carry out the fatigue evaluation in virtual human modeling and ergonomic application, especially for static and quasi-static cases. The fatigue effect of different muscle groups can be evaluated by fitting k from several simple static experiments for certain population.

Figure 3 :

 3 Figure 3: ICC diagram for MET shoulder models after regression

Figure 4 :

 4 Figure 4: ICC diagram for Hip/back shoulder models after regression

  using extended MET model in comparison with General static models f MVC Endurance Time in General models [min]

Figure 9 :

 9 Figure 9: MET prediction using extended MET model in comparison with shoulder static models

Table 1 :

 1 Parameters in Dynamic Fatigue Model

	Item	Unit Description
	MVC	N	Maximum voluntary contraction, maximum capacity of muscle
	F cem (t)	N	Current exertable maximum force, current capacity of muscle
	F load (t)	N	External load of muscle, the force which the muscle needs to generate
	k	min -1 Constant value, fatigue ratio, here k = 1
	%MVC		Percentage of the voluntary maximum contraction
	f MVC		%MVC/100

Table 2 :

 2 Static validation results r and ICC between the extended MET model and the other existing MET models in[START_REF] El Ahrache | Percentile values for determining maximum endurance times for static muscular work[END_REF]

	Model	MET equations (in minutes)	r	ICC 1	ICC 2
	General models
	Rohmert	MET = -1.5 + 2.1 f MVC -0.6 f 2 MVC	+ 0.1 MVC f 3	0.9717 0..9505 0.9707
	Monod and Scherrer	MET = 0.4167 ( f MVC -0.14) -2.4	0.6241	0.0465	0.4736
	Huijgens	MET = 0.865 1-f MVC f MVC -0.15	1/1.4	0.9036	0.8947	0.8916
	Sato et al.	MET = 0.3802 ( f MVC -0.04) -1.44 0.9973	0.8765	0.9864
	Manenica	MET = 14.88 exp(-4.48 f MVC )	0.9829	0.9357	0.9701
	Sjogaard	MET = 0.2997 f -2.14 MVC			0.9902	0.9739	0.9898
	Rose et al.	MET = 7.96 exp(-4.16 f MVC )	0.9783	0.6100	0.9573
	Upper limbs models
	Shoulder
	Sato et al. Rohmert et al.	MET = 0.398 f -1.29 MVC MET = 0.2955 f -1.658 MVC			0.9988 0.9993	0.5317 0.7358	0.9349 0.8982
	Mathiassen and Ahsberg	MET = 40.6092 exp(-9.7 f MVC )	0.9881	0.8673	0.9711
	Garg	MET = 0.5618 f -1.7551 MVC			0.9968	0.9064	0.9947
	Elbow
	Hagberg	MET = 0.298 f -2.14 MVC			0.9902	0.9751	0.9898
	Manenica	MET = 20.6972 exp(-4.5 f MVC )	0.9832	0.9582	0.9708
	Sato et al. Rohmert et al.	MET = 0.195 f -2.52 MVC MET = 0.2285 f -1.391 MVC			0.9838 0.9997	0.9008 0.2942	0.9688 0.9570
	Rose et al.2000	MET = 20.6 exp(-6.04 f MVC )	0.9958	0.9627	0.9708
	Rose et al.1992	MET = 10.23 exp(-4.69 f MVC )	0.9855	0.7053	0.9766
	Hand
	Manenica	MET = 16.6099 exp(-4.5 f MVC )	0.9832	0.9840	0.9646
	Back/hip models
	Manenica (body pull)	MET = 27.6604 exp(-4.2 f MVC )	0.9789	0.7672	0.9591
	Manenica (body torque)	MET = 12.4286 exp(-4.3 f MVC )	0.9804	0.8736	0.9634
	Manenica (back muscles) MET = 32.7859 exp(-4.9 f MVC )	0.9878	0.8091	0.9819
	Rohmert (posture 3) Rohmert (posture 4) Rohmert (posture 5)	MET = 0.3001 f -2.803 MVC MET = 1.2301 f -1.308 MVC MET = 3.2613 f -1.256 MVC			09655 0.9990 0.9984	0.4056 0.8356 0.1253	0.9482 0.9396 0.9263

Table 3 :

