Michael Günther

Syn Schmitt

A macroscopic ansatz to deduce the hill relation

Keywords: Hill equation, Muscle model, Contraction dynamics, Biomechanics

In this study, we derive the hyperbolic force-velocity relation of concentric muscular contraction, first formulated empirically by A.V. Hill in 1938, from three essential model assumptions: (1) the structural assembly of three wellknown elements -i.e. active, parallel damping, and serial -fulfilling a force equilibrium, (2) the parallel damping coefficient explicitly depending on muscle force output and three parameters, and (3) the kinematic gearing ratio between active and serial element being assigned to a parameter. The energy source within the muscle represented by the force of the active element is an additional fifth parameter. As a result we find the Hill "constants"

A and B as functions of our five model parameters. Using A and B values from literature on experimental data, we predict heat power release of our model. By calculating enthalpy rate and mechanical efficiency, we compare the model heat power to predictions from another Hill-type model, to

In 1938 Archibald Vivian Hill published his famous paper [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] in which he fixed the hyperbolic relation between muscular force and velocity during concentric contractions from heat measurements empirically.

The isometric force is one parameter of this relation, whereas he named the two remaining parameters the "dynamic constants of muscle". Later, he acknowledged that the force-velocity relation had to be modified for a better reproduction of the decrease in enthalpy rate found at high shortening velocities [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF]. Still, a hyperbolic relation remained with now five parameters necessary to align with experimental findings.

In 1957 Andrew Fielding Huxley presented the first comprehensive theoretical model [START_REF] Huxley | Muscle structure and theories of contraction[END_REF], based on the "sliding filament theory". He derived the known force-velocity relation of a muscle [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] from mapping the state of the art knowledge about the structure of the crucial molecules involved in the crossbridge cycling mechanism of muscular force production.

He needed to set nine microscopic parameters in order to fit the Hill relation [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF]. Assuming a two-stage cycle, a later refinement of Huxley's sliding filament model [START_REF] Huxley | A note suggesting that the cross-bridge attachment during muscle contraction may take place in two stages[END_REF] again succeeded in reproducing the enthalpy rate factored in by Hill's modification [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF]. The number of sliding filament model parameters remained the same.

When talking about models, we will label them in the following as either "Hill-type" or "Huxley-type". The latter denomination gains somehow double weight as Hugh Esmor Huxley and Jean Hanson [START_REF] Hanson | The structural basis of the cross-striations in muscle[END_REF][START_REF] Huxley | Changes in the cross-striations of muscle during contraction and stretch and their structural interpretation[END_REF] were on their way to put forward the sliding filament model simultaneously [START_REF] Huxley | Fifty years of muscle and the sliding filament hypothesis[END_REF][START_REF] Huxley | Memories of early work on muscle contraction and regulation in the 1950's and 1960's[END_REF] to Andrew Fielding Huxley and Ralph Niedergerke [START_REF] Huxley | Structural changes in muscle during contraction. Interference microscopy of living muscle fibres[END_REF].

The empirically derived Hill relation is not self-explanatory. For example, it might constitute a thermodynamical state equation or a fairly complex visco-elastic force law. In numerical neuro-musculo-skeletal model simulations Hill-type models are usually used in the force law sense [START_REF] Günther | Synthesis of two-dimensional human walking: a test of the λ-model[END_REF][START_REF] Günther | High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models[END_REF][START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF][START_REF] Kistemaker | Is equilibrium point control feasible for fast goal-directed single-joint movements?[END_REF]van Soest and Bobbert, 1993) interpreting it uni-directionally, with the force given and the respective velocity constituting the model response.

As real muscle can respond to different protocols demanding a variety of disturbances, including force and length steps, bi-directionality seems to be one important criterion, among others, for a predictive muscle model. Hill himself meant to read his relation bi-directionally [START_REF] Hill | First and Last Experiments in Muscle Mechanics[END_REF]. In contrast, to the best of our knowledge none of the Huxley-type models developed so far [START_REF] Barclay | A weakly coupled version of the Huxley crossbridge model can simulate energetics of amphibian and mammalian skeletal muscle[END_REF][START_REF] Chin | Mathematical simulation of muscle cross-bridge cycle and force-velocity relationship[END_REF][START_REF] Cooke | A model of the release of myosin heads from actin in rapidly contracting muscle fibers[END_REF][START_REF] Huxley | Muscle structure and theories of contraction[END_REF][START_REF] Huxley | A note suggesting that the cross-bridge attachment during muscle contraction may take place in two stages[END_REF][START_REF] Lan | Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation[END_REF]Piazzesi andLombardi, 1995, 1996) can be applied bi-directionally. We have found one study [START_REF] Baker | A thermodynamic muscle model and a chemical basis for A.V. Hill's muscle equation[END_REF] yet, deriving contraction velocity as a hyperbolic function of force with the muscle fulfilling thermodynamic equilibrium.

Our aim is to build a bi-directional Hill-type muscle model as simple as possible, i.e. based on less parameters than Huxley-type models, and of predictive character. Thereto, the Hill relation must be derived from basic principles, either at least one independent equation or microscopic mechanisms. Concretely, we derive the Hill relation from a force equilibrium of mechanical structures rather than exclusively from an explicit model of the source of mechanical power as Huxley did [START_REF] Huxley | Muscle structure and theories of contraction[END_REF][START_REF] Huxley | A note suggesting that the cross-bridge attachment during muscle contraction may take place in two stages[END_REF][START_REF] Huxley | Mechanics and models of the myosin motor[END_REF] follow-

A c c e p t e d m a n u s c r i p t

submitted to the Journal of Theoretical Biology 6 ing Hill's empirical path [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF]. Our model is meant to provide some insight into the mechanical behaviour of a muscle as a whole. In particular, we suggest a causal explanation for the origin of the Hill relation in terms of only three macroscopic mechanical parameters and one well-established internal degree of freedom which is resolved by a gearing ratio between internal and external contraction. Consequently, the model can be used to predict the force-velocity relationship for muscle with given architectural parameters, rather than having to measure the force-velocity relationship directly.

Methods

Our model coarsely maps the macroscopic features of an assembly of active muscle fibres incorporating three basic, physiologically motivated, structural elements (Fig. 1). In the following we will derive the Hill relation from one basic equation and two additional features. First, the serial arrangement of model elements implies the introduction of one degree of freedom, representing an internal length measure that occurs in addition to the length of the active muscle fibres. This is dealt with by applying a kinematic gearing ratio between the elements in series. Second, a simple force law for the parallel damper is assumed. The characteristics of the other two elements do not have to be specified.

The first element is the source, and possibly drain, of mechanical energy.

