Olaf Till
email: olaf.till@uni-jena.de

Tobias Siebert

Reinhard Blickhan

Lengths Olaf Till

A mechanism accounting for independence on starting length of tension increase in ramp stretches of active skeletal muscle at short half-sarcomere lengths

Keywords: accounting : for a tension increase independent muscle contractions half-sarcomere constant-velocity contractions of a skeletal muscle relation Skeletal :::::::: muscle :, contraction :, Muscle model, Simulation

Based on previous experimental results of a length-independent increase of tension with extension distance

Introduction

In physiologic motion contracting skeletal muscles may not only be shortened (= concentric contraction) or retain a constant length (= isometric contraction), but may also be lengthened (= eccentric contraction). It is well known that in maximally activated skeletal muscles, tension in isometric contractions is enhanced after a period of lengthening [START_REF] Abbott | The force exerted by active striated muscle during and after change of length[END_REF]. This effect of contraction "history" is called "force enhancement". Suggested physiological explanations are :::::::: altered : crossbridge-cycling characteristics (e.g. [START_REF] Mehta | Cross-bridge induced force enhancement?[END_REF], enhanced elastic stiffness (e.g. [START_REF] Pinniger | Crossbridge and non-crossbridge contributions to tension in lengthening rat muscle: forceinduced reversal of the power stroke[END_REF], and :::::: rapid :::::::::::::: lengthening :: of ::::::: some ::::::::::::::::: half-sarcomeres ::: to ::::::::: lengths ::

at ::::::: which :::::: they :::::::::: generate ::::: only ::::::::: passive :::::::: tension :::::::::::::: ("sarcomere :::::::::::: popping", [START_REF] Morgan | Tension changes in the cat soleus muscle following slow stretch or shortening of the contracting muscle[END_REF] ::: -:::: no ::::::::::::::: experimental ::::::::::: evidence :::: as ::::: yet, : [START_REF] Telley | Dynamic behaviour of half-sarcomeres during and after stretch in activated rabbit psoas myofibrils: sarcomere asymmetry but no 'sarcomere popping[END_REF] :

), :::::: and ::::: just heterogeneity of sarcomere lengths ::::::::::::: lengthening :::::::::: velocity :::::::::: without :::::::::::: "popping" [START_REF] Telley | Inter-sarcomere dynamics in muscle fibres. A neglected subject?[END_REF][START_REF] Telley | Dynamic behaviour of half-sarcomeres during and after stretch in activated rabbit psoas myofibrils: sarcomere asymmetry but no 'sarcomere popping[END_REF]. It is also known that tension can already be "enhanced" during lengthening in an extension-distance dependent way, in the sense that there is a tension increase with lengthening at constant velocity, even though purely isometric contractions at the corresponding muscle lengths would be roughly equal in tension [START_REF] Pinniger | Crossbridge and non-crossbridge contributions to tension in lengthening rat muscle: forceinduced reversal of the power stroke[END_REF][START_REF] Roots | Comparison of the tension responses to ramp shortening and lengthening in intact mammalian muscle fibres: crossbridge and non-crossbridge contributions[END_REF].

In a previous study [START_REF] Till | Characterization of isovelocity extension of activated muscle: A Hill-type model for eccentric contractions and a method for parameter determination[END_REF] we quantified the tension increase during lengthening of the M. gastrocnemius (Caput mediale) of the rat, assuming that the muscle consists of a tension-generating component (TGC) in series to an elastic component. Experimental results showed the tension gradient at constant lengthening velocity of the TGC to be independent of muscle length in the range up to optimal muscle length(ascending limb of the tension-length relation); tension traces of constant-velocity contractions at different starting muscle lengths within this length range were strikingly parallel(figure 2). Tension increase with TGC length change in eccentric constant velocity contractions starting from different muscle lengths, rat M. gastrocnemius (Caput mediale). This example of experimental data is taken from . Lengths are given relative to optimum fibre length (l f 0 ) and tension is normalized to cross-sectional area. Differences of starting lengths to optimum TGC length are -0.45 l f 0 , -0.33 l f 0 , -0.19 l f 0 , -0.13 l f 0 ,-0.07 l f 0 , and 0 l f 0 (1-6). Extension velocity of whole muscle was 0.77 l f 0 /s, extension distance was 0.15 l f 0 . In their straight part, the curves are nearly parallel. Parallelism is striking for curves 1-5, while at the longest starting length (curve 6) there is a slight deviation from parallelism. A Hill-type model was suggested under the usual assumption that tension of the TGC without effects of contraction "history" is the product f T GC = f im r v r l (f im : maximal isometric tension, r v : factor due to the tension-velocity relation, r l : factor due to the tension-length relation of the TGC : . ::::::::::::::: Parallelism ::: at ::::::::: different :::::::::: starting :::::::: lengths ::::: was :::::::: evident :::: for :::::::::: different ::::::::::: extension ::::::::::: velocities ::::::::::: (constant ::::::: within ::::: the ::::::: series), and a term was included to account for enhancement of tension due to eccentric contraction history. The term compensated for the variable gradient of the tension-length relation, to achieve a constant tension gradient at constant r v at different muscle lengths, in accordance with the experimental results.

Throughout the present paper, the designation "enhanced" will be used to indicate an extension-distance dependent increase in tension which is not due to ::::::

there :::: was :::: no :::::::::: evidence ::: for :: a :::::::::::::: dependence :: of :::: the ::::::::: tension :::::::::: gradient ::: on ::::::::::: extension :::::::::

velocity :::::::: (latter :::::: data ::::: not ::::::::::::: reproduced :::: in : the influence of the tension-length relation :::::::: current ::::::::: paper). :::: A ::::::::::::::::::::: phenomenological :::::::: model ::::: was :::::::::::: suggested, ::::::: with :: a :::::::

special ::::::: term ::: to ::::::::: account :::: for ::::: the :::::::::: observed ::::::::::::: parallelism. We now discuss the problems arising in attempts to explain the constant tension gradient in eccentric contractions of constant velocity with physiological mechanisms. Considering the events of the sliding filament mechanism which are well accepted to determine the tension-length relation [START_REF] Gordon | The variation in isometric tension with sarcomere length in vertebrate muscle fibres[END_REF][START_REF] Edman | Contractile properties of mouse single muscle fibers, a comparison with amphibian muscle fibers[END_REF], an important prediction on filament mechanics is derived. An example model, based on physiological mechanisms, is introduced and tested in simulations. The model can also account for enhancedtension after completion of an eccentric contraction :::::::::::::

Throughout :::: the ::::::::: present :::::::: paper, :::: the :::::::::::::: designations :::::::::::::: "enhanced" ::: or ::::::::::::::::: "enhancement" ::::

will ::: be :::::: used :::: to ::::::::: indicate :::: an ::::::::::::::::::::: extension-distance ::::::::::::: dependent ::::::::: increase ::: in ::::::::: tension :::::: which ::: is ::::: not :::: due :::: to :::: the ::::::::::: influence ::: of :::: the ::::::::::::::::: tension-length ::::::::: relation.

