## Abstract

How does a human lift a weight? Can we relate the dynamics of the lift to the molecular actin–myosin interactions responsible for muscle contraction? We address these questions with bench press experiments that we analyse with a theoretical model, based on the sliding filament theory. The agreement is fair, and we discuss its possible extension to medical diagnostics.

## 1. Introduction

The motion of vertebrates results from a coupling between bones-joints and muscles. Even if its study goes back to Aristotle's De motu animalium [1], the first quantitative analysis was produced in 1675 by Giovanni Borelli [2] (figure 1). A review on biomechanics can be found in [3,4]. At this macroscopic scale, the muscle is often characterized by the heuristic Hill's law [5–7] and modelled as an active viscoelastic structure, able to produce a force, to store a tension and to dissipate energy [8].

The precise mechanism of force generation has been itself a long scientific quest since antiquity and the work of Hippocrates of Cos [9,10]. One way to measure the difficulties associated with the understanding of the force generation mechanism consists of comparing the scale at which the force is used (typically the body scale, 1 m) to the scale at which the force is generated which is a topological conformation change of the myosin molecule by 10 nm [11]). Eight orders of magnitude thus separate the molecular origin of the force from its macroscopic function, namely the motion of organisms. Considering the scales involved, this quest has been subjected to the development of techniques from the early microscope (the sarcomere is visualized for the first time in 1674 by van Leeuwenhoek [10]) to the electron microscope [12]. The sliding filament model came out of this quest. According to Needham [9], *the first hints of the sliding-filament mechanism of contraction were given by the low-angle X-ray diffraction patterns obtained by H.E. Huxley with living and glycerol-extracted muscle* in 1953 [13]. A theory for the contraction based on this sliding-filament model was then proposed by A. F. Huxley in 1957 [14] and further completed in 1968 by Deshcherevskii [15]. The connection between the force generated and the precise molecular machinery is still an active field of research [16–18].

The goal of this article is to show how the sliding filament model can be coupled to a classical mechanical joint equation to account for motions observed in weightlifting. The experimental set-up is presented in §2, and the results in §3 prior to the model in §4 and the discussion on its possible extension to medical diagnostics in §5.

## 2. Experimental set-up

In order to minimize the number of joints involved in the motion, we have worked with the bench press configuration presented in figure 2*a*. This is an upper body exercise where the athlete lies on his back, lowers the barbell to chest level and then pushes it back up until the arms are straight.

The morphological characteristics of the athletes are gathered in the table of figure 2*b*. *H* is their height and *M*_{0} is their mass. The length of their arm, from shoulder to elbow is *L* and *S* is the section of the biceps. Finally, *M** is their record lift.

## 3. Experimental results

The athletes are asked to lift the weight once at maximal speed. They perform this explosive task for different masses of barbell from the lightest mass *M**. Between each lift, a resting period of several minutes is respected.

### (a) Qualitative results

The two snapshot sequences of figure 3 present qualitatively the dynamics of the lift. The athlete lifts the barbell from *z*=0 (bent arms) to *z*=2*L* (straight arms) in a time *T*_{L}. In the upper sequence, the athlete takes 0.4 s to raise a mass of 20 kg. In the lower sequence, it takes him 0.9 s to raise a 140 kg barbell.

### (b) Quantitative results

The precise dynamics of the barbell is recorded with an accelerometer Myotest acquiring at 500 Hz which provides velocity and acceleration. Its vertical location is measured using a video camera at 30 fps.

Figure 4 shows the evolution of the vertical position of the barbell *z*_{M}(*t*) and its velocity *M**, and for two different athletes with different morphologies: Michael (*M*_{0}=71 kg, *M**=105 kg) in figure 4*a*–*c* and Gregory (*M*_{0}=114 kg, *M**=230 kg) in figure 4*b*–*d*.

With no surprise, one observes that the dynamics of the barbell is slower for heavier barbells. The position curves show that the height travelled by the barbell decreases a bit when its mass increases. This is due to the holding location of the hands along the bar. The heavier the mass, the larger the distance between the hands and the lower the final height of the barbell.

Concerning the velocities, the barbell is initially at rest and recovers this state at the end of the lift. The initial and final velocities *c*–*d*. A further feature is the deformation of the velocity curves when the load increases. For light barbell, the velocity signal is rather symmetric, whereas for heavier barbell, a second bump systematically appears.

The dynamics of the barbell is presented for two athletes with different morphologies: one observes that the general trends are similar, but a careful analysis of the curves reveals some differences, particularly in the shape of the velocity curves. For heavy weights, the velocity profiles of Michael are rather ‘triangular’, whereas those of Gregory are more ‘rectangular’.