 3 Fatigue resistance m of different MET models

	Segment	m 1	m 2	m 3	m 4	m 5	m 6	m 7	m	σ m
	General Rohm.	Mono.	Hijg.	Sato.	Mane.	Sjog.	Rose
		0.8328	-	0.9514 0.6836	0.8019	1.1468	0.4647 0.8135 0.2320
	Shoulder	Sato.	Rohme.	Math.	Garg
		0.4274	0.545	0.698	1.3926				0.7562 0.4347
	Elbow	Hagb.	Mane.	Sato.	Rohm. Rose00 Rose92
		1.1403	1.1099	1.3461 0.2842	0.7616	0.5234		0.8609 0.4079
	Hand	Mane.
		0.8907							0.8907	-
	Hip	pull	torq.	back	pos3	pos4	pos5
		1.5986	0.7005	1.5931 3.2379	1.356	3.3345		1.9701 1.1476

Acknowledgments

This research was supported by the EADS and the Région des Pays de la Loire (France) in the context of collaboration between the École Centrale de Nantes (Nantes, France) and Tsinghua University (Beijing, PR China).


Z-H Huang

M Abkarian

A Viallat
email: annie.viallat@inserm.fr

Sedimentation of vesicles: from pear-like shapes to microtether extrusion

Keywords: 87, 16, D-47, 15, G-

We study the sedimentation of buoyant giant lipid vesicles in a quiescent fluid at velocities ranging from 5 to 20 µm/s. Floppy vesicles are deformed by the flow. Their bottom (upstream) part remains spherical while their top (downstream) part narrows down and elongates along the direction of motion, resulting in pear-like shapes or in the reversible formation of a micron-size tube at the vesicle top. The sedimentation velocity of vesicle is very similar to that of a rigid sphere. Using a thermodynamic approach, we show that the hydrodynamic force acting at the top of a floppy vesicle can exceed the critical force needed to draw a membrane tube. We predict that the tube radius scales as the power 1/3 of the ratio of the bending energy to the typical hydrodynamic stress, ηU/R where η is the fluid viscosity, U is the sedimentation velocity and R the vesicle radius. This result is consistent with the reported experimental data. The tensions of vesicles exhibiting a tube and of pear-like shape are deduced from the thermodynamic approach.

Introduction

The behavior of living cells submitted to an external force or a hydrodynamic flow is an important issue both for physiology and for in-vitro processing and manipulation. It has been much studied for cells and giant lipid vesicles. The latter are basic soft shells whose behavior is a physical reference for the cell 'passive' behavior. For instance, in a shear or an elongational flow, giant vesicles and red blood cells present a variety of motions and deformations whose commonalities and differences have been characterized and understood on the basis of the specificity of the mechanical properties of each particle [1,2,3]. Another example is the behavior of vesicle and cell membranes subjected to an external pulling point force. It is well known that there is a critical force for which a nanotube, called tether [4, 5,6,7,8,9] is drawn from the vesicle/cell and coexists with the spherical body of the particle. In-vivo, tether formation is involved in many biological processes such as intracellular trafficking [10,11,12] or cell migration [13]. In-vitro, tethers extruded from the membrane (by pulling a small bead adhering on the membrane) are used as sensors to gain information on vesicle/cell mechanical properties (cortical tension, coupling energy between the membrane and the cytoskeleton) [14,15,16,17]. Highly surprising, the behavior of vesicles and cells suspended in a fluid and submitted to a gravitational force is not thoroughly documented. One theoretical study [18] predicts that sedimenting vesicles should deform and present either pear-like shapes or bean-like shapes, with no experimental support. However, the question of the movement of deformable objects due to a body force in a viscous fluid, arises naturally in different situations such as centrifugation, magnetophoresis or cell sorting in microfluidics. For instance, recent observations of red blood cells during centrifugation show how the cells resume very peculiar pear-like shapes [19]. In a more general point on view, sedimentation and shape instability of simple viscous drops had also been observed. While a spherical drop remains spherical during sedimentation (the sphere is a steady solution to the Stokes equations [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF]), non-initially spherical drops are unstable [START_REF] Kojima | [END_REF] and prolate shapes can develop long tails at their rear [22,23,24]. Initially spherical droplets can also be destabilized when surfactants are present. In this case, the flow gathers the surfactants at the rear of the droplet reducing drastically the local surface tension and leading to tip streaming [25].