It is called the active element (AE) producing the force F AE . In macroscopic muscle models the chemical state, i.e. the relative number of actively forceproducing crossbridges, can be quantified by normalised muscle activation 0 ≤ q ≤ 1. The naming of any further model element is chosen due to its The second element is the serial element (SE) loaded with the force F SE .

It maps compliant structures within the muscle fibres [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF][START_REF] Hill | The series elastic component of muscle[END_REF][START_REF] Katz | The relation between force and speed in muscular contraction[END_REF][START_REF] Krueger | Myocardial sarcomere dynamics during isometric contraction[END_REF][START_REF] Kishino | Force measurements by micromanipulation of a single actin filament by glass needles[END_REF][START_REF] Levin | The viscous elastic properties of muscle[END_REF][START_REF] Lindstedt | Do muscles function as adaptable locomotor springs?[END_REF][START_REF] Maruyama | Connectin, an elastic protein of muscle: characterization and function[END_REF][START_REF] Neumann | Elastic properties of isolated thick filaments measured by nanofabricated cantilevers[END_REF][START_REF] Pandy | An optimal control model for maximum height human jumping[END_REF]Wakabayashi et al., 1994;[START_REF] Zajac | Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control[END_REF]. A recent approach (Denoth et al., 2002;Telley and Denoth, 2007), modelling the muscle as an inhomogeneous chain of many force generators coupled visco-elastically to each other both in parallel and in series, incorporates series elasticity as a key feature.

The third element is the parallel damping element (PDE) producing a frictional force F P DE . Energy dissipation within the muscle can be a passive [START_REF] Günther | High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models[END_REF][START_REF] Minajeva | Unfolding of titin domains explains the viscoelastic behavior of skeletal myofibrils[END_REF][START_REF] Wang | Viscoelasticity of the sarcomere matrix of skeletal muscles. The titinmyosin composite filament is a dual-stage molecular spring[END_REF] or an active [START_REF] Gasser | The dynamics of muscular contraction[END_REF][START_REF] Hartree | The thermoelastic properties of muscle[END_REF][START_REF] Hill | The maximum work and mechanical efficiency of human muscles, and their most economical speed[END_REF][START_REF] Levin | The viscous elastic properties of muscle[END_REF][START_REF] Lupton | The relation between the external work produced and the time occupied in a single muscular contraction in man[END_REF][START_REF] Lupton | An analysis of the effects of speed on the mechanical efficiency of human muscular movement[END_REF] phenomenon. Two studies point to a dependency of the PDE on muscle force [START_REF] Baker | A thermodynamic muscle model and a chemical basis for A.V. Hill's muscle equation[END_REF][START_REF] Pate | Simulation of stochastic processes in motile crossbridge systems[END_REF].

The kinematic relation for the lengths of the elements AE,PDE on the one hand and the SE on the other hand, arranged in series, writes

l AE = l P DE = l M -l SE (1)
with l M representing the muscle length (Fig. 1). The basic equation describing the mechanics of our muscle model is the equilibrium of the forces generated by the elements AE, PDE, and SE

F M = F SE = F AE + F P DE (2)
F P DE = d P DE • lPDE = d P DE • ( lM -lSE ) ( 3 )
whereat we took the first time derivative (dot symbol " ˙") of Eq. (1) into account. The PDE is the only element within this study to which we attribute a specific functional dependency. We assume that its force depends linearly on the contraction velocity lAE of the AE (see Eq. ( 3)) and a respective damping coefficient Furthermore, by defining the isometric ( lM = 0) muscle force as

d P DE (F M ) = D P DE,max • (1 -R P DE ) • F M F AE,max + R P DE (
F M,0 = F M ( lM = 0) , (5)
by working out the force equilibrium (Eq. ( 2)) in the isometric condition, with F P DE substituted by Eq. ( 3), next solving the equilibrium in isometry for

F AE = F M,0 + d P DE (F M,0 ) • lSE , (6)
and finally re-substituting Eq. ( 6) into the general force equilibrium (Eq.

(2)) we find In a nutshell, we reduced the manifold of all possible solutions of Eq. ( 2)

F M = F M,0 + d P DE (F M,0 ) • lSE + d P DE (F M ) • ( lM -lSE ) . (7)
for the relation between lM and F M to those with one specific value of the isometric force F M,0 as defined by Eq. ( 5). Thus, F M,0 is used as a parameter here. For a chosen isometric muscle force value F M,0 the related force F AE acting in the AE, and vice versa according to (Eq. ( 6)), is potentially a function of internal length re-distribution lSE .

We aim at relating the muscle force F M explicitly to muscle contraction velocity lM . With this in mind, Eq. ( 7) is a preliminary result, because F M still depends explicitly on both the external muscle contraction velocity lM and the contraction velocity of the internal degree of freedom lSE . By assuming that a gearing ratio

κ v = lSE lM (8)
between internal and external velocity can be specified, the force equilibrium as manifested by Eq. ( 7) transforms to

(F M -F M,0 ) -lM • (d P DE (F M ) • (1 -κ v ) + d P DE (F M,0 ) • κ v ) = 0 (9)
which is an equation relating lM to F M . In appendix Appendix A we suggest what κ v might depend on. Equation ( 9) is the force equilibrium expressed in Eq. ( 2) with the current distribution between lSE and lAE specifically expressed by one parameter κ v , while F M,0 represents another one.

There is no physical reason to favour either lM or F M as the independent variable within Eq. ( 9). The idea that muscle force and velocity are interchangeable as dependent and independent variables is a generally understood principle and in accordance with Hill's view [START_REF] Hill | First and Last Experiments in Muscle Mechanics[END_REF]. So far, this idea is not often applied in muscle modelling. At least in our approach, we may solve Eq. ( 9) either for F M = F M ( lM ) or for lM = lM (F M ), whenever appropriate.

Equation ( 9) is a bi-directional relation. Transformation between both views is uniquely determined if d P DE is a linear function of F M as assumed here (Eq. ( 4) with F M = F SE ). In this case, Eq. ( 9) constitutes a hyperbola

(F M + A) • lM = -B • (F M,0 -F M ) (10)
with the parameters A, B, F M,0 being positive and lM consistently being negative in the shortening (concentric) case.

