Modelling muscle energy-metabolism in anaerobic muscle

Meat Science 85 (2010) 134–148

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Modelling muscle energy-metabolism in anaerobic muscle I. Vetharaniam a,*, R.A. Thomson b, C.E. Devine c, C.C. Daly a,1 a

AgResearch Limited, Ruakura Research Centre, Private Bag 3123, Hamilton 3240, New Zealand Department of Chemistry, University of Waikato, Private Bag 3105, Hamilton 3240, New Zealand c Bioengineering, HortResearch Limited, Ruakura Research Centre, Private Bag 3123, Hamilton 3240, New Zealand b

a r t i c l e

i n f o

Article history: Received 27 May 2009 Received in revised form 7 December 2009 Accepted 11 December 2009

Keywords: Anaerobic muscle Muscle metabolism Muscle pH Modelling

a b s t r a c t A mathematical model of anaerobic muscle energy-metabolism was developed to predict pH and the concentrations of nine muscle metabolites over time. Phosphorous-31 Nuclear Magnetic Resonance was used to measure time-course data for some phosphate metabolites and pH in anoxic M. semitendinosus taken from three slaughtered sheep. Muscles were held at 35 °C during the experiment. Measurement commenced 25 min post mortem and concluded before rigor mortis. The model was fitted to these data within experimental error, by simultaneously varying model parameter values and initial substrate concentrations. The model was used to simulate the period from death until metabolic activity ceased, in order to predict the different stages of metabolic response to anoxia. The model suggested that alkalinisation would occur in all three muscles in the first few minutes after the onset of anoxia, followed by a steady decline in pH. For two of the muscles this decline continued until rigor, with final pH values of 5.60 and 6.07. For the other muscle, pH reached a low of 5.60 near rigor but then increased to a final value of 5.73. A rise in pH after rigor has been observed but not previously explained in the literature. The modelling results suggest it was caused by the alkalising effect of adenosine monophosphate deamination being greater at low pH than the acidifying effect of inosine monophosphate dephosphorylation. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Intracellular energy metabolism has, as its central outcome, the maintenance of cellular concentrations of adenosine triphosphate (ATP). In aerobic muscle the resynthesis of ATP, consequent to its enzymatic hydrolysis to liberate chemical energy, is dominated by the activity of the tricarboxylic acid cycle using carbohydrate, fatty acids or amino acids as substrates. In anaerobic muscle, energy metabolism is maintained by the activity of the glycolytic pathway, with glycogen as the initial substrate, and by the creatine kinase reaction (Lohmann reaction; Lohmann, 1934). Sustained anoxia and ischaemia results in depletion of ATP, leading to muscle stiffness as cross-bridges form between actin and myosin during the development of rigor mortis (Jeacocke, 1984). The decline and eventual disappearance of ATP during the pre-rigor period is associated with the accumulation of lactate (La), hydrogen ions (H+) and inorganic phosphate (Pi) which can be measured by 31P NMR. We followed the metabolism of muscle from the onset of anoxia using NMR in real time and developed a mathematical model of * Corresponding author. E-mail address: [email protected] (I. Vetharaniam). 1 Present address: Carne Technologies Ltd., 32 Lake Street, Cambridge, New Zealand. 0309-1740/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.meatsci.2009.12.017

how the individual ATP metabolites are influenced by each other. We used the model to show how there can be different outcomes in factors known to affect meat texture and flavour, such as pH (Chrystall, Devine, Davey, & Kirton, 1981; Chrystall, Devine, Snodgrass, & Ellery, 1982; Devine, Chrystall, Davey, & Kirton, 1983; Kirton, Sinclair, Chrystall, Devine, & Woods, 1981) and level of inosine production (see Farmer, 1994; Wood et al., 1999). The temperature used was 35 °C which is close to body temperature during life, and hence the onset of rigor mortis is more rapid than at the lower temperatures generally encountered in the literature or in slaughter plants.

2. Background Resynthesis of ATP during anoxic energy metabolism is initially by creatine phosphate (CP) hydrolysis but later dominated by glycogenolysis and glycolysis (Baker et al., 1994; Sahlin & Broberg, 1990; Welsh & Lindinger, 1993). Eventually CP disappears with a concomitant increase in Pi, and La accumulates in parallel with an increase in H+ concentration. As the initially stable ATP level declines, adenosine diphosphate (ADP) accumulates, followed by adenosine monophosphate (AMP) through myokinase activity. Accumulation of both inosine monophosphate (IMP) and ammonia results from AMP deaminase activity, and accumulation of inosine

I. Vetharaniam et al. / Meat Science 85 (2010) 134–148

(IN) occurs via IMP dephosphorylase activity (Goodman & Lowenstein, 1977; Idström, Soussi, Elander, & Bylund-Fellinius, 1990; Krause & Wegener, 1996; Smith, Donos, Bauer, & Eisner, 1993; Soussi, Lagerwall, Idström, & Scherstén, 1993). When anoxia is maintained, these reactions run to completion. The flow of adenonucleotides (ATP, ADP, AMP and IMP) to the nucleoside, In, is depicted in Fig. 1. The seven reactions involved are as follows: à1. ATP hydrolysis: ra is rate of ATP hydrolysis, 2

ATPn þ H2 O ! ADPðn1Þ þ cPi

1

þ ð1  cÞPi

þ cHþ ;

à2. Creatine kinase pathway: rc is rate of creatine production,

CP2 þ ADPðn1Þ þ Hþ  ATPn þ Cr; à3. Deaminase: rd is rate of AMP deamination,

AMPm þ Hþ ! IMPm þ NHþ4 à4. Glycolysis: rg is rate of ATP synthesis through glycolysis

1 2 1 ADPðn1Þ þ glycogen þ cPi þ ð1  cÞPi þ cHþ 3 2 2 ! ATPn þ H2 O þ La þ Hþ ; 3 3 à5. Dephosphorylase: ri is rate of IMP dephosphorylation, 2

IMPm ! Inðm1Þ þ cPi

1

þ ð1  cÞPi

þ cHþ

 6. Myokinase: rm is rate of myokinase ADP consumption,

ADPn 

1 1 ATPðnþ1Þ þ AMPðn1Þ ; 2 2

 7. Aerobic ATP synthesis: ro is rate of aerobic ATP synthesis, 2

ADPðn1Þ þ cPi

1

þ ð1  cÞPi

þ cHþ ! ATPn þ H2 O;

Energy metabolism has been extensively modelled for different muscles in a range of organisms. More recent studies addressing specific tissues include work on cardiac tissue using rat and pig data (Cortassa, Aon, Marban, Winslow, & O’Rourke, 2003) or pig data (Zhou, Salem, Saidel, Stanley, & Cabrera, 2005), and work on canine gastrocnaemius muscle (Lai, Saidel, Grassi, Gladden, & Cabrera, 2007). Models of generic skeletal muscle metabolism have also been developed, either parameterised from varied sources (Korzeniewski & Zoladz, 2001; Lambeth & Kushmerick, 2002), calibrated to in vitro experiments (Vinnakota, Kemp, & Kushmerick, 2006) or reflecting human metabolism (Dash et al., 2008a). These models generally contained considerable biochemical detail, and required many kinetic parameters typically measured in vitro. While such models provide useful frameworks for integrating knowledge on metabolism, the in vivo behaviour of enzymes and fluxes can vary considerably from predictions based on in vitro characteristics (Mulquiney, Bubb, & Kuchel, 1999; Mulquiney & Kuchel, 1999a, 1999b; Teusink et al., 2000). Thus for these models to predict biological outcomes, they must be parameterised to in vivo data which often may not be available. For example a model of skeletal muscle metabolism (Dash, Li, Kim, Saidel, & Cabrera, 2008b) required 91 parameters be estimated from a relatively sparse set of in vivo data, necessitating the imposition of additional constraints. Furthermore, reducing the number of variables and parameters can increase the stability of a model used for prediction. The approach we took was to construct a model with the minimum number of variables and parameters needed to predict the outcomes that we were interested in. The trade-off for the reduced biochemical detail was an increased complexity of the mathematics: to represent aggregated action, functions more complex than typical Michaelis–Menten kinetics were used. 3. Method

where c represents the fraction of free phosphate existing as PO2 4 in equilibrium with PO1 4 , with pKa = 6.70 (Kushmerick, 1997):

kP ; kP ¼ 106:70 ; c¼ kP þ ½Hþ 

135

ð1Þ

Eq. (1) shows that glycolysis (à4) is alkalising at pH > 7.0 and acidifying at pH < 7.0. Other Pi energy reactions are negligible since their pKa values are outside the pH range of anoxic muscle.

The methodology involved measuring the changes in phosphate metabolites and pH in anoxic muscle from three sheep, using 31P NMR (Section 3.1), and developing a mathematical model of muscle energy-metabolism under anoxic conditions (Section 3.2). The model was fitted to all of the NMR data, but separately for each of the three muscles. The fit was achieved by adjusting both the model parameters and initial substrate values (Section 3.3). For each muscle, the fitted model was then used to predict dynamics from slaughter (onset of anoxia) until well after rigor. This predicted the profiles (and final values) of pH and metabolites outside the period of data collection, and allowed differences between animals to be examined. 3.1. Materials and data collection

Fig. 1. Flow of adenosine in post-mortem muscle, showing the cycling between ATP and ADP, and the deaminase sink which converts adenosine to inosine.

