St= Si+ S 0 (t< L)
St= Si+ S 0 exp(k(tL))
(t> L) (10.3)where Stis the total residual fibre at time t,
Siis the indigestible fibre, S 0 is the initial
digestible fibre, kis a rate constant (units
time^1 ) and L is a discrete lag term, a time
during which no digestion occurs (Mertens
and Loften, 1980).
If we use this kind of expression to
model gas production, the equation must
be modified because we are now measuring
the appearance of a product rather than the
disappearance of a substrate. The modified
exponential equation becomes:
Vt= 0 (t< L)
Vt= Vf(1 exp(k(tL)))
(t> L) (10.4)where Vt= volume of gas at time t, Vf=
final asymptotic gas volume corresponding
to complete substrate digestion, and k, t
and L have the same meanings as in
Equation 10.3. Equation 10.4 produces
curves of the shape shown in Fig. 10.4.
This simple exponential equation can
be derived on the assumption that the rate
of gas production, at times beyond the lag
time, depends solely on the amount of
digestible substrate available. The micro-
bial population size is assumed not to limit
the rate at any stage. The discrete lag term
is something of an embarrassment but is
necessary to get a good data fit in many
cases. Some justification can be offered on
the grounds that the cellulolytic bacteria
must first attach to the fibre, and perhaps
also express some cellulase genes, before
digestion can begin. In principle, Equation
10.4 can be expanded to cover multiple
pools:Vt= VFn(1 exp(kn(tLn)))
where the subscript n would be the pool
number. However, summed exponential218 P. Schofield
Fig. 10.3.Typical gas curves recorded using the closed system depicted in Fig. 10.2. Note the different
curve shapes. Substrates (100 mg DM each) were lucerne, wheat straw, soy hulls and maize silage.