The Model

Rate of pH fall

The pH of the musculature at slughter was taken to be 7. In a carcass the rate of pH fall decreased somewhat with time until final pH is reached. Such non-linear fall would however be hard to model and would require the introduction of at least one other parameter besides the initial rate of pH fall to define it precisely. We therefore assumed that the pH fell linearly from its starting value to its final rigor value.

For pig carcasses the normal rate of pH fall is about 0.01 units/min, corresponding to a rigor time of about 150 min. A marginal case of PSE is usually considered to correspond to a pH at 45 min of 6, or about 0.02 units/min. In an extreme case, rigor is achieved in only 15 min corresponding to a rate of 0.1 units/min. For beef carcasses, rigor typically occures in about 24 h corresponding to a rate of pH fall of about 0.001 units/min. Rate of pH fall over a 100-fold range from 0.001 to 0.1 units/min were therefore considered.

The standard value for the final pH was taken to be 5.5. However, the effect on the amount of myosin denaturation of altering the final pH in the range of 5 to 6 was also included.

Time-course of cooling

In a continously convex body initially at a uniform temperature Ti and then chilled with air at a temperature Tf, there is a region between the centre and the surface where the temperature decays approximately exponentially to its final value Tf. This implies that the heat loss is propotional to the difference in temperature between that region and the air. It is useful to defin the time for half cooling, tc, as the time required for the temperature of that region to fall by half its maximum extent, that is to a temperature of (Ti + Tf)/2. A point nearer the surface cools initially faster than this region, but, after a lag period, cools at the same exponential rate, while a point nearer the centre will initially cool more slowely, but, after a lag period, again cools at the same exponential rate.

When a beef side was treated as four different thermal portions, the leg, loin, shoulder and flank, it was found that for each portion the average temperature-time curve closely followed the immediately exponential plot. For beef sides of 140 kg cooled in air of relative humidity 94% and temperature 0oC at a velocity of 0.5 m/s, half-cooling times of 687 min for the leg, 486 min for the shoulder, 300 min for the loin and 194 min for the flank were reported. Equivalent data is not available for the pig. For pig sides cooled in air of reltive humidity 92%, and temperature 4oC at a velocity of 0.33 m/s, the time required to reach 10oC were 13.9 h for the deep leg, 8.5 h for the surface leg, 7.3 h for the deep loin and 5.1 h for the surface loin. The averaege times for the leg or loin to reach 10oC must lie between the times for the centre and surface. If tp is the time required for the temperature to fall exponentially until the temperature difference is only a fraction p of the original temperature difference, the half-cooling time is given by:

tp x ln 2/ln (1/p).

Hence these values correspond to the half cooling times of 328, 202, 172 and 120 min respectively. We therefore consider half cooling times over the range 100 min to 700 min, but it must be appreciated that these relate to the average temperature for a thermal portion and there will be regions initially cooling more slowely. Where a single standard value for the half cooling time was neede, a value of 180 min was chosen, or occasionally to simulate beef leg, 700 min. The air temperature was taken to be 4oC.

It was assumed for most of the calculations that chilling started immediately post mortem and that there was no lag time. However for a real carcass, chilling seldom starts before about an hour post mortem. For one calculation (shown in Fig. 8) the effect of introducing a lag period of 60 min was therefore considered.

An additional complication is that in the pre-rigor period the musculature is metabolically active and generates heat, mainly due to the conversion of glycogen converted to lactic acid, but also due to the hydrolysis of creatine phosphate and ATP. It has been estimated that in pig carcasses sufficient heat is liberated from these reactions to raise the temperature of the carcass by 3oC. It was assumed that this heat was liberated at a constant rate during post-mortem glycolysis. This is equivalent to supposing that the temperature rise due to metabolic heat was proportional to the pH fall with a constant of proportionality of h (normally 2oC/pH unit).

Hence, taking into account both the heat loss due to the cooling air and the metabolic heat, if in the small time interval the pH fall was dpH, the tempareture fall is:

(ln 2)/tc x (T-Tf)dt - h dpH,

where tc is the half cooling time in the absence of metabolic activity.

Integrating this expression, the realtion between the temperature T and time t in the pre-rigor period is:

Eqn (1): T = Tf + (hytc/ln 2) + (Ti - Tf - (hytc/ln 2)) exp((-t/tc)ln 2))

where y is the rate of pH fall. The tempareture therefore rises from its initial value to a miximum temperature at rigor if (hytc/ln 2) > (Ti - Tf ). For h = 2, and a half cooling time of 180 min, this will occur if the rate of pH fall exceeds 0.067 pH units/min.