Muscle recording: Ovine M. semitendinosus from three sheep (referred to as muscles A, B and C) was used as a model tissue. Sheep were euthanised by an excess of pentobarbitone, the muscle dissected free and a tissue strip approximately 10 mm in diameter cut along the length of the muscle in the direction of the muscle fibre. The strips were mounted on stainless steel wire guides and inserted into 10 mm diameter NMR tubes, which were then filled with mineral oil and sealed with a plastic cap. Any oxygen dissolved in the mineral oil would be available to only the small surface area of muscle in contact with the oil (Leet & Locker, 1973) and would be rapidly exhausted. Thus the muscle can be regarded as in an anaerobic environment. One advantage of mineral oil is that all of the metabolites would be retained in the muscle and not dispersed, unlike if a physiological saline solution had been used (Young, Chi, & Lowry, 1986). Muscles were held at 35 °C during

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the experiment. Recordings commenced within 25 min of death. 31 P NMR recordings were made using a Bruker AC300, with a multi-nuclear probe tuned to 31P at 121.49 MHz, and a 60° tip angle with a 1 s relaxation delay. The 31P peaks were quantified using software in the instrument, according to the method of Vogel, Lundberg, Fabiansson, Rudérus, and Tornberg (1985) and Lundberg, Vogel, Fabiansson, and Rudérus (1987), using an intracellular volume to wet weight conversion of 0.58 ml/g. P NMR was used to measure CP, Pi, ATP and pH within experimental error. This error related mainly to resolving the base of each 31 P peak in the background noise of the spectra. Substrates such as ADP, AMP and IMP could not be measured since their 31P peaks were not sufficiently strong. La cannot be measured directly using NMR, and chemical measurement of La would require sequential removal from muscle samples (already small), disturb the NMR configuration, and undesirably expose tissue to oxygen. However, there is a strong, empirical relationship between pH and La, and all of the factors involved in La production and pH change are stoichiometrically-related, temperature dependent and known (Bendall, 1973, Chapter 5). This allowed La concentration to be estimated from pH, using the following equation which was derived from a parallel experiment on sheep (Daly, unpublished)

½La ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  7:079  pH  0:130 =10:283;

0:049dðpHÞ d½La ¼ 0:004 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 7:079  pH

pH 6 7:06;

Substrate

Symbols

Reaction (direction)

Chem

Conc

Creatine phosphate

CP

Cp

rc

Creatine

C

C

rc

Adenosine triphosphate

ATP

A3

rc ; ra ; rg ; ro ; rm

Adenosine diphosphate

ADP

A2

rc ; ra ; rg ; ro ; rm

Adenosine monophosphate

AMP

A1

Inosine monophosphate

IMP

I1

Inosine

IN

I0

Hydrogen ion

H+

H

Phosphate

Pi

P

Lactate

La

L

ðþÞ ðÞ rm ; rd ðþÞ ðÞ rd ; ri ðþÞ ri ðþÞ ðÞ ðÞ ðÞ ðþÞ ðÞ ra ; rc ; rd ; rg ; ri ; ro ðþÞ ðÞ ðþÞ ra ; rg ; ri ðþÞ rg

ðÞ ðþÞ ðþÞ

ðÞ

ðþÞ

ðþÞ

ðþÞ

ðÞ

ðþÞ

ðÞ

ðÞ

ðÞ

ð2Þ ð3Þ

where d[La] and d(pH) represent the uncertainties in La concentration and pH, respectively. Thus there was no experimental confirmation of predictions of levels of the other substrates. The uncertainty in La was large, reflecting the variability in the La-pH data used. 3.2. Model construction The model was based on the schema shown in Fig. 1 and its associated reactions (à1–à7). It included time (in minutes) as an independent variable in the model, and assumed constant muscle temperature. The concentrations of substrates consumed or produced by the reactions were included as dependent variables, all with units of mol L1. However, glycogen concentration was omitted as a variable from the model, based on the assumption that glycogen availability would not be a limiting factor of ATP synthesis in muscle from well-fed and unstressed animals. A set of differential equations was written expressing the rates of change of the substrate concentrations, in terms of the reaction rates. The reaction rates were quantified in terms of the substrate concentrations and model parameters. The differential equations were solved numerically to predict how substrate profiles changed over time from initial starting values. The substrates included in the model are listed in Table 1 along with their mathematical symbols and relevant reactions. From a modelling point of view, these ten substrates (C, CP, ATP, ADP, AMP, IMP, In, H+, Pi and La) are assumed to form a closed system over the period of interest, and this allows some substrates to be expressed in terms of others, as follows. Assuming that creatine and CP transform from one to the other, the sum of their concentrations in mol L1 (respectively C and Cp) will be a constant, v, say, with units mol L1. Thus creatine concentration can be expressed in terms of CP concentration:

C ¼ v  Cp:

Table 1 Important substrates, their chemical (Chem) symbols, mathematical symbols for concentration (Conc), and corresponding reactions. ra is the rate of ATP hydrolysis; rg is the rate of gylcolytic activity; rm is the rate of the myokinase reaction; ri is the rate of dephosphorylation activity; rd is the rate of deaminase activity; rc is the rate of the creatine kinase reaction; ro is the rate of aerobic ATP synthesis. The superscript after each reaction indicates the direction of the change in substrate caused by that reaction. A + or  indicates, respectively, production or consumption, while ± indicates that at times the reaction consumes the substrate and at other times produces it.

ð4Þ

Further, define ‘‘total phosphate” as the sum of Pi and phosphate bound in CP, ATP, ADP, AMP and IMP, denoting this Ptot with units mol L1. Since the system is closed, total phosphate will be a con-

stant, and thus IMP can be expressed in terms of Pi, CP, and adenonucleotides concentrations:

I1 ¼ Ptot  P  C P  A1  2A2  3A3 ;

ð5Þ

where I1, A3, A2, A1, are respectively the concentrations of IMP, ATP, ADP and AMP in mol l1 Similarly, the adenonucleotides and inosine transform from one to the other. Thus the sum of their concentrations will be a constant, Atot, say, with units mol L1. Thus inosine concentration, I0 (mol L1), can be expressed in terms of the adenonucleotide concentrations:

I0 ¼ Atot  A3  A2  A1  I1 :

ð6Þ

Thus the dimension of the system has effectively been reduced from ten to seven variables since C, I1 and I0 are always known from other substrate values by using Eqs. (4)–(6). The net flow of H+ into solution is obtained by summing the contributions of the participating reactions. Let H (mol L1) be the concentration of H+ in solution and RH (mol L1 min1) be the rate at which H+ is produced by chemical reactions. The net change in H+ is then

dH ¼ aRH ; dt

ð7Þ

where a reflects the effect of hydrogen buffers. The conventional measure of buffering, b (used for example by Vinnakota et al., 2006) is related by b = 2.3H/a. In a modelling context, a is the more natural measure to use. A significant source of hydrogen buffering will come from phosphates. Additionally we considered the effect of cytosolic buffering which is largely due to histidine residues but also in this context includes bicarbonates and other sources of buffering, such as ATP and ADP. Although ATP and ADP in muscle are predominantly bound to magnesium or potassium, their non-bound forms are potentially active as hydrogen buffers in the physiological pH range (Kushmerick, 1997). The contribution of water to hydrogen buffering is negligible in physiological pH ranges and can be ignored. An expression for a can be obtained by examining the equilibrium nature of hydrogen buffers:

1=a ¼ 1 þ

PkP 2

ðH þ kP Þ

þ

BkB ðH þ kB Þ2

;

ð8Þ

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where P (mol L1) is the concentration of phosphate, B (mol L1) is the effective ‘‘concentration of cytosole” acting as a buffer, and kP and kB (both mol L1) are respectively the hydrogen-buffer constants of Pi and cytosole. Thus the system can be captured by the following system of seven differential equations, which are expressed in terms of the rates of the seven reactions:

dC p ¼ r c ; dt dA1 1 ¼ rm  rd ; 2 dt dA2 ¼ r c  r g  r o  r m þ r a ; dt dA3 1 ¼ rc þ rg þ ro þ rm  ra ; 2 dt dP ¼ ra þ ri  rg  ro ; dt     dH 2 ¼ a c ra  rg  ro þ ri  rc  rd þ rg ; dt 3 dL 2 ¼ rg ; dt 3

ð9Þ ð10Þ ð11Þ ð12Þ ð13Þ

ð17Þ

The first term in each of Eqs. (16) and (17) represents the forward rate of the reaction (with rate constant subscripted with an ‘‘f”), and the second term represents the backward rate (with rate constant subscripted with a ‘‘b”). In isolation from other reactions, the forward and backward rates will reach a balance to give the following equilibria:

ð18Þ ð19Þ

where kc (mol1 L) and km (unitless) are respectively the equilibrium constants of the creatine kinase and myokinase reactions. Note that kc is related to the ‘‘observed” or ‘‘apparent” equilibrium constant, kobs by kobs = kcH (Veech, Lawson, Cornell, & Krebs, 1979). In the model we only use kc which does not vary with pH, in contrast to kobs which does. In a system of reactions, the above equilibria will be approximated only if their equilibrium reactions occur on a very much faster time scale than the other reactions involved. Veech et al. (1979) found that the in vivo activity of the myokinase reaction is near equilibrium, and is close to in vitro activity. The creatine kinase equation, too, can be treated as close to equilibrium (Korzeniewski & Zoladz, 2001). Thus Eqs. (18) and (19) can be used to express ADP and AMP in terms of other substrates:

A3 C ; kc C p H

       1 n  2 rc ¼ 1  1 ra  ro  rg þ n2 r g þ cr i  ðn2 þ n3 Þr d ; f 3 f

where

kmf A22

A2 ¼

reducing the dimension of the system to five differential equations by eliminating the need for of Eqs. (10) and (11). Expressions (16) and (17) would always equate to zero under approximations (20) and (21), and thus are no longer suitable for expressing the rates of the creatine kinase and myokinase reactions. The typical approach in this situation would be to eliminate rc and rm from the set of differential equations by judicious definition of new variables. This is straightforward for Eqs. (10)–(12), but problematic for Eq. (9) and (14). Thus, an alternative approach was taken. Eqs. (18) and (19) were differentiated and Eqs. (4), (9)–(15) substituted in, to express rc and rm in terms of the other reaction rates:

ð15Þ

ð16Þ

kcf A3 C ¼ ; kcb C p A2 H kmf A3 A1 ¼ 2 ; km  kmb A2

ð21Þ

ð14Þ

kcf C p A2 H  kcb A3 C;

kc 

km A22 : A3

      1 2 g1 ra  ro  rg  g2 rg  cri þ ðg2 þ g3 Þrd ; rm ¼ f 3

3.2.1. Equilibrium reactions Two of the reactions are (reversible) equilibrium reactions in vitro. The first of these is the creatine kinase reaction in which CP buffers ATP (Chase & Kushmerick, 1995; Connett, 1988; Soussi et al., 1993). The second is the myokinase reaction, involving the resynthesis of ATP and the production of AMP under conditions where ADP is accumulating. Typically, the rates of equilibrium reactions are modelled using a mass-balance approach. Thus the rate of the creatine kinase and myokinase reactions would be represented as, respectively,

 kmb A3 A1 :

A1 ¼

ð20Þ

ð22Þ ð23Þ

1 f ¼ QV  HðA2 þ 2A3 Þ2 ; 2 n1 ¼ V n1 ;

n2 ¼ aA2 A3 V;

n3 ¼

1 A2 þ 2A3 HA23 ; 2 k m A2

g1 ¼ ðA2 þ 2A3 Þn1 ; g2 ¼ aA2 A3 ðA2 þ 2A3 Þ; g3 ¼ 2

ð24Þ

ðC þ C p ÞA3  aA2 A3 ðc  1Þ; n1 ¼ kc C 2p V¼

QA23 ; k m A2

1 1 A23 ; A2 þ 2A3 þ 2 2 km A2

Q ¼ ðA2 þ A3 ÞH þ aA2 A3 þ

ðC þ C p ÞA23 kc C 2p

;

with A2 and C given by respectively (20) and (4). 3.2.2. ATP hydrolysis Resting, living, normoxic muscle has an energy requirement for maintenance, and this ensures a basal rate of ATP hydrolysis. One might assume that unexercised anoxic muscle will have similar energy requirements. However, an assessment of the rate of ATP hydrolysis from the rate of La accumulation and ATP disappearance confirmed previous observations (Bendall, 1973) that ATP hydrolysis is initially increased and subsequently declines in response to the disappearance of ATP. The majority of the ATPase activity in resting anoxic muscle appears to be myosin ATPase (Hamm, Darymple, & Honikel, 1973). Therefore it is reasonable to assume that the increased ATP hydrolysis is associated with increased free intracellular Ca2+ concentration (Fralix, Murphy, London, & Steenbergen, 1993; Marban et al., 1990). The increased Ca2+ may originate from the sarcoplasmic reticulum (Ruff & Weissman, 1995; Westerblad & Allen, 1993; Wolosker et al., 1997) or from mitochondria which act as a low affinity, high capacity Ca2+ sink (Altschuld, 1996; Gunter, Buntinas, Sparagna, & Gunter, 1998). In either case, the accumulating Pi offers the most likely metabolic change responsible for increased Ca2+, either through effects on the sarcoplasmic reticulum Ca2+ATPase (Steele, McAinsh, & Smith, 1996; Stienen, Papp, & Zaremba, 1999; Zhu & Nosek, 1991) or on the mitochondrial permeability transition (Gambassi et al., 1993; Gunter & Pfeiffer, 1990; Zoratti & Szabo, 1995). The subsequent decline of ATPase activity, leading ultimately to its cessation at rigor mortis, was attributed to

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declining ATP concentration (Glyn & Sleep, 1985) and to acidosis (Blandchard, Pan, & Solaro, 1984; Chase & Kushmerick, 1988), although the latter appears to be minimal at 37 °C (Pate, Bhimani, Franks-Skiba, & Cooke, 1995; Wiseman, Beck, & Chase, 1996). To model this behaviour, the rate of ATP hydrolysis was written as a ‘‘basal” Michaelis–Menten term multiplied by pH, plus a second, Ca2+-associated term proportional to both the fourth power of ATP concentration and Pi concentration above a Pi threshold. Better model performance was obtained if this second term was zero below that Pi threshold. The use of a fourth power was an arbitrary choice which ensured a rapid decline as ATP levels fell. The rate of ATP hydrolysis was represented as

ra ¼

EA43 P

!

aA3 ; logð1=HÞ þ am þ A3 1 þ ðap =PÞ8

  35 gp ðPPi Þ egL   ðr a  r o Þ; rg ¼ 1  e 38 1 þ ðL=g Þ8

ð27Þ

l

ð25Þ

a (mol L1 min1) governs the magnitude of the basal term; am (mol L1) is the Michaelis–Menten constant; E (mol4 L4 min1) governs the magnitude of the calcium efflux; the denominator of the second term effects an on-off, phosphate-dependent switch with ap (mol L1) being the threshold phosphate level. The denominator of the second term switches the term ‘‘on” as Pi increases above (or ‘‘off” as Pi decreases below) ap, with the switching rate governed by the arbitrarily chosen eighth power. 3.2.3. Aerobic ATP synthesis Under normoxic conditions in resting muscle the rate of aerobic ATP synthesis will equal the rate of ATP hydrolysis. With anoxia caused by ischaemia or death, the rate of aerobic ATP synthesis will rapidly decline to zero with the fall in oxygen levels. We assumed that this decline will follow the pattern of a slow-moving switch, as the system moves to an anoxic state. Letting the switching time be tox minutes post mortem we write

r o ¼ r a =ð1 þ expðt  t ox ÞÞ:

Bergström, & Hultman, 1988; Spriet, 1990). As phosphate levels increase, glycolysis can increase up to the difference in ra and ro. The decline and eventual cessation of glycolysis, preceding rigor mortis, is attributed to the accumulation of La. La dehydrogenase equilibrium serves to regenerate nicotinamide adenine dinucleotide (NAD+), but the eventual accumulation of La forces a depletion of NAD+, depriving the glyceraldehyde 3-phosphate dehydrogenase reaction of the necessary reducing agent. Combining these considerations gives the following expression for glycolysis:

ð26Þ

Eq. (26) would allow the equivalent of tox + ln (1 + exp(tox)) minutes of post mortem aerobic ATP synthesis. Bendall (1973) calculated that the oxygen present in normoxic, resting muscle is sufficient for buffering about 2 min worth of the resting-rate ATP hydrolysis. Rather than using Bendall’s figure, tox was estimated by fitting to data. 3.2.4. Glycolysis The glycolytic pathway consists of a sequence of ten reactions involving a number of substrates and enzymes. To represent these mechanistically would require ten equations, plus the inclusion of many variables and parameters to represent the substrates and enzyme actions involved. We obviated this need by taking an empirical approach to specify the rate of glycolysis, rg. Since glycolysis functions to meet the shortfall between the rate of ATP hydrolysis, ra and aerobic ATP synthesis, ro, we assumed that rg will be proportional to the shortfall ra  ro by some function whose construction is developed below. Glycolysis and aerobic ATP synthesis both share the Embden– Myerhoff pathway which gives a net generation of three moles of ATP and two moles of pyruvate per mole of glycogen. Under normoxic conditions these two moles of pyruvate are oxidised via the Krebs cycle, generating a further 35 moles of ATP. Thus glycolysis, rg, is 3/38 as efficient as aerobic ATP synthesis, ro, in terms of ATP resynthesis. Since O2 levels do not affect the Embden–Myerhoff pathway (Arthur, Hogan, Bebout, Wagner, & Hochachka, 1992) it is postulated that for every unit decline in ro caused by falling O2, there will be an increase in rg of 3/38. Furthermore, it is postulated that phosphate availability is a limiting factor in the phosphorylase reaction converting glycogen to glucose-1-phosphate (Chasiotis, 1983, 1985; Ren, Chasiotis,