Influence of the tension-length relation

Figure 1A shows the approximate shape of the tension-length relation, as it is determined from isometric contractions of short segments of muscle fibres [START_REF] Gordon | The variation in isometric tension with sarcomere length in vertebrate muscle fibres[END_REF][START_REF] Edman | The sarcomere length-tension relation determined in short segments of intact muscle fibres of the frog[END_REF][START_REF] Edman | Contractile properties of mouse single muscle fibers, a comparison with amphibian muscle fibers[END_REF], and as it is thought to be representative for a single half-sarcomere. The filaments of a sarcomere are schematically illustrated in figure 1B. The ascending limb consists of two ranges with different gradients (1 and 2). The cause of the gradient of range 1 was originally identified with a compression of the ends of opposite thin filaments against each other and/or an overlap of the contralateral thin filament with the ipsilateral thick filament active zone, the so called "double overlap" [START_REF] Gordon | The variation in isometric tension with sarcomere length in vertebrate muscle fibres[END_REF]. The data of [START_REF] Elmubarak | Temperature sensitivity of tension development in a fast-twitch muscle of the rat[END_REF] :::::::: (whole :::: rat :::::::::: muscle) ::::: and : [START_REF] Edman | Contractile properties of mouse single muscle fibers, a comparison with amphibian muscle fibers[END_REF] :::::::: (mouse :::::::: muscle :::::: fibre :::::::::::: segments) : make the latter cause more probable, which could lead to an inactivation of cross-bridges and/or to a counteracting force, possibly by cross-bridge formation with the contralateral thin filament. In range 2, the gradient is further increased by a compression of the thick filament, leading to a counteracting spring-force and/or an inactivation of cross-bridges. If the tension increase ::::::::: gradient : in eccentric constant-velocity contractions of a half-sarcomere would be :::::: were solely caused by the mechanisms responsible for the tension-length relation, and if these mechanisms remain :::::::::: remained unchanged compared to isometric contractions, the tension gradient would be higher in length range 2 than in length range 1. In eccentric constant-velocity contractions of a whole skeletal muscle in , however, the tension gradient was constant over a length range corresponding to isometric forces from 15 of optimal to optimal, i.e. a range including range 1 and most of range 2.

:::: not ::: be :::::::::::: constant, ::: as ::::::::::: observed :::::::::::::::::: experimentally, ::::: but ::::::: would :::::::: follow :::: the :::::::::: variable :::::: slope ::: of :::: the ::::::::::: isometric ::::::::::::::::: tension-length :::::::::: relation. :: Although in the experiments of the cited paper additional enhancement of tension in an extension-distance dependent way was shown to occur, it seems unlikely that this effect compensates for the differences in the mechanisms :::::::::

variable :::::: slope : of the tensionlength relation in ranges 1 and 2 to achieve a length-independent tension gradient ::::::::

tension :::::::::: gradient ::::::::::::::: independent :::: of :::::::::: starting :::::::: length. To take more plausible assumptionscompatible with the experimental findings : , ::::::::::::: compatible :::::

with :: a ::::::::::: constant ::::::::: tension :::::::::: gradient ::: in ::::::::::: eccentric :::::::::::::::::::: constant-velocity :::::::::::::: contractions ::

in :::::::: ranges :: 2 ::::: and :: 1 ::: of ::::: the ::::::::::: ascending ::::::: limb, : we hypothesize that:

1. The extension-distance dependent enhancement of tension during eccentric constant-velocity contractions is independent of half-sarcomere length ::::: over :::: the :::::::: whole ::::::: range ::: of ::::::::::::::::: half-sarcomere ::::::::: lengths ::::::::: covered :::: by ::::: the ::::::::::::::

experimental :::::: data. 2. The tension-length relation in eccentric contractions is different from that in isometric contractions and has a constant gradient over its whole ascending limb.

To account for assumption 2, we note that compression of the thick filament is confined to range 2 of the ascending limb, while the mechanism underlying range 1 is in effect over the whole ascending limb, probably with the same gradient as in range 1 (dashed line I in figure 1A). So the absence of effects of thick filament compression in eccentric contractions would yield the desired constant gradient of the ascending limb. This would imply that a possible counteracting compression force does not diminish gradually, but gets zero instantaneously at the start of extension -e.g. because the thick filament does not extend again as long as the muscle is activated -and that possibly inactivated cross-bridges in the compressed filament either remain inactivated or all get reactivated at once at the start of decompression. In an eccentric contraction starting from the short end of the ascending limb, a counteracting compression force instantaneously getting zero or instantaneously reactivated cross-bridges would lead to an ascending limb as dashed line I in figure 1A, while cross-bridges remaining inactivated would produce dashed line II, which is parallel to dashed line 1 and starts at tension zero. A mixture of these effects might yield a line in between and parallel to the dashed lines.

To extend the length range of constant extension-distance dependent enhancement of tension to include the plateau region :::::::::::::::::: tension-gradient ::: in ::::::::::

eccentric :::::::::::::: contractions :::: to ::::::::: include ::::::::: regions ::::::::: outside :::: the :::::::::::: ascending :::::: limb : of the isometric tension-length relation, one could assume that even the mechanism underlying range 1 in figure 1A is not effective in eccentric contractions . This could be due to cross-bridges inactivated by "double overlap" remaining inactive even if the double overlap is removed during an eccentric contraction. Such an additional hypothesis, however, is not further pursued here since available data for comparing tension gradients of eccentric contractions within

Origin of the extension-distance dependent enhancement of tension

The three possible explanations of enhanced tension after :::::: after a stretch suggested in the literature and mentioned in the introduction were considered as causes of the extension-distance dependent enhancement of tension during :::::::: during : an eccentric contraction on the ascending limb of the tension-length relation. Sarcomere length inhomogeneities were ::::::: which ::: is, ::: as :::

we :::::::::::::: hypthesized :::::::: above, ::::::::::::::: independent ::: of ::::::::::::::::: half-sarcomere :::::::: length ::::: over ::::: the ::::::: whole :::::: range ::: of :::::::::::::::::: half-sarcomere ::::::::: lengths ::::::::: covered :::: by ::::: the :::::::::::::::: experimental :::::: data. ::::::::: Rapid ::::::::::::: lengthening ::: of :::::: some ::::::::::::::::::: half-sarcomeres ::: to ::::::::: lengths ::: at ::::::: which :::::: they :::::::::: generate :::::: only :::::::: passive ::::::::: tension :::::::::::::: ("sarcomere :::::::::::: popping", : [START_REF] Morgan | Tension changes in the cat soleus muscle following slow stretch or shortening of the contracting muscle[END_REF] :

