Why exercise builds muscles: Titin mechanosensing controls skeletal muscle growth under load
WWhy exercise builds muscles: Titin mechanosensing controls skeletal muscle growth under load
Neil Ibata and Eugene M. Terentjev ∗ (Dated: February 2, 2021)Muscles sense internally generated and externally applied forces, responding to these in a coordinated hier-archical manner at different time scales. The center of the basic unit of the muscle, the sarcomeric M-band, isperfectly placed to sense the different types of load to which the muscle is subjected. In particular, the kinasedomain (TK) of titin located at the M-band is a known candidate for mechanical signaling. Here, we developthe quantitative mathematical model that describes the kinetics of TK-based mechanosensitive signaling, andpredicts trophic changes in response to exercise and rehabilitation regimes. First, we build the kinetic modelfor TK conformational changes under force: opening, phosphorylation, signaling and autoinhibition. We findthat TK opens as a metastable mechanosensitive switch, which naturally produces a much greater signal afterhigh-load resistance exercise than an equally energetically costly endurance effort. Next, in order for the modelto be stable, give coherent predictions, in particular the lag following the onset of an exercise regime, we haveto account for the associated kinetics of phosphate (carried by ATP), and for the non-linear dependence of pro-tein synthesis rates on muscle fibre size. We suggest that the latter effect may occur via the steric inhibitionof ribosome diffusion through the sieve-like myofilament lattice. The full model yields a steady-state solution(homeostasis) for muscle cross-sectional area and tension, and a quantitatively plausible hypertrophic responseto training as well as atrophy following an extended reduction in tension. I. INTRODUCTION
Why does exercise build skeletal muscles, whereas long pe-riods of immobility lead to muscle atrophy? The anecdotalevidence is clear, and the sports and rehabilitation medicinecommunity has amassed a large amount of empirical knowl-edge on this topic. But the community has not as yet ad-dressed and understood two key phenomena which underlyhypertrophy and atrophy: how does the muscle ‘know’ that itis being exercised (when it is certainly not the tactile sense,processed via the nervous system, that is at play in this), andhow does it signal to provoke a morphological response to anincrease or a lack of applied load? Here we develop a quanti-tative theoretical model which seeks to explain both of theseprocesses. In order to be useful, the model must build on therelevant knowledge accumulated from studies of the anatomyand physiology of muscles, as well as the biological physicsof molecular interactions and forces.Muscles, their constituent cells, and the structure of theirmolecular filament mesh must respond mechanosensitively – i.e. in a manner which depends on the changes in the magni-tude of the forces and stresses that arise during the contractionand extension of the muscle – at many different timescales.At the fastest time scales (tens or hundreds of milliseconds),skeletal muscles can produce near maximal force for jump-ing or for the fight-or-flight response. Most muscles also gothrough cycles of shortening and lengthening with a periodof the order of a second in the vast majority of sprint or en-durance exercise (running, climbing, etc.) At a much longertimescale of many days, a muscle must also be able to mea-sure changes in its overall use in order to effect adaptive mus-cle hypertrophy/atrophy – ultimately helping to prevent injuryon the scale of months and years. ∗ Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue,Cambridge, CB3 0HE, U.K.; [email protected]
How the muscle cell keeps track of the history of its loadand stress inputs within a number of intracellular output sig-nals (which then go on to stimulate or inhibit muscle proteinsynthesis), is inherently an incredibly complex biochemicalquestion. With the help of recent theoretical insights into thefolding and unfolding rates of mechanosensor proteins un-der force, we hope to gain insights into the first part of thispuzzle for the specific case of muscle hypertrophy. To makeprogress, we use a simple model for force-induced transitionsbetween the different conformations of the titin kinase (TK)mechanosensor. If the conformational change helps createan intracellular signal, we can model the signal’s strength interms of the duration and intensity of the mechanical inputs(external force on the TK domain in our case).
Force chain
The individual sub-cellular, cellular and super-cellularcomponents of a muscle act in concert to scale up a vast num-ber of molecular force-generating events into a macroscopicforce. The hierarchical structure of the muscle (see Fig. 1)allows the macroscopic and microscopic responses to mirroreach other [9].The sarcomere is the elementary unit of the muscle celland the basic building block of the sliding filament hypoth-esis [10, 11]. Its regular and conserved structure, sketched inFig. 2 for the vertebrate striated muscle, allows for a seriestransmission of tension over the whole length of the muscle.In vertebrates, six titin molecules are wrapped around eachthick filament [12, 13] on either side of the midpoint of thesarcomere: the M-line.During active muscle contraction, myosin heads (motors)bind to actin and ‘walk’ in an ATP-controlled sequence ofsteps [14] along the thin filaments. When a resistance is ap-plied, the myosin motor exerts a force against it. During slowresistance training in both concentric and eccentric motions,tension is passed along the sarcomere primarily through the a r X i v : . [ q - b i o . T O ] F e b N m F f thick filament (myosin bundle) sarcomere .. . ... . . . . . .. . . myofibrilmuscle fibre(muscle cell) N c FIG. 1. The ‘textbook’ hierarchy in the anatomy of skeletal muscle.The overall muscle is characterised by its cross-section area (CSA),which contains a certain number ( N c ) of mucle fibers (the musclecells with multiple nuclei, or multinucleate myocytes). A given mus-cle has a nearly fixed number of myocytes: between N c ≈ N c > N m ) of parallel myofibrils (organelles), each of which can bedivided into repeated mechanical elements called sarcomeres. Thetypical length of a sarcomere is ca. 2 µ m, so there are ca. 10 ofthese elements in series along a fiber in a typical large muscle [2].Each sarcomere contains a number of parallel thick filaments (heli-cal bundles of myosin, red) whose constituent myosins pull on theactin polymers in the thin filaments (F-actin, blue) to generate force.Within the myofibrils, the spacing between neighbouring myosin fil-aments is ca. 0.046 µ m at rest [3, 4]. The typical cross-sectionalarea of a single muscle fibre substantially varies between individu-als and muscle types, but is of the order of 4000 µ m [5]. Together,this means that a typical muscle fibre has some ca. 2,000,000 par-allel filaments across, between which the macroscopic force F mustbe divided. Rather than using this awkward number, we will expressour results in terms of the single muscle fibre CSA. A chemicallyactivated muscle fibre with a CSA of 4000 µ m shows a force inthe vicinity of 300-1000 µ N for untrained individuals (with a verylarge individual variation) [6], which