where gp (mol1 L) governs the increase in rg with phosphate concentration, Pi (mol L1) is the initial phosphate concentration; g (mol1 L) reflects the damping effect of accumulating La on glycolysis, and gl (mol L1) is the La threshold for cessation of glycolysis. 3.2.5. AMP deaminase AMP deaminase allows for maximum energy production by preventing the accumulation of AMP, thereby relieving the inhibition of the myokinase reaction by AMP (Coffee & Solano, 1977). The deaminase reaction is therefore the predominant route for adenine nucleotide depletion (Idström et al., 1990; Sahlin & Broberg, 1990). The stimuli for activation of AMP deaminase may include cellular acidosis (Dudley & Terjung, 1985; Soussi et al., 1993) or elevations in ADP or AMP (Dudley & Terjung, 1985; Sahlin & Broberg, 1990). However, Coffee and Solano (1977) found AMP deaminase activity to be insensitive to the total adenylate pool size, and concluded that the most important regulatory factors of AMP deaminase in vivo are the relative levels of the adenine nucleotides. Certainly, expressing deaminase activity as the ratio of AMP to ADP gave better behaviour than the other alternatives, and thus the rate for AMP deaminase was written as

rd ¼ dA1 =ðdm þ A2 Þ

ð28Þ

with A2 and A1 being obtained from (20) and (21). Here d has units of mol L1 min1 and dm (mol L1) acts in a Michaelis–Menten fashion to prevent rd being too high for very small A2. 3.2.6. Dephosphorylase While restitution of normoxia allows the conversion of IMP back to AMP via the purine nucleotide cycle, prolonged anoxia results in irreversible damage caused by the dephosphorylation of IMP to inosine, further increasing phosphate levels. This reaction is expected to behave in a Michaelis–Menten fashion (Spychala, Madrid-Marina, Nowak, & Fox, 1989; Tanaka, Hasan, Hartog, van Herk, & Wever, 2003) and additionally is likely to be caused by extreme acidosis

ri ¼ iI1 H2 =ðim þ I1 H2 Þ;

ð29Þ 2

2

1

2

2

1

where i has units mol L min ; im (mol L min ) is the Michaelis–Menten parameter for the equation, and the square of H+ concentration ensures that the rate is relatively insignificant for higher pH values when compared with lower pH values. I1 is obtained by substitution of Eq. (5). 3.3. Numerical solution and fitting The post mortem concentrations over time of CP, Pi, H+, ATP and La can be obtained by solving the five differential equations (9) and (12)–(15) in conjunction with Eqs. (22)–(29) which specify the rates of the reactions. The identities (4)–(6), (20) and (21) will give concentrations for creatine, IMP, In, ADP and AMP, and additionally are used in the equations for the reaction rates.

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This system of equations is not analytically solvable, and thus was solved numerically, using Hindmarsh’s LSODE package (Hindmarsh, 1983). The model was fitted to data for phosphate, ATP, CP, H+ and La concentrations over time, separately for the muscles A, B and C. The data used for fitting had been taken at seven minute intervals, starting from 15.5 to 24.5 min post mortem (depending on the animal). The number of data values for A, B and C were respectively 110, 100, and 70 (measured at 22, 20 and 14 time points). For each muscle, model simulation was performed over the period from death to the time of the last measurement and used to predict values for the measured concentrations at the corresponding time points. Modelling the pre-data period was for the sake of completeness since the aim of the model was to predict the effects of anoxia from its onset. A genetic algorithm was used to find values for the initial concentrations and for parameters in the rate equations that minimised the sum of squares of the differences between the measured and predicted concentrations. Many of the parameters in the model would in principle show biological variation between animals, animal condition and muscle fibre types. This is because these parameters govern the rates of chemical reactions which are dependent on enzymes or catalysts whose concentrations are variable. Thus they along with initial concentrations for the five measured metabolites were expected to be different for different data sets, and fitted separately. However, the phosphate buffering constant, kP, and the total creatine concentration, v, were assumed to be constant across data sets and were set using values from the literature, and are given in Table 2. The equilibrium constants for the creatine kinase and myokinase reactions vary considerably with the concentration of free Mg2+ (Lawson & Veech, 1979), which itself will vary during the post mortem period. Thus while authors such as Korzeniewski and Zoladz (2001) specified a priori values for these equilibrium constants, we chose to obtain them through data fitting. Our values will therefore imply an effective average for free Mg2+ concentration over the data period. Additionally it was assumed that the initial concentrations of In and IMP (respectively I0i and I1i) were zero for each data set. For each data set, initial values for AMP and ADP (respectively A1i and A2i) were calculated from the fitted initial values of CP, H+, ATP (respectively CPi, Hi and A3i) using Eqs. (4), (20) and (21). The fitted initial values for La and Pi (respectively Li and Pi) together with I0i = 0, I1i = 0, CPi and A3i were used to calculate Atot and Ptot from Eqs. (5) and (6) for each data set. These estimates of Atot and Ptot were then subsequently used to estimate I0 and I1during the simulation. 4. Results For each of the three muscles, the model was fitted to the NMR data within experimental error (Figs. 2–4). It should be noted that the relatively fast reaction rates are a consequence of maintaining the muscle at a relatively high temperature of 35 °C which is close to that of the live muscle. Values for the fitted initial conditions are shown in Table 3, and fitted parameter values are shown in Table 4. Initial values for sub-

Table 2 Model parameters set constant across the data sets, with values obtained from Kushmerick (1997). Symbol

v kP

Value

Interpretation 1

4.4e2 mol L 2.00e7 mol L1

Total creatine Phosphate H-buffering constant

139

strates varied between the sets, as would be expected owing to variation in the animals’ physiology and stress levels (the relatively low pH and high La for muscles B and C indicate a degree of agitation before death). The fitted parameter values for each muscle were used to simulate its muscle energy-metabolism until 360 minutes post mortem, well after the data collection period. The simulated reaction rates are shown in Figs. 5–7. The rates of ATP hydrolysis, the creatine kinase reaction, glycolysis and aerobic ATP synthesis (respectively ra, rc, rg and ro) were, qualitatively, roughly similar for all three muscles, although the peaks in ra and rg were less pronounced for muscle C. In quantitative terms, muscle A had the higher reaction rates and muscle C the lower. While the rates of deaminase activity, dephosphorylation activity and the myokinase reaction (respectively rd, ri and rm) showed defined peaks for muscle A, they more or less plateaued for muscle B, with muscle C being intermediate in this regard. rd and ri tracked each other very closely for muscle B and were reasonably close for muscle A. However for muscle C, ri was up-regulated significantly later than rd, after which it rapidly increased and became significantly greater. There was considerable difference between the muscles in the slow-down of reactions around rigor (based on an arbitrary cutoff of 105 mol L1 of ATP): Muscle A had a relatively slow decline while muscle B showed an abrupt cessation of reactions. Muscle C was similar to muscle B in this respect except for ri which persisted for about 45 min after the other reactions had ceased. Simulated adenosine nucleotide concentrations for the three muscles are shown in Figs. 8–10. In the simulations the pool of adenonucleotides was largely (98%) in the form of ATP, with small amounts (roughly 2%) of ADP. By the end of the simulation all adenonucleotides had largely vanished, having been converted into inosine. However the adenonucleotide profiles between these two time points differed vastly between the three animals. For muscle A, ADP and AMP had peaks in concentration of similar magnitude (though AMP lagged ADP), and significantly dominated IMP which had a much flatter profile which peaked 15–20 min before ADP. For muscle B, ADP dominated AMP which in turn dominated IMP, with the peaks in ADP and AMP concentration occurring within a few minutes of each other, and IMP peaking about 40 minutes before ADP. In contrast for muscle C, while ADP dominated AMP, IMP reached a much higher concentration than (and peaked about 20 min before) ADP, with AMP showing no discernible peak. For muscle B, all the adenonucleotides had been essentially depleted at rigor. For muscle C, IMP was the only significant adenonucleotide remaining at rigor and was steadily dephosphorylated to inosine within an hour. By contrast for muscle A, AMP was near its peak concentration at rigor, with much lesser amounts of ADP and IMP present; in this case the adenonucleotides persisted until the end of the simulation, and inosine was still asymptoting to its final concentration. The simulated time-course for pH (including ultimate pH) is shown in Fig. 11 for all three muscles. For all three muscles very slight alkalinisation was predicted to occur for the first few minutes after the onset of anoxia – this is likely due to released phosphates binding with H+. The pH for muscle C had a relatively low initial value of 6.70, and then a more-orless linear decline for a period of about 2.5 h before abruptly plateauing at 5.60. The pH for muscle B had an initial value of 6.81 and gradually approached its (relatively high) final value of 6.07. Surprisingly in the simulation for muscle A, pH (initially 7.02) reached a low of 5.60 at 215 min post mortem and then increased slightly before plateauing at a final value of 5.73 about 415 min post mortem.

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Metabolite and pH profile over time: Muscle A CP

Phosphate 50 40

8

30

6 20 4 10

2 0

0 pH

7

Lactate 100

pH

6.8 6.6

80

6.4

60

6.2

40

6

Lactate (mM)

ATP (mM)

10

CP & Phosphate (mM)

ATP 12

20

5.8 5.6

0 0

20

40

60

80

100

120

140

160

Time post mortem/minutes Fig. 2. Model prediction versus experimental data for muscle A. NMR-measured values for ATP, CP, pH and phosphate, and estimated lactate (from pH) are indicated within experimental uncertainty by the vertical bars. Curves intersecting these bars represent the model predictions of the corresponding variable.