Another interesting feature of all these signals is the initial rising phase. The time evolution of the height is shown on a log–log scale in figure 5 for Michael (*a*) and Gregory (*b*) and different mass. In all the cases, this presentation reveals that the height initially increases as the third power of time: *z*_{M}∼*t*^{3}, which implies that the velocity is initially quadratic in time.

## 4. Model

### (a) Joint equation

We model the weightlifter by a single joint as presented in figure 6. The elbow of radius *r* and mass *M*_{a} is connected to two massless bones of identical length *L*. The shoulder is motionless and defines the origin *z*=0. During a single lift, the weightlifter extends his arms by contracting his muscles (red). The barbell goes from *z*_{M}=0, with arms bent (*ψ*=*π*/2) to the vertical position *z*_{M}=2*L* (*ψ*=0). During the motion, the barbell is in the weightlifter's hands at the location *F* which enables the lift.

To derive the equation of motion *ψ*(*t*), we first apply Newton's equation in *z*_{M}. Along the vertical direction *z*, this equation writes *C*_{1} is the force exerted by the bone on the mass. This equation gives an expression for *C*_{1}
*B*, the equation of motion along the direction *x* takes the form: *C*_{2} stands for the force exerted by the lower bone on the joint, the expression of which can be deduced
*C*_{1} and equation (4.2) for *C*_{2}, we get

Using

Equation (4.4) must be solved with the initial conditions: *ψ*(0)=*π*/2 and *F*_{0} is the maximal force that the athlete can develop, the maximal mass that he can lift is

For a given force and athlete mass, *M*^{⋆} is maximized by short segments and large articulations. Elongated and thin bodies are thus disadvantaged in weightlifting.

### (b) Force relation

At this point, it must be underlined that the differential equation (4.4) can be integrated only if the force *F* is known. Determining this closure relation is the purpose of this section. Using *F*_{0}, the closure term can be written as
*M*_{a} can be evaluated using classical physiological studies as the sum *M*_{a}=*M*_{arm}+*M*_{forearm} where the mass of the two arms is *M*_{arm}=0.08×*M*_{0}−1.31 and the mass of the two forearms is *M*_{forearm}=0.04×*M*_{0}−0.23 [19]. For Gregory, we find *M*_{a}=12.1 kg and deduce *M*^{⋆}/*M*_{a}=19. For Michael, we find *M*_{a}=7 kg and deduce *M*^{⋆}/*M*_{a}=15.

#### (i) Constant force: F ¯ = 1

We first naively consider the muscle as an ideal force generator, producing always the maximal force *F*=*F*_{0}, whatever the load or the velocity of contraction. We then integrate numerically equation (4.4) with *M*^{⋆}/*M*_{a}=15 and present the dynamics of weightlifting in figure 7. Figure 7 shows the results obtained for the mass height (*a*) and mass velocity (*b*) for different masses of the barbell ranging from *M*/*M*^{⋆}=0.1 to *M*/*M*^{⋆}=0.9.

Several features differ from the experimental observations presented in figure 4: the initial rise of the velocity is linear here, whereas it seems quadratic experimentally. At low masses, the experimental velocity is almost symmetric, whereas it is always dissymmetric in the numeric. Finally, no secondary peak appears at large mass in the numeric. We conclude that the experimental results cannot be approached with a constant force model. A deeper understanding of muscle contraction must be introduced in order to improve our model.

#### (ii) Presentation of the sliding filament model

A scheme of the contractile machinery is presented in figure 8*a*: muscle cells extend from one tendon to another, which connect to bones. The muscle cell is composed of nuclei and of myofibrils, a linear assembly of sarcomeres, the elementary contractile unit. A sarcomere itself is made of thin actin filaments connected to thick myosin filaments via myosin heads (figure 8*b*(i)(ii)). When a neuron stimulates a muscle cell, an action potential sweeps over the plasma membrane of the muscle cell. The action potential releases internal stores of calcium that flows through the muscle cell. Actin and myosin filaments are juxtaposed but cannot interact in the absence of calcium (relaxed state *b*(i)). With calcium, the myosin-binding site is open on the actin filaments, which allows the myosin motors to crawl along the actin, resulting in a contraction of the muscle fibre *b*(ii) [21,12].

A theory for the contraction based on this sliding-filament model was then proposed by Huxley in 1957 [14] and further completed in 1968 by Deshcherevskii [15]. We present the main steps of this later model and then establish the connection with weightlifting.