translates to an average fila-ment force of 150-500pN (see Supplementary A.6). Training canincrease the neural activation level [7] as well as the number of ac-tive myosin heads and the maximum voluntary contraction force perfilament ( e.g. by stretch activation [8]). Because of this, we wouldexpect resistance training to lead the filament forces to tend towardsthe upper end of the range ( ≈ thin filament, myosin heads [15, 16], the thick filament, andinto the cross-bridge region of the sarcomere where thick fila-ments are crosslinked with their associated M-band proteins.The load in each of the sarcomere components ultimatelydepends on the relative compliance of elements. The relativeload on the thick filament and the M-band segments of titinwhen the filament is either under internal (contracting) or ex-ternal (extending) load is discussed in Supplementary A.4. Itis well-known that titin is under load when the sarcomere isextended [17, 18]. Recent X-ray diffraction experiments [19]suggest that that the thick filament may be more compliantthan originally thought; if so, M-line titin is likely substan-tially extended and loaded titin when the muscle actively gen-erates force. Others disagree [20] and attribute the change inline spacing in diffusion experiments to a mechanosensitiveactivation of the entire thick filament at low forces. Eitherway, M-band titin is under some tension during active musclecontraction. This situation is sketched in Fig. 2.We estimate the force in each filament both macroscopi-cally and microscopically (see the full discussion in Supple- Open TK can be phosphorylated
Z-disk M-band A-band Z-disk
Titin PEVK ff FIG. 2. A sketch of the mechanically active elements of sarcomere.The thick filaments are crosslinked across the M-line, with six titinmolecules bonded to these filaments, on each side of the M-line. Thefull filament is under the measurable microscopic force identified inFig. 1, shown by the blue arrow in the middle filament. At the molec-ular level, the force is borne by the individual titin and myosin fila-ments. If we assume that the thick filament and titin extend by thesame amount during muscle contraction, then the graphical relation-ship between titin force and force in the thick filament is illustratedin Figure 3 in the Supplementary Material. This figure illustrates anadditional possibility: if titin wraps around the thick filament (top),then TK can lengthen substantially more than we consider in thiswork, for titin extending with the thick filament (bottom). The forcein TK would be much higher, making TK bear more load and createa greater mechanosensitive signal. mentary Part A.2 and A.3). We divide the force in the entiremuscle by the number of active myofilaments (see Fig. 1) tofind a large variation in force per filament in untrained individ-uals (150 − ≈ TK is a mechanosensor of the ‘second kind’
Cells sense and respond to the mechanical properties oftheir environment using two main classes of force receptors.The first type of mechanosensor responds immediately underforce [22, 23]. Mechanosensitive ion channels are the archety-pal example of such a sensor and have been proposed to playa role in tactile signaling (transforming a mechanical signalinto chemical) [22, 24]. However, the ions which they usein signaling are rapidly depleted, making it difficult for thesesensors to signal in response to a sustained force.The other type of mechanosensor, dubbed of the ‘2nd kind’by Cockerill et al. [25], can either indirectly ‘measure’the response coefficients, or time-integrate an external forceacting on the molecule. The focal adhesion kinase (FAK)mechanosensor [26, 27] is a good example: it can sense sub-strate stiffness by measuring the tension in the integrin-talin-actin force chain, which binds a cell to its extracellular ma-trix (ECM). FAK and the TK domain both unfold under force,can be phosphorylated, and appear pivotal to mechanosensi-tive signalling, they also has many structural similarities. TKhas already been suggested to act as mechanosensor [28–30],and although recent experimental work has focussed mainlyon other regions of the titin molecule, we believe that it isworth returning to the TK domain to examine it as a time-integrating mechanosensor. In the Results section below, wesee that the metastability of the TK open state, when the mus-cle is under steady-state passive tension, can indeed allow forthe TK domain to help produce increased signal levels longafter the end of an exercise session.
TK domain opens under force
Many signaling pathways use a molecular switch to ini-tiate a signaling cascade. One of the most common post-translational modifications of proteins involves the reversibleaddition of a phosphate group to some amino acids (mainlytyrosine); this addition alters the local polarity of the targetprotein, allowing it to change its shape and to bind a new sub-strate [31]. Phosphorylation can form the basis for signalingif an input changes the protein’s conformation, from a nativefolded conformation which cannot bind to a phosphate group(often called ‘autoinhibited’), to an ‘open’ conformation inwhich the geometry of the molecule allows phosphate groupsto be donated to the phosphorylation site [26]. The phospho-rylated protein can then bind to a third substrate molecule, andcan either directly catalytically affect or indirectly activate asignaling pathway.Protein unfolding under force has been analysed exten-sively, beginning with studies of in the titin Ig domain [32,33]. These experiments show characteristic force-extensioncurves, which can help deduce the transition energies betweenconformations for the molecules in question. We note thatthe Ig domains unfold under quite a high force [32, 34, 35]and could initially appear to be candidates for mechanosen-sors. However, very few phosphorylation sites have beenfound on the Ig domains, compared with the remainder of themolecule [36], suggesting that they do not contribute to force-induced signaling, but rather help control the length of the titinmolecule and avoid immediate sarcomere damage under highload.Titin kinase was initially thought to be the only catalyticdomain on titin [37]. Bogomolovas et al. [38] suggest that TKacts as a pseudokinase, simply scaffolding the aggregation ofa protein complex when it is phosphorylated, and allowingfor another protein to be allosterically phosphorylated. Com-putational and experimental studies of TK have shown thatits force-length response also follows a characteristic step-wise unfolding pattern, but with much smaller steps than those observed for the Ig domains. In particular, AFM experi-ments [28] show that the presence of ATP (an energy supply)changes the conformational energy landscape of the moleculeas it is stretched. This shows that the molecule possesses along-lasting denatured conformation, in which it can accom-modate the recruitment of signaling molecules upstream of amechanosensitive signaling pathway. Being the largest knownmolecule in vertebrates, titin interacts with an unsurprisinglylarge number of molecules [39]; Linke et al. [40] summa-rized this knowledge in a protein-protein interaction network(PPIN), shown in their Figure 2, where in particular the nbr1and MuRF pathway (localized in M-band) is shuttled into thenucleus, leading to SRF and transcription of new actin.
