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A kinetic model that explains the effect of inorganic phosphate on the mechanics and energetics of isometric contraction of fast skeletal muscle

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A kinetic model that explains the effect of inorganic phosphate on the mechanics and energetics of isometric contraction of fast skeletal muscle
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  A kinetic model that explains the effect ofinorganic phosphate on the mechanicsand energetics of isometric contractionof fast skeletal muscle Marco Linari 1,2 , Marco Caremani 1,2 and Vincenzo Lombardi 1,3, * 1 Laboratorio di Fisiologia, Dipartimento di Biologia Evoluzionistica, and   2 CNISM, Universita` di Firenze, Italy 3 CRS SOFT-INFM-CNR, Universita` di Roma ‘La Sapienza’, Italy A conventional five-step chemo-mechanical cycle of the myosin–actin ATPase reaction, which impliesmyosin detachment from actin upon release of hydrolysis products (ADP and phosphate, Pi) and bindingof a new ATP molecule, is able to fit the [Pi] dependence of the force and number of myosin motorsduring isometric contraction of skeletal muscle. However, this scheme is not able to explain why the iso-metric ATPase rate of fast skeletal muscle is decreased by an increase in [Pi] much less than the number of motors. The question can be solved assuming the presence of a branch in the cycle: in isometric contrac-tion, when the force generation process by the myosin motor is biased at the start of the working stroke,the motor can detach at an early stage of the ATPase cycle, with Pi still bound to its catalytic site, and thenrapidly release the hydrolysis products and bind another ATP. In this way, the model predicts that in fastskeletal muscle the energetic cost of isometric contraction increases with [Pi]. The large dissociationconstant of the product release in the branched pathway allows the isometric myosin–actin reaction tofit the equilibrium constant of the ATPase. Keywords:  chemo-mechanical cycle in muscle; myosin–actin ATPase; skinned fibre mechanochemistry;kinetic model of myosin motor 1. INTRODUCTION ( a )  The myosin working strokein isometric contraction During muscle contraction, the globular head of themyosin molecule (M) extending from the thick filamentcyclically attaches to the actin site (A) on the thin filamentand undergoes a structural working stroke accounted forby the energy released by the hydrolysis of one ATP mol-ecule (Huxley 1969; Huxley & Simmons 1971; Lymn & Taylor 1971). According to the crystallographic model(Dominguez  et al  . 1998; Geeves & Holmes 2005) the working stroke consists of a 70 8  tilting of the light chaindomain of the myosin head (the lever arm) about a ful-crum in the catalytic domain firmly attached to actin,corresponding to an axial movement ( D ) of 10 nmbetween the catalytic domain and the attachment of thelever arm to the myosin filament. A similar amount of fila-ment sliding has been found in single muscle fibres whenthe force of the half-sarcomere is suddenly reduced tosynchronize the working stroke in the actin attachedmyosin motors (Huxley & Simmons 1971; Piazzesi  et al  .2002; Reconditi  et al  . 2004). The biochemical step asso-ciated with the working stroke is the release of the hydrolysis product orthophosphate (Pi), while therelease of ADP follows the execution of the working stroke(Bagshaw & Trentham 1974; Sleep & Hutton 1980; Ferenczi  et al  . 1984; Hibberd  et al  . 1985).The probability of the completion of the ATPase cycleis reduced when the myosin motors act under high load,with respect to low load, as proven by the reduction of the rate of energy liberation (Fenn 1923; Hill 1938) and ATP hydrolysis (Kushmerick & Davies 1969). Recently,it has been shown that the average strain ( s ) in themyosin motors in isometric contraction is one order of magnitude smaller than the size of the working stroke  D (Decostre  et al  . 2005; Piazzesi  et al  . 2007). The energyrequired to strain the motor elastic element during an iso-metric working stroke is  12  1 D 2 , where  1  is the stiffness of the myosin cross-bridge. In a single fibre of frog skeletalmuscle,  1  is approximately 3.2 pN nm 2 1 (Decostre  et al  .2005; Piazzesi  et al  . 