The numbers A, B, F M,0 are functions of our five model parameters. This becomes clear by comparing the terms in the Hill relation Eq. ( 10) solved

for the contraction velocity lM = B • (F M -F M,0 )/(F M + A) to Eq. (9) solved for lM = F M -F M,0 (1 -κ v ) • d P DE (F M ) + κ v • d P DE (F M,0 ) (11)
which becomes lM = 1

D P DE,max F AE,max • (1 -R P DE ) • F M -F M,0 (1 -κ v ) • F M + R P DE (1-R P DE ) • F AE,max + κ v • F M,0 (12)
expressed explicitly in terms of our model parameters. Thereto, the linear transforming operations

d P DE (x 1 -x 2 ) = d P DE (x 1 ) -d P DE (x 2 ) + D P DE,max • R P DE d P DE (x 1 -x 0 ) -d P DE (x 2 -x 0 ) = d P DE (x 1 ) -d P DE (x 2 ) .
must be used for d P DE (F M ) after Eq. ( 4).

Finally, the Hill parameters emerge as explicit functions of our model parameters

A = R P DE (1 -R P DE ) • (1 -κ v ) • F AE,max + κ v 1 -κ v • F M,0 (13)
D P DE,max F AE,max • (1 -R P DE ) • (1 -κ v ) . ( 14
)
Accordingly, the maximum shortening velocity writes

lM,max = B A • F M,0 = F AE,max D P DE,max • F M,0 κ v • (1 -R P DE ) • F M,0 + R P DE • F AE,max . (15)
The unloaded muscle (F M = 0) would contract concentrically with lM = -lM,max .

In the limit case of neglected minimum (passive) internal parallel damping (R P DE = 0) the Hill parameters arise from Eqs. ( 13),( 14) as

A = κ v 1 -κ v • F M,0 (16)
B = F AE,max (1 -κ v ) • D P DE,max = κ v 1 -κ v • lM,max , (17)
with the maximum shortening velocity (compare Eq. ( 15))

lM,max = F AE,max κ v • D P DE,max . (18)
Note that a concurrent parameter variation fulfilling B/A = const meets the constraint lM,max = const, whereat solely the curvature is changed. In our model this is equivalent to

κ v • D P DE,max = const.
From hereon, we will think of the hyperbolic Eq. ( 10 The concentric contraction velocity v was a positive number in the original formulation of the Hill relation [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF])

(P + a) • v = b • (P 0 -P ) . (19)
Thus, a comparison to Eq. ( 10)

gives v = -lM , P = F M , P 0 = F M,0 , a = A, b = B.

Results

Empirically, both mechanics and heat release during concentric muscle contractions are represented by all three parameters of the Hill relation.

Therefore, we choose appropriate parameters in accordance with literature data first. Second, in order to validate our model generally, its predicted mechanical and heat power are compared to force-velocity and enthalpy rate measurements. Third, the model allows for an estimation of the mechanical power distribution between AE, PDE, and SE.

An estimate of model parameters

A median piglet muscle [START_REF] Günther | High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models[END_REF] would be represented by

F M,0 = F AE,max = 30 N, κ v = 0.2, D P DE,max = 1000 Ns/m (i.e. lM,max | R P DE =0 = 0.15 m/s, A rel | R P DE =0 = 1/4
), and R P DE = 0.01 (i.e. lM,max | R P DE =0.01 = 0.144 m/s). On the one hand, this muscle can be compared to more recent experimental findings [START_REF] Barclay | Energetics of fast-and slow-twitch muscles of the mouse[END_REF][START_REF] Curtin | Efficiency of energy conversion during sinusoidal movement of white muscle fibres from the dogfish Scyliorhinus canicula[END_REF][START_REF] Barclay | Mechanical efficiency of fast-and slow-twitch muscles of the mouse performing cyclic contractions[END_REF][START_REF] Barclay | Mechanical efficiency and fatigue of fast and slow muscles of the mouse[END_REF]. On the other hand, the κ v ,A rel choice (see Eqs. (13,16))

A c c e p t e d m a n u s c r i p t

submitted to the Journal of Theoretical Biology 13 accords with Hill's original [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] and later [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF] work. In order to validate our model, we follow the same path as a recent computer simulation study [START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF] in which the muscular enthalpy rate along with the mechanical efficiency was predicted by modelling muscular heat rates as functions of the AE state variables q, l AE during isovelocity contractions, while assuming a fully activated muscle (q = 1).

Maintenance heat rate, enthalpy rate, mechanical efficiency: a model validation

The PDE is the only explicitly formulated element within our model (Eq. ( 4)). Thus, we can calculate the minimum dissipative loss ḣ =

d P DE (F M ) • l2 AE = d P DE (F M ) • ((1 -κ v ) • lM ) 2 during a concentric contraction.
With that, we can predict the enthalpy rate

Ḣ = -F M • lM + ḣ + ḣ0 , (20)
knowing the contraction velocity lM from Eq. ( 11) or ( 12), respectively.

The first term is the net mechanical power output of the muscle, the second term ḣ is its shortening heat rate, and the third term ḣ0 is its maintenance heat rate. The latter is an unavoidable energy loss owing to the fact that an active muscle, loaded isometrically, produces heat at the rate ḣ0 due to crossbridge cycling, even though it does not do any mechanical work at all.

Taking these three terms into account, the assumed enthalpy rate Ḣ (Eq. 20) is the theoretical minimum power the muscle has to generate to gain its net mechanical power output. This is equivalent to assuming that all other hypothetical internal elements work without loss and no other heat generating process does exist.

A c c e p t e d m a n u s c r i p t

submitted to the Journal of Theoretical Biology

14

For example, in order to keep the model as simple as possible, we neglected the activation heat, released due to any Ca 2+ -flux into and out of the sarcoplasmatic reticulum as a response to stimulation -also called excitation -from action potentials. Whereas maintenance heat primarily depends on filament overlap and free Ca 2+ -concentration (state of activation q), activation heat depends on stimulation frequency. Maximum activation heat rate is roughly half as high as maintenance heat rate (Barclay et al., 2007;[START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF]. Therefore, the heat contributions to enthalpy rate Ḣ are systematically under-estimated within this study, while the mechanical efficiency as defined by the ratio

ε M = -F M • lM Ḣ (21)
is over-estimated.

As part of the enthalpy rate the maintenance heat rate ḣ0 must be known.