Deshcherevskii considers three main states in the actomyosin cycle (figure 8*c*): a myosin head is either 'free (A), or developing an active force (B) or detaching (with a breaking force) (C). Each myosin head within a sarcomere moves from one state to the other following the sequence A–B–C–A–B–C–…. Denoting *n* and *m* as the number of myosin heads in the states (B) and (C) and *α*_{D} the total number of myosin heads in half a sarcomere, Deshcherevskii expresses the rate of change of both populations via two kinetic equations
*τ*_{1} is the characteristic time of the transition from state A to B and *τ*_{2} is the characteristic opening time of bridges which leads to a transition from C to A; ℓ is the mean value of conformational transformation of the myosin head during the power stroke (ℓ≈10 nm). If *v*_{s} is the relative velocity between the actin and myosin filaments, ℓ/*v*_{s} represents the time after which one active head switches from stretched (active) to compressed (resisting). In the isometric limit where there is no net contraction of the sarcomere, the characteristic time of the transition B–C reduces to the characteristic time *τ*_{nm} of the topological switch of the myosin molecule. One could thus write the velocity *v*_{s}=*v*_{t}+*v*_{i} as the sum of a translation velocity *v*_{t} and intrinsic velocity *v*_{i}=ℓ/*τ*_{nm}. This later term becomes dominant only in the isometric limit. Because we focus here on the dynamic, this term is going to be neglected in the following discussion where we assume *v*_{s}≈*v*_{t}.

Assuming that the active and braking force developed by the myosin head are identical and equal to *f*^{1} , the force developed by a sarcomere is equal to
*N*_{s}. If *L*≈0.3 m is the length of the muscle and ℓ_{s}≈3 *μ*m the length of a sarcomere, then one gets *N*_{s}≈10^{5}. This number connects the sliding velocity *v*_{s} and the contraction velocity *v* via the relation: *v*=*N*_{s}.*v*_{s}. From Wilkie [6], we get that, in humans, the contraction velocity is in the range *v*≈ 1–10 m s^{−1}. This implies that the sliding velocity in the sarcomere is in the range *v*_{s}≈ 10–100 *μm* *s*^{−1}. The next step consists of evaluating the number of sarcomeres placed in parallel in a muscle, *N*_{p}. If *S* (*S*≈*π*(0.1)^{2}/4≈10^{−2} m^{2}) stands for the cross section of the muscle and *S*_{s} (*S*_{s}≈*π*(0.5×10^{−6})^{2}/4≈2.5×10^{−13} m^{2}) for the cross section of a sarcomere, one gets *N*_{p}≈4×10^{10}. The total force of the muscle *F* is then related to the force developed by a single sarcomere *F*_{s} via the relation: *F*=*N*_{p}.*F*_{s}. Because *F*≈500 N, one deduces the number of simultaneously active myosin heads *DC*.*α*_{D}≈*F*/*f*.*N*_{p}≈3000. In this expression, *DC* is the fraction of active to the total number of myosin heads which can be evaluated as the duty factor which is of the order of 0.2 [26,27]. Which leads to *α*_{D}≈15 000. This global evaluation can be compared with a direct counting of the number of myosin heads: using the work of Huxley [28], one evaluates eight myosin heads every 43 nm of myosin filament. Because the distance between the thick myosin filament is 50 nm, we deduce that in a sarcomere (500 nm in diameter), the total number of heads is *α*_{D}≈(500/50)^{2}×(1000/43)×8≈18 400. Both evaluations are compatible.

#### (iii) The steady limit and Hill's equation

In the steady regime where *v*_{s} is a constant, the populations of the three stages A,B,C remain constant on average, and one deduces *τ*_{1} *v*_{s} *n*/ℓ=*α*_{D}−(*n*+*m*) and *τ*_{2} *v*_{s} *n*/ℓ=*m*.

These equations enable us to get the force–velocity relationship

This hyperbolic relation between the force and the velocity was first proposed heuristically by Hill in 1938 [5]. Here, it appears as the quasi-steady limit of the Deshcherevskii's kinetic model. Using the previous estimates, we get *τ*_{2}. The velocity ℓ/*τ*_{nm} is thus lower, and we deduce *τ*_{nm}>*τ*_{2}. Moreover, from Wilkie [6], we find *F*_{0}/*a*≈5 in humans which implies *τ*_{1}≈4 *τ*_{2}. These orders of magnitude are compatible with those proposed by Deshcherevskii [29], who evaluate *τ*_{1}≈3−6 *τ*_{2} and *τ*_{2}≈6 ms.

Because the contraction velocity *v* is linked to the angle of the joint *ψ* through the relation *G*≪1, this expression of the force reduces to the constant limit *G*=0.3 and *τ*_{1}/*τ*_{2}=3 is shown in figure 9 for both the height (*a*) and the rising velocity (*b*).