In this paper, we study the deformation of giant unilamellar vesicles (GUVs) sedimenting in a quiescent fluid. We deal with very deformable floppy vesicles. We show that, depending on their sedimentation velocity, the vesicles present either original pear-like shapes, or, above a critical hydrodynamic force, develop at a micro-size lipid tube, which coexists with a quasi-spherical shape. A striking point is that the tube extraction does not require the application of a point force. The distributed hydrodynamic stress, which is much weaker than point forces generally used to extract tethers, is however sufficient to pull micron-size tubes on floppy vesicles. We show that a free energy analysis describes such a behavior. We first detail the materials and method before describing the regimes of motion and the deformations of the vesicles. Then we show that vesicle velocities are similar to that of rigid spheres translating in a quiescent fluid. We propose an approach to describe and understand the formation of membrane tubes. Finally, we estimate and discuss the values of the surface tensions of sedimenting vesicles.

Experimental

Giant Unilamellar Vesicles (GUVs) preparation

GUVs are prepared by the electroformation method [26]. Dioleoyl-phosphatidylcholine (DOPC, Sigma) dissolved in chloroform and methanol solutions (9:1 volume ratio) at 2mg/ml and 10 µl of the solution is spread on the conductive faces of two transparent glass plates coated with a film of indium tin oxide (ITO). After drying (2h under primary vacuum), about 2ml of a sucrose solution (with concentration 300mM) is injected in a chamber formed by the ITO plates facing each other, separated by Teflon spacers and connected to an AC generator (frequency 10 Hz). The potential is very slowly increased from 0.2 to 0.8 V and kept constant for one night. The frequency and the voltage are finally decreased to 4 Hz and 0.2 V to detach vesicles from the ITO plates. The suspension of vesicles we obtain is then diluted in a binary sucrose/glucose solution with the following protocol: 100 µl of the initial sucrose suspension of vesicles is diluted in 1 ml of the chosen binary sucrose/glucose solutions for 1 hour. Then, 100 µl of this suspension is rediluted in 1 ml of the binary sucrose/glucose solution for another hour. Finally 100 µl of this suspension is diluted again in 450 µl in the binary sucrose/glucose solution and is gently introduced in the observation chamber (volume 0.550 ml). Only few vesicles are present in the observation chamber in order to limit hydrodynamic interactions. The characteristics of the vesicles are listed in Table 1 andTable 2.

Method of observation

We use a parallelepiped chamber from Hellma (Mullheim, Germany) which has four optically transparent faces (width x-direction:10 mm, length y direction: 55 mm, height z-direction: 1 mm). Sedimenting vesicles and their reflections on the bottom of the chamber are observed in the vertical yz plane, from a microscope (LEICA IRB) tipped to the horizontal [27]. We let the vesicles settle at the bottom of the chamber during 20 min. Then, we turn the chamber upside down and observe the sedimentation of one vesicle using a CCD camera (COHU 4910, 25 fps). In order to prevent vesicle adhesion on the chamber wall, we first incubated a casein solution in the chamber before introducing the vesicles. The vesicles were found to behave similarly in the chamber rinsed in Millipore water and in the chamber treated with casein. We therefore perform most experiments by simply using the chamber rinsed with Millipore water. Movies are saved using a digital videocassette recorder SONY DSR25. Vesicle tracking, contour and center of gravity determinations as well as shape analysis are carried out with the software IDL. The volume V and area A of the vesicles are measured at rest from the shapes of the settled vesicles. The effective radius R is deduced as (3V/4π) 1/3 .

Observations

Full experiment

Typical experiments are shown in figure 1. First, the vesicle departs from the top of the chamber and progressively elongates and deforms while its velocity increases. Then a stationary regime is reached: the vesicle, located halfway the observation chamber, has a constant velocity and a stable shape. Finally, upon approaching the bottom wall, the vesicle velocity decreases and its shape progressively changes (no more vertical elongation). Then, upon landing on the bottom of the chamber, the vesicle flattens down with a typical gravity-induced shape deformation. When a tube is observed at the top of the vesicle, it retracts and it is swallowed by the main body of the vesicle during the landing stage, as shown by the last images of each sequence represented in Fig. 1.