Methods used in modelling
We model a resistance training repetition as a piecewisefunction for force. During the loading phase (start at t = f ( t = ) and asymp-totically approaches the maximal force per filament f max dur-ing the repetition, with a rate k f ≈ s − . The full-musclerate of force development is substantially lower, at ca. 5 s − [84], but we assume that there is a lag due to the macroscopicmuscle providing some slack before macroscopic force de-velopment. It therefore seems likely that the molecular rateof sarcomere force development (which impacts the rate oftitin being placed under force) is closer to the much faster rateof force increase during muscle tetani. During the unloadingphase, the muscle force decreases with a fast rate (same rateas force development for tetani, a bit slower for twitches, butultimately insignificant relative to the timescales of a musclerepetition). The force per titin as well as the muscle openingand closing rates k − and k + are calculated at every time step.Because the TK conformations quickly change during exer-cise, the next time step of the numerical integration is adap-tively calculated at each time step as a fraction of the greatestfractional change in all of the molecular species in the model.Several repetitions make up a set, and several sets make up anexercise session. The exercise regime is assumed to be adap-tive, such that the repetition force on TK remains constant asthe muscle CSA increases. II. THE MODEL
Here we explain why we believe that the kinetic processesschematically shown in Fig. 3 are the necessary elements forany TK-based treatment of mechanosensing of the secondkind and of subsequent mechanosensitive intracellular signal-ing. Our model can be divided into three parts:• The opening and phosphorylation of the TK domain.This stage is highly non-linear because TK opens as amechanosensitive switch and because the mechanosen-sitive complex binds allosterically. The open state ismetastable if the muscle is under a steady-state load.• The creation and degradation of signaling molecules, ofnew ribosomes, and of structural proteins. All of theserates can be approximated as linear, apart from a sizefeedback term, which arises because ribosomal diffu-sion is sterically hindered in large cells (see discussionbelow).• Exercise can only be so hard before the muscle depletesits short-term energy supplies. The balance between en-ergy generation from oxidative phosphorylation and thedepletion of short-term energy stores has to be consid-ered to correctly model the dynamic response.
A. Opening and phosphorylation of TK domain
The energy barrier for the transition between the ‘closed’native domain conformation, and the ‘open’ conformationwhich supports ATP-binding and phosphorylation is the keydeterminant of the kinetic transition rates between the two TKstates. AFM data collected by Puchner et al. [28] is essentialhere; we match the relevant TK conformations to their dataand explain how to extract several important model parame-ters in Parts A.4 and A.5 of the Supplementary Information.In the absence of any signaling, the concentration of total(free+bound) ATP is constant, and the transitions from closedto open to phosphorylated TK domain conformations are sim-ple and reversible:• Closed ↔ Open: TK can open under force with a force-dependent rate constant k + ( f ) and likewise close witha force-dependent rate constant k − ( f ) . Here we use theframework of [27] to derive these two rate constants.The concentrations of the closed and open conforma-tions are n c and n o , respectively, cf. Fig. 3.• Open ↔ Phosphorylated: the open state of TK can bephosphorylated with a rate constant k p , the total rate ofthis process depends on both the concentration of ATPand of the open TK: [ATP] and n o . The phosphory-lated state with the concentration n p can also sponta-neously de-phosphorylate with a rate constant k r , butcannot spontaneoulsy close until then.This cyclic reaction, illustrated in the TK section of Fig. 3 isdescribed by the kinetic equations for the evolution of n c , n o ,and n p : dn c dt = − k + n c + k − n o (1) dn o dt = k + n c − k − n o − k p n o [ATP] + k r n p (2) dn p dt = k p n o [ATP] − k r n p (3) n c + n o + n p = n titin (constraint) , (4)where the last condition encodes the total concentration of TKunits; this is equal to the concentration of titin and remainsconstant on the time-scale of signaling. These equations are GTPGDP p + Glucose (slow) k O k A T P → C P k C P → A T P Creatine phosphate (fast)
Titin Kinase
ATP CP p + Phosphate TransferNucleus
Closed OpenPhosphorylated k − ( f )k + ( f ) k p n p = ( n o ·p + ) n o k r k s n c nbr1 n s = ( n p · nbr1 α ) k dn s Open n o k − ( f ) k + ( f ) Signal
Ribosome biogenesisk sr a Actin k a ∅ k ds ∅ k dr ∅ k da SRF
FIG. 3. Sketch of the kinetic processes which link titin kinase open-ing and phosphorylation, mechanosensing complex formation, signalactivation, ribosome biogenesis and the increased synthesis of struc-tural proteins (of these, only actin is listed for simplicity). examined in Supplementary Information, Part A.6 where theyare shown to adequately reproduce the phosphorylation kinet-ics of TK measured by Puchner et al. [28], providing an aposteriori justification for their use.
B. Signal generation from phosphorylated TK
The phosphorylated TK domain can bind the zinc fin-ger domain protein nbr1 [41], and begin to form an aggre-gate; the concentration of the signaling complexes n s mustbe introduced with a new separate kinetic equation. Themechanosensing complex identified in the most general for-mulation by Lange [41] is a multispecies aggregate, which weconsider in more detail in Supplementary Part A.7.SRF, the mechanosensitive signaling molecule in the Langemodel, is known to undergo activation by phosphorylation[42, 43]. There are many phosphorylation sites on nbr1, p62and some on MuRF [44], which suggests that SRF could beactivated by phosphate transfer originating from TK. An acti-vation would most likely irreversibly alter the conformationof the signaling complex, and result in the disassembly ofthe complex every time a new signaling molecule was acti-vated. Assuming that the complete mechanosensing complexhas a time-independent probability to disassemble, with a rate k dn s n s , we estimate the corresponding rate constant k dn s fromexperiments [45] that show the increase in phospho-SRF (ac-tivated signal) after exercise. They find that the level of acti-vated SRF binding to DNA increases by a factor of 2 an hourafter skeletal muscle cell contraction, and reaches half of itsmaximum increase after 10 minutes of exercise. This meansthat the degradation rate of the mechanosensing complex oc-curs with a half-life of ca. 10 minutes ( k dn s ≈ /