2007). In this case, for a single work-ing stroke of 10 nm, the energy would be 160 zJ, twicethat released by the hydrolysis of one molecule of ATP(Pate & Cooke 1989 a ) and approximately 40  k b Q  (where  k b  is the Boltzmann constant and  Q   the absolutetemperature). The corresponding equilibrium constantof the reaction would be too low to explain the isometricforce on the basis of motors that have undergone thewhole working stroke transition. Note that the argumentremains valid also assuming that the stiffness of themyosin motor is as low as approximately 1.7 pN nm 2 1 ,the value found in skinned fibres from rabbit psoas(Linari  et al  . 2007; Lewalle  et al  . 2008), as also in thiscase the energy for the working stroke transition in theisometric condition would be greater than 20  k b Q  . Apossible solution to the problem is suggested by recentX-ray experiments (Reconditi  et al  . 2004; Piazzesi  et al  .2007), demonstrating that the 10 nm working strokeoccurs only at low load, while, for loads greater than or *  Author for correspondence (vincenzo.lombardi@unifi.it).Electronic supplementary material is available at http://dx.doi.org/10.1098/rspb.2009.1498 or via http://rspb.royalsocietypublishing.org. Proc. R. Soc. B doi:10.1098/rspb.2009.1498 Published online Received   19 August 2009  Accepted   14 September 2009  1  This journal is  q 2009 The Royal Society  on September 23, 2016http://rspb.royalsocietypublishing.org/ Downloaded from  equal to 0.5  T  0 , the working stroke is approximately 6 nm.These results can be explained assuming that the workingstroke is made up of a series of steps and the detachmentprobability increases sharply for motor movementsabove 6 nm. The average strain per motor in isometriccontraction is approximately 1.7 nm at 5 8 C and isaccounted for by the first of a series of steps of approxi-mately 2.8 nm between motor states that have the samestiffness (Decostre  et al  . 2005). The energy for the firsttransition is ( 12 .3.2.2.8 2 ¼ ) 12.5 zJ, that is approximately3 k b Q  , a value that is compatible with the possibility thatthe transition occurs by thermal fluctuation.(  b )  The effect of Pi on mechanics and energeticsof isometric contraction In Ca 2 þ -activated skinned muscle fibres, addition of Pihas been found to increase the rate of force generationfollowing photolysis of caged ATP starting from rigor(Hibberd  et al  . 1985). Accordingly, the rate constantof development of isometric force following a period of unloaded shortening ( r  F ) and the rate constant of theforce transient elicited by a jump in [Pi] ( r  Pi ) are fasterat higher [Pi] (Millar & Homsher 1990; Dantzig  et al  .1992; Walker  et al  . 1992; Regnier  et al  . 1995; Tesi et al  . 2000, 2002; Caremani  et al  . 2008). At any [Pi]above 3 mM,  r  Pi  is larger than  r  F  and both rate constantsshow saturation at large [Pi]. The steady isometricforce ( T  0 ) developed by a muscle fibre is decreased byincrease in [Pi] (Brandt  et al  . 1982; Hibberd  et al  .1985; Pate & Cooke 1985; Kawai  et al  . 1987; Cooke  et al  .1988; Pate & Cooke 1989 b ; Millar & Homsher 1990;Fortune  et al  . 1991; Kawai & Halvorson 1991; Dantzig et al  . 1992; Martyn & Gordon 1992). Also the ATPase rate of isometrically contracting fibres is reduced by Pi,but less than in proportion to the reduction of force(Bowater & Sleep 1988; Potma & Stienen 1996). These findings were explained by assuming that the effect of Pidepends on the strain of the myosin motor (Pate & Cooke1989 a ) as it is the case if Pi release is associated with thetransition to higher force-generating states of the myosinmotor.However,thisideaiscontradictedbytherecentfind-ingthat the Pi-dependent reduction inforceisexplainedbya proportional decrease in the number of myosin motorswithout any effect of Pi on the force per motor (Caremani et al  . 2008). This finding makes existing conventionalmodels unable to explain the reduced effect of Pi on theATPase rate (Woledge  et al  . 2009). 2. RESULTS AND DISCUSSION The models described here are intended to explain thekinetics of the chemo-mechanical coupling in isometricconditions; therefore, the myosin motors experienceonly a relatively narrow range of strains and a detaileddescription of the dependency of the rate constants con-trolling state transitions on the axial position of themotors is not necessary. The rate constants for forwardand backward transitions and the corresponding equili-brium constants are defined as  k x ,  k 2 x  and  K  x ,respectively. Time-dependent distributions of myosinmotors among the various states following Pi jump orunloaded shortening were calculated by solving a systemof linear differential equations as reported in theelectronic supplementary material.