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Double-substrate interactive model
om9-mFj93 s6.w + 0.00 8 1993 F’ewmon Press Ltd
INTRODUcTION In biochemical reactions, a living microorganism such as a type of bacteria or a cell. is fed with a substrate S. As a consequence, the cell grows and the number of cells increases. The possible mechanism to express these observations was first postulated by Michaelis and Menten (1913) and based on their mechanism, a kinetic equation was first proposed by Monod (1949), which has a certain resemblance with the so-called Langmuir monolayer adsorption kinetics (Langmuir, 1916). Other single-substrate models have also been proposed elsewhere (Blackman, 1905; Tessier, 1936; Moser, 1958). These models may give a better fit to experimental data, but are not frequently used because of the mathematical complexities which one frequently would prefer to avoid. Microorganisms usually take up a number of different nutrients such as carbon, nitrogen, and mineral substances from the environment to grow. Under certain conditions, it has become apparent that the growth rate of an organism may be simultaneously limited by two or more substrates (Panikov, 1979; Court and Pirt, 1981; Harder and Dijhuizen, 1983). Bader (1978,1982) classified the double-substrate limitedgrowth into two categories; namely, noninteractive and interactive modes. The former implies that the growth rate of the organism can only be limited by one substrate at a time. The latter is based on the assumption that if two substrates are present in less than saturation concentrations (addition of more substrate would increase its growth rate), then both must affect the overall growth rate of the organism. Pavlou and Fredrickson (1989) extended a cybernetic modeling concept, developed by Ramkrishna and his coworkers (Ramkrishna, 1983; Ramkrishna et al., 1984; Kampala et al., 1984, 1986 Hhujati et al., 1985) to situations where nutrients (substrates) are not necessarily substitutable (Baltzis and Fredrickson, 1988). In their study of Lactobacilfus cnsei, McGee et al. (1972) proposed a double-substrate limited interactive growth model by simply multiplying two Monad single-substrate limited models together to give
cc _=_ Pm
Sl
S2
(K, + S,)(K2 + S,)
(1)
where p,,, is the model parameter showing the maximum specific growth rate, S1 and Sz are concentrations, and K, and K, are constants, respectively, for substrate 1 and 2. The quantities Si and Sz are what Pavlou and Fredrickson (1989) call complementary. Inhibition by substrate is less common but for some microbial systems the growth rate decreases at high substrate concentrations even though Michaelis-Menten kinetics are obeyed at low substrate concentrations. In this note, the characteristics of a double-substrate interactive model with inhibition effect(s) are presented for a continuously stirred tank bioreactor(CSTR). The model is formulated by extending McGee’s double-substrate limited interactive growth model with substrate inhibition effects us-
ing Andrews’ substrate (Andrew, l%R), which
inhibition gives
for each substrate
s,s2
P
-=
P,
model
Wi
+
SI + S:/Ki.
1W2
+ S2 + S,2/&,2)
(2)
In eq. (Z), KiV1 and &.z are inhibition coefIicients for substrate 1 and 2, respectively, and, when both become infinite, the substrate inhibition effects drop out and eq. (2) reduces to McGee’ formula, eq. (1).
ANALYTICAL FORMULATION A CSTR is fed with two limiting substrate concentrations S,i and SF2 with the dilution rate D. Then governing equations on the cell and two limiting substrates are
$ =(p dS, -= dt dsz = dt
where X is concentrations growth rate cells by the steady state, reduce to
D)X
fX + D(SF, -
S,)
I
gx +D(Sp2 -
S,)
the concentration of oells, S, and Sz are the of the limiting substrate 1 and 2, p is a specific of cells, and Yi and Yz are the yield factors of two limiting substrates 1 and 2, respectively. At the governing equations in dimensionless forms (6)
(4-d)y=O 6 -5y+d(a,-a)=0
- $J The dimensionless defined as
+ d(&
variables
a = SIlKI,
(7 - B) = 0.
are for steady-state
B f S21Kz,
d = D/L, YI = KI K/X,, kt = KifK,.,>
Y = XIX,
(8) values and Pa*)
4 = P/P,
(loa, b)
YZ = Kz K/X,
(1 la, b)
k, = WKc,.