Figure 1: Vesicle sedimentation a: vesicle (2.2), pear shape, full process from departing to landing, t= 0s, 0:57 s, 2:22 s, 4:46 s, 5:08 s, 6:28 s, shapes are fluctuating. b: vesicle (12.1), short tube, full process, t=0s, 0:24s, 1:04s, 1:49s, 2:51s, 4:38s, 6:14s. c: vesicle radius 6 µm, landing process from stationary shape to settled vesicle, t=4:40s, 5:32s, 7:04 s, 8:30s, 9:12s, 12:00s. d: departing of a vesicle (radius 49 µm) and tube growth (time between image = 1s) e) landing of the vesicle with tube retraction (60s, 76 s, 84s, 90s, 92s, 93s).

Stationary shapes

Vesicles, which do not present large excess area cannot significantly deform and thus remain spherical during their sedimentation. However, most vesicles are osmotically deflated and are therefore floppy and deformable. We now describe their stationary shape. The vesicles exhibit a spherical upstream (bottom) region and a vertically elongated downstream (top) part with a radius of curvature, r, smaller than the spherical part. Weakly deformable vesicles present an egg-shape. More deformable vesicles look like pears with, at their top, a spherical cap of radius r < R and an elongated region of length L≤ r (figure 2). Some vesicles present a spherical part, which coexists with a tube. The tube radius, r << R, ranges from less than one micron to 2-3 microns. It is slightly pinched at its basis and, as seen in figure 1b and in figure 2, a catenoid shape is observed at the junction between the spherical part and the tube. Its length L is longer than r. Most of these vesicles present large membrane undulations (figure 1a), disclosing that the tension is very low. Some vesicles, which are characterized by the highest sedimentation velocities, exhibit several tens of microns long pearling tubes of submicron radius (figure 1d). When vesicles are submitted to two or three successive experiments of sedimentation, similar shape deformations and sedimenting velocities are retrieved. Finally it is worth noting that there is no lipid flow on the membrane surface as attested by small lipid aggregates bound to the membrane, which do not move relatively to the center of mass of the vesicle. Indeed, the driving force of the motion being vertical, the vesicles present a vertical axis of symmetry. This symmetry prevents the apparition of a lipid recirculation on its surface. Moreover, contrarily to a fluid drop, the two-dimensional confinement of lipid molecules and the symmetry of the experiment prevent any momentum transfer to the inner fluid and therefore no flow recirculation is observed.

Velocities

We show in this section that the velocity field around a sedimenting vesicle is given by the flow produced around a rigid sphere with a radius equal to the effective radius of the vesicle and shows that at the leading order, the deformation of the vesicle does not modify the velocity field of the equivalent sphere. The evolution of the vertical velocity of the center of mass of the vesicles as a function of time is illustrated in figure 3a for one pear-like shape vesicle. The existence of the stationary regime is clearly observed (Fig. 3a). The two transient regimes correspond to the departing and to the landing of the vesicle and the associated velocities are represented in figure 3b as a function of the distance h to the departing or the landing wall of the chamber. In comparison, we also plot the sedimentation velocity U of the equivalent sphere given by the following law:

€ U = 2g 9 Δρ R 2 η 1+ R h ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , ( 1
)
where Δρ is the difference of density between the inner fluid of the sphere and the outer suspending fluid and g is the gravitational acceleration. U is derived from the approximated expression of the drag force of a rigid sphere moving close to a wall perpendicular to the direction of the sphere motion [START_REF] Guyon | Hydrodynamique Physique[END_REF].

One can clearly see that departing and landing velocities are pretty similar and are well described by (1). This result shows that the deviation of the vesicle shape from a sphere does not affect the velocity field around the particle.

To confirm this result, we also plot the stationary velocity U observed far from the walls for all studied vesicles, whatever their shapes and for an spherical bead of agarose (R= 36 µm) as a function of

€ g 9
Δρ R 2 η . All data lie on a single line as shown in figure 4. This result confirms that the vesicle behave like rigid spheres in a Stokes flow while its surface is an incompressible two-dimensional fluid. Because the lipids cannot leave the interface, the divergence of the velocity field is exactly zero. This strong constrain does not exist for a liquid drop where the molecules can leave the interface and transfer momentum to the volume even in an axisymmetric case. A liquid drop has therefore a lower friction force than the fluid vesicle. As a consequence, we will consider in a first approximation, that the drag force that applies on a sedimenting vesicle far from the walls is equal to the Stokes drag acting on a rigid sphere of equivalent radius: 6 π η R U.