600 s − ).We can now rewrite our kinetic equations to add the forma-tion and degradation rates of the signaling complex as well asthe activation of the SRF signal: dn c dt = − k + n c + k − n o (1) dn o dt = k + n c − k − n o − k p n o [ATP] + k r n p + k dn s n s (2b) dn p dt = k p n o [ATP] − k r n p − k s n p (3b) dn s dt = k s n p − k dn s n s (5) n c + n o + ( n p + n s ) = n titin ( constraint ) (4b) dn SRF dt = k dn s n s − k ds n SRF . (6)The concentration of ATP is expressed in number per titin: to-tal phosphate is assumed to scale proportionately to the sizeof the myofibril and the number of titin molecules. In Supple-mentary Part C, we also track the kinetics of ATP depletionduring intense exercise. The additional equations are mathe-matically more complicated, and do not help understand thefull model, but are included in the numerical simulations inthe Results section.These are the core equations which describe the relativelyfast activation of a signaling molecule during muscle load-ing. We show in the Results section below that they displaya very pronounced switching behavior: in other words, smallchanges in tension result in large changes to the signal concen-tration. We also find that these equations support an increasein the concentration of signal (possibly SRF) for a substantialtime of the order of a couple of days, which could help accountfor the immediate increase in protein synthesis post-exercise.But we shall see in the next section that a simple one-step sig-nal cannot by itself account for the observed time-dependenceof hypertrophy. C. Muscle protein synthesis after mechanosensor signaling
The constituent molecules of most signaling pathways havea short lifetime relative to that of the structural proteins. It isalso well documented that a few bouts of exercise do not havea tangible effect on muscle volume, and that muscle takes atleast a few of weeks to begin to show visible hypertrophicadaptations. The debate on whether true hypertrophy is soondetected, or whether initial post-exercise changes in muscleCSA are the signs of muscle micro-damage, is a rather fraughtone [46–49]. Three weeks of resistance training appears tobe a consensus time, after which true hypertrophy is actu-ally detected. This means that there has to be a way of ‘in-tegrating’ the signal over such a long period of time – be-yond the scope of the simple force-integration supported bya metastable open state of TK. Here we combine the abovemodel of mechanosensitive signaling with a simple model ofprotein synthesis from a signaling molecule, and propose amechanism by which this integration may occur.Based on a review and discussion of the current literaturein the Supplementary Information, Part B.2, we conclude thatit is likely an increase in ribosome biogenesis (rather than the temporary increase in mRNA transcript number) which allowsfor this ‘time integration’ of the signal. Its effect would be tosuppress fluctuations in the concentration of TK conforma-tions or signaling molecules, smoothly increasing the concen-tration of the structural muscle proteins over the time similarto the half-life of ribosomes. We suggest that this effect couldhelp explain the delay of a few weeks between starting re-sistance exercise and the first detection of measurable musclegrowth, as noted by trainers and rehabilitation specialists.New experiments show that sarcomeric proteins are synthe-sised in situ at the sarcomeric Z-line and M-band [50]. As faras we are aware, ribosomal subunits can only move by diffu-sion, whereas mRNA can be actively transported to the syn-thesis site. The inhibition of the diffusion of ribosomal sub-units by the myofilament lattice [51] could reduce the synthe-sis of new sarcomeric proteins by a sizeable amount (5 − − α n titin , where the coefficient α depends on the ribosome diffusion constant, the lattice spac-ing and the rate of lysosomal degradation. This term has sev-eral important consequences: it provides a bound on musclegrowth or shrinkage, and it affects the speed of muscle sizeadaptations. We examine this point in more detail in Supple-mentary part B.4.We use the number of titin molecules n titin in the muscle fi-bre cross-section as a proxy for the muscle fibre CSA, becausethe hierarchical sarcomere structure is well-conserved in mostmuscles at rest. When necessary, one can convert from one tothe other as in Fig. 1. The above equations are combined asfollows (more details in Supplementary Part B): dn c dt = − k + ( f ) n c + k − n o (1) dn o dt = k + n c − k − n o − k p n o [ATP] + k r n p + k dn s n s (2b) dn p dt = k p n o [ATP] − k r n p − k s n p (3b) dn s dt = k s n p − k dn s n s (5) n c + n o + ( n p + n s ) = n titin (4b) dn SRF dt = k dn s n s − k ds n SRF (6) dn rRNA dt = k sr n SRF − k dr n rRNA (7) dn titin dt = k st n rRNA ( − α n titin ) − k dt n titin (8)In the Supplementary Part E, we consider the possibility thatthe force produced by the muscle does not scale linearly withmuscle size. It is unclear exactly how much active muscleforce scales with muscle size. Krivickas et al. [6] find thatforce increases slower at larger muscle CSA, whereas Akagiet al. [52] do not see a substantial non-linearity between forceand myofibre volume. So in the main body of this paper weproceed with the simplest assumption of the linear scaling. TABLE I. Values of rate constants, directly obtained in experimentsor simulations or extrapolated from the data presented.Constant Value (s − ) Source k p − [28] k r k s − − − [53] k dn s k ds − [54] k st − [57–59] k dt · − [60] k sr k dr ca. 9 · − [62, 63] III. RESULTS
The steady-state load required for the muscle to maintainhomeostasis can be obtained analytically. Once we have ‘ze-roed’ our problem by checking that this value makes sense interms of steady-state tension (muscle tone), in sections B andC, we consider the dynamics of equations (1-5) for TK only,to show that it does indeed open as a metastable mechanosen-sitive switch. Following that, we will proceed to study whateffects different types of resistance exercise have on musclefibre CSA, and compare them with reports from the literature.
A. Steady State
The steady-state solution to equations (1-8) is obtained inSupplementary Part D. We find the following tension per in-dividual TK domain: f = ∆ G u max + k B Tu max ln (cid:16) ( k r + k s ) k p [ ATP ] (cid:0) ζ − − k s k dns − k r + k s k p [ ATP ] (cid:1) (cid:17) (9) where the shorthand ζ is the ratio of synthesis to degradationcoefficients: ζ = k st ( − α n titin ) k s k sr k dt k dr k ds . (10)The first key result here is that the force on the TK domain,which maintains a steady state muscle fibre CSA, is deter-mined almost exclusively by two parameters: the energy bar-rier ∆ G between the closed and open conformations of theTK domain, and the unfolding distance u max . It is clear thatchanging any of the coefficients in the logarithm in (9) wouldonly have a minor effect on the steady-state force. The typicalresting muscle forces are plotted in Fig. 4 as a function of ∆ G and u max (illustrated in Fig. 5 of the Supplementary Material).The typical homeostatic force experienced by a TK domain isof the order of 2-10pN.The other key point is that a small change in the musclesteady-state force (perhaps supported by an increase in ten-don tension, which lengthens the sarcomeres) can maintaina large change in muscle size. The fractional change in thesteady state muscle tone as a function of the fractional changein muscle size is plotted in Fig. 5. G u m a x ( n m ) FIG. 4. Steady-state force (expressed in pN, labelled in contour lines)from (9) as a function of the TK activation energy ∆ G and the open-ing distance of the mechanosensor u max . ∆ G is expressed in di-mensionless units scaled by the thermal energy β = / k B T , with T = K . The values of rate constants are given in Table I, andthe following typical concentrations were used: p + = st = . σ = .