Metabolite and pH profile over time: Muscle B CP

Phosphate 50 40

8

30

6 20 4 10

2 0

0 pH

7

Lactate 100

pH

6.8 6.6

80

6.4

60

6.2

40

6

Lactate (mM)

ATP (mM)

10

CP & Phosphate (mM)

ATP 12

20

5.8 5.6

0 0

20

40

60

80

100

120

140

160

Time post mortem/minutes Fig. 3. Model prediction versus experimental data for muscle B. NMR-measured values for ATP, CP, pH and phosphate, and estimated lactate (from pH) are indicated within experimental uncertainty by the vertical bars. Curves intersecting these bars represent the model predictions of the corresponding variable.

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Metabolite and pH profile over time: Muscle C CP

Phosphate 50 40

8

30

6 20 4 10

2 0

0 pH

7

Lactate 100

pH

6.8 6.6

80

6.4

60

6.2

40

6

Lactate (mM)

ATP (mM)

10

CP & Phosphate (mM)

ATP 12

20

5.8 5.6

0 0

20

40

60

80

100

Time post mortem/minutes Fig. 4. Model prediction versus experimental data for muscle C. NMR-measured values for ATP, CP, pH and phosphate, and estimated lactate (from pH) are indicated within experimental uncertainty by the vertical bars. Curves intersecting these bars represent the model predictions of the corresponding variable.

Table 3 Initial values (mol L1) for substrates in the model, for each data set A–C. Initial values for the five ‘‘independent” substrates (CPi, Pi, Hi, A3i and Li) were obtained by fitting, while those for the four ‘‘dependent” substrates were estimated or calculated. Symbol

A

C pi Pi Hi pHi A3i Li I1i I0i A2i A1i

2.35e02 2.32e02 3.09e03 3.69e03 9.66e08 1.55e07 7.02 6.81 1.03e02 9.29e03 8.53e03 3.43e02 0 by assumption 0 by assumption Calculated from Eq. (22) Calculated from Eq. (23)

B

C

Substrate

2.02e02 3.01e03 1.99e07 6.70 9.56e03 4.69e02

Creatine phosphate Phosphate H+ ions pH ATP Lactate IMP Inosine ADP AMP

5. Discussion The surprising post-rigor increase in pH for muscle A occurred during a period when the only proceeding reactions which affect H+ concentration were AMP deamination (rate rd) and IMP dephosphorylation (rate ri). Although the rates of these reactions were very close to each other, the H+ consumption of AMP was not matched by the net H+ production of IMP dephosphorylation. This was because the low post-rigor pH ensured that a large fraction of the phosphate liberated by IMP dephosphorylation was released as 2 + PO1 4 rather than PO4 , thus binding much of the H which was also produced. This can be seen by examining Eq. (1). Small pH rises have sometimes been observed in meat during storage (Carse & Locker, 1974; Moore & Gill, 1987) but not explained. Taking into account temperature (and thus reaction-rate) differences, the modelling results may explain this occasionally observed rise. In the construction of this model, a number of simplifications and assumptions were made in both the conceptual and mathe-

matical representations of anoxic muscle energy-metabolism. Functional forms for the rates of some reactions were obtained by appealing to general patterns based on experimental evidence in the literature. Well-established mathematical relationships for equilibrium reactions were used for other reactions. The exercise of parameterising the model was to calibrate the model to reproduce a range of different metabolic responses by muscle to anoxia, rather than to estimate ‘‘true” parameter values within a confidence interval. Indeed, taking averages can sometimes obscure individual effects. While we used only three animals in the experiment, we found quite a range in the behaviour of their muscles, which the model was able to reproduce. Of course the use of more animals would have revealed a greater variety of behaviours and further tested the model’s robustness. The glycogen level depends on the nutritional status of the animals (Immonen & Puolanne, 2000; Immonen, Rasmussen, Hissa, & Puolanne, 2000). The muscles used in the experiment were assumed to have a high glycogen content typical of well-fed and unstressed sheep, and thus glycogen concentration was omitted as a limiting variable from the model. However, the high final pH of muscle B (see Fig. 11) was characteristic of a low glycogen level at slaughter. While the data looked superficially the same (Figs. 2–4), there were differences in the underlying dynamics (cf Figs. 5–7) and consequently the long-term metabolite profiles (cf Figs. 8–10). A major reason for these differences was that muscle B had a notably smaller value of im (Table 4) which allowed IMP dephosphorylase to activate and drain adenonucleotides at a much higher pH. While it is possible this was experimental variation, it is more likely that the muscle B had a much lower starting glycogen than muscles A and C, with this difference being expressed by some of the variation in the fitted parameters. The differences in pH between muscles were likely to be responsible for their different times in achieving rigor mortis – muscles A and C took longer than muscle B which was already

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Table 4 Parameter values obtained by data fitting. Abbreviations and symbols in the Interpretation column are: Ca is calcium; ATP is adenosine triphosphate; Pi is phosphate; La is lactate; rd is the rate of adenosine monophosphate deamination; rg is the rate of glycolysis; ri is the rate of inosine dephosphorylation. Symbol

Data set

B mol L1 E mol4 L4 min1 a mol L1 min1 am mol L1 ap mol L1 d min1 dm mol L1 g mol1 L gl mol L1 gp mol1 L i mol L1 min1 im mol3 L3 kB mol L1 kc mol1 L km tox

Interpretation

A

B

C

1.33e01 7.29e+06 7.57e05 3.45e05 4.87e03 9.29e04 3.47e02 2.19e03 1.08e01 8.92e+01 2.79e04 2.78e15 1.24e07 4.32e+08 4.37e01 1.60e+00

9.48e02 1.03e+07 4.17e05 3.44e05 2.53e03 4.35e04 1.44e04 4.89e02 9.02e02 8.00e+01 2.88e04 6.52e17 1.42e07 3.20e+08 5.38e01 1.38e+00

6.13e02 5.03e+06 8.90e05 7.36e05 8.68e03 9.68e04 6.66e04 1.21e03 1.08e01 1.00e+02 2.11e04 8.78e15 3.94e07 3.03e+08 1.01e+00 1.75e+00

Cytosole ‘‘concentration” Ca efflux ATP hydrolysis rate ATP hydrolysis rate limiting constant Pi threshold for Ca efflux rd rate constant rd limiting constant La inhibition of rg La cut-off for rg Pi threshold for rg ri rate constant ri limiting constant Cytosolic-buffering constant Creatine kinase constant Myokinase constant Aerobic–anaerobic switch time

Metabolite Flux (mM/min)

Simulated reaction rates: Muscle A ra rc

rigor

1.5

rg r

o

1

0.5

0

rigor

Metabolite Flux (mM/min)

0.25

r

d

ri

0.2

rm 0.15 0.1 0.05 0 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 5. Predicted reaction rates for muscle A extrapolated past the rigor event. ra is rate of ATP hydrolysis; rg is gylcolytic activity; rm is the rate of the myokinase reaction; ri is dephosphorylation activity; rd is deaminase activity; rc is the rate of the creatine kinase reaction; ro is the rate of aerobic ATP synthesis.

becoming asymptotic at 50 min (Fig. 11). Additionally pH will have affected the nature of the phosphate compounds at the entry into rigor mortis. These effects were reflected in the predicted adenosine nucleotide levels, in particular the early onset of inosine at elevated pH. The elevated pH would have also had consequences on the development of attributes such as flavour of meat, since IMP has a favourable flavour and further degradation produces hypoxanthine that has a bitter flavour characteristic of high pH meat (Honikel, 2004; Lawrie, 1998). The conversion of IMP to hypoxanthine is faster with elevated pH (Honikel, 2004; Lawrie, 1998). While the time period covered in this study was short, it occurred at a higher temperature than normally encountered and early in-

creases in production of IMP (and inosine) can be seen in Fig. 9 where the ultimate pH is elevated. Muscle fibre type affects the rate of glycolysis (Devine, Ellery, & Averil, 1984) and so influences the time course of rigor mortis. Most muscles used for meat are mixed fibre types (Davies, 1989) and additionally the large numbers of individual muscle fibres involved will exhibit natural variation in their initial glycogen content. Thus for some individual muscle fibres, as must always occur in a population, the reactions would already have run to completion before rigor mortis was exhibited in the whole muscle. The model did not address differences between fibres within a muscle, and while the model parameters and endpoints depend on the variation between fibres they are representative of an average.

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Metabolite Flux (mM/min)

Simulated reaction rates: Muscle B r

a

rigor

1.5

r

c

r

g

ro

1

0.5

0

Metabolite Flux (mM/min)

0.25

rigor

rd ri

0.2

r

m

0.15 0.1 0.05 0 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 6. Predicted reaction rates for muscle B extrapolated past the rigor event. ra is rate of ATP hydrolysis; rg is gylcolytic activity; rm is the rate of the myokinase reaction; ri is dephosphorylation activity; rd is deaminase activity; rc is the rate of the creatine kinase reaction; ro is the rate of aerobic ATP synthesis.

Metabolite Flux (mM/min)

Simulated reaction rates: Muscle C r

a

rigor

1.5

rc r

g

ro

1

0.5

0

rigor

Metabolite Flux (mM/min)

0.25

rd ri

0.2

r

m

0.15 0.1 0.05 0 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 7. Predicted reaction rates for muscle C extrapolated past the rigor event. ra is rate of ATP hydrolysis; rg is gylcolytic activity; rm is the rate of the myokinase reaction; ri is dephosphorylation activity; rd is deaminase activity; rc is the rate of the creatine kinase reaction; ro is the rate of aerobic ATP synthesis.