Shape deformation: detailed description

The sedimentation of a vesicle generates an excess pressure upstream and a lower pressure downstream, at its top. This difference of pressure is at the origin of the pear-shape deformation. The viscous forces exerted by the fluid on the vesicle membrane redistribute a part of the membrane excess area toward the top part of the vesicle and under some conditions are able to draw a membrane tube.

Using the pressure field around a rigid translating sphere [29], we can express the variation of the pressure p on the surface of the vesicle at a distance R from the center of mass:

€ p = p ∞ - 3 2 η  U .  n R , ( 2
)
where p ∞ is the pressure in the fluid far from the vesicle and €  n is the normal to the membrane. The difference of pressure between the top and the bottom part of the vesicle is therefore 3 η U/R, which is typically of the order of 10 -3 Pa in our study.

Hydrodynamic tube extrusion

We consider the tube observed on the top of the vesicles in the stationary regime, i. e. when r <<R and L > r. We write the free energy of the tube membrane by using the approach developed by Waugh et al. [7,30,31], Evans et al. [32], Derényi et al. [33], Hochmuth et al. [16], Brochard-Wyart et al. [8] and Fournier [34]. It is based on the Helfrich Hamiltonian, which describes the membrane as a fluid, incompressible surface with a homogeneous surface tension σ and a bending rigidity κ. For a fixed tension and a fixed pressure, the tube free energy F of radius r and length L can be written as

€ F = κ 2r 2 +σ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2π r L -Δp π r 2 L -f L , ( 3
)
where Δp is the pressure difference across the tube membrane and f is the pulling force, which elongates the tube on a length L. We consider that the thickness of the membrane is constant. Then, the area difference between the lipid inner monolayer and the outer monolayer of the membrane changes when a tether is pulled out. The change in area difference creates a spontaneous curvature of the membrane calculated in reference [35]. The energetic cost for this curvature was estimated in [36] to ≈ κ L 2 /R 2 , where R is the vesicle radius. This contribution is significant only for tube lengths of more than 100 microns so that we omit it in the free energy.

The tube radius and the pulling force are calculated at equilibrium by taking € ∂F ∂r = 0 and ∂F ∂L = 0.

The first derivative gives the equilibrium Laplace law for a cylinder:

€ σ r - κ 2r 3 = Δp (4)
We estimate the pressure difference across the tube membrane by considering that the hydrostatic pressure out of the tube is roughly equal to the pressure upstream at the vesicle top:

Δp = p in -(p ∞ -3 η U/2R)
, where p in is the pressure within the vesicle. Here, the vesicles are initially tensionless since they have been deflated. In this case, the difference of pressure p in -p ∞ ≈ 0 << 3 η U/2R. Therefore, (4) rewrites as

€ σ r - κ 2r 3 = 3ηU 2R (5)
The surface tension of the vesicle is induced by the flow and is therefore of the order of the 'hydrodynamic' tension σ ≈ ηU ≈ 10 -8 N/m. The pressure term in the right-hand term of ( 5) is one order of magnitude smaller (≈ 10 -3 ), than the tension and the bending term (≈ 10 -2 ) since tubes radii are much smaller (r ≈ 1 µm) than R. The right-hand term of pressure of (5) plays therefore only the role of a small corrective factor. If the vesicle is initially tense (σ >> ηU), the pressure terms in (5) are not dominant. One retrieves the classical frame [8,33,34] r= (κ/2σ) 1/2 , which yields a typical size for r of the order of 40 -50 nm for a standard value of σ ≈ 10 -5 N/m.