) ::::: was : ruled out since they are :: it ::: is : not believed to enhance tension :::: take ::::::: place : on the ascending limb [START_REF] Morgan | Tension changes in the cat soleus muscle following slow stretch or shortening of the contracting muscle[END_REF][START_REF] Herzog | Force enhancement following stretch of activated muscle: critical review and proposal for mechanisms[END_REF], ::::: and ::: at :::::: least ::: at ::::: the :::::::: shorter ::::::: muscle ::::::::: lengths ::: of ::::: our ::::::::::::::: experimental :::::: data ::: all :::::::::::::::::: half-sarcomeres ::::: can ::: be ::::::::::: expected ::

to :::: be :::: on ::::: the :::::::::::: ascending ::::::: limb. :::::::::::::: Similarly, :::::::::::::::: heterogeneity :::: of ::::::::::::::::: half-sarcomere ::::::::::::: lengthening :::::::::: velocity : [START_REF] Telley | Dynamic behaviour of half-sarcomeres during and after stretch in activated rabbit psoas myofibrils: sarcomere asymmetry but no 'sarcomere popping[END_REF] :: is :::::::::: unlikely ::: as ::: a ::::::: cause, ::::::: since :::::: with ::: all :::::::::::::::::: half-sarcomeres ::: on :::: the :::::::::::: ascending ::::: limb ::::: any ::::::::: slowing :::::: down ::: of ::::::::::: extension ::: of ::: an ::::::::::: individual ::::::::::::::::: half-sarcomere ::::::::: relative ::: to :::: the ::::::: others ::::: will :::::::::: decrease :::: this :::::::::::::::::: half-sarcomeres :::::::: tension ::::::::: relative :::: to :::: the :::::::: others ::::: and :::::: thus ::::::::::::: counteract :::: the ::::::::::::: slow-down. The extent of any effects of altered cross-bridge kinetics on tension is likely to be proportional to the currently available number of active cross-bridges. Thus, since :::::::::: However, : this number is not constant over any length rangeon :::::::::: probably including : the whole ascending limb of the tension-length relation. So we suggest the third possibility, enhanced passive stiffness, as the reason for the extension-distance dependent enhancement of tension.

The sarcomere protein titin is known to be associated with passive muscle tension in response to stretch (e.g. [START_REF] Magid | Myofibrils bear most of the resting tension in frog skeletal muscle[END_REF][START_REF] Higuchi | Changes in contractile properties with selective digestion of connectin (titin) in skinned fibers of frog skeletal muscle[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]. It is a giant protein (≈3.5 MDa), tightly bound to the thick filament and freely extending through the so called I-band of the sarcomere to the Z-disk, where its N-terminal part is fixed (see e.g. [START_REF] Tskhovrebova | Role of titin in vertebrate striated muscle[END_REF] for a review). The stiffness of I-band titin could be enhanced by an elevated calcium concentration during activation [START_REF] Labeit | Calcium-dependent molecular spring elements in the giant protein titin[END_REF]. Also, titin stiffness could be indirectly enhanced in activated muscles by adhesion of parts of I-band titin to the much stiffer thin filament : , ::: as :::::::::::: suggested ::::: e.g. :::: by : [START_REF] Pinniger | Crossbridge and non-crossbridge contributions to tension in lengthening rat muscle: forceinduced reversal of the power stroke[END_REF]. For the latter suggestion, available evidence is scarce [START_REF] Kellermayer | Calcium-dependent inhibition of in vitro thin-filament motility by native titin[END_REF]; there is even evidence that a calcium concentration elevated to still physiological values (10 μM) inhibits binding of an I-band titin fragment to a thin filament constituent [START_REF] Kulke | Interaction between PEVKtitin and actin filaments: origin of a viscous force component in cardiac myofibrils[END_REF]. Nevertheless, we assume that binding of Iband titin to the thin filament at muscle activation is the cause of enhanced passive stiffness, since, as detailed in the next section, this can naturally explain the constancy of the extension-distance dependent enhancement of tension over all half-sarcomere lengths.

Thus, since titin is fixed at the end of the thick filament but must be able to generate tension even in eccentric contractions over length range 2 of the ascending limb (where the thick filament is shortened by compression), a prediction inherent to the above conclusions is that the thick filament, as long as the muscle is activated, withstands this tension even in a compressed state, i.e. tension will not lead to a re-elongation of the compressed thick filament. This prediction makes the conclusions of the previous subsection more precise. Except for this indirect reasoning, to our knowledge there is no evidence as yet for such a behavior, nor any known mechanism which could lead to such an internal lengthening-resistance of the compressed thick filament.

Example model of a half-sarcomere

The above conclusions do not define each model detail. For an example model, we choose that in range 1 of the ascending limb, tension is diminished by overlap of the contralateral thin filament with the ipsilateral thick filament active zone ("double overlap"), which completely abolishes active tension generation in the double overlap zone, but generates no counteracting force. In range 2 of the ascending limb, compression of the thick filament is thought to generate only negligible counteracting spring force, and is assumed to reduce tension generation by inactivation of cross-bridges in such a way that active tension is always proportional to the length l o of the (possibly compressed) free part of the thick filament active zone (i.e. the part which is overlapped by the ipsilateral thin filament, but not overlapped by the contralateral thin filament). These choices are in accordance with [START_REF] Edman | Contractile properties of mouse single muscle fibers, a comparison with amphibian muscle fibers[END_REF], who achieved a good fit to an experimental tension length curve of mouse M. flexor digitorum brevis with equivalent numerical ::::::::::::: numerically :::::::::::: equivalent assumptions (although, differently to the current assumptions, diminution of tension by compression of the thick filament was not related to cross-bridge inactivation, but to a counteracting force). In addition, active tension f a depends on the sliding velocity v between the thin and thick filamentand : , on a factor r act due to activation with 0 ≤ r act ≤ 1, :::: and :::: on :: a ::::::: factor :::: r av :::::: with ::::::::::::: 0 ≤ r av ≤ 1 ::::: due ::: to ::::::::::: "delayed ::::::::::::::: availability" ::: of ::::: the ::::: thin ::::::::::: filament ::::::::: coming ::::: into :::::::: contact :::::: with :::: the ::::::: thick :::::::::: filament :::: (as ::::::::::: discussed ::: in ::::::::: section ::: 2), : thus

f a = f im r v (v)r act r av ::: l o /l a , ( 1
)
where f im is the maximum isometric tension, r v (v) is a factor due to the sliding velocity v with r v (0) = 1, and l a is the uncompressed length of the thick filament active zone.

To account for compression and decompression, the thick filament active zone is modelled as a bidirectional visco-elastic spring which is practically inextensible over the uncompressed length l a and very compliant for compression to below l a . Viscous resistance of the spring linearly depends on r act . At high activation (large r act ), viscous resistance to lengthening is high, practically not allowing (re-)extension. At low activation (small r act ), viscous resistance is negligible. For shortening, viscous resistance is always negligible, independently of activation.