Ten substrates were considered important players in anaerobic muscle energy-metabolism in this model. The NMR tech-

niques that were used could not measure five of them, leaving no data for concentrations of ADP, AMP, IMP and ino-

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Simulated adenonucleotide concentrations: Muscle A 10

mM

8 6

ATP Inosine

rigor

4 2 0 3.5

ADP IMP AMP

3

mM

2.5

rigor

2 1.5 1 0.5 0 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 8. Predicted adenosine nucleotide levels for muscle A extrapolated past the rigor event. Arrows indicate time events.

Simulated adenonucleotide concentrations: Muscle B 10

mM

8 6

ATP Inosine

rigor

4 2 0 3.5

ADP IMP AMP

3

mM

2.5 2

rigor

1.5 1 0.5 0 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 9. Predicted adenosine nucleotide levels for muscle B extrapolated past the rigor event. Arrows indicate time events.

sine. However, by considering the metabolic system as closed, by appealing to principles of conservation, and by using equilibrium reaction identities, we were able to express to ADP, AMP, IMP and inosine in terms of the other substrates and reduce the

dimension of the system to the five variables that had been measured. Inosine will further degrade to hypoxanthine (Tikk et al., 2006) but this process was not included in the model because of the lack

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Simulated adenonucleotide concentrations: Muscle C 10

mM

8 6

rigor

ATP Inosine

rigor

ADP IMP AMP

4 2 0 3.5 3

mM

2.5 2 1.5 1 0.5 0 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 10. Predicted adenosine nucleotide levels for muscle C extrapolated past the rigor event. Arrows indicate time events.

Simulated pH: Muscles A, B and C 7.2 7 A B C

6.8 6.6

pH

6.4 6.2 6 5.8 5.6 5.4 5.2 0

50

100

150

200

250

300

350

400

Time post mortem/minutes Fig. 11. Predicted pH in muscles A–C, extrapolated past the rigor event. Arrows indicate time events.

of information. Thus inosine in the model may be considered a proxy for a combined inosine-hypoxanthine pool. Logistical difficulties prevented NMR measurement of data for the immediate post mortem period. This and the absence of data

for times preceding and following rigor prevented rigorous testing of the model over the whole range. The equilibrium ratios for the creatine kinase and myokinase reactions were treated as constant, though some changes in their

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values can be expected from the increased ionic strength that develops in muscle during prolonged ischaemia (Kushmerick, 1997). Creatine kinase is sensitive to changes in free [Mg2+] (Lawson & Veech, 1979) which can be expected to increase as ATP becomes depleted (Lamb & Stephenson, 1992; Westerblad & Allen, 1994) and Mg2+ bound to ATP is released. Furthermore the model assumed that all the measured ATP was in a salt form available to the enzymes; in reality, the ATP will be in bound to a variety of cations besides Mg2+ as well as in the free form, though the Mg2+ form can be expected to dominate. The buffering capacity of the muscle tissue was estimated as a single parameter. This value will include primarily the histidine residues present in tissue proteins and dipeptides (Curtin, 1986; Hochachka & Mommsen, 1983) as well as bicarbonate which accumulates during the initial period of ischaemia. Buffering effects associated with changes in metabolites were limited to changes in the concentrations of Pi. Although ATP buffers pH through the exchange of the Mg2+ ligand for H+, the effects are small since most of the ATP is Mg2+-bound, and the accumulating H+ will, to some extent, be offset by accumulating Mg2+. Direct measurement of changes in Mg2+ during metabolic inhibition did not reveal substantial changes until near complete depletion of ATP (Jeacocke, 1993). The detailed examination of the instantaneous changes in the rigor process using NMR and modelling is of value when one wants to understand how such processes can affect the quality of meat, in order to improve it. The present study is merely a start following from early studies of the rigor process using NMR (Vogel et al., 1985). That pre-rigor factors affect meat quality has been established by studies on electrical stimulation that accelerates glycolysis (Chrystall & Devine, 1978; Devine, Hopkins, Hwang, Fergusson, & Richards, 2004; Hollung et al., 2007; Hwang, Devine, & Hopkins, 2003; Toohey, Hopkins, Stanley, & Nielsen, 2009), and the tenderness of meat clearly depends on many factors modifying the rigor process. Measurement of meat quality using techniques such as near infrared spectroscopy are predicated on the changes throughout the rigor and ageing processes (McGlone, Devine, & Wells, 2005; Rosenvold et al., 2009) that can be modified by various processing technologies. While it is known that meat toughens through cold shortening (Locker, Davey, Nottingham, Haughey, & Law, 1975) and that this can be avoided by accelerating rigor mortis using electrical stimulation, further studies have shown that this is only part of the improvement, as the accelerated glycolysis increases tenderness above that of non-stimulated muscle (Rosenvold et al., 2008) even under conditions where cold shortening cannot occur. Breed differences in tenderness also diminish dramatically with electrical stimulation and accelerated glycolysis (Ferguson, Jian, Hearnshaw, Rymill, & Thompson, 2000; Hearnshaw et al., 1998). The tenderisation process is mediated through proteinases such as calpains (Kemp, Sensky, Bardsley, Buttery, & Parr, 2010). The calpain activity in turn appears to be influenced by the rate at which muscle enters rigor mortis, and at the accelerated rates from electrical stimulation is an important contributor to tenderness (Rosenvold et al., 2008). Understanding the kinetics of the rigor process and how it influences glycogen levels is therefore important to direct further improvements in processing. This study showed that during anoxia at the relatively high temperature of 35 °C, significant changes in ATP levels occurred within 100 min, leading rapidly to the production of other metabolites. Such rapid changes would occur in living muscles with ischaemia, and indeed would be more rapid if there were low glycogen before the ischaemia started. While the present study used skeletal muscle, parallel changes would take place in cardiac or other tissue. Hence trends and differences relating to muscle condition and ultimate pH are broadly transferable to a medical setting, for example in the case of a cardiac infarction or a stroke.

With ATP unavailable in these fibres, ongoing sarcoplasmic elevations of calcium would activate catabolic systems, permanently destroying most of these fibres (Kemp et al., 2010). Thus the potential for their recovery after re-oxygenation would be low.

Acknowledgements We wish to thank S. Lovatt, M. North, P. O’Callaghan, D. Pulford and K. Rosenvold for helpful comments, suggestions and editorial contributions. This work was funded by The Foundation for Research, Science and Technology, New Zealand, Contract MRI602.

References Altschuld, R. A. (1996). Intracellular calcium regulatory systems during ischemia and reperfusion. In M. Karmazyn (Ed.), Myocardial ischemia: Mechanisms, reperfusion, protection (pp. 87–97). Basel/Switzerland: Birkhäser-Verlag. Arthur, P. G., Hogan, M. C., Bebout, D. E., Wagner, P. D., & Hochachka, P. W. (1992). Modelling the effect of hypoxia on ATP turnover in exercising muscle. Journal of Applied Physiology, 73(2), 737–742. Baker, A. J., Brandes, R., Schendel, T. M., Trocha, S. D., Miller, R. G., & Weiner, M. W. (1994). Energy use by contractile and noncontractile processes in skeletal muscle estimated by 31P-NMR. American Journal of Physiology, 266(Cell Physiology 35), C825–C831. Bendall, J. R. (1973). Postmortem changes in muscle. In G. H. Bourne (Ed.), The structure and functioning of muscle (2nd ed., Vol. II, pp. 243–309). NY: Academic Press. Blandchard, E. M., Pan, B.-S., & Solaro, R. J. (1984). The effect of acidic pH on the ATPase activity and troponin Ca2+ binding of rabbit skeletal myofilaments. Journal of Biological Chemistry, 259(5), 3181–3186. Carse, W. A., & Locker, R. H. (1974). A survey of pH vales at the surface of beef and lamb carcasses, stored in a chiller. Journal of the Science of Food and Agriculture, 25, 1529–1535. Chase, P. B., & Kushmerick, M. K. (1988). Effects of pH on contraction of rabbit fast and slow skeletal muscle fibers. Biophysical Journal, 53, 935–946. Chase, P. B., & Kushmerick, M. J. (1995). Effect of physiological ADP concentrations on contraction of single skinned fibres from rabbit fast and slow muscles. American Journal of Physiology, 268(Cell Physiology 37), C480–C489. Chasiotis, D. (1983). The regulation of glycogen phosphorylase and glycogen breakdown in human skeletal muscle. Acta Physiologica Scandinavica, 518(Suppl.), 1–68. Chasiotis, D. (1985). Activation of glycogen phosphorylase by electrical stimulation of isolated fast-twitch and slow-twitch muscles from rat. Acta Physiologica Scandinavica, 123, 43–47. Chrystall, B. B., & Devine, C. E. (1978). Electrical stimulation, muscle tension and glycolysis in bovine sternomandibularis. Meat Science, 2, 49–58. Chrystall, B. B., Devine, C. E., Snodgrass, M., & Ellery, S. (1982). Tenderness of exercise-stressed lambs. New Zealand Journal of Agricultural Research, 26, 331–336. Chrystall, B. B., Devine, C. E., Davey, C. L., Kirton, A. H. (1981). Animal stress and its effect on rigor mortis development in lambs. In: D. Hood, P. V. Tarrant (Eds.), The problem of dark cutting in beef. Current topics in veterinary medicine and animal science. (Vol. 10, pp. 269–282). The Hague: Martinus Nijhoff Publishers. Coffee, C. C., & Solano, C. (1977). Rat muscle 50 -adenylic acid aminohydrolase. The Journal of Biological Chemistry, 252(5), 1606–1612. Connett, R. J. (1988). Analysis of metabolic control: New insights using scaled creatine kinase model. American Journal of Physiology, 254(Regulator, Integrative and Comparative Physiology 23), R949–R959. Cortassa, S., Aon, M. A., Marban, E., Winslow, R. L., & O’Rourke, B. (2003). An integrated model of cardiac mitochondrial energy metabolism and calcium dynamics. Biophysical Journal, 84, 2734–2755. Curtin, N. A. (1986). Buffer power and intracellular pH of frog sartorius muscle. Biophysical Journal, 50, 837–841. Dash, R. K., Li, Y., Kim, J., Beard, D. A., Saidel, G. M., & Cabrera, M. E. (2008a). Metabolic dynamics in skeletal muscle during acute reduction in blood flow and oxygen supply to mitochondria: In-silico studies using a multiple-scale, topdown integrated model. PLoS One, 3(9), e3168. Dash, R. K., Li, Y., Kim, J., Saidel, G. M., & Cabrera, M. E. (2008b). Modeling cellular metabolism and energetics in skeletal muscle: Large-scale parameter estimation and sensitivity analysis. IEEE Transactions on Biomedical Engineering, 44(4), 1298–1318. Davies, A., 1989. Structure and function of carcass tissues in relation to meat production. In R. W. Purchas, B. W. Butler-hogg, A. S. Davies (Eds.), Meat production and processing (pp. 43–59). New Zealand Society of Animal Production, Occasional Publication, number 11. Devine, C. E., Chrystall, B. B., Davey, C. L., & Kirton, A. H. (1983). Effects of nutrition in lambs and subsequent post mortem biochemical changes in muscle. New Zealand Journal of Agricultural Research, 26, 53–57. Devine, C. E., Ellery, S., & Averil, A. (1984). Responses of different types of ox muscle to electrical stimulation. Meat Science, 10, 35–51.