The second derivative € ∂F ∂L = 0 gives the classical value of the equilibrium pulling force exerted on the tube of radius r:

€ The combination of the two equations ( 5) and ( 6) sets the equilibrium radius r 0 of the tube coexisting with a quasispherical vesicle and the associated equilibrium pulling force f 0 .

f c = 3πκ 2r + π r σ (6)
In the case of a sedimenting vesicle, the force exerted on the vesicle surface is the drag force induced by the flow. The pulling force, which initiates the tube is exerted on the top of a spherical cap of area S:

€  f H = - 3η  U 2R S . (7)
When this hydrodynamic force is equal to the extrusion force f 0 it is possible to draw a tube. The condition f H = f 0 is

€ 3η U 2R S = 3π κ 2r 0 + π r 0 σ (8)
Noteworthy, equation 8 compares the critical force of extraction at a tension and for hydrodynamic conditions that are not exactly the one of the stationary regime. However, since we see a tube emerging for a value of the velocity close to 90% of the stationary regime, we believe that our approximation is valid. We assume that the radius of this spherical cap is of the order of magnitude of the tube radius plus the crossover region that relates the quasi-spherical vesicle to the tube (red line in figure 5). As pointed out in [37] the crossover region is an exponentially damped sinusoid with a decay length of the order of r 0 . We therefore write S ≈ π (α r 0 ) 2 with α being the only adjustable parameter, close to 2. By using ( 5) and (8) to eliminate σ, one obtains a relation for the equilibrium tube radius

€ r 3 α 2 -1 ( ) = 4κ R 3η U (9)
Figure 6 shows a plot of the parameter € r 3 as a function of € 4 R 3η U measured for 13 vesicles of different sizes R (from 16 to 48.5 µm) and velocity U ranging from 4 to 12.5 µm/s. Data are fitted with (9) with the slope, κ/(α 2 -1), equal to 8.11 10 -21 J. By setting that the bending rigidity κ ≈ 10 kT = 4 10 -20 J, which is a value currently accepted for DOPC membranes, we find that α = 2.45, which is in good agreement with the prediction of the crossover region given in [37].. From the experimental measurements of the equilibrium tube radius and of the sedimenting velocity, we have estimated the tension of the 13 vesicles during their sedimentation from equations ( 5) and (8). They are very low (Table 2) as expected for floppy vesicles. Noteworthy, the force f 0 required to draw a tube from a floppy vesicle is quite small, of the order of 10 -13 N.

For tensed vesicles, where the pressure term in equation ( 4) is neglectible, the same power law is retrieved,

€ r = κ R 6η U ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 3
and the tension scales as

€ σ = κ 1 3 2 6η U R ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3
. The velocity required to draw a tube from a vesicle of tension σ and bending energy κ is

€ U = R 2σ ( ) 3 2 6η κ 1 2
.

In figure 7 we show the theoretical curves of variation of the ratio of the tube radius to the vesicle radius and of the adimensioned tension, σ R 2 /κ as a function of the adimensioned capillary number η U R 2 /κ for a vesicle drawing a tube.

Figure 7 Theoretical curves (small dots) of variation of the ratio of the tube radius to the vesicle radius and of the adimensioned tension, σ R 2 /κ as a function of the adimensioned capillary number η U R 2 /κ for a vesicle drawing a tube. Filled circles correspond to pearlike vesicles (Table 2) where the tension is calculated from equation 12. Squares correspond to vesicles drawing a tube (Table 1), where the tension is calculated from equation 5. The bending energy set equal to 4 10 -20 J

In order to get an idea of the evolution of the vesicle tension during sedimentation, we measure its apparent surface area. When the tension increases, the apparent vesicle area A also increases unfolding the submicron thermal fluctuations of the membrane. We plotted in figure 8 the parameter R Area = (A/4π) 1/2 as a function of time. While decelerating just before landing the vesicle reduces its area, which reaches a minimum, revealing a minimal surface tension. At this stage, the vesicle flattens and swallows the tube. Then, the area increases again when the vesicle settles onto the substrate and reaches a value similar to that observed during the steady sedimentation stage. Vesicle tensions falling steadily in a fluid or settled onto a substrate are similar. This is not a surprise since the vesicle is submitted in both cases to the same gravitational force. In one case the reaction is the drag force while in the other case it is the reaction of the substrate. The vesicle area and volume are similar in both cases but the repartition of the force is different, yielding different shapes. defined as the ratio of the volume of the vesicle relatively to the volume of the sphere with the same surface area (ν ≤ 1, ν = 1 for a sphere), where V is the vesicle volume. It is deduced from azimuthal integration over the contour of the vesicle settled on the substrate by applying the Papus-Guldin theorem (assuming an axisymmetric shape) [START_REF] Abkarian | Theorem of Pappus[END_REF]. The length of the tube L can be then estimated by expressing the overall volume spherical part + the tube part as V= 4πR 3 /3 + π r 2 L, and the surface area as A=4 πR 2 + 2πrL and by replacing V and A in the expression of the reduced volume. This yields