According to literature, which indicates diverse findings about maintenance heat rate in various muscle preparations, the maintenance heat rate ḣ0 may generally be rather proportional to than equal to

A • B (= 1/16 • F M,0 • lM,max if A = 1/4 • F M,0
). Thus, in the first instance, we assume phenomenologically

ḣ0 = c ḣ0 (T -T 0 , params) • F M,0 (T = T 0 , params) • lM,max (T = T 0 , params) (22)
to adopt the view that ḣ0 runs approximately in proportion to the number of currently active crossbridges [START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF]. This implies, as a tion [START_REF] Huxley | Muscle structure and theories of contraction[END_REF][START_REF] Huxley | A note suggesting that the cross-bridge attachment during muscle contraction may take place in two stages[END_REF]. There is another, pure mathematical, argument [START_REF] Mcmahon | Muscles, Reflexes, and Locomotion[END_REF] in favour of ḣ0 = A • B. For the addition of A • B to both sides of Eq. ( 10) or Eq. ( 19) provides an immediate transformation of the Hill relation to the force-velocity relation derived earlier from mere mechanical measurements (Fenn and Marsh, 1935). Later modifications of the hyperbolic lM (F M ) relation introduced by Hill himself [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF] demonstrate that the Hill parameter A may incorporate two different heat portions, one depending on load and one on velocity. Thus, the product A • B is somehow unlikely to be exactly equal to the heat rate at zero velocity.

Yet, by accepting ḣ0 = A • B for a start, utilising the approximation R P DE = 0, and comparing Eqs. (16,17,18) to Eq. ( 22), we predict

c ḣ0 = (κ v /(1 -κ v )) 2 . ( 23
)
The lowest and highest c ḣ0 values (0.01, 0.3) extracted from literature [START_REF] Barclay | Mechanical efficiency of fast-and slow-twitch muscles of the mouse performing cyclic contractions[END_REF][START_REF] Barclay | Mechanical efficiency and fatigue of fast and slow muscles of the mouse[END_REF][START_REF] Barclay | Energetics of fast-and slow-twitch muscles of the mouse[END_REF]Elzinga et al., 1987;[START_REF] Günther | High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models[END_REF][START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF][START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF][START_REF] Wank | Muscle growth and fiber type composition in hind limb muscles during postnatal development in pigs[END_REF]

correspond to κ v = 0.09, 0.35. A median value κ v = 0.2 corresponds to A = A rel •F M,0 = 1/4•F M,0
(for R P DE = 0: Eq. ( 16)) with the belonging factor The value A rel = 1/4 was both originally found by Hill [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] for a frog sartorius muscle at T = 0 o C and represents the average value of further studies [START_REF] Barclay | Mechanical efficiency of fast-and slow-twitch muscles of the mouse performing cyclic contractions[END_REF][START_REF] Barclay | Mechanical efficiency and fatigue of fast and slow muscles of the mouse[END_REF][START_REF] Barclay | Energetics of fast-and slow-twitch muscles of the mouse[END_REF][START_REF] Günther | High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models[END_REF][START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF]. The Eqs. (23,24) result in

A rel = κ v /(1 -κ v ) . (24)
c ḣ0 = A 2 rel , (25)
i.e. A rel = 1/4, κ v = 0.2, and c ḣ0 = 0.0625 correspond to each other.

Therefore, in Fig. 2 the enthalpy rate in excess of maintenance heat rate and the mechanical efficiency predicted by this median muscle (κ v = 0.2) can be easily compared to (i) Hill's original direct measurements of mechanical power and heat rates [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] and to (ii) his more accurate data published later [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF]. Additionally, we compare enthalpy rate and efficiency to (iii) the prediction of ḣ calculated from another simple mechanical model which assumes F M,0 to represent the energy source and in which a non-linear parallel damper is chosen to reproduce the Hill relation exactly [START_REF] Winters | Hill-based muscle models: a system engineering perspective[END_REF]. Consistently, for these three comparative model cases the parameter values A,B of our median muscle are adopted (for details see legend of Fig. 2). Enthalpy rate and mechanical efficiency are both plotted versus muscle force and contraction velocity, because in some cases force-plots (compare Fig. 3) and in other cases velocity-plots (see the subtle lM,max differences between both Hill-cases in Fig. 2(D)) are more case-sensitive. Moreover, either data presentations are used in literature.

Our model reasonably approximates Hill's measured enthalpy rates for

F M > F M,0 /10 (Fig. 2(A)) or | lM | < 2/3 • lM,max , respectively (Fig. 2(B)).
Furthermore, our model predicts a maximum in enthalpy rate which is located nearby force and velocity values also reported by Hill [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF], Hux- ima in enthalpy rate may be distinguishable for some muscle preparations [START_REF] Barclay | Energetics of fast-and slow-twitch muscles of the mouse[END_REF][START_REF] Barclay | Mechanical efficiency of fast-and slow-twitch muscles of the mouse performing cyclic contractions[END_REF][START_REF] Barclay | Mechanical efficiency and fatigue of fast and slow muscles of the mouse[END_REF] or unobservable in others [START_REF] Barclay | Energetics of fast-and slow-twitch muscles of the mouse[END_REF][START_REF] Curtin | Efficiency of energy conversion during sinusoidal movement of white muscle fibres from the dogfish Scyliorhinus canicula[END_REF][START_REF] Barclay | Mechanical efficiency and fatigue of fast and slow muscles of the mouse[END_REF]. Note, if we included activation heat into enthalpy rate (Eq. ( 20)) and normalised the latter to the sum of maintenance and activation heat rate, the relative enthalpy rate values as predicted in Fig. 4(A) would be about one third lower.

Generally, our model predicts enthalpy rates considerably better than the non-linear damper model [START_REF] Winters | Hill-based muscle models: a system engineering perspective[END_REF]) not incorporating a SE. Hereby, the discrepancies become more apparent by looking at enthalpy rate (Fig. nance heat rate (Fig. 3). Note again that we slightly over-estimate efficiency due to neglecting activation heat rate.

The mechanical power output

The net mechanical power released by the muscle during a concentric contraction is -F M • lM , here defined as a positive number with a pulling The required mechanical energy must have been stored within the SE during the contraction history preceding the experimentally analysed contraction.

In appendix Appendix C we illustrate how our model can cover such storing situations that load the SE.

Note that the force contributions of AE and PDE always fight each other as the signs of F AE and F P DE are counter-directed. For themselves, they are usually distinctly higher than their net contribution -

F M • (1 -κ v ) • lM .
For instance, in case of κ v = 0.09 the SE power is 9% of net muscle power output. Here, exemplarily choosing a muscle velocity of lM ≈ -0.25 • lM,max with a belonging force F M ≈ 0.2•F M,0 , the power produced by the AE itself is

F AE
F M ≈ 0.75 0.2 = 3.75 times higher than net muscle power output. F AE is always higher than F M as a result of comparing Eq. ( 6) to Eq. ( 7). Here, the SE contributes only about 9% 3.75 = 2.4% of current AE power to net muscle power output. With lM,max kept constant, in a less curved case of κ v = 0.2 [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] the SE would contribute approximately 10% of AE power to net muscle power output when contracting at a moderate velocity of about -lM,max /3.