Rigor formation occures at a time trig given by:

Eqn (2): trig = (pHinitial - pHfinal)/y

After rigor, no more metabolic heat is evolved and the temperature falls merely due to cooling:

Eqn (3): T = Tf + ((hytc/ln 2)exp((trig/tc)in 2) + Ti - Tf - (hytc/ln 2))exp((-t/tc)ln2)

The time-course of the average temperature was taken to be 39oC, although it is possible that for carcasses destined to become PSE, the temperature at death is already above 39oC due to an increased aerobic metabolism immerdiately before death caused by stress.

Rate constant of myosin inactivation

It was assumed that the denaturation event in myosin ultimately responsible for the enhanced drip loss, which we belive to be the shrinking of the myosin heads, to be identical to that causing loss of enzymatic activity. This assumption will need to be tested critically in the future because it is conceived that these events are not identical and may have a different dependency on pH and temperature. In 1M KCl it has been shown that the kinetics of activition of rabbit myosin ATPase were frist order and a plot of log10k, the rate constant of activition, against pH was a straight line of gradient -1.3. This implies that k is proportional to 10-1.3pH. In 1M KCl, the Arrhenius activation energy was 47.6 kcal/mol and in a medium of ionic strength 0.16M, a condition which more nearly resembles that in musclce, it was 43.5 kcal/mol. The dependence of the inactivation on pH at the lower inoic strenght was not measured, but, assuming that it is the same as that at the higher ionic strength, the rate of inactivation at absolute temperature T in degrees K can be expressed as:

Eqn (4): k = k0 exp(-43500/RT) 10-1.3pH s-1

where R, the gas constant, is 1.987 cal/K/mol.

It has been found that k = 1.2 x 10-3 s-1 at 35oC and pH 5.7 in 0.16M KCl, thus giving k0 = 2.28 x 1035.

At pH 7.5, ATP reduces the rate of inactivation to 9.35% of its value without ATP. It was assumed that the same factor applied at lower pH values and that in the presence of ATP the dependence of the activation of myosin in pre-rigor muscle was taken to be:

Eqn (5): k = 2.13 x 1034 exp(-43500/RT) 10-1.3pHs-1

It was assumed that the same relationship holds for myosin from all meat species.

Time-course of myosin inactivation

Calculations of the amount of myosin denaturation were made by finite-element analysis, that is by considering the changes occuring in small interval of time dt. For the time-course these intervals were chosen to be 1 min, and for the effects of cooling rate, rate of pH fall and final pH on the otal amount of myosin denatured at rigor, they were chosen to be 1/100th of the total time for rigor. Checks were made that these time intervals were adequately small. The temperature at the beginning and end of each time interval could be calculated from Eqn (1). From these and the pH values, the rate constant of myosin inactivation at the beginning and end of each small time interval could be calculated using Eqn (5), and the mean value during this interval, km, determined. If the fraction of myosin native at the beginning of this interval was xi, the fraction native at the end of the time interval will be xi exp(- km dt). By using this value at the beginning of the next interval, etc., the accumulated amount of myosin denatured ocer the complete period to rigor could be calculated. The calculation could then be repeated for different rates of pH fall, different cooling rates and different values of the final pH.

When the muscle enters the rigor state, the myosin will be protected against denaturation by combination with actin. The extent of this protection is best estimated by comparing the rate of inactivation of isolated myosin with that in rogor myofibrils, where a high fraction (>95%) of the myosin heads are attached to actin in the overlap region of the thick and thin filaments. The dependence of the rate of inactivation of myosin in myofibrils on pH is different from that in solution, but at pH 5.4 and 37.5oC the rate constant of denaturation of myosin in rigor myofibrils is one-hundredth that of isloated myosin. This may be an underestimate of the pretective effect of actin since firstly the sarcomere length of the myofibrils was not rigorously controlled and any myosin not overlapped by actin would not be pretected. Secondly, isolated myofibrils can shorten much than in intact muscle and it has been shown that in myofibrils with very short sarcomere lengths, many of the myosin heads are not bound to actin and would therefore be vulnerable to denaturation. It was therefore assumed that once an intact muscle anters the rigor state, the myosin overlapped by actin filaments binds to actin and denaturation of these myosin molecules ceases.

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