5, these are from the appendix and willconfuse the reader. The circle marks the ‘sweet spot’ where the likelyvalues of u max and ∆ G should be. Combining supplementary part A.2, A.3 (maximum thickfilament force) and A.4 (titin force in terms of thick filamentforce), we estimate the force per titin during a contraction atthe maximum voluntary contraction (MVC) to be ≈ B. Titin kinase as a metastable mechanosensitive switch
In Fig. 6, we see that TK obeys switching kinetics: abovea critical load, its closed conformation is no longer favoured. Δ f st / f st -0.0050.0050.010 Δ n titin / n titin G = FIG. 5. Fractional change in steady state muscle force (vertical axis)vs fractional change in muscle size (horizontal axis), from (9). Notethat ∆ n titin = − n titin (the left limit of the axis) represents a completedegradation of the muscle. The values of opening energy ∆ G arelabelled on the plot. The values of rate constants are given in Table I,and the following typical concentrations were used: p + = st = .
002 per titin, σ = .
5. The maximum opening dis-tance of TK was taken as u max = nm . The values for ∆ G and u max were estimated from AFM data and molecular dynamics simulationsconducted by Puchner et al. [28] in Supplementary Part A.5. However, the low TK opening and closing rates k + and k − plotted in Fig. 7 do not allow TK to quickly change betweenits conformations at physiological loads. If resistance exer-cise increases the number of open TKs, their number willremain elevated up to days after exercise; in other words,the TK open/phosphorylated/signalling complex-bound stateis metastable. We use numerical simulations to explore thispoint further in the next section.TK signaling increases linearly with exercise duration (bar-ring the effects of fatigue), whereas opening rates (and sig-nalling) increase exponentially with force in TK. While TKforce scales roughly linearly with myosin force (see Fig. 3in the Supplementary Material), this allows mechanosensitivesignaling to increase much faster than the corresponding en-ergetic cost at high exercise force. At very high forces, how-ever, it appears that TK force increases much more slowlythan myosin force, leading to a plateau in the efficiency ofmechanosensitive signalling (Fig. 4 in the Supplementary Ma-terial). Excluding mechanosensitive signalling at the steady-state force (which is efficient because thick filament force islow, but does not do much to change muscle CSA), signal-ing in response to resistance training is always more effec-tive as the load increases until at least about ≈
70% of theMVC force. Our current model does not extend to how mus-cle fatigue induces changes in muscle stiffness [68, 69], whichcould alter TK signaling kinetics at high forces as well.
C. Long-term mechanosensitive signaling and response
In order to compare with experimental data in the literature,we consider a ‘typical’ resistance exercise session consistingof 3 sets of 10 repetitions (more details in the Methods sectionbelow). This mimics a common resistance training program(see e.g. DeFreitas et al. [46], who set up resistance training f (pN) titin G higher G lower n /n titini } n c n s n p n o FIG. 6. Log-plot of the steady-state concentration of the TK confor-mations (blue - closed, red - open, orange - phosphorylated, green- bound to the mechanosensor complex) as a function of the steadystate force per titin, from (9). As the steady-state force increases,the preferred conformation of TK switches from closed to a fixedratio of open, phosphorylated and signaling complex-bound. Thisplot is for ∆ G = k B T . Note that the molecule switches frombeing preferably closed to preferably open/phosphorylated/signalingslightly above the steady-state force of a few pN. But even though thesteady-state conformation may be favoured at forces even slightlyabove the resting muscle tension, titin takes a long time to openenough to actually signal in large numbers, because the opening rate k + is much less than 1 s − at low- and medium forces (see Fig. 7 foran illustration of this behaviour).
20 40 60 80 100 f titin ( p N ) k ± G = ( s ) -1 FIG. 7. Log-plot of the closing rate k − (brown), and opening rates k + for different values of activation barrier ∆ G , as labelled on the plot.Even when TK opening is favoured, at k + > k − , the opening rates aremuch less than 1 s − , meaning that TK opens linearly with increasingtime under load, and exponentially with increasing force. We suggestthat this behaviour is the basis for high intensity resistance training:doubling the force increases mechanosensitive signaling by severalorders of magnitude. sessions with 8-12 repetitions to failure over 3 sets). Choos-ing a specific value of repetition force is not straightforwardbecause while most force studies consider MVC force, mosthypertrophy programs compare the training load to the single-repetition maximum load for a given exercise. The muscleforce during one full repetition is necessarily smaller than theinstantaneous force. Determining the corresponding force perTK might be further complicated because titin is under moreload when the muscle is stretched (passive force) than when itis actively contracting. Nevertheless, our choice of 20 pN pertitin seems to be supported by several factors discussed here -2 0 2 4 6 8 10 12 14Time (min) F o r c e ( p N ) o n F r a c t i on o f open , pho s pho r y l a t ed T K p n p + n o n (a)(b) + n s FIG. 8. Simulation of an exercise session involving three sets of 10ten-second repetitions. (a) All repetitions are performed at the sameforce per titin, but their duration is cut short upon reaching exhaus-tion. As the number of titins increases, we assume that the trainingregime adapts by proportionately increasing the repetition force. (b)The depletion of ATP leads to a temporary drop in phosphorylatedTK during exercise. However, the sum of open, phosphorylated andsignaling complex-bound TK steadily increases during the exercise.Since the closing rate of TK is quite low (of the order of 10 − s − , de-pending on the number of attempts at crossing the energy barrier andthe barrier height ∆ G ), the baseline concentrations of phosphory-lated and signaling TK conformations remain elevated after exercise. and in the Supplementary Information.We simulate a typical exercise session as a fixed numberof repetitions at a given force, grouped into a fixed numberof sets, as shown in Fig. 8(a) (more details in the Methodssection). During each repetition, the opening rate k + of TKbecomes much greater than its closing rate, which decreasesthe proportion of closed TK and increases its propensity tosignal. Because the muscle is under a combination of passiveand active tension at rest, the closing rate of titin is small afterexercise, even though it is greater than the opening rate (seeFig. 7). This allows TK to revert to its steady-state confor-mation after a time of the order of hours to days, in a mannerwhich depends on the number of attempts at crossing the en-ergy barrier between the closed and open conformations (seeSupplementary Part A.5), as well as the height of the activa-tion barrier ∆ G . The metastability of the open state at steady-state tension would then naturally allow the muscle to producea mechanosensitive signal long after the end of exercise. Thismight account for the increase in myofibrillar protein synthe- Time (min) F o r c e ( p N ) F r a c t i on o f open T K , n o FIG. 9. The first set of 10 ten-second repetitions from Fig. 8. Notethat the repetitions become shorter as ATP runs out during the periodof high load: as the ATP level falls below a critical value (whichwe set to a half of the homeostatic level), the muscle can no longersustain the load and the only possibility is to drop the weight andreturn to the steady-state force. So the period of loading becomesshorter than the prescribed period, shon in dashed line in the plot andarrows marking the prescribed period. sis in the two days following exercise, specifically resistancetraining [70, 71].The important aspect of exercise, naturally reflected in ourmodel, is the effect of fatigue. To make it more clear, weplot the same data as in Fig. 8, zooming in to just one (thefirst) set of repetitions in Fig. 9. Both myosin motors increasetheir ATP consumption under the high load, and the freshlyopen TK domains require ATP for phosphorylation. Duringthe high-intensity loading, the level of ATP could drop be-low a critical value, after which the muscle would no longerbe able to maintain the force: the only option is to drop theweight and return to the steady-state force recovery stage. Wesee that this effect of fatigue occurs after a few repetions inFig. 