A sketch of the model is given in figure 3. Adhesion of the central part j of titin to the thin filament is modelled by assigning a viscous resistance to this part. If j moves away from the Z-disk, resistance linearly depends on r act , being high for full activation and negligibly small if there is no activation. For movement of j towards the Z-disk, we have to account for shortening (concentric) contractions. In shortening, the central part j of titin is supposed to move freely towards the Z-disk, to avoid the generation of counteracting forces by adhesion of j to the thin filament. To achieve this, resistance is assumed to be negligibly small even during activation if j moves towards the Z-disk. Physiological explanations for this could be a drop of calcium concentration during concentric contractions [START_REF] Vandenboom | Effects of rapid shortening on rate of force regeneration and myoplasmic [Ca2+] in intact frog skeletal muscle fibres[END_REF] or a direct dependence of resistance on the direction of tension.

The distal part of titin is supposed to generate extension-distance dependent passive tension in eccentric contractions and is therefore modelled as a linear spring. The length of the spring at zero tension is assumed to be very short and is set to zero for simplicity. This short force-less length is necessary to let generation of tension start as soon as the eccentric contraction begins, so that no upwards bent in the tension trace is produced some time after the start of extension. Since passive tension of the inactive half-sarcomere is ::::::: muscle ::::: was : negligible in the length range of the ascending limb of the tension-length relation of the activated muscle ::::::::::::::

experimental :::::: data ::::::::

refered ::: to, no attempt is made to realistically account for this passive tension and proximal titin is arbitrarily modelled as a linear spring with negligible tension and a length of zero at zero tension. We previously published a model which, among others, detailed the way I-band titin could increase its stiffness in eccentric contractions [START_REF] Rode | Titin-induced force enhancement and force depression: A "sticky spring" mechanism in muscle contractions?[END_REF]. Briefly, an elastic central region of titin (the so called PEVK region) was assumed to attach at multiple binding sites of the thin filament at activation. The single attachments were assumed to withstand a certain disruptive force and to rapidly re-attach after a disruption. Within the physiological length range, this was shown to yield a nearly linear tension-elongation behavior at activation. This is the same effect as achieved by the units j and dt together in the current model (figure 3). However, since the multiple attachments in the previous model do not involve distal titin, a slack distal titin would cause an initial delay of elastic tension development in eccentric contractions at the ascending limb of the tension-length relation. This in turn would lead to an upwards bent in the tension trace some time after the start of extension(not shown), which is not seen in our experimental contractions.

Simulation

Simulations of eccentric contractions with the example model are performed in order to test whether they are indeed in accordance with experimental results. Since simulations will be :::::::::::::: qualitatively : compared with tension traces of experiments with a whole muscle, the modelled conditions are partially adapted to those within a real muscle by including an elastic component in series to the half-sarcomere into the model. :::::::::: Without ::::: this, :::::::::: imposed ::::::::: velocity ::::::::: changes ::: of ::::: the ::::::: whole :::::::: muscle :::::::: would ::: be ::::::::::: identical ::: to :::::::::: velocity ::::::::: changes ::

of ::::: the :::::::::::::::::::::: tension-generating :::::::::::::: component, :::::::: which :::::::: would :::::: lead ::: to ::::::::::::::::: unrealistically :::::: sharp :::::::::: velocity ::::::::::: gradients :::::: and :::::::::::::::: corresponding ::::::::: tension ::::::::::::: gradients. :: The length of this "series elastic component" (SEC) at zero tension is unimportant for our purpose and is set to zero. The development over time of the length l := l s + Δl SEC , where Δl SEC is the extension of the SEC, is an input of the model. ::::::::

Length ::: is ::::::: given ::: in ::::::: units ::: of :::::::::: optimal :::::::: length ::::: l opt , :::::::::: defined ::: as Four ::::: Five : coupled differential equations were used to describe the development over time of, respectively, the half-sarcomere length l s , the distance l pt of j from the middle of the Z-disk, the length l thick := l act + l b /2 of half the thick filament, and the state of activation r act , ::::: and ::::: the ::::::: factor :::: r av ::::: due ::: to :::::::::: "delayed :::::::::::::: availability" ::: of ::::: the ::::: thin :::::::::: filament : -see figure 3 for explanation of lengths:

dls dt = r -1 v (f SEC (l-ls)-f dt (ls-l thick -lpt))la f im ract(l sovl -l dovl ) (f SEC (l-ls)-f dt (ls-l thick -lpt))la f im ractrav(l sovl -l dovl )
::::::::::::::::::::::::::::: The : functions r v , f SEC (tension of SEC), f dt (tension of distal titin), v j (velocity of j ), f pt (tension of proximal titin), v t (extension velocity of thick filament active zone), and f thick (tension of modelled spring in thick filament active zone) are specified as follows:

dlpt dt = v j (f dt (l s -l thick -l pt ) -f pt (l pt ), r act ) dl thick dt = v t (f dt (l s -l thick -l pt ) -f thick (l thick ),
The tension-velocity relation is defined according to [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] and van Soest and Bobbert (1993) as

r v (v) = arv+b b-v if v ≤ 0, B(1-A) B-v + A if v > 0, (3)
where a r is defined as a/f im , and a and b are parameters of Hills equation; B is defined as b(1 -A)/(κ(a r + 1)) and parameters A and κ are a tension limit relative to f im and the slope change of the tension-velocity curve at v = 0, respectively. f SEC is modelled according to [START_REF] Winters | Hill-based muscle models: a systems engineering perspective[END_REF]:

f SEC (Δl SEC ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ f SEC1 e k sh -1 e k sh Δl SEC d SEC1 -1 if Δl SEC ≤ d SEC1 , f SEC1 + s(Δl SEC -d SEC1 ) if Δl SEC > d SEC1 , 0 i fΔ l SEC < 0, (4)
where parameters d SEC1 , f SEC1 , and k sh are the SEC elongation and the corresponding tension at which the relation changes from exponential to linear, and a shape factor, respectively. s is the slope at elongation d SEC1 and can be calculated from the three parameters. Distal and proximal titin are modelled as linear springs of length zero at zero tension with coefficients of stiffness k dt and k pt , respectively. The bidirectional spring of the thick filament active zone was modelled with the functional equation

f thick (l thick ) = k lth (l thick -l 0th ) if l thick ≥ l 0th , k sth (l thick -l 0th ) if l thick < l 0th , (5)
where k lth , k sth , and l 0th are the coefficients of stiffness for lengthening and shortening and the length at zero tension, respectively. For the velocities v j and v t the equations

v s (f s ) = f s /k fr-min if f s ≤ 0, f s /k fr (r act ) if f s > 0 (6) and k fr (r act ) = k fr-min + r act (k fr-max -k fr-min ) ( 7 )
were used with v s = v j or v s = v t and suitable values for k fr-min and k fr-max , which are the minimal and maximal coefficient of friction, respectively. Filament related lengths (see figure 3) are taken from : l thin = 1.10 μm, l a (defined by l act in the uncompressed state of act) = 0.73 μm, l b = 0.16 μm, l z = 0.05 μm, and are normalized to optimal half-sarcomere length l opt := l thin + (l b + l z )/2. The parameter l 0th (half thick filament length at zero spring force) is defined by l a + l b /2. The parameters of equations 3 and 4 are those experimentally determined in for rat M. gastrocnemius (Caput mediale) (rat no. 4), normalized to l opt and f im : a r = 0.26, b = 2.26 s -1 , A = 1.59, κ = 3.0, k sh = 1.79, d SEC1 = 0.56, f SEC1 = 0.4. The remaining parameters are arbitrarily set: k on = 0.0001, k of f = 0.001; k dt = 3 (distal titin); k pt = 0.01 (proximal titin); k lth = 10.000, k sth = 0.1 (thick filament active zone); k fr-min = 0.00015, k fr-max = 150 (junction between proximal and distal titin); k fr-min = 0.0001 s, k fr-max = 10.000 s (thick filament active zone).