€ ν = R 3 + 3 4 r 2 L ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1+ rL 2R 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ - 3 2 R -3 (10)
After linearization, we obtain the expression of the tube length,

€ L = 4 3 R 3 (1-ν ) r (νR -r) ( 11
)
Figure 9 shows the plot of L measured as a function of the reduced parameter

€ 4 3 R 3 (1-ν ) r (νR -r)
for the 13 vesicles, which exhibit micron-size tubes. The agreement with the theoretical curve, without any adjustable parameter is quite good.

Figure 9 Experimental tube length versus a reduced parameter (see text). The straight line of slope 1 is the theoretical prediction.

Egg and Pear-like shape vesicles

When the sedimenting velocity of a vesicle is too small, the hydrodynamic force is smaller that the critical force required to extrude a tube. This happens for the couples (σ, ηU) located above the curves displayed in figure 7. The vesicle deforms but no tube is extruded at its rear. Because the external pressures at the bottom and at the top part of the vesicle are different, the curvature radii at the top and at the bottom of the vesicles are different, generating egg or pear-like shapes.

We propose to calculate the tension of little extended objects like egg and pear-like vesicles. With a good approximation we can describe the bottom and the top of the vesicle by two spherical caps of radius R and r top respectively. In this case, we can roughly estimate the vesicle surface tension by writing the Laplace law on both spherical parts: p in -p ∞ -3 η U/2R = 2 σ bot /R and p in -p ∞ + 3 η U/2R = 2 σ top / r top . The flow imposes a pressure gradient inside the membrane surface, which compensates the applied viscous force per unit area. The tension difference between bottom and top endcaps is equal to 3ηU: σ botσ top =3ηU, leading to the following relation between the velocity, the radii and the tension:

€ σ top = 9η U r top 2 R -r top ( ) (12)
We measured values of R and r top on 18 egg and pear-like vesicles and we estimated the membrane surface tension of the vesicles, which are reported in Table 2.. In figure 7b, we plot the reduced tension of these vesicles as a function of the capillary number η U R 2 /κ to obtain a phase diagram. The domain of coexistence at equilibrium of a tube and a spherical vesicle is on the black curve. The pear like regime lies in the highest part of the diagram, above the curve of coexistence tube-vesicle. Below the coexistence curve, the vesicles are initially floppy and are not in equilibrium: a tube is initiated, which grows and increases the vesicle tension. The length of the tube stops when the tension reaches the equilibrium coexistence tension. In our case, most vesicles exhibit large excess area but are slightly tensed due to the gravity.

Conclusion

We have shown that the drag force is able to draw a tube from the top of a vesicle sedimenting in a quiescent fluid while it is not a point force as usually described in the literature for membrane tethers extrusion. In our study we used vesicles, which were osmotically deflated before the experiments, and were characterized by very small (vanishing) surface tensions. In this case the critical force to extrude a tube is very small, of the order of 10 -13 N, and a vesicle moving at a few µm/s can develop tubes with micron-size radii. Noteworthy, no force barrier seems to exist for the formation of tubes, contrarily to the case of tube formation by application of a point force, where the force barrier grows linearly with the size of the area the force is exerted on 39. With a classic thermodynamic approach, we proposed a basic model, which predicts that the tube radius varies as

€ r = κ R 6η U ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 3
consistently with reported experimental data. For large tensions, we predict that the velocity required to draw a tube increases with the power 1.5 of the surface tension.

These results may be of importance when GUVs are subjected to centrifugation, as it is the case in the recent method called double emulsion used to prepare vesicles. In this case, the vesicle velocity can reach several hundreds of µm/s and can generate the formation of membrane tubes. Finally we hope that our work will stimulate numerical more detailed theoretical descriptions able to predict all the observed shapes of vesicles, from the existence of tubes to pearlike shapes, within a single frame.

Figure 2

 2 Figure 2 Typical stationary vesicle shapes. Top, pear-like shapes: (1.2) Radius 26 µm, (14.2) Radius 13 µm, (6.2) Radius 32 µm ; bottom, vesicles with tubes: (11.1) Radius 30 µm, (13.1) Radius 16 µm ; Radius 16 µm .

