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If we assume that microbial growth is governed by a single rate
limiting enzyme, then we can interpret kas the specific growth rate
constant andEas a temperature characteristic. If this is the case andA
andEare constant with temperature, then a plot of lnkagainst 1/T(the
absolute temperature) would give a straight line. In fact a concave
downward curve is obtained indicating that the activation energy E
increases with decreasing temperature.
To improve the fit with observed behaviour, the basic equation has
been modified by Davey to include a quadratic term:

lnk¼C 0 þC 1 =TþC 2 =T^2 ð 3 : 28 Þ

This can be further modified to include other parameters affectingksuch
as pH andaw.For example:

lnk¼C 0 þC 1 =TþC 2 =T^2 þC 3 awþC 4 a^2 w ð 3 : 29 Þ

The Schoolfield equation is another variation of the Arrhenius model
where additional terms have been added to the basic equation to account
for the effects of high-and low-temperature inactivation on growth rate.
Terms describing the effect ofawand pH can also be incorporated here to
give a considerably more complex equation.
An alternative, rather simpler, approach which has met with some
success is the square root model to describe growth at sub-optimal
temperatures:

k

p

¼bðT
TminÞð 3 : 30 Þ

wherekis the rate of growth,Tthe absolute temperature (K), andTminis
a conceptual minimum temperature of no physiological significance since
it is usually below the freezing point of microbiological media.
Application of this expression to describe microbial growth was first
described by Ratkowsky, although it is now recognized as a special form
of the Beˇlehra ́dek power function originally described nearly 70 years ago.
A plot ofOkagainstTshould give a straight line with an intercept on
theTaxis atTminand this has been observed and reported by a number
of authors monitoring growth in both laboratory media and foods
(Figure 3.18).
To include the effects of other constraints on growth the square root
equation has been extended separately to give similar equations includ-
ing anawterm and a pH term.

k

p

¼c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðaw
awminÞ

q

ðT
TminÞð 3 : 31 Þ

and

k

p

¼d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðpH
pHminÞ

p

ðT
TminÞð 3 : 32 Þ

Chapter 3 59