Discussion

We demonstrated that the empirically formulated concentric force-velocity relation lM (F M ) [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] of an activated muscle can generally be derived from a simple structural arrangement fulfilling an independent equation: the respective internal force equilibrium of three force bearing elements (AE, PDE, SE). Isolated function and connective arrangement of these elements (Fig. 1) have been confirmed both on macroscopic and microscopic levels for many decades, again and again. Eventually, just two other specific assumptions had to be superimposed: (i) the damping coefficient of the PDE must depend linearly on the force produced by the muscle, which is the only non-linearity of the model, and (ii) the gearing ratio κ v between contraction velocity of the solely modelled internal degree of freedom and the whole muscle has to be introduced as a parameter. With this reduced approach Hill's empirical hyperbolic muscular force-velocity relation [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF][START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF] can be transformed to a macroscopic mechanical model of the muscle. The

The gearing ratio κ v : concentric contractions and maintenance heat rate

Originating in this conjoining character of κ v , our model suggests a mechanism behind an observation as empirical and as old [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] as the Hill relation itself: the product A • B of the Hill parameters is assumed to equate, or at least to be in proportion to, the maintenance heat rate ḣ0 (compare Sect. 3.2). Now, our model provides one possible causal explanation for that. For the parameter κ v does not only shape all parameters of the Hill relation (Eqs. (16,17,18)) but also scales the maintenance heat rate ḣ0 in proportion to the square of the ratio κ v /(1κ v ) (Eqs. (22,23)), i.e. of the ratio between SE velocity and AE,PDE velocity. Remember that we also 16)). Therefore, we suggest that both maintenance heat rate and all parameters of the Hill relation (Eqs. (16,17,18)) may be traced back directly to the same set of four parameters, with κ v constituting a crucial one.

find κ v /(1 -κ v ) = A rel = A/F M,0 (Eq. (
Former studies [START_REF] Baker | A thermodynamic muscle model and a chemical basis for A.V. Hill's muscle equation[END_REF][START_REF] Pate | of the USA 90 (6), 2451-2455. actin and myosin filaments during muscle contraction[END_REF][START_REF] Woledge | The energetics of tortoise muscle[END_REF] have already suggested that interactions among crossbridges compro- dissipation should be force-dependent. Crossbridge interaction, i.e. some myosin heads stroking concentrically against other attached crossbridges loaded eccentrically, is also present in the isometric condition, which can be concluded from two studies [START_REF] Daniel | Compliant realignment of binding sites in muscle: transient behavior and mechanical tuning[END_REF][START_REF] Pate | Simulation of stochastic processes in motile crossbridge systems[END_REF].

In the [START_REF] Pate | Simulation of stochastic processes in motile crossbridge systems[END_REF] study the interaction between a halfsarcomere and a microscopic force sensor made of an elastic needle in series was simulated. The authors aimed at the sensor compliance to be optimised within an experimental setup in order to measure sensitively rather than to suppress ("elastic damping") the fluctuation amplitudes of the microscopic processes that occur within a real sarcomere. They described the interaction between a half-sarcomere and serial elasticity, the latter connected to a fixed point, as follows: any sudden stochastic change in the number of attached crossbridges, i.e. the pulling force of the half-sarcomere, requires a belonging adjustment in the length of the serial elastic needle. In turn, all remaining attached crossbridges will respond on their part by a displacement being partly concentric and partly eccentric. These permanent, stochastic, small-amplitude fluctuations are always present causing heat losses, even in the isometric condition in which source and drain balance to zero mechanical power output.

In the [START_REF] Pate | Simulation of stochastic processes in motile crossbridge systems[END_REF] paper the effect of serial elasticity was 1994; [START_REF] Kishino | Force measurements by micromanipulation of a single actin filament by glass needles[END_REF]Wakabayashi et al., 1994), myosin [START_REF] Huxley | X-ray diffraction measurements of the extensibility of actin and myosin filaments in contracting muscle[END_REF][START_REF] Neumann | Elastic properties of isolated thick filaments measured by nanofabricated cantilevers[END_REF]Wakabayashi et al., 1994), crossbridge (Ford et al., 1981;[START_REF] Goldman | Actin compliance: are you pulling my chain?[END_REF][START_REF] Huxley | Filament compliance and tension transients in muscle[END_REF], and titin [START_REF] Maruyama | Connectin, an elastic protein of muscle: characterization and function[END_REF] connecting Z-disc and M-line. Using actin somehow as a representative for all these structures, the [START_REF] Daniel | Compliant realignment of binding sites in muscle: transient behavior and mechanical tuning[END_REF] study modelled the effect of internal serial compliance, distributed along the actin, on interactions among crossbridges. Amongst others, they could quantitatively reproduce two findings from experiments: force responses to rapid short range changes in length (Piazzesi et al., 1997;Seow et al., 1997) and inflections in the force-velocity relation at low shortening velocities (Edman et al., 1976). Serial compliance is the crucial factor for both findings.

In our model, the coupling of muscle force production to internal serial compliance is mapped in the simplest possible way: one internal degree of the SE is not just a force transmitter but also an integral catalytic converter generating the macroscopic energy drain PDE by mechanically feeding back any crossbridge work stroke to other crossbridges. As a consequence, our model backs the view that the mechanics of concentric contractions and the origin of maintenance heat rate are indeed due to the same process [START_REF] Huxley | Muscle structure and theories of contraction[END_REF]. Note, that this feedback character of the SE reappears in items (i:

d P DE (F M )
) and (ii: κ v ) crucial for the Hill relation. Up to now we have suggested a qualitative explanation for item (ii). According to item (i), the specific damping coefficient d P DE depends on muscle force output F M . However, as F M always equates the force F SE in the SE (Eq. ( 2)), the damping coefficient may also be symbolised by d P DE (F SE ) (compare Eq. ( 4)). Furthermore, damping in proportion to muscle force may not only be qualitatively representative of stochastic feedback but also quantitatively representative of heat power release assumed to run approximately in proportion to the number of attached crossbridges doing work and, thus, releasing heat. Therefore, our model analysis based on the gearing ratio κ v may help to connect Hill's heat measurements [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF]), Huxley's sliding filament theory [START_REF] Huxley | Muscle structure and theories of contraction[END_REF] with a "stiffness" parameter k transforming crossbridge distribution to force output, modern compliant Huxley-type models [START_REF] Daniel | Compliant realignment of binding sites in muscle: transient behavior and mechanical tuning[END_REF], and a more recent micro-mechanical muscle model (Denoth et al., 2002;Telley and Denoth, 2007).

