The Model

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.

In a continously convex body initially at a uniform temperature
*T _{i}* and then chilled with air at a temperature

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 0^{o}C 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 4^{o}C at a
velocity of 0.33 m/s, the time required to reach 10^{o}C
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 10^{o}C must lie between
the times for the centre and surface. If *t _{p}* is the time
required for the temperature to fall exponentially until the temperature
difference is only a fraction

*t _{p} 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 4^{o}C.

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 3^{o}C.
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 2^{o}C/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 *d*pH, the tempareture fall is:

*(ln 2)/t _{c} x (T-T_{f})dt
- h dpH*,

where *t _{c }*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 = T_{f} + (hyt_{c}/ln
2) + (T_{i} - T_{f} - (hyt_{c}/ln 2)) exp((-t/t_{c})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 *(hyt _{c}/ln 2)* >

Rigor formation occures at a time *t** _{rig}*
given by:

**Eqn (2):** t_{rig}*= (pH*_{initial} - pH_{final})/y

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

**Eqn (3):** T = Tf + ((hyt_{c}/ln 2)exp((t_{rig}/t_{c})in
2) + T_{i} - T_{f} - (hyt_{c}/ln 2))exp((-t/t_{c})ln2)

The time-course of the average temperature was taken to
be 39^{o}C, although it is possible that for
carcasses destined to become PSE, the temperature at death is already above
39^{o}C 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 log_{10}*k*, 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 = k_{0}
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 35^{o}C
and pH 5.7 in 0.16M KCl, thus giving *k** _{0 }*=
2.28 x 10

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 10^{34}
exp(-43500/RT) 10-^{1.3pH}s^{-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 *d*t. 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, *k** _{m}*,
determined. If the fraction of myosin native at the beginning of this interval
was

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 37.5^{o}C 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.