Even if the numerical integrations seem to get closer to the experiments, the initial phase is much too quick and exhibits a behaviour

#### (iv) The unsteady limit of Descherevskii's model

In a non-dimensional form, the kinetic model of Deshcherevskii (4.7), (4.8) and (4.9) takes the form

Equation (4.16) must be solved with initial conditions that are going to be discussed below. Once integrated, one gets the time evolution of

We now discuss the initial conditions used to integrate equation (4.16). Initially, the athlete holds the weight in the position *ψ*=*π*/2. The weight neither falls nor rises (*c* and described by equations (4.7)–(4.9), the population *m* is fed by the population *n* which has produced the force *f* during the time ℓ/*v*_{s} and is depleted over the characteristic time *τ*_{2}. In other words, if all the active myosin heads (state B) are initially active, then the population *m* reduces to 0. This is the condition with which we start our calculation: *M*^{⋆}/*M*_{a} which has already been discussed in §4b, we take the values of Deshcherevskii: *τ*_{2}=6 ms (*τ*_{1}=6*τ*_{2}. The only parameter is *G*, and we chose it constant for a given athlete in order to approach the velocity signal at all mass.

An example of numerical integration of equations (4.16), (4.15) and (4.14) with the parameters *G*=0.17, *z*_{M}/2*L* and barbell speed

We first observe in figure 11*a* that the barbell height increases as *t*^{3} as shown experimentally in figure 5. Concerning the velocity, the non-dimensionalized experimental results associated with Michael (figure 4) are presented in figure 11*b* and can be compared with the numerical results presented in figure 11*c*. For the same range of lifted masses *M*/*M*^{⋆}, the rising time is similar, and the range of velocity seems to differ by a factor 1.3. The main difference between the two sets of curves seems to be in the final fall of the velocity which is systematically sharper in the numerics than in the experiments. This effect is probably due to the fact that both hand and shoulder positions are assumed to be fixed in the numerics while they slightly change in the experiments.

Using the characteristics of the other athlete Greg (*M*^{⋆}/*M*_{a}=19) and similar values of the muscle properties (*G*=0.33, *b*) and the numerical integration of the joint–muscle system (equations (4.4), (4.16), (4.15) and (4.14)). Again, the initial rise is cubic in time and the numerical signal evolves towards a more rectangular shape, as does the experimental one. The characteristic time for the rise are similar, whereas the absolute value of the velocity still differs by a factor 1.4.

## 5. Discussion and medical perspectives

We have just shown that the microscopic understanding of muscle contraction is essential to account for the motions observed in weightlifting. The model we propose to describe bench press lifting is, indeed, composed of two different parts, one for the articulation (equation (4.4)) and one for the force generation (equations (4.12)–(4.14)). This model involves three different constants, two for the ratio between the biochemical kinetics and the macroscopic timescale (*G*). Up to now, the first two constants have been fixed to the values estimated in Deshcherevskii's studies. Only *G* was slightly changed in the simulations presented in figures 11 and 12 from 0.17 to 0.33. If one explores numerically, then the whole phase diagram *G* leads to the transition already observed from a triangular behaviour of the velocity in time to a rectangular one. At larger values of *G*), or concentrated in the first phase of the rise (large *G*). This suggests that bench press experiments could be used as a muscle rheometer to probe characteristic times of muscle contraction or to identify muscle pathologies.

To check these ideas, Loïc Auvray has agreed to work with us and performed the same bench press experiments as the other athletes. Loïc developed, eight months ago, an amyotrophic lateral sclerosis. This muscle pathology is a rapidly progressive neurological disease that attacks the nerve cells (neurons) responsible for controlling voluntary muscles. The experimental results obtained with Loïc are presented in figure 13*b*. There, the velocity oscillations are clearly visible and contrast with the smooth evolutions observed in figure 4.

These different behaviours suggest that the analysis of bench press lifts can indeed be used as a muscle rheometer, able to distinguish regular behaviours from pathological ones. This perspective of using bench press lifts as a diagnostic tool is a very promising and stimulating perspective of this work.

## Ethics

Most of the data presented in the article have been obtained in a sports centre with informed consent members. The data presented in figure 13*b* were obtained with Loïc Auvray who is an author of the article.

## Data accessibility

The experimental data presented are accessible upon request to the authors.

## Authors' contributions

C.C. carried out the experiments and participated in both the data analysis and the elaboration of the model; B.D.T. was involved in the elaboration of the model, G.L. selected the athletes, carried out the experiments and participated in the data analysis. L.A. performed experiments and participated in the discussion on the model; C.C. analysed the data, developed the model and drafted the manuscript. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

This work was supported by the LadHyX.

## Acknowledgements

We obviously thank here all the athletes who agreed to perform in order to get the quantitative data. This work would not have existed without their help. We also thank Etienne Reyssat with whom we discussed the joint equation and the physics of the lift. Etienne is always able to shed light on a problem from a different angle and this is always very precious.

## Footnotes

- Received July 10, 2015.
- Accepted November 20, 2015.

- © 2015 The Author(s)