( a )  Predictions of a conventional cycle The effects of Pi on the isometric force under steady-stateand transient conditions can be explained by a five-stepreaction scheme (scheme 1) for the myosin–actinATPase cycle similar to that described in the 1990s(Fortune  et al  . 1991; Kawai & Halvorson 1991; Dantzig et al  . 1992).ATP binds to the actomyosin (AM) complex (step 1),promotes rapid dissociation of myosin from actin (incor-porated in step 1) and then is hydrolysed (step 2). Themyosin with the hydrolysis products is in rapid equili-brium with the weakly bound A-M.ADP.Pi state. Actinattachment by the closure of the actin binding cleft of the myosin head (Geeves & Holmes 2005) forms thestrongly bound, stiffness generating, AM 0 .ADP.Pi statethat, without significant delay undergoes an interdomainstructural change leading to the strained conformationresponsible for isometric force (step 3). Thus, step 3 isthe combination of two processes, formation of stronglybound cross-bridges and force generation, with the kin-etics determined by the much slower attachmentprocess. This definition of step 3 is in agreement with afast kinetics for force generation (Huxley & Simmons1971) and is supported by the recent evidence thatforce rises in proportion to number of myosin motorsduring isometric force development (Caremani  et al  .2008). Pi is released in a rapid reaction (step 4, Dantzig et al  . 1992), without any further contribution to force.Thus both the AM 0 .ADP.Pi and the AM 0 .ADP statesare a mixture of different force-generating states of cross-bridges in rapid equilibrium and the sum of theoccupancies of these two states constitutes the fraction(  f   ) of actin attached motors. Therefore each of the twostates contributes equally to the fibre force and stiffness,in agreement with the finding that an increase in Pidecreases the ensemble force and the number of motorsby the same amount (Caremani  et al  . 2008). The sub-sequent ADP-release step (step 5) occurs at a rate that,in the isometric contraction is low (Nyitrai & Geeves2004; Sleep  et al  . 2005; West  et al  . 2005). The cycle isrepeated as long as ATP is available and the fibreis activated.In the scheme, the development of isometric force israte limited by both the ATP hydrolysis step (step 2)and the attachment of cross-bridges (step 3), while thesteady-state flux through the whole cycle (the ATPaseactivity) is limited by the rate of the ADP release (step 5).The constraint is removed, according to the view that AM AMA-M·ADP·Pi AM ¢ ·ADP·PiADPATPAM ¢ ·ADP1 2 3Pi4 5M·ATP Scheme 1. Conventional chemo-mechanical cycle. 2 M. Linari  et al. Isometric contraction mechanochemistry Proc. R. Soc. B  on September 23, 2016http://rspb.royalsocietypublishing.org/ Downloaded from  the rate of ADP release becomes very rapid following thestroke-dependent conformational change (Nyitrai &Geeves 2004; Geeves & Holmes 2005), to obtain the state distribution during the period of unloaded shorten-ing that precedes the development of isometric force.During this period, the rate constant for ADP release( k 5 ) is set to 1000 s 2 1 .The reference experimental data are from Caremani et al  . (2008), for the development of isometric force andfrom Dantzig  et al  . (1992), for the force transientfollowing a Pi jump.(i)  Rate constants Unless differently specified, the values of the rate constantsof the transitions (listed in table 1) have been chosen inagreement with those reported in previous kinetic studieson rabbit psoas actomyosin both in solution and in skinnedfibres, as specified in the electronic supplementarymaterial. Rapid equilibrium reactions that follow akinetically relevant step are incorporated in the step itself.(ii)  Equilibrium constant of ATP hydrolysis The product of the equilibrium constants of all stepsreported in table 1 gives an equilibrium constant forATP hydrolysis (  K  ATP ) in isometric contraction of 2.5 M, which is smaller than that reported in literature(2–5  10 5 M, Guynn & Weech 1973) by a factor of approximately 10 5 . This corresponds to a difference inthe associated energy of approximately 50 zJ, consistentwith the energy of the working stroke (White & Taylor1976). This difference is accounted for by the differencein the combined equilibrium constant of steps 3 and 4(isomerization and Pi release steps) between unloadedconditions and isometric conditions, which explains whysteps 3 and 4 are reversible in the isometric contractionand no net work is done.