(1% b)
The quantity X0 is a scaling factor usually taking the maximum number attainable in the process. Then the dimensionless specific growth rate, eq. (2), becomes
The washout steady-state solution to the above three simultaneously coupled equations, where y = 0, c+ = a, and
2169
2170
Shorter Communications
BP = 8, is a trivial one which is a washout. solution, eq. (6) becomes
For the nontrivial
d=d
(14)
and from eqs (7) and (S), we obtain
(BP-B) -@F- N By rearranging
Yl
z
UW
eqs (15a) and (7), we obtain
(15’3 and
Equation (13) in the a-p phase plane with constant d (= 4) is the growth contour curves and eq. (15b) is the stoichiometric line with the slope of yl/y2 through the feed point (c+, BP). The multiplicity is determined by the number of intersection point(s) between these two curves from eqs (13) and (15b). The criterion of a local stability is that the eigenvalues of the Jacobian matrix evaluated at the respective steady state have negative real parts (Verhulst, 1990). The elements of the Jacobian matrix are
Substrate inhibition for one substrate only When only one of the substrates d( causes an inhibition effect, then k2 = 0. The growth contour curves for three dilution rates, 0.45,0.50, and 0.55, with k, = 0.1 are shown in Fig. 2(a). This figure shows that the minimum of the U-shape curves moves upward and the width decreases as the dilution rate increases. At the limiting dilution rate, dli, = 0.6125, the two vertical portions in the U-shape curve merge to form a line. If the inhibition effect is due to the substrate 2 or /I (k, = 0), the U-shape profiles in the phase diagram will show exactly the same profiles except for interchanging the coordinates a with 8. The growth contour curve, d, = 0.45, and the stoichiometric line (y, = yr = 0.8; CQ= 9, BF = 12) intersect at point A (0~= 1.6, p = 4), which is the only stable steady state for this set of feed substrate concentration. When the dilution rate increases from dl = 0.45 to dz = 0.50, the stoichiometric line intersects the growth contour curve dl twice at B (a = 2.15, /? = 5.20) and C (a = 6.50, fi = 9.50). Thus, there are two steady states in which B is stable and C is unstable steady state, respectively. The bifurcation diagram for the case is presented in Fig. 2(b) showing the hysteresis effect. The state with the lowest substrate concentration is the desired state between the two possible steady states. Thus, B is the desired stable steady state.
wherefis are the left-hand side of eqs (6)<8) and zjs are the variables, y, a, and p, respectively. 6.00
-
a
k,=k,=O
RESULTS
Numerical simulations for the following three cases: (1) without substrate inhibition for both substrates, (2) with substrate inhibition for one substrate only, and (3) with substrate inhibition for both substrates, are performed to obtain the growth contour curves in the c+_Bphase plane and the stoichiometric line as a function of dilution rate, d. Both substrates without substrate inhibition For the case k, and k2 are zero, eq. (13) reduces to the case of Bader (1978, 1982) given in eq. (1). The growth contour curves in the m-p plane are hyperbola as shown in Fig. l(a) for three dilution rates, d = 0.24,0.34, and 0.54, in which the curves move upwards as the dilution rate increases. Depending upon the initial feed concentration, (a,, BF), and the yield coefficient ratio, yl/yz. there are two possible classes of steady states: (i) stat& which are formed by an intersection of a stoichiometric line and a contour curve as shown at the points A and B, respectively, on the d, and d2 contour lines, and (ii) a state in which the contour curve is always above the feed point (aF, /$), where a stoichiometric line and a contour curve never intersect with each other as shown for the d3 contour line. The former case is the nontrivial steady state. As the dilution rate increases, the contour curve moves upbards, while the stoichiometric line remains unchanged. It shows that the steady state moves from state A to state 3, resulting in a shift of the nontrivial steady state along the stoichiometry line as the dilution rate increases. The latter case is the trivial steady state, which is the washout condition. The concentration profiles of y, a, and fi with a feed concentration given in Fig. 1 (a) as a function of dilution rate are presented in Fig. l(b). As shown in the figure, there is only one steady state and it is a stable one. The dilution rate at y = 0 is the washout steady state. The Poincark-Bendixson theorem (Verhulst, 1990) states that the washout steady state has to be stable, since it is impossible to have a limit cycle around the washout point (the cell cannot be created after the cell is gone!). Figure l(b) is also known as a bifurcation diagram, in which solid lines stand for stable steady region and dotted lines represent unstable steady region.