I. Vetharaniam et al. / Meat Science 85 (2010) 134–148 Devine, C. E., Hopkins, D. L., Hwang, I. H., Fergusson, D. M., Richards, I. (2004). Electrical stimulation. In W. Jensen, C. E. Devine, M. Dikeman (Eds.), Encyclopedia of meat sciences (pp. 413–423). Oxford: Academic Press. Dudley, G. A., & Terjung, R. L. (1985). Influence of acidosis on AMP deaminase activity in contracting fast-twitch muscle. American Journal of Physiology, 248(Cell Physiology 17), C43–C50. Farmer, L. J. (1994). The role of nutrients in meat flavour formation. In Proceedings of the nutrition society (Vol. 53, pp. 327–333). Ferguson, D. M., Jian, S.-T., Hearnshaw, H. R., Rymill, S. R., & Thompson, J. M. (2000). Effect of electrical stimulation on protease activity and tenderness of m. longissimus from cattle with different proportions of bos indicus content. Meat Science, 55, 265–272. Fralix, T. A., Murphy, E., London, R. E., & Steenbergen, C. (1993). Protective effects of adenosine in the perfused rat heart: Changes in metabolism and intracellular ion homeostasis. American Journal of Physiology, 264(Cell Physiology 33), C986–C994. Gambassi, G., Hansford, R. G., Sollott, S. J., Hogue, B. A., Lakatta, E. G., & Capogrossi, M. C. (1993). Effects of acidosis on resting cytosolic and mitochondrial Ca2+ in mammalian myocardium. The Journal of General Physiology, 102, 575–597. Glyn, H., & Sleep, J. (1985). Dependence of adenosine triphosphate activity of rabbit psoas muscle fibres and myofibrils on substrate concentration. Journal of Physiology, 365, 259–276. Goodman, M. N., & Lowenstein, J. M. (1977). The purine nucleotide cycle. The Journal of Biological Chemistry, 252(14), 5054–5060. Gunter, T. E., Buntinas, L., Sparagna, G. C., & Gunter, K. K. (1998). The Ca2+ transport mechanisms of mitochondria and Ca2+ uptake from physiological-type Ca2+ transients. Biochimica et Biophysica Acta, 1366(1–2), 5–15. Gunter, T. E., & Pfeiffer, D. R. (1990). Mechanisms by which mitochondria transport calcium. American Journal of Physiology, 258(Cell Physiology 28), C755–C786. Hamm, R., Darymple, R., Honikel, K. (1973). On the post-mortem breakdown of glycogen and ATP in skeletal muscle. In Meat research workers European 19th meeting, Paris (Vol. 1, pp. 73–86). Hearnshaw, H., Gursanksy, B. G., Gogel, B., Thompson, J. M., Fell, L. R., Stephenson, P. D., et al. (1998). Meat quality in cattle of varying Brahman content: The effect of post slaughter processing, growth rate and animal behaviour on tenderness. In Proceedings 44th international congress of meat science and technology, Barcelona, Spain (pp. 1048–1049). Hindmarsh, A. C. (1983). Odepack, a systemised collection of ode solvers. In R. S. Stapleman, et al. (Eds.), Scientific computing: Application of mathematics and computing to the physical sciences (pp. 55–64). North-Holland Pub. Co. Hochachka, P. W., & Mommsen, T. P. (1983). Protons and anaerobiosis. Science, 219, 1391–1397. Hollung, K., Veiseth, E., Frøystein, T., Aass, L., Langsrud, Ø., & Hildrum, K. I. (2007). Variation in the response to manipulation of post-mortem glycolosis in beef muscles by low-voltage stimulation and conditioning temperature. Meat Science, 77, 372–383. Honikel, K. O. (2004). Conversion of muscle to meat – Ageing. In W. J. Jensen, C. Devine, M. Dikeman (Eds.), Encyclopedia of meat science (pp. 330–338). Oxford: Academic Press. Hwang, I. H., Devine, C. E., & Hopkins, D. L. (2003). The biochemical and physical effects of electrical stimulation on beef and sheep meat tenderness – a review. Meat Science, 65, 677–691. Idström, J. P., Soussi, B., Elander, A., & Bylund-Fellinius, A.-C. (1990). Purine metabolism after in vivo ischaemia and reperfusion in rat skeletal muscle. American Journal of Physiology, 258(Heart and Circulatory Physiology 27), H1668–H1673. Immonen, K., & Puolanne, E. (2000). Variation of residual glycogen–glucose concentration at ultimate ph values below 5.75. Meat Science, 55, 279–283. Immonen, K., Rasmussen, M., Hissa, K., & Puolanne, E. (2000). Bovine muscle glycogen concentration in relation to finishing diet, slaughter and ultimate ph. Meat Science, 55, 25–31. Jeacocke, R. E. (1984). The kinetics of rigor onset in beef muscle fibres. Meat Science, 237–251. Jeacocke, R. E. (1993). The concentrations of free magnesium and free calcium ions both increase in skeletal muscle fibres entering rigor mortis. Meat Science, 35, 27–45. Kemp, C. M., Sensky, P. L., Bardsley, R. G., Buttery, P. J., & Parr, T. (2010). Tenderness – An enzmatic view. Meat Science, 84, 248–256. Kirton, A. H., Sinclair, D. P., Chrystall, B. B., Devine, C. E., & Woods, E. G. (1981). Effect of plane of nutrition on carcass composition and palatability of pasture-fed lamb. Journal of Animal Science, 52, 285–291. Korzeniewski, B., & Zoladz, J. A. (2001). A model of oxidative phosphorylation in mammalian skeletal muscle. Biophysical Chemistry, 92, 17–34. Krause, U., & Wegener, G. (1996). Exercise and recovery in frog muscle: Metabolism of PCr, adenine nucleotides, and related compounds. American Journal of Physiology, 270(Regulator, Integrative and Comparative Physiology 39), R811–R820. Kushmerick, M. J. (1997). Multiple equilibria of cations with metabolites in muscle energy. American Journal of Physiology, 272(Cell Physiology 41), C1739–C1747. Lai, N., Saidel, G. M., Grassi, B., Gladden, L. B., & Cabrera, M. E. (2007). Model of oxygen transport and metabolism predicts effects of hyperoxia on canine muscle oxygen uptake dynamics. Journal of Applied Physiology, 103, 1366–1378. Lambeth, M. J., & Kushmerick, M. J. (2002). Computational model for glycogenolysis in skeletal muscle. Annals of Biomedical Engineering, 30, 808–827.