9. We also find, in this simulation of model exercise, thatsubsequent sets of repetitions have this fatigue-driven cutoffof the later loading periods becoming less pronounced, be-cause the overall level of ATP marginally increases during thesession.In Figs. 10 and 11, and afterwards, we return to measuringthe muscle ‘size’ directly by the CSA of a fibre (by convertingto that from the measure of titin molecules, which is equiva-lent but carries less intuitive appeal). Since the volume of amyonuclear domain is close to 16000 µ m , and remains con-served in a developed adult muscle [72], and the density oftitins is also an approximate constant (ca. 3000 per µ m ), seeFig. 1 – or in an alternative equivalent estimate: the density oftitins across the unit area of CSA (ca. 6000 per µ m ) – allowsquantitative measure of CSA as our output.Also note, that since in this test we are applying a constantforce per titin, and the CSA increases with time, this meansthat the actual exercise load to the whole muscle must be in-creasing proportionally (in our current simplified model therelation between CSA and n titin is linear) to achieve the opti-mal growth.In Fig. 10 we test the long-term consequences of a regu- Time (days)500 1000 1500 2000 2500 3000 35000 M u sc l e fi b r e C SA ( m ) G = Time (days)200 400 600 800 10000 G = M u sc l e fi b r e C SA ( m ) (a)(b) FIG. 10. Time course of muscle growth in response to a regular re-sistance training program (exercise of Fig. 8, every 3 days). (a) Thetotal muscle load F is kept constant, so the force per titin f effec-tively diminishes as the CSA increases. (b) The force per titin f ismaintained constant (20 pN, as discussed before), which effectivelyimplies that the total muscle load F increases in proportion with CSA(vertical axis). Several curves for different values of the energy bar-rier ∆ G are labelled on the plot. As might be expected, muscleCSA changes are faster and greater in magnitude if the energy bar-rier ∆ G is smaller ( i.e. TK opens faster during exercise, and signalsto a greater extent). We overlay the predictions of our model withmeasurements of fractional changes in muscle CSA over an 8-weekperiod, measured by De Freitas et al. [46] (red crosses, same valuesin both prlots). An initial force per titin of 20 pN matches well withreal data, showing a ca. 1% growth per week. lar resistance training program (the standard model exerciseas in Fig. 8(a) repeated every 3 days). Several curves are pre-sented, showing the final homeostatic saturation level, and thetime to reach it, dependent on the key model parameter: theenergy barrier ∆ G for TK opening. The earlier discussionbased on the data obtained by Puchner et al. [28], and thestructural analogy between TK and FAK [26, 73], suggest that ∆ G could be around 30 k B T (or ca. 75 kJ/mol).The comparison between plots (a) and (b) in Fig. 10 is im-portant. As our model relies on the value of force per titin f , the total load on the muscle is distributed across filamentsin parallel across CSA. So if one maintains the same exerciseload, the effective force per titin diminishes in proportion tothe growing CSA, the result of which is shown in plot (a). Incontrast, one might modify the exercise by increasing the to-tal load in proportion with CSA – plot (b) shows the resultof such an adaptive regime. In the non-adaptive case, the fi-nal saturation is reached in about a year and the total CSA in- Time (days)500 1000 15000 M u sc l e fi b r e C SA ( m ) Time (days)5 10 15 20 25 30 350 M u sc l e fi b r e C SA ( m ) t = ex t = ex t = ex t = ex t =
12 days ex t = ex t = ex t = ex t = ex t =
12 days ex (b)(a) FIG. 11. Time course of muscle response for different exercise fre-quencies. Here we take β ∆ G =
35 (see Fig. 10), and a representativevalue of ribosome diffusion inhibition α n titin = . ≈
1% CSA changes per weekin response to high intensity resistance exercise [74]. This simula-tion shows a similar rate of CSA change, which means that a TKmaximum force of ≈
20 pN during high-intensity resistance exercisecould produce an adequate signal for muscle hypertrophy to occur.Because of the switch-like nature of TK, it is unlikely that this max-imum load on TK be too different from 20 pN. This force value isconsistent with a picture where the myosins bear most of the loadduring active muscle contraction and titin acts as a parallel stretchsensor. crease is about 30% (assuming ∆ G = k B T ). In the adaptiveexercise, the final saturation is reached much slower, but thetotal CSA increase is about 88%: almost doubles the musclevolume in about 2 years time. It is reassuring that the experi-mental measurement of De Freitas et al. [46] of CSA growthover a period of 8 weeks, in a similar exercise regime, quanti-tatively agrees with our prediction of ca. 1% CSA increase aweek in the initial period.The regularity of the exercise has a strong effect: the long-term magnitude of hypertrophy predicted by the model is af-fected by what happens on the daily basis. Fig. 11(a) com-pares the long-term results when the interval between themodel exercise ∆ t ex varies from frequent, to very sparse bouts0(the ∆ t ex = D. Adaptations to resistance training exercise
We showed in section 2.A above that constant titin kinasemechanosensing at the steady state muscle tension allows themuscle to maintain its size. In order to consider dynamicchanges in muscle size, we must first assure ourselves thatit reaches a new steady-state; secondly, that it predicts thatmuscles grow with the correct time-dependence; and finally,we must check whether the model predictions for the mag-nitude of change in muscle size are in the reasonable range,given that we have no free parameters (all rate constants andconcentrations are independently known).In Fig. 12, we see that both muscle growth during the ex-ercise program, and muscle detraining after exercise programends, are strongly dependent on the feedback from the slowdiffusion of ribosomes across the large and sterically hinderedsarcoplasm. Greater muscle fibre CSA at the start of train-ing implies more ribosomal diffusion blocking, hence a higherhindrance term α n titin – resulting in a faster, lower magnituderesponse to the same training load. This behaviour is qualita-tively observed in the literature: strength trained athletes re-spond to a much lesser degree to a resistance training regime,see e.g. [74].After stopping a resistance training programme, muscleCSA slowly decreases, eventually returning to its pre-trainedhomeostatic value. The time course of detraining is harderto investigate. Low values of two months [7, 75] for skeletalmuscle, to several years for recovering hypertrophic cardiacmuscle [76], have been reported. In our model, we observereasonable time-courses which match this range for detrain-ing for a 5-10% degradation of ribosomes before they arriveat the sarcomere, or for very low force feedback in the rangeof 0 . < µ < .