::::

The :::::::: model ::::::::::::: parameters :::::: used ::: in ::::::::::::: simulation :::: are ::::::: given ::: in :::::: table ::: 1. :: Equations 2 are numerically solved with the 'LSODE' software [START_REF] Hindmarsh | Two new initial value ordinary differential equation solvers[END_REF] distributed with the numerical computing language Octave, whose algorithm uses the backward differentiation formula and adaptive time stepping. The initial value ::::::: values : of r act is ::::

and :::: r av ::::: are : zero. For l s , l pt , and l thick , initial values are iteratively solved for using the condition f SEC = f pt = f dt = f thick , or are calculated from the sums of the respective spring lengths if tension is zero.

Tension traces of simulated eccentric constant-velocity contractions at half-sarcomere lengths up to optimal (figure 4) show the typical time course of experimental eccentric contractions: a first phase with a higher slope, followed by a second phase with an almost linear tension increase, corresponding to a constant extension velocity of the tension-generating component [START_REF] Till | Characterization of isovelocity extension of activated muscle: A Hill-type model for eccentric contractions and a method for parameter determination[END_REF], which in the case of the current model corresponds to the half-sarcomere.

As intended by model design, the traces are parallel in their last phase ::::: over ::: all ::::::: parts ::: of :::: the ::::::::::: isometric ::::::::::::::::: tension-length ::::::::: relation ::: up :::: to :::: the ::::::: start ::: of :::: its ::::::::::::: descending :::::: limb. :::::::: Note ::::: that :::: no :::::::::: attempt ::::: was :::::: not ::::::::: covered :::: by ::::: the ::::::::: current :::::::: model. Figure 5 shows a simulation which in-cludes an isometric phase immediately after an eccentric contraction. In this phase, tension is still elevated. This means that the current model can also explain "force enhancement" after completion of a stretch :::::

Summary

Previous experimental results of a length-independent increase of tension ::::

the :::::::::::::::: independence ::: on :::::::::: starting :::::::: length ::: of ::::: the ::::::::: tension :::::::::: increase : with extension distance in eccentric constant-velocity contractions, together with well accepted assumptions of the sliding filament theory, restrict the possibilities to suggest a suitable mechanism underlying the tension increase during eccentric contractions. It is plausible to assume that the extension-distance dependent enhancement of tension (defined as the component of extension-distance dependent tension increase not explainable by the tension-length relation) is constant at each half-sarcomere length and that the tension-length relation in eccentric contractions is different compared to that in isometric contractionsand has a constant gradient over its whole ascending limb. Furthermore, the constancy of the extension-distance dependent enhancement of tension favors the hypothesis that enhanced titin-stiffness during activation is its cause.

These assumptions in turn lead to the prediction that the thick filament retains a shortened length in eccentric contractions starting at half-sarcomere lengths with thick filament compression, even though tension of titin acts to re-extend the thick filament. It is worth noting that this prediction probably can be checked experimentally, since there are fixation techniques which can preserve the state of filaments during a contraction for electron-microscopical examination [START_REF] Sosa | Ultrastructure of skeletal muscle fibers studied by a plunge quick freezing method: myofilament lengths[END_REF]. This example of experimental data is taken from . Lengths are given relative to optimum fibre length (l f 0 ) and tension is normalized to cross-sectional area. Differences of starting lengths to optimum TGC length are -0.45 l f 0 , -0.33 l f 0 , -0.19 l f 0 , -0.13 l f 0 ,-0.07 l f 0 , and 0 l f 0 (1-6). Extension velocity of whole muscle was 0.77 l f 0 /s, extension distance was 0.15 l f 0 . In their straight part, the curves are nearly parallel. Parallelism is striking for curves 1-5, while at the longest starting length (curve 6) there is a slight deviation from parallelism. 3: ::::::: elastic :::::::: tension :: of :::::: distal ::::: titin ::: in :::: the ::::::::: eccentric :::::::::::: contraction. After the end of eccentric movement, the elevated tension decays towards the purely isometric tension.

  figure ::::: 2B, ::::: the :::::: slope ::: of ::::: the ::::::::::::::::: tension-length ::::::::: relation :::: of :::: the :::::::: whole :::::::: muscle ::::: was :::: still :::::::::::::::: non-constant. ::::::: Thus, ::::: the :::::: range ::: of ::::::::::::: parallelism :::::::::: included :: a ::::: part ::: of :::: the ::::::: whole ::::::::: muscles :::::::::::::::: tension-length ::::::::: relation :::::: with ::::::::: variable ::::::: slope, :::::::::: probably ::::::::::::: comprising ::::::::: lengths ::::: with :::::::: muscle ::::::: fibres ::: at ::::::: ranges :: 2 ::::: and :: 1 ::: of :::: the ::::::::::: ascending ::::::: limb, ::: at :::: the ::::::::: plateau, ::::: and ::::: even ::: at :::: the :::::: start ::: of :::: the ::::::::::::: descending :::::: limb ::: of :::: the ::::::: fibres :::::::: tension :::::::: length ::::::::::: relations.

  :::::::::::: additionally :::::::::: assumed :::::: that ::::::: parts :: of ::::: the ::::: thin :::::::::: filament ::::::::: getting ::::: into ::::::::: contact ::::: with ::::: the :::::: thick ::::::::::: filament :::::::: during ::::::::::: eccentric ::::::::::::::: contractions :::: are ::::: not ::::::::::::::: immediately :::::::::: available :::: for ::::::::::::: crossbridge ::::::::::::: formation, ::::: but ::::::::: change :::::: their ::::::: state ::: to :::: get ::::::::::: available :::::::::::depending :::: on :::::::::: contact :::::: time ::: in :::: an :::::::::::::: exponential :::::::::: manner. ::::::: This :::: has ::::: the ::::::: effect ::::: that ::: at ::: a ::::::::::::: sufficiently ::::::: large :::::::::::: extension :::::::::: velocity, ::::: the :::::::::: amount :::: of ::::::::::::: "available" ::::: thin :::::::::: filament ::: (if ::::::: some ::::: was ::::::::::: available :::: at ::::: all) ::::::::::: decreases ::::::::: already :::: at ::::: the :::::: start ::of :::::::::::::: elongation, ::::::::: making ::::: the :::::::::: gradient ::: of ::::: the :::::::::: effective ::::::::::::::::: tension-length :::::::::: relation :::::::::: negative, :::::: even :: at ::::: the :::::::::::: "ascending ::::::: limb" ::: of :::: the ::::::::::: isometric :::::::::::::::: tension-length :::::::::: relation. :::::::::Without ::::::::::::: attempting :: a ::::::::::::::: physiological :::::::::::::: explanation ::: for ::::: the :::::::::: "delayed ::::::::::::::: availability", ::it :: is :::::::: shown ::::::: below :::: in ::::::::::::: simulations :::::: that ::::: this ::::::::::::: assumption ::::::::: suffices ::: to ::::::::: extend :::: the :::::: range ::: of :::::::::::::: parallelism ::::: into : the plateau region to those within the ascending limbis scarce :::: and ::::: the :::::: start ::: of :::: the ::::::::::::: descending :::::: limb.