(iii)  Simulation of the effect of Pi on the isometric forceduring steady state and transient conditions The effect of [Pi] on the occupancyof the various states of the reaction scheme (except the AM state that is not sig-nificantly populated at the physiological [ATP]) duringsteady isometric contraction are plotted in figure 1 a .With a concentration of myosin heads in skeletal muscleof 0.15 mM (He  et al  . 1997; Sun  et al  . 2001), the concen-tration of motors bearing stiffness and force in isometriccontraction in control solution (no added Pi, correspond-ing to [Pi] ¼ 1 mM) is approximately 0.052 mM, whichgives a fractional number of (0.052 mM/0.15 mM ¼ )0.34, in agreement with that fraction of motors estimatedfrom mechanical measurements under the same con-ditions (Zhao & Kawai 1991; Linari  et al  . 2007; Piazzesi et al  . 2007). The number of motors (and thus the forceof the motor ensemble in each half-sarcomere) decreaseswith increases in [Pi] (figure 1 b , blue line), in agree-ment with the experimental data up to a minimum of 35 per cent of the value in control solution (circles,from Caremani  et al  . 2008; similar data for the force–Pirelation are reported by Bowater & Sleep 1988;Millar & Homsher 1990; Potma  et al  . 1995; Potma &Stienen 1996).To simulate the rate of isometric force developmentfollowing unloaded shortening, the starting distributionof myosin states induced by unloaded shortening isobtained by increasing  k 5  to 1000 s 2 1 , and 15 ms afterthe jump in  k 5  (a time similar to that of unloaded short-ening in Caremani  et al  . 2008) the motor states withsignificant occupancy are only M.ATP and AM.ADP.Pi.The rise in the number of actin-attached motors when k 5  is re-assigned the isometric value is shown infigure 1 d   (data relative to the steady isometric value atthe respective [Pi]). The rate constant of force rise ( r  F ),estimated by the reciprocal of the time from the force attime 0–63% of the steady isometric force (see Caremani et al  . 2008), increases with [Pi] (blue continuous line infigure 1  f   ) in agreement with the experimental results(filled circles from fig. 6C of  Caremani  et al  . 2008).The same set of rate constants was used to reproducethe force transient elicited by a Pi jump. The simulatedtime course of the number of attached cross-bridges inresponse to a Pi jump from 0 to 11 mM Pi (figure 1 e ,dashed line), is faster than that following unloaded short-ening in the presence of 11 mM Pi (continuous line). Thecalculated relation between the rate of the force transientelicited by a Pi jump ( r  Pi ) and [Pi] (blue dashed line infigure 1  f   ) reproduces quite satisfactorily the observedrelation (open squares from fig. 6 of  Dantzig  et al  .1992) and both lie above the  r  F  –Pi relations (filled circlesand continuous line) as a consequence of the fact that theweight of each process in approaching the new distri-bution is determined by the position of the perturbationin the cycle (see also Caremani  et al  . 2008). Table 1. Rate constants for the forward ( k x ) and the backward ( k 2 x ) transitions and corresponding equilibrium constants(  K  x ) in schemes 1 and 2.  x  represents the step number, as reported in brackets in the first row. For the second-order rateconstants, the apparent rate constants are calculated assuming [MgATP] ¼ 5 mM (step 1), [ADP] ¼ 20  m M (step 5) and [Pi]according to the experimental conditions (step 4). The unloaded shortening condition is simulated by rising  k 5  from 8 to1000 s 2 1 .