Figure 3 a

 3 Figure 3 a : velocity versus time, the steady regime is reached in the middle of the chamber, b : velocity during () departing of the vesicle from the top substrate, () landing of the vesicle to the bottom substrate and (⎯ ) equation 1; h is the distance between the vesicle membrane and the top wall (during departing) or between the vesicle membrane and the bottom wall (during landing).

Figure 4 .

 4 Figure 4. Variation of the translation velocity observed for different vesicles in the steady state versus Δρ g R 2 /η, square: velocity of an agarose rigid bead (R= 36 µm, Δρ=1.0466).

Figure 5

 5 Figure 5 Schematic of a sedimenting vesicle.

Figure 6 :

 6 Figure 6: Variation of a parameter scaling as the cube of the tube radius versus the reduced variable 4R/3ηU. The straight line is the theoretical prediction from (9) with a slope equal to 8.11 10 -21 J.

Figure 8 :

 8 Figure 8: Variation of (A/4π) 1/2 () and the velocity (•) versus time. Top: pictures of the vesicle at various times of the sedimentation process.

Table 1

 1 Characteristics of vesicles with a tubeTable 2 Characteristics of pear or egg like vesicles

										Calculated
	#	Vesicle radius R (µm)	Tube radius r (µm)	Length tube (µm)	Velocity U (µm.s -1 )	Viscosity η (10 3 Pa.s)	Density Δρ (g.l -1 )	Reduced volume ν	tension from eq. 5 σ (10 8
										N.m -1 )
	1.1	30.5		2.71	7	8.11	1.33	7.4		0.985	0.42
	2.1	25		3	11	5.48	1.33	7.4		0.96	0.35
	3.1	25.8		3.1	16.4	6.18	1.33	7.4		0.96	0.36
	4.1	29.5		3.6	18.9	4.93	1.36	4.4		0.28
	5.1	48.1		3.1	9	12.4	1.36	4.4		0.99	0.37
	6.1	48.5		3.3	11	10.2	1.37	3.2		0.99	0.33
	7.1	34		3.7	14.8	5.97	1.33	4.2		0.97	0.28
	8.1	28.6		2.7	12.6	12.4	1.29	11		0.97	0.5
	9.1	17.5		3.6	23.5	4	1.26	10		0.31
	10.1 30.3		3.6	17	5.87	1.33	4.2		0.959	0.29
	11.1 28.5		3.2	19.8	7.64	1.29	6.7		0.95	0.35
	12.1 25.3		3.6	13.5	5.3	1.36	4.4		0.95	0.3
	13.1 16		2.8	15	9.5	1.17	18		0.9	0.5
	#		Bottom radius R (µm)	Top radius r (µm)	Velocity U (µm.s -1 )	Viscosity η (10 3 Pa.s)	Density Δρ (g.l -1 )	Calculated tension from eq. 12 σ (10 8 N.m -1 )
	4-1.2		25.4		8.5	10.63	1.29		11	3
	5-2.2		26.4		12.6	4.66	1.33		4.2	2.7
	6-3.2		27.2		14	4.8	1.33		4.2	3
	7-4.2		44.8		24.4	20.9	1.3		7	13.2
	8-5.2		23.5		15.5	4	1.31		4	4.5
	2-6.2		30		8.9	6.2	1.33		4.2	1.5
	10-7.2		24.3		22	5.6	1.33		7.4	33
	11-8.2		48		13	20.27	1.29		11	4.5
	13-9.2		25.7		18.2	10.7	1.29		11	15
	14-10.2	21		14.75	6.5	1.29		11	9
	18-11.2	10.7		2.2	2.9	1.17		18	0.3
	19-12.2	12.8		4	3.8	1.17		18	0.9
	20-13.2	13.7		3.8	6	1.17		18	1.2
	21-14.2	12.9		3.3	7.1	1.17		18	1.2
	23-15.2	23.8		8	15.6	1.17		18	4.2
	0522 V17	15		5	3.2	1.26		10	1.3
	16.2
	0521 10	32.2		19.2	7.91	1.30		7	6.9
	17.2
	070520		15		4	3	1.33		7.4	0.9
	5 18.2

Acknowledgments: We want to thank J. B. Fournier and J. M. Allain for stimulating discussions. The labs belong to the CNRS consortium CellTiss.