  :::: not :::::::::: constant :::::::: either ::: in ::::::: range ::: 1 ::: of : the ascending limb :::: (the ::::::::::: assumed :::::::: double :::::::: overlap ::::: will :::::::::::: inactivate ::::::::::::::: cross-bridges ::: or ::::::::::: introduce ::::::::::::::: cross-bridges :::::::: acting :::: on :::: the ::::::::: "wrong" :::::: thin ::::::::::: filament) :::: or ::: in ::::::: range :: 2 ::::: (in :::: the :::::::::::::: compressed :::::: part ::: of ::::: the :::::: thick ::::::::: filament :::::::::::::::: cross-bridges, ::: if ::::: not :::::::: wholly :::::::::::::: inactivated, ::::: are :::::::: denser ::::::::: spaced ::::: and ::: so ::: are ::::::: likely ::: to :::::: have :::::::: altered :::::::: overall :::::::::: activity :::::::: against ::::: the ::::::::: binding :::::: sites ::: at :::: the ::::: thin :::::::::::filament). ::::::: Thus, cross-bridge kinetics are unlikely as cause of an extensiondistance dependent enhancement of tension which is constant over : a ::::::: range ::::::::::

  Note :::::: that ::::: the ::::::: above :::::::::::: modelling :::: of :::::::::: adhesion :::: of :::::::: central :::::: titin :::: to :::: the :::::: thin ::::::::: filament :::::: with ::: a ::::::::: viscous :::::::::::: resistance :::::::: makes ::::::::: passive ::::::::: tension ::::::::: depend ::::: not :::::: only ::: on ::::::::::: extension ::::::::::: distance, ::::: but :::::: also ::: on :::::::::::: extension ::::::::::: velocity: ::: at ::::::::: higher ::::::::::: extension ::::::::: velocity, ::::::::: tension :::::::::::: elevation ::: is :::::::: faster, ::::::::: leaving ::::: less ::::::: time :::: for ::::::::::: "sliding" ::: of ::::: the :::::::: central :::::: titin ::: -:::: so ::::::::: tension ::::::::::: elevation :::::::: needs ::::: less :::::::::::: sarcomere :::::::::::: extension, ::::::: since :::: less ::::::::: central :::::: titin ::::::::::: "sliding" :::: has :::: to :::: be ::::::::::::::: compensated ::::: for. ::::::: This ::: is ::: in :::::::::: contrast :: to ::::: our :::::::::::::::: experimental ::::::: data, :::::::: where :::: we :::::::: found ::: no ::::::::::: evidence :::: for ::: a :::::::::::::: dependence :: of ::::: the ::::::::: tension ::::::::::: gradient :::: on :::::::::::: extension :::::::::: velocity. ::::::::::::: However, :::: for ::::::: large :::::::: values ::of ::::::::: viscous :::::::::::: resistance ::::: the :::::::::::::::::::::::: velocity-dependence ::: of ::::: the ::::::::: tension :::::::::: gradient :::::: may ::: be ::::::: small ::::: and ::::: not ::::::: easily ::::::::: evident :::: in ::::::::::::::: experimental ::::::: data. :::::::: Also, ::::: the :::::::::: assumed ::::::::: negative ::::::::::: gradient ::: of :::: the :::::::::: effective ::::::::::::::::: tension-length :::::::::: relation ::: in ::::::::::::::: contractions ::: of :::::::::: sufficient ::::::::::: extension :::::::::: velocity ::::: (see :::::::::::::: assumption ::: of :::::::::: "delayed ::::::::::::::: availability" ::: of :::: the ::::: thin :::::::::: filament, ::::::::: section ::::: 2), :::::::::: together :::::: with ::::: the :::::::::: positive :::::: slope :::: of :::: the ::::::::::: eccentric ::::::::::::::::: tension-velocity ::::::::::: relation, ::::::::::::: counteracts ::::: this :::::::::::::::::::::::: velocity-dependence. :

:

  ::::::::::::::::::::::::: l opt := l thin + (l zl b )/2 :::: (= ::::::: short ::::: end :: of ::::: the ::::::::::: plateau). :