(1) (2) (3) (4) (5) (6) (7)scheme 1 k x  10 6 M 2 1 s 2 1 18 s 2 1 21 s 2 1 500 s 2 1 8 s 2 1 k 2 x  0.1 s 2 1 15 s 2 1 100 s 2 1 40  10 3 M 2 1 s 2 1 10 5 M 2 1 s 2 1  K  x  10 7 M 2 1 1.2 0.21 12.5  10 2 3 M 8  10 2 5 Mscheme 2 k x  10 6 M 2 1 s 2 1 14 s 2 1 21 s 2 1 500 s 2 1 8 s 2 1 24 s 2 1 k 2 x  0.1 s 2 1 9 s 2 1 90 s 2 1 33  10 3 M 2 1 s 2 1 10 5 M 2 1 s 2 1 8  10 2 3  K  x  10 7 M 2 1 1.56 0.23 15.2  10 2 3 M 8  10 2 5 M 3  10 3 5  10 2 M Isometric contraction mechanochemistry  M. Linari  et al.  3 Proc. R. Soc. B  on September 23, 2016http://rspb.royalsocietypublishing.org/ Downloaded from  [Pi] (mM)    A   T   P  a  s  e  r  a   t  e   (  r  e   l  a   t   i  v  e  u  n   i   t  s   ) 00.40.81.2[Pi] (mM)    f  o  r  c  e   (  r  e   l  a   t   i  v  e  u  n   i   t  s   ) 00.40.81.2[Pi] (mM)   m  y  o  s   i  n   h  e  a   d  s   (  m   M   ) 00.040.080.12( a )( d  )( c )( b )time (s)    f  o  r  c  e   (  r  e   l  a   t   i  v  e   t  o       T    0   ) 00.40.81.221 mM1 mMtime (s)   a   t   t  a  c   h  e   d  m  y  o  s   i  n   h  e  a   d  s   (  m   M   ) 00.020.040.060.08time (s)0.30.20.10   a   t   t  a  c   h  e   d  m  y  o  s   i  n   h  e  a   d  s   (  r  e   l  a   t   i  v  e  v  a   l  u  e  s   ) 00.40.81.2( e )[Pi] (mM)5 10 15 20 25 305 10 15 20 25 300.1 0.2 0.30.1 0.2 0.35 10 15 205 10 15 25 3020   r  a   t  e   (  s  –   1   ) 020406080100 r  Pi r  F (  f  ) Figure 1. Responses of the simulation. ( a ) Effect of [Pi] on the occupancy of the cross-bridge states (as defined by the colours)during isometric contraction, calculated with scheme 1. The attached AM state is not significantly populated and is omitted.Pink line, M.ATP; green line, A-M.ADP.Pi; blue line, AM 0 .ADP.Pi; red line, AM 0 .ADP. ( b ) Pi-dependence of isometric force(relative to the force in control solution, 1 mM Pi). Observed relation, filled circles (data from figs 1  f   and 6 a  of  Caremani  et al  .2008). Simulated relations: blue line (scheme 1) and red line (scheme 2). ( c ) Pi-dependence of ATPase rate: observed relation,open symbols (circles, Bowater & Sleep 1988; triangles, Potma  et al  . 1995; diamonds, Potma & Stienen 1996). Simulated relations: blue line (scheme 1) and red line (scheme 2). Values relative to those in control solution. ( d  ) Time course (calculatedwith scheme 1) of the number of attached cross-bridges (the sum of the occupancy of AM 0 .ADP.Pi and AM 0 .ADP) followingthe end of the unloaded shortening in control solution ([Pi], 1 mM) and at different [Pi] (3.5, 6, 11, 16 and 21 mM). Values onthe ordinate are normalized for the steady isometric value at the respective [Pi]. ( e ) Superposition of the time courses (calcu-lated with scheme 1) of the number of attached cross-bridges following either a step of [Pi] from 0 to 11 mM (dashed line) or aperiod of unloaded shortening in 11 mM Pi (continuous line). For facilitating the comparison, in the inset, the data are plottedafter making them relative to their respective maximum change and applying the operation (1 2  y ) to data (  y ) for the rise innumber of cross-bridges following unloaded shortening. (  f   ) Pi dependences of   r  F  and  r  Pi .  r  F  relation: observed (filled circles)from fig. 6 c  of  Caremani  et al  . (2008); simulated: scheme 1 (blue continuous line) and scheme 2 (red continuous line).  r  Pi relation: observed (open squares) from fig. 6 of  Dantzig  et al  . 1992; simulated: scheme 1 (blue-dashed line) and scheme 2 (red-dashed line). The simulated  r  Pi  relations lie slightly below the observed relation probably because of the different pro-cedure to estimate the rate constant of the force transient. While in the experimental records of  Dantzig  et al  . (1992), theinitial lag does not contribute to the estimate of the rate constant, in our simulation the lag is incorporated in the estimate. 4 M. Linari  et al. Isometric contraction mechanochemistry Proc. R. Soc. B  on September 23, 2016http://rspb.royalsocietypublishing.org/ Downloaded from  (iv)  Simulation of the effect of Pi on the isometric ATPase rate In control solution, the calculated ATPase rate is0.38 mM s 2 1 , in agreement with the values reported forfast skeletal muscle (Potma & Stienen 1996). Withscheme 1, the increase in [Pi] reduces the ATPase rate(blue line in figure 1 c ) in proportion to the reduction of the occupancy of the AM 0 .ADP state (red line infigure 1 a ). Thus, the Pi-dependent decrease in theATPase rate is larger than that of isometric force (blueline in figure 1 b ), as the force results from the occupancyof both the decreasing AM 0 .ADP state and the increasingAM 0 .ADP.Pi state (blue line in figure 1 a ). In contrast tothe simulated relation, the observed relation (open sym-bols in figure 1 c ) obtained from several studies atcomparable temperature (11–15 8 C, Bowater & Sleep1988; Potma  et al  . 