A mathematical model for the simple batch fermentation process,. S + RgC5 .... Haldane lower bound in the enzyme model.6 afforded by Theorem 1 and ...
Parameter estimation and mathematical properties of a model for batch fermentation Philip S. Crooke Department
of Mathematics,
Vanderbilt University, Nashville, Tennessee
37235,
USA
Robert D. Tamer Department of Chemical Engineering, Tennessee 37235, USA (Received December
A mathematical S + RgC5
Vanderbilt University, Nashville,
1978; revised June 1980)
model for the simple batch fermentation
process,
2 R, is studied. It is shown that that the system of nonlinear
ordinar; differential equations which model the chemical process can be conveniently handled by the introduction of the specific growth rate variable, i.I = (dX/dt)/X where X = K[R + Cl. In particular, the mathematical analysis of the associated initial-value problem for the reaction in the p - S phase plane leads to techniques which can be employed to estimate the parameters k4, k5 and kg.
Introduction In this paper we study certain mathematical properties nonlinear system of ordinary differential equations: dS
;=-L,RS+k,C
dR it=
S(O)=S*
- k4RS + (ks t 2k6) C
dC z = k4RS - (ks + k6) C dX =k6KC dt
The above initial-value fermentation process:
(la> R(0) = R*
C(0) = 0
X(0)=X*
(lb)
(lc)
(Id)
problem is a model for the batch
StR>&2R k, X=K[R
of the
(2a) +C]
(2b)
In the reaction S denotes the concentration of the substrate, R the concentration of the lumped cell machinery variable (related to RNA and DNA), C the concentration of the lumped cell machinery-substrate complex, X the cell concentration, typically given in optical density units, K is a conversion factor and ki (i = 4, 5,6) are rate constants. Typically, except for X, the concentrations are given in the units, [g.mol.l-‘1, in which case the rate constants would
haveunits: K4= [l.s-‘.g-1.mol-‘],k5,k6= K = [(optical density units) g-‘(mol/l)-‘I.
[s-‘],and
Fermentation processes can be represented by enzymekinetic models since on a fundamental basis, biological reactions are reducible to a series of enzyme catalysed reactions. The above simple model minimizes the difficulties associated with models which incorporate multiple intermediate reactions. Hence, one could view this model as a first approximation for intractable complex systems. The reaction (2) can serve as the growth portion of a model for the fermentation of glucose by Pseudomonas ovalis to gluconic acid via gluconolactone at constant pH and temperature.’ Simple models for batch fermentation processes are often used to describe continuous fermentations.’ Typically, the batch model is used to extract kinetic data from batch experiments, while the continuous model is employed to extract the flow parameters using known kinetic parameters. The four differential equations in (1) can be reduced to two by using the conservation equations: 2Ci-R
tS=R*+S*
@a)
R-(X/K)+C=O
W)
which are readily obtainable from (1). In particular, the system can be reduced to two differential equations in the dependent variables, S and C, and the independent variable t. In the literature3 a composite variable, the specific growth rate, is often introduced; that is, we define: __ldX #u’=x;
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376
Appl. Math. Modelling,
1980,
Vol 4, October
0 1980 IPC Business Press
Model
which represents the physiological changes of the system on a per cell basis. The reason that this variable is introduced is in the phase plane of c versus S, the trajectory of the initialvalue problem forms a hyperbola. The shape of this curve is very much like that of the enzyme-substrate complex versus substrate for the single intermediate enzyme modeL4 E+S>&P+E * where E denotes the enzyme, S the substrate, C the enzymesubstrate complex and P the product. Hence, one hopes that the techniques and information gathered for the previous enzyme model studies4 would carry over directly to the fermentation model. This analogy has had limited success.’ One of the difficulties is that one no longer had differential equations involving & and S only. In the paper we will establish these differential equations. Having the differential equations will lead to several new relations between C, S, X, R, and p. Developing these relationships is important in estimating the parameters kg, k5, and k6 from experimental data. These parameters, in turn, are important in building continuous fermentation models.*
New system of ordinary differential equations In this section we derive a system of two differential equations in the dependent variables, c and S. By definition and (2b), (Id), we have: 1 dX ,-=_--=_ Xdt Differentiating have :
k6C
for batch fermentation:
Tanner
k6 and positive parameters: e=-
R* S *’
k6 Y=-
ks +k, h= __ k4S*
and
kJ*
so that : 1 --= kJ*
ds
(1 + E - s)(-s
dt
+ /.LS+ (X - y) p)
s(0) = 1
1.1+1
1 dp
--=s-ps-A/,-yp2 k4S* dt
(7a) p(O)=0
(3)
Properties of the system In this section we present some of the basic facts (theorems) about the system (equations (7)). It will be shown in the next section how these results can be used for parameter estimation. However, these results can also be used to determine whether the proposed model for batch fermentation is physically sound for the situation it is being used to study. Our first results are concerned with upper bounds for p(s) in the p - s phase plane of (7). Theorem I: If-P max =
max sEIO,il
C(max< ’
p(s), then:
- (1 + X) + J(1 + X)” t 4-r < ] . 2Y
Root From the nondimensional analogue of (6a), we see that the maximum value of P(S), 0 < s < 1, must lie on the curve r:
From (3b), (1 d) and (2b), one can show that:
yfi2 + x/J + (jJ - 1) s = 0 fi
and R. 0.
&
Rtc with respect to t and using (1 b) and (1 c), we
P. S. Crooke
= (1 - c/k6) S
On r, we find by implicit differentiation that:
(8) with respect to s
so that: Y/J2+ XI.1
dp ;
ds
= k4k6S - k4,i7S - (ks + k6) i; - ji’
The initial condition
for ji(t) is: i(O) = 0
Using similar arguments, that:
one can show with some difficulty
(5)
The system, (4) and (S), is autonomous and hence, we can view ji as a function of S, F(S), so that we have the initialvalue problem:
$s*)
=0
(6a) (6b)
Before we analyse the new system of ordinary differential equations, we introduce dimensionless variables: S SE-S*
o =- k,
Si
0, we have from Theorem which implies that:
1 that 0 G p(.$) f 1
x=p
Then:
1+e-s
e(1+ P)
1-s
ProoJ: From Theorem 2, rewriting p(t) in terms of x(t),
E
we have :
This proves the results:
In [s]
Corollary I :
ForO
