Fed-Batch Bioproduction of Spectinomycin
41
by the Abbott Laboratories and in part by Sigma Xi, Grants in Aid of Research.
The authors also acknowledge the facilities and opportunity provided by IIT,
Delhi, India in completing this manuscript. The critical comments of Dr. P.K.
Roychoudhury and Dr. U.S. Agarwal which helped in shaping the final form of
this manuscript are sincerely appreciated.
10 Appendix
Mathematical Basis for Exponential Structure
The Monod equation is a phenomenological model for representing the growth
of microorganisms. If we expand the Monod equation in a polynomial form by
performing the division we obtain the following solution
Km+S
#,.S
Km
K 2
-
-
+
-
'
(A.1)
Where p,, is the maximum specific growth rate, K~ is the Monod constant and
S is the substrate concentration. Now, comparing this series form of the Monod
equation with the exponential series
K~ _ 1 Ke
K 2
K~ a
~(- 1)"n!S,
---~- + 2!S---- 5 -3!$3 + .......
e -KJs
(A.2)
o
we immediately notice that the factorial components present in Eq. (A.1) are
absent in Eq. (A.2). Hence the exponential equivalent of the Monod equation is
#(S) =/~mexp (- ~)
(A.3)
Where K e is the equivalent of the Monod constant Kin. The factorial components influence the shape of the generated curve in the lower range of substrate
S values. At higher S values the difference between the Monod and the exponential kinetics becomes negligible. Figure A.1 presents the difference between the
exponential and Monod structures with Ke = Kin. The exponential kinetic
structure exhibits a delay in the initial slope resembling the prolonged lag phase
observed in spectinomycin biosynthesis. Furthermore, the change of slope at
S -- Ke -- Km is more sensitive to Ke (exponential structure) than Km (Monod
structure). Notice that both the Monod and the exponential structures saturate
at #m. However, the constants Km and Ke exhibit some difference. When Km= S
we obtain the ratio #/#m = 2 for the Monod equation. In the case of the
exponential kinetics ~t/#. = e.
41
by the Abbott Laboratories and in part by Sigma Xi, Grants in Aid of Research.
The authors also acknowledge the facilities and opportunity provided by IIT,
Delhi, India in completing this manuscript. The critical comments of Dr. P.K.
Roychoudhury and Dr. U.S. Agarwal which helped in shaping the final form of
this manuscript are sincerely appreciated.
10 Appendix
Mathematical Basis for Exponential Structure
The Monod equation is a phenomenological model for representing the growth
of microorganisms. If we expand the Monod equation in a polynomial form by
performing the division we obtain the following solution
Km+S
#,.S
Km
K 2
-
-
+
-
'
(A.1)
Where p,, is the maximum specific growth rate, K~ is the Monod constant and
S is the substrate concentration. Now, comparing this series form of the Monod
equation with the exponential series
K~ _ 1 Ke
K 2
K~ a
~(- 1)"n!S,
---~- + 2!S---- 5 -3!$3 + .......
e -KJs
(A.2)
o
we immediately notice that the factorial components present in Eq. (A.1) are
absent in Eq. (A.2). Hence the exponential equivalent of the Monod equation is
#(S) =/~mexp (- ~)
(A.3)
Where K e is the equivalent of the Monod constant Kin. The factorial components influence the shape of the generated curve in the lower range of substrate
S values. At higher S values the difference between the Monod and the exponential kinetics becomes negligible. Figure A.1 presents the difference between the
exponential and Monod structures with Ke = Kin. The exponential kinetic
structure exhibits a delay in the initial slope resembling the prolonged lag phase
observed in spectinomycin biosynthesis. Furthermore, the change of slope at
S -- Ke -- Km is more sensitive to Ke (exponential structure) than Km (Monod
structure). Notice that both the Monod and the exponential structures saturate
at #m. However, the constants Km and Ke exhibit some difference. When Km= S
we obtain the ratio #/#m = 2 for the Monod equation. In the case of the
exponential kinetics ~t/#. = e.