  r act ) dract dt = (Sr act (t))/k act drav dt ::::::::::::::::::::::.(2) In the first equation, r -1 v is the reverse function of r v (f a ) (tension-velocity relation), l sovl = min{l thin +l thick +l z /2-l s , l thick -l b /2} is the length of single overlap, and l dovl = max{0, l thin + l z /2l sl b /2} is the length of double overlap. In the fourth equation, which models activation as an exponential function of time, during stimulation S = 1 and k act = k on , while after stimulation S = 0 and k act = k of f . The :::: fifth ::::::::::: equation :: is ::::: the ::::::: result ::: of :: a :::::::::::: restriction :: of ::::: the :::::::: model ::: to :::::::::: simplify :::::::::::::: simulation: :::: i) ::::::::::: Eccentric ::::::::::::::: contractions ::::::: must :::::: start ::::: with ::::::::: r av ≈ 1 :::::::::::::: (sufficiently :::::: long :::::::::::: isometric :::::::::::::::::::: pre-contraction). :::: ii) :::::: The ::::::::: current :::::: lower :::::::: border ::: of ::::: the ::::::::: contact ::::::: range ::: of :::::: thin ::::: and :::::: thick :::::::::: filament ::::::: must ::::::: never ::: be :::::: larger :::::: than :::: the :::::::: initial ::::::: upper :::::::: border ::: of ::::: that :::::::: range. ::::: The ::::::::::::: compliance :::::: with :::::: these :::::::::::: restrictions ::::: has :::::: been ::::::::: checked ::: in :::: the ::::::: result :: of :::::: each ::::::::::::: simulation. ::::::: Also, :::: the ::::: fact :: is ::::: used :::::: that :::: due ::: to ::::: the :::::::::: assumed :::::::::::::: exponential :::::::::::::::::::: time-dependence, :::: the ::::::::: change ::::: over ::::: time ::: of ::::::: mean ::::::::::::: availability ::: in :: a :::::::::: constant ::::::::::::::::: length-interval ::: of :::::::::::: thin-thick :::::::::: filament :::::::: contact :::: in ::::::::::: activated ::::::::: muscle :: is ::::::::::::::: proportional :::: to ::::: the ::::::::::: difference :::: of ::::: this ::::::: mean :::::::::::: availability :::: to ::::: one. ::::::::::: ∂r av /∂t, ::::: the :::::::::::: derivative ::: of :::: r av :::::: over :::::: time ::: in ::::: the ::::::::: current :::::::::::::::: length-interval ::: of ::::::::::: thin-thick :::::::::: filament :::::::::: contact, :: is :::::::::::: calculated ::: as :: a :::::: linear :::::::::: function :: of :::::: r act , :::::::::: ranging :::::: from ::::::::::: -r av k uav ::::: for :::::::::: r act = 0 ::: to ::::::::::::::::: (r av -1) * k av ::::: for :::::::::: r act = 1. ::::::::: ∂r av /∂b l :::::: and :::::::::: ∂r av /∂b u ::::: are ::::: the ::::::::: changes :::: of :::: r av ::::: due :::: to ::::::::: changes ::: of ::::: the ::::::: lower :::: (b l ) ::::: and ::::::: upper :::::: (b u ) :::::::: border ::: of ::::: the ::::::::: contact ::::::::: interval ::::: and ::::: are ::::::: easily :::::::::::: calculable :: in ::::::::::: eccentric :::::::::::::: contractions :::::: (due ::: to ::::: the ::::::: above :::::::: model :::::::::::::: restrictions) :::::: from ::::::::: current ::: r av ::::: and :::::::::: interval :::::::: length; ::::: the ::::::::::: velocities ::::::: db l /dt ::::: and ::::::::: db u /dt :::: are :::::::::::: calculated :::::: from ::::::: dl s /dt ::::: and :::: the :::::::::: filament ::::::::: lengths. :: ::::

  made ::: to :::::::: adjust ::::::: model :::::::::::::: parameters ::: to ::::::::: achieve ::::::::: tension ::::::: traces ::::::::: exactly :::::::::: identical :: to :::::::: those ::: of ::::: the :::::::::::::::: experimental :::::: data ::::::::: (figure :::: 2). ::::::: The :::::::::: reasons ::::: are :::::: that, :::: as :::::::::: discussed ::: in : Till et al. (2008) : , ::::: the :::::::::::::: non-straight ::::: first :::::: part ::: of :::: the ::::::::::::::: experimental ::::::: traces :: is :::::::::::: influenced :::: by :::: an :::::::::::: inaccurate ::::::::::::::::: determination ::: of :::::: SEC ::::::::::::: elongation, ::::: and :: in ::::: the :::::::::: straight :::::: part ::: of ::::: the :::::::::::::::: experimental :::::::: traces :::: the ::::::::::: absolute ::::::: value ::: of ::::: the :::::::: tension :::::::::: gradient ::: is ::::::: likely ::: to :::: be :::::::::::::: additionally ::::::::::::: influenced ::: by :::::::::: fatigue, ::::::: which ::: is ::::

  seen ::: to :::::: decay :::::::::: towards :::: the :::::::: purely ::::::::::: isometric :::::::::: tension, :::: due ::: to ::: a ::::::: decay ::: in ::::::: elastic ::::::::: tension ::of ::::::: distal ::::::: titin. :::::::: Note :::::: that :::::::::::: simulation :::: of ::::::::::: "delayed :::::::::::::: availability" :::: of :::: the :::::: thin ::::::::: filament :::::: (see :::::::: section :::: 2) :::::::::::: obviously :::::: does ::::: not ::::::: much ::::::: affect ::::: the ::::: late :::::::: phase ::: of ::::::: overall :::::::::: tension :::::::: decay, ::::::: since ::::::::::: decaying :::::::: "extra ::::::::::: tension" ::: is ::::::::: roughly :::::::: equal ::: to :::: the ::::::::: plotted ::::::::::::: distal-titin :::::::::: tension. ::::::: The ::::::: decay :::: in ::::::::::::: distal-titin ::::::::: tension ::::: can :::: be ::::::::::: attributed :::: to ::: a ::::::::::: "sliding" ::: of ::::: the :::::: titin ::::::::::: junction :: j ::::::: along ::::: the :::::: thin ::::::::::: filament.:::::: Thus, :::::: long :::::::: lasting ::: or ::::::::::::::: "permanent" :::::::::: residual :::::: force ::::::::::::::: enhancement :: is ::::: not ::::::::: present. ::::: This :: is ::: in ::::::::::::: accordance :::::: with ::::::::::::::: experimental :::::::: results ::: of : Julian and Morgan (1979) :::: and : Edman et al. (1982) ::::: with :::::::: single :::::::: muscle ::::::: fibres :::: at ::::::::: lengths ::::::::: shorter :::::: than :::: the ::::::::::::: descending :::::: limb ::: of :::: the ::::::::::::::::: tension-length :::::::::: relation. ::::: To ::::::::: achieve :::::: long :::::::: lasting ::::: force :::::::::::::::: enhancement, ::: as :::::::::::: described ::: for :::::: fibre ::::::::: lengths ::: at :::: the ::::::::::::: descending ::::: limb :::::: (and ::::: even ::: at ::::::::: shorter :::::: fibre ::::::::: lengths ::: -: Peterson et al. (2004) : ), :::::::: elastic :::::::: forces ::: in :::: the :::::::::: proximal :::::: titin ::: at :::::::: longer ::::::::::::::::: half-sarcomere ::::::::: lengths ::::::: could :::::: slow :::::: down ::::: the :::::::: sliding :: of :::: the :::::: titin :::::::::: junction ::: j . ::::::::::: However, :::::::::::: additional ::::::::::::::: assumptions ::::::: would :::: be ::::::::::: necessary :: to :::::::::: explain ::::::::::::::: "permanent" ::::::: force ::::::::::::::::: enhancement. ::::::::::::: According :::: to ::::: this ::::::: view, ::::: the :::::::::::::::: half-sarcomere :::::::: length ::::::::: beyond :::::::: which :::::::::: residual :::::: force :::::::::::::::: enhancement ::: is ::::::::: present :: is :::: not ::::::::: caused ::: by ::::::::::::::::: characteristics ::: of ::::: the ::::::: active ::::::::::::::::: tension-length :::::::::: relation. :::: In :::: the :::::::: current :::::::: paper, :::::::: elastic ::::::::: tension ::: in :::: the ::::::::::: proximal :::::: titin ::: at :::::::: longer ::::::::::::::::: half-sarcomere :::::::: lengths ::::: has :::: not :::::: been ::::::::::::: attempted ::: to ::: be :::::::::: modeled :::::::::::::: realistically.