In Sects. 3.1,3.2 we had indirectly inferred realistic κ v values from literature data on Hill parameters. Two recent studies [START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF][START_REF] Loram | The passive, human calf muscles in relation to standing: the short range stiffness lies in the contractile component[END_REF] allow for more direct estimates from measuring finite shortening steps. At first, [START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF] [START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF]), a ratio lAE : lSE ≈ 2 : 5, and therefore κ v ≈ 0.7, can be estimated (see appendix Appendix A). This would be in striking contradiction to our prediction κ v ≈ 0.2 ( lAE : lSE ≈ 4 : 1).

However, two aspects should be taken into account.

First, the most compliant serial elastic contribution dominates the net outcome. In the aforementioned muscle preparations [START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF] the tendons were at least partially included. According to the κ v estimation mentioned above, we conclude that their "serial elastic component" (SEC) was dominated by tendinous (external) material, thus, hiding an about (4/1)/(2/5) = 10 times stiffer fibre (internal) SE. Interestingly, [START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF] calculated almost exactly the same numbers.

They compared their data to the stiffness of crossbridges and contractile filaments (Piazzesi and Lombardi, 1995). [START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF] estimated that this internal SE should be about 13 times stiffer than their SEC stiffness. Thus, our prediction of a realistic mean value κ v ≈ 0.2 matches literature data (Piazzesi and Lombardi, 1995;[START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF] to a high degree. Also consistently, another recent study [START_REF] Loram | The passive, human calf muscles in relation to standing: the short range stiffness lies in the contractile component[END_REF] concluded from directly visualising macroscopic elongations of fibres and tendons that the ratio of internal (fibre) to external (tendon) stiffness may range from about 30 to 1, depending on the load and elongation amplitudes, with an average ratio of 12 in case of short elongations typical for quiet human Second, we would like to annotate that dissipative coefficients might significantly contribute to or even dominate the gearing ratio κ v in particular contraction modes. That is, κ v might even be more representative of a distribution between parallel and serial damping than of a stiffness distribution.

Whether this might both be compatible with a hyperbolic force-velocity relation and allow for an improved prediction of the enthalpy rate could be verified by introducing a serial damping element (SDE) into the current model, which would also imply history effects.

Perspective

Our model might be seen as the most reduced mechanical muscle model to derive the hyperbolic Hill relation from a basic principle. To that, it starts from one equation (force equilibrium) and requires minimally four mechanical parameters F M,0 (q, l M ), F AE,max , D P DE,max , κ v . Mathematically, the ratio D P DE,max /F AE,max is actually relevant, i.e. just those three independent parameters that are necessary to uniquely determine a hyperbola would appear within our model.

It is not surprising that there are deviations between this extremely reduced model and real muscle. First, there is a discrepancy between our enthalpy-velocity relation and those from experiments, particularly at higher velocities. Second, there are deviations of the real muscle from a forcevelocity hyperbola at high contraction velocities nearby lM,max [START_REF] Claflin | Shortening velocity extrapolated to zero load and unloaded shortening velocity of whole rat skeletal muscle[END_REF][START_REF] Julian | The maximum speed of shortening in living and skinned frog muscle fibres[END_REF]Reggiani, 2007) which are, however, difficult to discern experimentally. Third, there are indications that the forcevelocity relation may generally be described by functions other than the , 1997). There and later [START_REF] Daniel | Compliant realignment of binding sites in muscle: transient behavior and mechanical tuning[END_REF], a microscopic Huxley-type model, rather than a Hill-type model, was used. This might be the better choice for trying to calculate model responses that are meant to reproduce the fine structure of the force-velocity relation.

The deficient prediction of enthalpy rate at high velocities is a crucial challenge. For this purpose, PDE dependency on whole muscle force should be scrutinised: should we not rather conceive a dependency on F AE , i.e. on the number of active crossbridges rather than on net muscle output force F M ? For a clear discrepancy between our prediction and measurements of enthalpy rate occurs when approaching maximum shortening velocity lM,max at which F M vanishes, but F AE does not vanish at all. Next, we neglected the significant contribution (Barclay et al., 2007;[START_REF] Houdijk | Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction[END_REF] of activation heat from Ca 2+ -flux through the sarcoplasmatic reticulum to enthalpy rate. Though, as it depends on the rate of action potentials rather than on the number of active crossbridges, it might be treated in the simplest way by adding another heat rate parameter to the enthalpy rate (Eq. 20). This would generally result in an about one third reduction of predicted relative enthalpy and in a slight decrease in efficiency (compare second paragraph of Sect. 3.2).

We developed our model for two reasons. First, we aimed at more insight 3). Generally, R P DE = 0.01 was chosen, therefore ḣ0 (Eq. ( 22))

depends slightly on κ v , along with lM,max (Eq. ( 15)).

Figure 4:

Comparison of enthalpy rates (A, including maintenance heat rate, normalised to the latter) and mechanical efficiencies (B) from experiments [START_REF] Barclay | Energetics of fast-and slow-twitch muscles of the mouse[END_REF][START_REF] Barclay | Mechanical efficiency and fatigue of fast and slow muscles of the mouse[END_REF] to those predicted by our model. The parameters of our model (identical to Fig.

2) represent a median piglet muscle [START_REF] Günther | High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models[END_REF]

  the muscle l SE length of the SE (internal degree of freedom) lSE contraction velocity of the SE κ v = lSE / lM ; gearing ratio of internal to external contraction velocity l AE = l Ml SE = l P DE ; length of the AE (and PDE) lAE = lM -lSE = lPDE ; contraction velocity of the AE (and PDE) q normalised muscle activation F M force generated by the muscle F M,0 isometric ( lM = 0) force of the muscle F SE = F M ; force of the SE F AE force of the AE F P DE force of the PDE F AE,max maximum force of the AE A Hill parameter: asymptote F M ( lM ) = -A, i.e. parallel to velocity-axis B Hill parameter: asymptote lM (F M ) = B, i.e. parallel to force-axis lM,max = B A • F M,0 ; absolute value of concentric contraction velocity at F M = 0 A rel = A/F M,0 normalised Hill parameter l CE,opt optimal length of assembly of active muscle fibres (not a parameter in our model; literature data) B rel = B/l CE,opt normalised Hill parameter d P DE (F M ) damping coefficient of the PDE linearly depending on F M D P DE,max maximum value of d P DE (F M ) R P DE minimum value of d P DE (F M ) normalised to D P DE,max ḣ shortening heat rate ḣ0 maintenance heat rate c ḣ0(T -T 0 , params) temperature (and other parameters) dependent factor of ḣ0 T temperature (reference value T 0 ) Ḣ = -F M • lM + ḣ + ḣ0 enthalpy rate of the muscle ε M = -F M • lM Ḣ

  respect to the AE.