147

Lamb, G. D., & Stephenson, D. G. (1992). Importance of Mg2+ in excitation– contraction coupling in skeletal muscle. News in Physiological Science, 7, 270–274. Lawrie, R. A. (1998). Lawrie’s meat science (6th ed.). Abington, Cambridge, UK: Woodhead Publishing. Lawson, J. W. R., & Veech, R. L. (1979). Effects of pH and free Mg2+ on the Keq of the creatine kinase reaction and other phosphate hydrolyses and phosphate transfer reactions. The Journal of Biological Chemistry, 254(14), 6528–6537. Leet, N. G., & Locker, R. (1973). A prolonged pre-rigor condition in aerobic surfaces of ox muscle. Journal of the Science of Food and Agriculture, 24, 1181–1191. Locker, R. H., Davey, C. L., Nottingham, P. M., Haughey, D. P., & Law, N. H. (1975). New concepts in meat processing. Advances in Food Research, 21, 157–222. Lohmann, K. (1934). Über die enzymatische Aufspaltung der Kreatinphosphorsaure; zugleich ein Beitrag zum Chemismus der Muskelkontraktion [Enzymic disruption of creatinephosphoric acid; the chemistry of muscle contraction]. Biochemische Zeitschrift, 271, 264–277. Lundberg, P., Vogel, H. J., Fabiansson, S., & Rudérus, H. (1987). Post-mortem metabolism in fresh porcine, ovine and frozen bovine muscle. Meat Science, 19, 1–14. Marban, E., Kitakaze, M., Koretsune, Y., Yue, D. T., Chacko, V. P., & Pike, M. M. (1990). Quantification of [Ca2+] in perfused hearts. Critical evaluation of the 5F-BAPT and nuclear magnetic resonance method as applied to the study of ischaemia and reperfusion. Circulation Research, 66(5), 1255–1267. McGlone, V. A., Devine, C. E., & Wells, R. W. (2005). Detection of tenderness, post rigor age and water status changes in sheep using near infrared spectroscopy. Journal of Near Infrared Spectroscopy, 13, 277–285. Moore, V. J., & Gill, C. O. (1987). The pH and display life of chilled lamb after prolonged storage under vacuum or under CO2. New Zealand Journal of Agricultural Research, 30, 449–452. Mulquiney, P. J., Bubb, W. A., & Kuchel, P. W. (1999). Model of 2,3bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations: In vivo kinetic characterization of 2,3bisphosphoglycerate synthase/phosphatase using 13c and 31p nmr. Biochemical Journal, 342, 567–580. Mulquiney, P. J., & Kuchel, P. W. (1999a). Model of 2,3-bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations: Computer simulation and metabolic control analysis. Biochemical Journal, 342, 596–604. Mulquiney, P. J., & Kuchel, P. W. (1999b). Model of 2,3-bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations: Equations and parameter refinement. Biochemical Journal, 342, 581–596. Pate, E., Bhimani, M., Franks-Skiba, K., & Cooke, R. (1995). Reduced effect of pH on skinned rabbit psoas muscle mechanics at high temperatures: Implications for fatigue. Journal of Physiology, 486(3), 689–694. Ren, J. M., Chasiotis, D., Bergström, M., & Hultman, E. (1988). Skeletal muscle glucolysis, glycogenolysis and glycogen phoshporylase during electrical stimulation in man. Acta Physiologica Scandinavica, 133, 101–107. Rosenvold, K., Micklander, E., Hansen, P. W., Burling-Claridge, R., Challies, M., Devine, C. E., et al. (2009). Temporal, biochemical and structural factors that influence meat quality measurement using near infrared spectroscopy. Meat Science, 82, 379–388. Rosenvold, K., North, M., Devine, C. E., Micklander, E., Hansen, P., Dobbie, P., et al. (2008). The protective effect of electrical stimulation and wrapping on beef tenderness at high pre-rigor temperatures. Meat Science, 79, 299–306. Ruff, R. L., & Weissman, J. (1995). Iodoacetate-induced skeltal muscle contracture: Changes in adp, calcium, phosphate and pH. American Journal of Physiology, 268(Cell Physiology 37), C317–C322. Sahlin, K., & Broberg, S. (1990). Adenine nucleotide depletion in human muscle during exercise: Causality and significance of AMP deamination. International Journal of Sports Medicine, 11(Suppl. 2), S62–S67. Smith, G. L., Donos, P., Bauer, C. J., & Eisner, D. A. (1993). Relationship between intracellular pH and metabolite concentrations during metabolic inhibition in isolated ferret heart. Journal of Physiology, 472, 11–22. Soussi, B., Lagerwall, K., Idström, J.-P., & Scherstén, T. (1993). Purine metabolic pathways in rat hindlimb perfusion model during ischaemia and reperfusion. American Journal of Physiology, 264(Heart and Circulatory Physiology 34), H1074–H1081. Spriet, L. L. (1990). Anaerobic ATP provision, glycogenolysis and glycolysis in rat slow-twitch muscle during tetanic contractions. Pflügers Archive, 417, 278–284. Spychala, J., Madrid-Marina, V., Nowak, P. J., & Fox, I. H. (1989). AMP and IMP dephosphorylation by soluble high- and low-Km 50 -nucleotidases. American Journal of Physiology – Endocrinology and Metabolism, 256, E386–E391. Steele, D. S., McAinsh, A. M., & Smith, G. L. (1996). Comparative effects of inorganic phosphate and oxalate on up take and release of Ca2+ by the sarcoplasmic reticulum in saponin skinned rat cardiac trabeculae. Journal of Physiology, 490(3), 565–567. Stienen, G. J., Papp, Z., & Zaremba, R. (1999). Influence of inorganic phosphate and pH on sarcoplasmic reticular ATPase in skinned muscle fibres of xenopus laevis. Journal of Physiology, 518(3), 735–744. Tanaka, N., Hasan, Z., Hartog, A. F., van Herk, T., & Wever, R. (2003). Phosphorylation and dephosphorylation of polyhydroxy compounds by class A bacterial acid phosphatases. Organic and Biomolecular Chemistry, 1, 2833–2839. Teusink, B., Passarge, J., Reijenga, C. A., Esgalhado, E., van der Weijden, C. C., Schepper, M., et al. (2000). Can yeast glycolysis be understood in terms of

148

I. Vetharaniam et al. / Meat Science 85 (2010) 134–148

in vitro kinetics of the constituent enzymes? Testing biochemistry. European Journal of Biochemistry, 267, 5313–5329. Tikk, M., Tikk, K., Tørngren, M. A., Meinert, L., Aaslyng, M. D., Karlsson, A. H., et al. (2006). Development of inosine monophosphate and its degradation products during aging of pork of different qualities in relation to basic taste and retronasal flavor perception of the meat. Journal of Agriculture and Food Chemistry, 54, 7769–7777. Toohey, E. S., Hopkins, D. L., Stanley, D. F., & Nielsen, S. G. (2009). The impact of new generation pre-dressing medium-voltage electrical stimulation on tenderness and colour stability in lamb meat. Meat Science, 79, 683–691. Veech, R. L., Lawson, J. W. R., Cornell, N. W., & Krebs, H. A. (1979). Cytosolic phosphorylation potential. The Journal of Biological Chemistry, 254(14), 6538–6547. Vinnakota, K., Kemp, M. L., & Kushmerick, M. J. (2006). Dynamics of muscle glycogenolysis modeled with pH time course computation and pH-dependent reaction equilibria and enzyme kinetics. Biophysical Journal, 91, 1264–1287. Vogel, H. J., Lundberg, P., Fabiansson, S., Rudérus, H., & Tornberg, E. (1985). Postmortem energy metabolism in bovine muscles studied by non-invasive phosphorous-31 nuclear magnetic resonance. Meat Science, 13, 1–18. Welsh, D. G., & Lindinger, M. I. (1993). Energy metabolism and adenine nucleotide degradation in twitch-stimulated rat hindlimb during ischaemia-reperfusion. American Journal of Physiology, 264(Endocrinology and Metabolism 27), E655–E661. Westerblad, H., & Allen, D. G. (1993). The influence of intracellular pH on contraction, relaxation and [Ca2+]i in intact single fibres from mouse muscle. Journal of Physiology, 466, 611–628.

Westerblad, H., & Allen, D. G. (1994). Relaxation, [Ca2+]i and [Mg2+]i during prolonged tetanic stimulation of intact, single fibres from mouse skeletal muscle. Journal of Physiology, 480(1), 31–43. Wiseman, R. W., Beck, T. W., & Chase, P. B. (1996). Effect of intracellular pH on force development depends on temperature in intact skeletal muscle from mouse. American Journal of Physiology, 271(Cell Physiology 40), C878–C886. Wolosker, H., Roacha, J. B. T., Engelender, S., Panizzutti, R., De Miranda, J., & de Meis, L. (1997). Sarco/endoplasmic reticulum Ca2+-ATPase isoforms: diverse responses to acidosis. Biochemical Journal, 321, 545–550. Wood, J. D., Enser, M., Fisher, A. V., Nute, G. R., Richardson, R. I., Sheard, P. R. (1999). Manipulating meat quality and composition. In Proceedings of the nutrition society (Vol. 58, pp. 363–370). Young, D. A., Chi, M. M.-Y., & Lowry, O. H. (1986). Energy metabolism of skeletal muscle biopsies stimulated anaerobically without load in vitro. American Journal of Physiology, 250(Cell Physiology 19), C813–C820. Zhou, L., Salem, J. E., Saidel, G. M., Stanley, W. C., & Cabrera, M. E. (2005). Mechanistic model of cardiac energy metabolism predicts localization of glycolysis to cytosolic subdomain during ischemia. American Journal of Physiology - Heart and Circulatory Physiology, 288, H2400–H2411. Zhu, Y., & Nosek, T. M. (1991). Intracellular milieu changes associated with hypoxia impair sarcoplasmic reticulum Ca2+ transport in cardiac muscle. American Journal of Physiology, 261(Heart Circulation Physiology 30), H620–H626. Zoratti, M., & Szabo, I. (1995). The mitochondrial permeability transition. Biochimica et Biophysica Acta, 1241(2), 139–176.