Figure 1 Figure 2 :

 12 Figure1: A: Approximate shape of the tension-length relation of a half-sarcomere. Diminution of tension at half-sarcomere lengths greater than optimal (descending limb) is thought to result from diminished overlap of the thin and thick filaments. For lengths shorter than optimal (ascending limb), two ranges with different gradients are marked 1 and 2. These and the dashed lines I and II are explained in the text. B: Scheme of the interaction of thin and thick filaments within a sarcomere. The dashed vertical line separates the two half-sarcomeres. The solid vertical lines represent the so called Z-disks at the sarcomere ends. thin: thin filaments; act : active zones of the thick filament, forming cross-bridges with the thin filament in the zone of filament contact, where active tension is generated; bare: bare zone of the thick filament, unable to form cross-bridges.

  Figure2: Tension increase with TGC length change in eccentric constant velocity contractions starting from different muscle lengths, rat M. gastrocnemius(Caput mediale). This example of experimental data is taken from . Lengths are given relative to optimum fibre length (l f 0 ) and tension is normalized to cross-sectional area. Differences of starting lengths to optimum TGC length are -0.45 l f 0 , -0.33 l f 0 , -0.19 l f 0 , -0.13 l f 0 ,-0.07 l f 0 , and 0 l f 0 (1-6). Extension velocity of whole muscle was 0.77 l f 0 /s, extension distance was 0.15 l f 0 . In their straight part, the curves are nearly parallel. Parallelism is striking for curves 1-5, while at the longest starting length (curve 6) there is a slight deviation from parallelism. :: A: :::::::: Tension :::::::: increase ::::: with ::::: TGC ::::::: length ::::::: change :: in ::::::::: eccentric :::::::: constant :::::::: velocity :::::::::::: contractions :::::::: starting :::: from ::::::::: different ::::::: muscle :::::::: lengths, ::: rat :::::::::::::::::::::::::::::::::: M. gastrocnemius (Caput mediale).

  This ::::::::: example ::: of ::::::::::::: experimental ::::: data ::: is :::::: taken ::::: from :[START_REF] Till | Characterization of isovelocity extension of activated muscle: A Hill-type model for eccentric contractions and a method for parameter determination[END_REF] : . ::::::::: Lengths :::: are :::::: given ::::::: relative ::: to ::::::::: optimum ::::: fibre ::::::: length :::::: (l f 0 ) :::: and :::::::: tension :: is ::::::::::: normalized ::: to ::::::::::::::: cross-sectional ::::: area.

:

  ::::::::Extension :::::::: velocity ::: of :::::: whole ::::::: muscle :::: was ::::: 0.77 ::::: l f 0 /s, :::::::::: extension ::::::::: distance :::: was :::: 0.15 :::: l f 0 . ::::::: Traces ::: are :::::::::: numbered ::: in ::: the :::::: order ::: of :::::::::: increasing :::::::: starting ::::::: length. ::: In ::::: their :::::::: straight :::::: part, ::: the ::::::: curves ::: are ::::::: nearly :::::::: parallel. :::::::::::: Parallelism ::: is :::::::: striking :::: for ::::::: curves :::: 1-5, :::::: while ::: at :::: the :::::::: longest :::::::: starting :::::: length ::::::: (curve :: 6) :::::: there :: is :: a :::::: slight ::::::::: deviation ::::: from :::::::::::: parallelism. ::: B: ::::::: Length ::::::: ranges ::::::::::: (horizontal ::::: bars) :: of :::: the :::::::::::: contractions :: in :: A ::::: with :::::::: respect :: to :::: the ::::::: muscles :::::::::: isometric :::::::::::::: tension-length :::::::: relation.:::::::::: Differences ::: of :::::::: starting ::::::: lengths :: to ::::::::: optimum :::::: TGC :::::: length :::: are :::::::::: -0.45 l f 0 , :::::::::: -0.33 l f 0 , :::::::::: -0.19 l f 0 , ::::::::::-0.13 l f 0 , :::::::::: -0.07 l f 0 , :::: and ::::: 0 l f 0 ::::: (1-6 ::: in :::: A).

Figure 3 :

 3 Figure3: Example model of a half-sarcomere. The dashed vertical lines are the borders of the half-sarcomere. thin: thin filament; act : active zone of the thick filament; bare: bare zone of the thick filament; pt : proximal titin; dt : distal titin; j : junction between proximal and distal titin, able to adhere to the thin filament. Please see text for explanation.

Figure 4 :Figure 5 :

 45 Figure4: Simulated contractions with constant extension velocity. Tension is given relative to maximum isometric tension. The different starting lengths, relative to optimal length l opt (defined as the border between the ascending limb and the plateau of the tension-length relation), are indicated at the traces. Each eccentric contraction starts after an isometric pre-contraction of 0.5 ::: 1.0 s (not shown) and covers a length interval of 0.1 l opt at a velocity of 1 l opt /s ::::::: (length ::::::: change ::::: and :::::::: velocity, ::::::::::::: respectively, :: of :::::: whole :::::::: muscle, ::::::::: including :::: the :::::: series :::::: elastic ::::::::::::: component). ::::::: Traces :::: are :::::::::: numbered :: in :::: the :::::: order :: of ::::::::::: increasing :::::::: starting ::::::: length. ::::: The :::::: dotted ::::: lines :::::: show :::: the ::::: parts ::: of :::: the ::::::::: isometric :::::::::::::: tension-length :::::::: relation ::::::::::::::: corresponding :: to :::: the :::::: length ::::::::: intervals ::: of :::::::::: extension; :::::::::::: simulations :: 1 :::: and :: 3 :::: are ::::::::: confined ::: to :::::: range :: 2 :::: and ::: 1 :: of :::: the ::::::::: ascending :::::: limb, :::::::::::: respectively, ::::::::::: simulation :: 2 :::::::: crosses :::: the ::::::: border :::::::: between :::::: these :::::::: ranges, :::: and ::::::::::: simulations :: 4 :::: and :: 5 ::::: cross :::: the :::::: lower :::: and :::::: upper ::::::: border ::: of :::: the ::::::: plateau :::::::: region, :::::::::::: respectively. The last parts of the ::::::::: simulated ::::::: tension : traces are apparently parallel.Parameter values are given in the text.

Acknowledgments

:::::

This :::::: work ::::: was :::::::::::: supported :::: by ::::::: grants ::::: no. ::: SI ::::: 841 :: / ::::: 2-2 ::::: and ::: SI :::: 841 ::: / :::: 2-3 ::: of :::: the :::::: DFG :::::::::::: (Deutsche :::::::::::::::::::::::::::: Forschungsgemeinschaft, ::::::::::::: Germany). :

length where this filament is compressed. An example model is implemented and checked with simulations.

Model ::::::::::: parameters ::::: used :: in :::::::::::: simulation.

filament related lengths (see figure 3 l a : ::::::::: defined ::: by :::: l act ::: in ::::: the :::::::::::::::: uncompressed ::::::: state :: of :::::: act. :