  ) as lM = lM (F M ), naming it the "Hill relation". The emergence of the Hill relation lM (F M ), according to the two velocity-dependent terms in Eqs. (7,9) and the specific function Eq. (4), might become more transparent by being explained appendix Appendix B). In total, we have introduced five parameters, four of which (F M,0 , κ v , D P DE,max , F AE,max ) essentially forming the hyperbola and one (R P DE ) being an optional modifier.

  first guess, that ḣ0 scales with [ overlap • activated volume / time = overlap • activation • cross sectional area • length / time ], therefore with the product F M,0 • lM,max . Additionally, ḣ0 should increase with temperature A c c e p t e d m a n u s c r i p t T , while T 0 means a reference temperature. This and other dependenciesnot explicitly modelled here (params) -are represented by the coefficient c ḣ0 (T -T 0 , params). In Hill's original experiments on frog sartorius muscle at T = 0 o C (Hill, 1938) approximate numerical equality ḣ0 = A • B was found, which was also exactly implied by Huxley in order to fit his muscle model to the Hill rela-

  , 1973), and another, more recent, Hill-type muscle model[START_REF] Lichtwark | A modified Hill muscle model that predicts muscle power output and efficiency during sinusoidal length changes[END_REF]. However, beyond the just mentioned boundaries, an unrealistically steep decrease to almost (note: R P DE = 0) maintenance heat rate is predicted for very low forces and high velocities, respectively. When taking more recent measurements into account (Fig.4(A)), unincisive max-

  2(A,B)) than by looking at mechanical efficiency (Fig.2(C,D)). The maximum deviation of measured efficiencies calculated in our model from those determined by Hill's experiments is about 10% (Fig.2(C,D)). Almost 100% deviations, occurring for a wider range of muscle preparations (Fig.4(B)), can be explained by variations in properties of the examined muscles, e.g. the curvature of the force-velocity relation lM (F M ) or the course of the mainte-

  > 0 and lM < 0 in case of shortening. Remembering Fig.1with l M = l AE + l SE (Eq. (1)) and the force equilibrium (Eq. (2); F P DE corresponding to Eq. (3)) the net muscle power output rewrites -F M • lM = -F M • ( lAE + lSE ) = -(F AE + F P DE ) • lAE -F SE • lSE . Now we can infer fromEq. (8) that the relative power contribution of the SE unloading is κ v • 100% and the net contribution of AE plus PDE is (1κ v ) • 100%. Therefore, our model predicts the SE to contribute 9 . . . 29% (compare Sect. 3.1: κ v = 0.09, 0.29) of the net muscle power output during concentric contractions.

  19

  Hill parameters A and B immediately arise as functions of minimum four model parameters. Our approach provides us with a platform to discuss how both the force-velocity relation and the maintenance heat rate may originate from the same transparent, physiologically based parameters, yet, without explicitly modelling detailed microscopic processes for a start.Our model is based on five parameters. Three of them, representing the energy drain PDE (D P DE,max , F AE,max , R P DE ), are of straight mechanical, thus transparent, character, with R P DE being ancillary. The fourth, the active force F AE , is more entangled, first of all, because it is formally connected to the isometric force F M,0 by an equation (Eq. (6)). Second, it conjoins all the muscular energy source, e.g. the dependency of the contractile machinery on sarcomere overlap and activation. The gearing ratio κ v as a fifth parameter is, on the one hand, the most crucial for the force-velocity relation and, on the other hand, as pooling as F AE and F M,0 , respectively. As proposed in appendix Appendix A, we would interpret it as the clumped effect of stiffnesses and damping of AE, PDE, and SE. These transparent material properties determine the relative contribution κ v of SE shortening to the current concentric contraction velocity of the muscle, and to the corresponding power output.

  output. Treating the muscle as a thermodynamic instead of a mechanical system,[START_REF] Baker | A thermodynamic muscle model and a chemical basis for A.V. Hill's muscle equation[END_REF] found additionally that one specific characteristic of muscular damping leads exactly to the Hill relation:

  examined as a parameter external to the sarcomere. Physiologically based compliance internal to the sarcomere adds up from actin(Huxley et al.,

  freedom resolved by one macroscopic parameter κ v . Comparing our model to the situation within the Pate and Cooke (1991) study, the AE replaces the half-sarcomere and the SE replaces the needle. And in comparison to the Daniel et al. (1998) study, the AE maps the net work generation of crossbridges, whereas the SE maps net muscle internal serial compliance, whether it comes from actin, crossbridges, or other elasticities within a sarcomere.According to our estimate, net serial compliance would contribute about 10 . . . 30% to sarcomere length changes per time and, as compared to AE power, a 2.5 . . . 10% share in net mechanical power output during quasistationary concentric contractions. Thereby, the SE transmits any change in force to a change in length, and vice versa, of all crossbridges within a sarcomere and of other sarcomeres in series. From this point of view, therefore,

  figure 5 in[START_REF] Barclay | The mechanics of mouse skeletal muscle when shortening during relaxation[END_REF]), a ratio lAE : lSE ≈ 2 : 5, and there-

  (see for example[START_REF] Abbott | The relation between velocity of shortening and the tension-length curve of skeletal muscle[END_REF] Edman et al. (1976 Edman et al. ( , 1978));Edman (1988);Edman et al. (1997); Fenn and Marsh (1935);[START_REF] Guschlbauer | The extensor tibiae muscle of the stick insect: biomechanical properties of an insect walking leg muscle[END_REF]). In particular, Edman and co-workers highlighted repeatedly that the concentric force-velocity relation of real skeletal muscle deviates from one overall hyperbola at forces nearby the isometric condition F M > 0.8F M,0 (Edman et al.

  mechanics behind macroscopic muscle contraction represented by the Hill relation. Second, the model was supposed to be bi-directional and to be as reduced as possible in order to be maximally transparent and feasible. Due to its bi-directional character caused by its derivation from force equilibrium, any structure of the muscle-tendon complex (e.g. serial and parallel elasticity, serial damping, tendon, aponeurosis) can be incorporated or coupled to the model transparently and explicitly as a force law. Therefore, as a predictive model it forms an extensible basis for modelling the active muscle in more detail. Furthermore, bi-directionality allows for coupling the muscle model to external dynamics, thus, using it in direct dynamic computer simulations.

Figure 1 :

 1 Figure 1: The macroscopic model of the muscle consisting of the active (AE), parallel damping (PDE), and serial (SE) element. Not being part of the model, parallel elastic element (PEE), tendon, and aponeurosis are depicted additionally (greyed element symbols). The term "muscle" denotes the arrangement of all activatable muscle fibres. The damping coefficient of the PDE depends linearly on muscle force F M (Eq. (4)) which in turn both equates (force equilibrium) the force in the SE and the sum of the forces of AE plus PDE (Eq. (2)). The muscle length is l M and the internal length degree of freedom is represented by l SE .