1995; Potma & Stienen 1996) shows that the ATPase rate is reduced less than the isometricforce (and the fraction of attached cross-bridges): anincrease of [Pi] to 15–20 mM produces a reduction inthe isometric ATPase of only approximately 20 per cent.This demonstrates that the conventional reactionscheme is not adequate to explain the energetics of theisometric contraction.(  b )  Unconventional cycle with the release of hydrolysis products after motor detachment  The contradiction can be solved by allowing a substantialATPase activity to occur also at high [Pi]. This is providedin scheme 2 by the branched pathway that allows theAM 0 .ADP.Pi force-generating state to detach before therelease of hydrolysis products (step 6).The detached M 0 .ADP.Pi state derives from a stronglybound state that has undergone the conformationalchanges that generates the stiff and strained conformationresponsible for force, and therefore it is structurally andkinetically different from the M.ADP.Pi state in rapid equi-librium with the weakly bound A-M.ADP.Pi state. Wehypothesize that the M 0 .ADP.Pi state, undergoes a rapidsequence of events consisting of the completion of boththe structural change normally associated to the executionof the 10 nm working stroke in the attached myosin headand the ATPase cycle by an almost irreversible release of Pi and ADP and binding of a new ATP (step 7). Rapidrelease of the hydrolysis products from myosin in theabsence of actin has not been observed in solution(White  et al  . 1997). However, in fibres, the releaseoccurs from a myosin state that, as a consequence of thereaction with actin, has assumed a strained conformationthat responds to detachment with the immediate openingof the nucleotide binding pocket and product release.In support of this view, recent X-ray experiments on intactfrog muscle fibres suggest that the rate of events that termi-natetheAMinteraction(ADPrelease,ATPbindingandtheensuing detachment of myosin from actin) is controlledspecifically by the conformation of the myosin head(Piazzesi  et al  . 2007). Since it is not possible to investigatethe hypothesized statein solution, there isnodirect conflictbetween our hypothesis and solution studies.(i)  Rate constants Unless otherwise specified, the values of the rate con-stants of the transitions (listed in table 1) are the sameas those selected for the simulation with scheme 1.Details for the selection of rate constants of the steps 6and 7 are given in the electronic supplementary material.The combined equilibrium constant of steps 6 and 7 is1.5  10 6 M, so that the equilibrium constant of theunconventional cycle (  K  2 .  K  3 .  K  6 .  K  7 ) is 5.3  10 5 M.Thus, in contrast to the conventional cycle, the uncon-ventional cycle accounts for the equilibrium constant of ATP hydrolysis also in isometric contraction. As a corol-lary, it must be noted that the unconventional cycleprovides a straightforward explanation for the findingthat in isometric contraction the free energy of the ATPhydrolysis is mostly released as heat.(ii)  Simulation of the effect of Pi on the isometric forceand on the ATPase rate With scheme 2, the effects of [Pi] on the occupancy of therelevant states of the reaction scheme (figure 2 a ) are sub-stantially similar to those in scheme 1 (figure 1 a ), becausethe new M 0 .ADP.Pi state is only a transient intermediateand its occupancy is relatively low even at the highest [Pi].As shown in figure 2 b , in control solution the ATPaserate (continuous line) is 0.47 mM s 2 1 , similar to thatobserved (Potma & Stienen 1996) and is 80 per centaccounted for by the flux through the conventional path(dot-dashed line) and 20 per cent by the flux throughthe branched path (dashed line). An increase in [Pi]alters the fluxes through the two pathways by massaction in opposite ways: the flux through the conventionalpathway is reduced, while that through the branchedpathway is increased. At 25 mM Pi, the ATPase rate hasdropped only to 0.33 mM s 2 1 and is 82 per centaccounted for by the flux through the branched pathway. AM AMA-M·ADP·Pi AM ¢ ·ADP·PiADPATPAM ¢ ·ADP1 2 3Pi4 5M ¢ ·ADP·Pi6M·ATPM·ATP7PiADP ATP Scheme 2. Unconventional chemo-mechanical cycle. Isometric contraction mechanochemistry  M. Linari  et al.  5 Proc. R. Soc. B  on September 23, 2016http://rspb.royalsocietypublishing.org/ Downloaded from
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