005 (see Fig. 12).There are some exceptions to this: career athletes main-tain significantly higher muscle CSA a long time after retiring[77], and the body maintains a memory of prior resistancetraining events [78] by changing its methylome. It seemslikely that the body can develop and maintain a higher restingmuscle tone if chronic resistance training changes the molec-ular architecture of the muscle. This complication is beyondthe scope of our model.
Time (days)1000 2000 30000 M u sc l e fi b r e C SA ( m ) initial n = titin n = titin n = titin n = titin n = titin n = titin FIG. 12. Time course of muscle growth and loss (starting after 600days of hypertrophy) in response to a regular resistance training pro-gram (every 3 days) with 3 sets of 10 repetitions at 20 pN per titin(our estimate of ≈
70% 1RM), followed by detraining. The diffusivefeedback depends on the degree of sarcoplasmic titin degradation,which in turn increases with myonuclear domain size and lysosomalactivity. Slow detraining may combine with an initial fast loss dueto atrophic conditions (see below). In this case a 5 −
10% ribosomedegradation en-route to the titin synthesis sites (0 . < α n titin < . E. Atrophy and recovery from bedrest or microgravity
When the body is subjected to bed-rest, microgravity [79],famine [80], or as the consequence of several pathologies [81],muscle size can very rapidly decrease. Any mechanism whichincreases degradation rates (SRF, ribosomes, titin degradationrates in our model, see Table I) will necessarily cause atrophy,and our model confirms this (see Supplementary Part E fordetail).Extended periods of bed-rest and microgravity are the moreinteresting atrophy-inducing conditions to study in the contextof mechanosensing, as it is the sudden lack of tension, whichpromotes muscle degradation. In other words, the steady-stateforce applied to the muscle (the homeostatic tone) is suddenlydecreased, and the muscle metabolism responds. We find aquick decrease in muscle CSA after a series of drastic param-eter changes at the start of our simulations, but it is the kinet-ics of muscle recovery after atrophy which appear to be moredependent on the type of feedback in the model. In practice,muscle is seen to recover relatively rapidly after very substan-tial atrophy, with most of the recovery occurring over a 1-2week period [82]. Fig. 13(a) shows our model predictionswith the simplifying assumption that there was no feedbackrelationship between muscle force per fibre and the CSA inthe case of hypertrophy. The curves show a response to a verysmall decrease of steady-state tone (maximum 0.5% in blackcurve), and recopvery when f st returns to its value prescribedby the (9) after 120 days. A very slow recovery of homeostaticmuscle CSA is found, not in agreement with observations.However, once we include the feedback, when the forceper filament decreases with an increasing CSA, the rate of re-sponse becomes much more realistic, see Fig. 13(b). Here amuch greater force increase is applied (up to 5% in the black1 Time (days)100 200 3000 M u sc l e fi b r e C SA ( m ) M u sc l e fi b r e C SA ( m ) (a)
400 5006543210 f f st st f st st f f st st f f st st f f st st f f st st f f st st f f st st f f st st f f st st (b) FIG. 13. Time course of muscle atrophy as the steady state force f st (discussed in (9) and Fig. 4) is suddenly diminished from the steadystate value to a lower value. In this simulation, after 120 days, theforce is brought up to its steady state value again. The recovery speeddepends on exactly how the muscle force scales with muscle CSAduring atrophy, the ‘force feedback’ discussed in Supplementary PartB.4. (a) The case of negligible force feedback ( µ = . µ = .
02) leads to a muchmore reasonable recovery rates, which we consider close to clinicalobservations. curve), and we see both the atrophy onset and the recoveryreaching the saturated steady state values within 60 days. Thissuggests that a reasonable force feedback scale (with the pa-rameter µ ∼ .
02 or even higher, see Supplementary Part Efor detail) is a required feature of our model, if quantitativepredictions are to be obtained.An unexpected feature of plots in Fig. 13(b) is the muscle‘overshoot’ during the fast recovery after atrophy. It seemslikely that the several intrinsic processes have low rate buthigh sensitivity, resulting in muscle keeping a memory of itsprevious architecture during atrophy, much like the career-trained athletes whose muscle CSA remains higher than nor-mal after retirement. This would translate into a correspond-ing increase in the muscle force at smaller muscle CSA.