z
6.00
-
4.00
-
b 2 z z
0.00 ,,.,.,....,......... 0.00 2.00
(a)
4.60
Concentration
(b)
Dilution
of
= . . . . . . . . .d, . . . . .0.26 . ..I 6dO 6
Substrate.
Rate.
3
c I-I
d I-I
Fig. 1. The case of both substrates without substrate inhibition: (a) growth contour curves in the substrate concentrations phase diagram; and (b) bifurcation diagram for concentrations of cell and substrates as a function of dilution rate.
Shorter
Communications
-
I J
5e
5000.00
2 B
2.50 I
I1
105000.00
& drm
=
5
0.6125
03 = 2.00
2 d
1.50
1 b .-s
40.00
P T
II
1 ~~~
1.00
8 E 0
0.50
__
0.00 c
0.00
2.00
4.00
Concentration
@I
0.11
e
50.00 1
2171
6. 0
8.00
of Substrate.
10.00
12.00
)
2.w
4.00
s.00
Concentration
(a)
I.00
of
lo.00
Sub&rote.
0
12.00
zz
P
,2,m :___-_-__________-I
----_---__-___-__
10.00
IT----+ \\ ,
r \
-
,
II
/J
1
’ ‘I , 1
v
4.00
14
-t-t
=H
76.00 ,
6.00
12.00
I
I
0.00
0.20
(W
0.40 Dilution
Rote,
0.60
0.00
0
Fig. 2. Tbe case of substrate inhibition for one substrate only (k, = 0): (a) growth contour curves in the substrate concentrations phase diagram; and (b) bifurcation diagram for concentrations of cell and substrates as a function of dilution rate.
A range
of controlling
dilution
rates may be defined
d(y = 0) < d < d(bd/ba
= 0).
t
(b)
d 1-1
as (17)
For this example, it is obtained as: 0.46 < d < 0.53. Within narrowly defined region as given in eq. (17). there are three possible steady states which may exist at a given dilution rate. Two out of three are stable steady states, which include the washout steady state. Proper initial conditions are necessary to ensure that the system will attain the desired steady state (Liu and Hsu, 1991). a
Substrate inhibirion for both substrates The contour curves of eq. (13) for four dilution rates, 0.11, 0.13, 0.15, and 0.16, with kL = 0.1 and k2 = 2.0 in the a-@ phase plane are a family of closed loops, as shown in Fig. 3(a). When the dilution rate increases, the closed-loop contour curve reduces its size and finally approaches a point, where the dilution rate is the limiting dilution rate. The characteristic number of steady states depends on the stoichiometric line and the feed concentration (ag, pp). For the case where the stoichiometric line M with a low feed concentration (aF = 1.89, pF = 0.41) and slope yI/yz = 0.18/1.4, in the region where the contour curves resemble the
Dilution
Rote.
d I-1
Fig. 3. The case of substrate inhibition for both substrates only (k, = 0.1 and k2 = 2.0): (a) growth contour curves in the substrate concentrations phase diagram; and (b) bifurcation diagram for concentrations of cell and substrates as a function of dilution rate.