Figure 3 :

 3 Figure 3: Enthalpy rate Ḣ = -F M • lM + ḣ + ḣ0 (top graphs A,C; note the different ordinate offset values) and mechanical efficiency ε M = -F M • lM / Ḣ (bottom graphs B,D) as functions of muscle force F M for F M,0 = F AE,max with varied curvature of lM (F M ), i.e. κ v • D P DE,max = const = 198 N s/m (labels κ v = 0.09, 0.20, 0.29), and two coefficients c ḣ0 of maintenance heat rate ḣ0 , respectively (left graphs A,C: c ḣ0 = 0.03 and right graphs B,D: c ḣ0 = 0.3). Generally, R P DE = 0.01 was chosen, therefore ḣ0 (Eq. (22))

  : F M,0 = F AE,max = 30 N , κ v = 0.20, D P DE,max = 1000 N s/m (A rel | RP DE =0 = 1/4), R P DE = 0.01 ( lM,max = lM,max | RP DE =0.01 = 0.144 m/s).

Figure 5 :FIGURES

 5 Figure 5: Graphical representation of the force-velocity (Hill) relation Eq. (7) or Eq. (9), respectively, for an assumed gearing ratio κ v = 0.09 between internal length l SE and muscle length l M contraction velocities lSE , lM (Eq. (8). A detailed explanation is given in appendix Appendix B.

  4) depending linearly on the current force F M of the muscle which equates the force in the SE (Eq. (2)), whereat D P DE,max is the maximum d P DE (F M ) value (at F M = F AE,max with conceiving of F AE,max as maximum isometric force) and R P DE is the minimum d P DE (F M ) value (passive damping in an inactive muscle) normalised to D P DE,max . Note that this force-dependency of d P DE is the only non-linearity that we introduce explicitly into our model.

Acknowledgements

Preparation of the manuscript accompanied us all through the year 2007 during which MG initially was supported by the "Deutsche Forschungsgemeinschaft" (DFG) grant MU1766/1-2, in 2008 and 2009 by grant MU1766/1-3. MG almost completely contributed at various places in Jena, Germany. There, he thanks André Seyfarth who really helped in rough seas during elapsing 2007. SS would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. We are very grateful to Tobias Siebert, Christian Rode, Olaf Till, and Reinhard Blickhan in Jena for an invaluably inspiring discussion giving the manuscript a decisive push.

Even more, many thanks to Reinhard for a most attentive and committed reading of early manuscript drafts, for asking vital questions, and giving ex-

A c c e p t e d m a n u s c r i p t

submitted to the Journal of Theoretical Biology 28 tremely incisive hints, again and again, during the whole preparation of this manuscript. Finally, MG cordially thanks all those creating the "Bohème" spirit that can turn work into purest joy. Above all, Mona and Lena may be awarded forevermore. Last but not least, atmosphere exuded by Evi and the subsequent charming lot at "Fairtrade Kontor" are, and will be, esteemed.

Jessie, you did a really good job.

A c c e p t e d m a n u s c r i p t

submitted to the Journal of Theoretical Biology 30 Denoth, J., Stüssi, E., Csucs, G., Danuser, G., 2002 a brilliant method to study kinetic properties of acto-myosin interaction.

The Journal of Physiology 583 (1), 5-7. (shortening heat rate) either as mechanical dissipation (damping) or as "extra" heat [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] used "extra heat" which is termed "shortening heat" here):

1. our model (solid line [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] (dashed line): ḣ = -A • lM with A due to Eq. ( 13), 3. "extra" heat according to [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF]; [START_REF] Huxley | A note suggesting that the cross-bridge attachment during muscle contraction may take place in two stages[END_REF]; [START_REF] Mcmahon | Muscles, Reflexes, and Locomotion[END_REF] (dotted line) with B after Eq. ( 14), α FM,0 = 0.16, α FM = 0.18, γ = 0.135, i.e. A =

4. non-linear damper according to [START_REF] Winters | Hill-based muscle models: a system engineering perspective[END_REF]

Muscle parameters according to Sect. [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF] with [START_REF] Hill | The effect of load on the heat of shortening of muscle[END_REF]. Note that the effective B and therefore lM,max are reduced. An alternative Hill-type model (pure F AE , i.e. without SE which always implies F AE = F M,0 ) after [START_REF] Winters | Hill-based muscle models: a system engineering perspective[END_REF] The gearing ratio κ v (Eq. ( 8) can be calculated by modelling specific contraction situations. If, e.g., we arranged two non-linear elastic elements in series the contraction velocity of the serial arrangement would be distributed according to the stiffnesses of the elements. Thus, if all damping forces were neglected in the force equilibrium of Eq. ( 2) it would write

Applying the first time derivative to this approximative force equilibrium, assuming the forces borne by all elements to depend on their respective lengths (F AE on filament overlap and additionally on the muscle activation q), and using the chain rule we get

After substituting lAE from the time derivative of the kinematic equation l AE = l Ml SE into Eq. (A.1), further simplifying that the muscle activation q does not change (e.g. fully activated), and remembering the definition of κ v (Eq. ( 8)) we find a specific gearing ratio Appendix B. Equations (7,9) as a sketch (Fig. 5)

The current muscle force F M is the sum of three terms in Eq. ( 7). The first two terms equal the current AE force F AE which is, in turn, F M,0 plus a linear function in lSE (Eq. ( 6)). The current force in the SE is always (1-R P DE )•F M,0 +R P DE •F AE,max = lM,max (compare Eq. ( 15)). The maximum contraction velocity of the muscle lM,max is much more sensitive to R P DE > 0 than lM,F AE =0 is, with lM,max < lM,F AE =0 .

Following the path of the depicted arrows in Fig. 5 The resulting asymptotes A,B (Hill parameters,see Eqs. (13,14)) and maximum contraction velocity -lM,max (Eq. ( 15)) are also depicted. Length and activation dependency of F AE reflect the chosen isometric force F M,0 and vice versa (Eq. ( 6)). The difference between F AE and F M,0 (Eq. ( 6 according to (Eq. ( 2)). Therefore, in case of calculating the internal length change for given muscle kinematics (e.g. isokinetic), lM = 0 would be just an additional term (compared to the isometric condition characterised by Eq. ( 6)) due to the assumption that the load is always applied to the active part of the muscle (AE and PDE) via a serial (visco-elastic) element with known characteristics.