IV. DISCUSSION
In this work, we developed a kinetic model combining theintracellular mechanosensor of the 2nd kind, the signallingchain pathway (admittedly one of several). with the riboso-mal kinetics of post-transcription synthesis – to examine howmuscles sense and respond to external load. The importantfactor of limitations to ATP supply, which affects both theMVC level due to myosin activation and the signalling dueto phosphorylation, is included in the background (see Fig. 3and Supplementary Part A). The primary marker of morpho-logical response for us is the cross-section area (CSA) of anaverage muscle fibre, which is directly and linearly mappedonto the number of titin molecules per fibre. We suggest thatthe titin kinase (TK) domain has the right characteristics toplay the role of the primary mechanosensor within the musclecell. By looking at how TK unfolds under force, we found thatit acts as a metastable switch, by opening rapidly only at highforces, but opening and closing slowly within a range of phys-iological forces. The muscle is known to apply a low-leveltensile force, and be under a steady-state passive tension atrest, which we compare with the steady-state force predictedby our model. We find that the two forces are of the same mag-nitude, which suggests that long-term muscle stability is due acombination of the active muscle tone and the passive muscleload stored in elastic sarcomere proteins – notably titin. Wefind that small changes in the steady-state force allow the mus-cle to maintain its size after the end of a resistance trainingprogramme, and we suggest that this change in steady-statemuscle tension might account for some of the ‘memory’ thatmuscle develops after long-term training [77, 78].Given the switch-like nature of TK, it seems likely thatdifferent individuals will have slightly different predisposi-tions towards applying somewhat more or less muscle tone inhomeostasis, and therefore can maintain muscle mass muchmore or much less easily. This low-level steady-state tensileforce will crucially depend on the number of available myosinheads and on the steady-state ATP concentration in the cell, aswell as sarcomere and tendon stiffness.Our model shows qualitatively reasonable time courses forhypertrophy, developing during a regular exercise regime, fol-lowed by detraining as well a muscle atrophy followed by re-covery. Although it is not explicitly included in the currentmodel, long-term changes in muscle architecture (slightly in-creasing the muscle tone with the same CSA), as well as in-creases in myonuclear number after chronic hypertrophy (in-creasing the synthesis rates in the model), could cooperate toincrease the steady-state muscle CSA. This could then providea rationale for the observed permanent increase in muscle sizeafter just one bout of resistance training in the past [83]. Themodel uses no free-fitting parameters, since all its constantsare independently measurable (indeed, Table I gives examplesof such measurements). Obviously, there would be a largeindividual variation between these parameter values, and soapplying the quantitative model predictions to an individual isprobably optimistic. However, we are excited to offer a soft-ware to implement the model and make specific predictionsin response to any chosen ‘exercise regime’, which could be2used and adapted to practitioners.We saw that a successive integration of the initial mechan-ical signal is necessary in order for muscle cells to displaytrophic responses at the right time scales. Indeed, a several-week lag is observed in the increase of structural proteins con-tent after the start of an exercise regime; in our model, thisarises due to a lag in ribosome number, because ribosometurnover and synthesis is relatively slow. Research into thesterically hindered diffusion of ribosomes in muscles so farappears to be very much in its infancy, despite its obviousoverarching implications in muscle development. This wouldbe an exciting avenue for future research in this area.To further improve the model, we could include more de-tails about the viscoelastic properties of muscle. Its effectswould be twofold: first, the switching kinetics between thetitin kinase conformations would change somewhat (see Sup-plementary Part E.2 for more details); secondly, it would al-low for a treatment of how muscle fatigue affects the com-pliance and therefore the mechanosensitivity of the mus-cle’s structural proteins. We expect the plateauing of themechanosensitive efficiency at high forces explored in Sup-plementary Part A.4 (in particular see Fig. 4 in the Supple-mentary) to be even more pronounced. In fact, because thesarcomere structural proteins likely change their mechanosen-sitive properties at high forces, we expect there to be an opti-mal force at which the exercise should be carried out. How-ever, this would substantially increase the mathematical com-plexity of the model, reducing its present intuitive clarity, andmake an analytical solution for homeostasis less tractable.This is the next stage of model development that we hope topursue. In summary: how intracellular signaling in muscle cells or-ganises a trophic response is a central question in exercisescience and in the study of conditions, which affect musclehomeostasis (including development and ageing, as well asnumerous pathologies). Cells have been shown to use time-integrated mechanical stimuli to initiate signaling cascades,in a way which depends on the strength and duration of thesignal ( i.e. mechanosensitively). This work provides a quanti-tative analytical rationale for a mechanosensitive mechanismfor trophic signaling in muscle, and gives an additional pieceof evidence that the titin kinase domain is a good candidatefor hypertrophic mechanosensing. We expect advances in tar-geted exercise medicine to be forthcoming, specifically if theexact structure of the mechanosensing complex bound to theTK domain and its downstream signaling cascade are studiedin more detail.
ACKNOWLEDGMENTS [1] Enoka RM (2015)
Neuromechanics of Human Movement . (Hu-man Kinetics Europe Ltd, Manchester).[2] Cutts A (1988) The range of sarcomere lengths in the musclesof the human lower limb.
J. Anat.
Proc. R. Soc. B
Biophys. J.
J.Anat.
Exp. Physiol.
J. Strength Cond. Res.
J. Struct. Biol.
Med. Biol. Eng. Comput.
Nature
Nature
Glob. Car-diol. Sci. Pract.
J. Mol. Cell. Cardiol.
J. Cell Sci.
J. Mol. Biol.
J. Muscle Res. Cell Motil.
J. Exp. Biol.
Front. Physiol. Skeletal Muscle.
Biophys. J.
Biophys.J.
Int. J. Mol. Sci.
J. Cell Sci.
AIMS Biophys.
Circ. Res. β Mechanism of Cell Sensing theSubstrate Stiffness.
PLoS One
Biophys. J.
Soft Mat.
Proc. Natl. Acad. Sci.
Curr. Biol.
Pflügers Arch. - Eur. J. Physiol.
Int. J. Mol. Med.
Science (80-. ).
Proc. Natl. Acad. Sci.
Sci-ence (80-. ).
Biophys. J.
Biophys. Rev.
Nature
Open Biol.
J.Biol. Chem.
Cardiovasc. Res.
Science (80-. ).
EMBO J.
Cell
Nucleic Acids Res.
J. Appl.Physiol.
Eur. J. Appl. Physiol.
J. Physiol.
Eur. J. Appl.Physiol.
Eur. J.Appl. Physiol.
Proc. Natl. Acad. Sci.
Biophys. J.
Age Ageing
J. Cell Sci.
Mol. Cell. Biol.
Autophagy
Elife
J. Bacteriol.
Adv. ProteinChem. (Academic Press Inc.), pp. 1–67.[59] Amos LA, Amos WB (1991)
Molecules of the Cytoskeleton .(Macmillan Education UK, London).[60] Isaacs WB, Kim IS, Struve A, Fulton AB (1989) Biosynthe-sis of titin in cultured skeletal muscle cells.
J. Cell Biol.
J. Neurochem. tal tissue protein in skeletal muscle and heart. J. Biol. Chem.
J. Biol. Chem.
J.Bodyw. Mov. Ther.
Antioxid. Redox Signal.
Front. Physiol.
J. Physiol.
J. Appl. Physiol.
Front. Physiol.
Am. J. Physiol. Integr.Comp. Physiol.
J. Physiol.
Int. J. Dev. Biol.
PLOS Comput. Biol.
Eur. J. Appl. Physiol.
Eur. Rev. Aging Phys. Act.
Am. J. Hypertens.
J.Bone Miner. Res.
Sci. Rep.
Extrem. Physiol. Med.
Mol. Cell. Biol.
Dis. Model. Mech.
J. Physiol.
Proc. Natl. Acad. Sci.