case of no inhibition, the concentration profiles of y, u, and B as a function of dilution rate will be the same as that for no inhibition as shown in Fig. l(a). As the feed concentration increases to (aF = 10, BP = LO), the stoichiometric line with the same slope becomes the H-line and intersects with the contour curve d, twice at B (a = 5.18, /3 = 0.36) and C (a = 8.65, p = 0.80) points. The bifurcation diagram, Fig. 3(b), with a high feed concentration as given in Fig. 3(a), resembles the case of single inhibition. The characteristics are exactly the same as the case of single inhibition. The range of controlling dilution rate is 0.12 < d -=c0.14. For generalization, the limiting dilution rate as a function of inhibition coefficient kl with three values of kl, 0.0, 0.1, and 0.5, is presented in Fig. 4. It shows that the lower the inhibition coefficients the higher the limiting dilution rate. DISCUSSION The numerical
AND CONCLUSION
values used in the simulation are typical cases to exemplify the effect of double substrates. The higher the feed concentration (CL=.pr) the wider the choice of clilution rate is, regardless to whether inhibition e&et(s) is present with the substrate. However, the effect of feed concentration on the “attainability” of the optimal stable steady state
2172
Shorter Communications 1.20
kp = a.00
,
1.00
Inhibition
2.60
Coefficient,
3.00
0.5 4.
k,
Fig. 4. The limiting dilution rate as a function of inhibition coefficient, kl, with parameter kz = 0, 0.1 and 0.5.
has to be taken into consideration, which has been discussed by Liu and Hsu (1991). The main effect of substrate inhibition is to generate multiple steady states showing bifurcation phenomena. The characteristics are to limit the concentration of substrate above which the cells cannot grow and to reduce the operable dilution rate. These are shown in the contour curves in the concentration phase diagram, Figs 2(a) and 3(a) for one substrate inhibition only and for both substrates, respectively, With a double inhibition effect, the contour curve becomes enclosed. Further increase in the dilution rate, causes the enclosed area to reduce its size and finally approach a point. With one or two substrate inhibition, the cell concentration profile no longer monotonically decreases with dilution rate and has alimiting dilution rate as shown in Figs 2 and 3 for the two cases and eq. (17). Acknowledgement-The authors gratefully acknowledge the support of the National Science Council (NSC SO-0402EOO2-23), Republic of China. HWAI-SHEN LIU SHENG-SHIH WANG HSIEN-WEN HSU’ Department of Chemical Engineering National Taiwan University Taipei, Taiwan, ROC
‘Present address: Department of Chemical Engineering, University of Tennessee, Knoxville, TN 37996-2200, U.S.A.
REFERENCES Andrews, J. F., 1969, Biotechnol. Bioengng 10, 707. Bader, F. G., 1978, Biotechnol. Bioengng 20, 183. Bader, F. G., 1982, Kinetics of double-substrate limited growth, in Microbial Population Dynamics (Edited by J. Mandelstam). CRC Press, Bofa Raton, FL. Baltzis, B. C. and Fredrickson, A. G., 1988, Biotechnof. Bioeng&j 31, 75. Blackman, F. F., 1905, Ann. Bot. 19, 28. Court, J. R. and Pirt, S. J., 1981, J. &em. Technol. Biotechno!. 31, 235. Hhurjati, P. S., Ramkrishna, D., Flbkinger, M. C. and Tsao, G. T., 1985, Biotechnol. Bioengng 27, 1. Harder, W. and Dijhuizen, L., 1983, Ann. Rev. Microbial. 37, 1. Kompala, D. S., Ramkrishna, D. and Tsao, G. T., 1984b, Biotechnol. Bioengng 26, 1272. Kompala, D. S., Ramkrishna, D., Jansen, N. B. and Tsao, G. T., 1986, Biorechnol. Bioengng 28, 1044. Langmuir, I., 1916, J. Am. them. Sot. 38,222l. Liu, H.-S. and Hsu, H.-W., 1991, Chem. Engng Sci. 46, 2551. McGee, R. D. III, Drake, J. F., Fredrickson, A. G. and Tsuchiya, H. M., 1912, Can. J. Microbial. 18, 1773. Michaelis, L. and Menten, M. M. L., 1913, Biochem. Z. 49, 333. Monod, J., 1949, Ann. Rev. Microbial. 3, 371. Moser, H.. 1958, The Dynamics of Bacterial Popalatiop Maintained in the Chemostat. Carnegie Institute of Washington, Washington, DC. Panikov, N., 1979, J. them. Technol. Biotechnol. 29, 442. Pavlou, S. and Fredrickson, A. G., 1989, Biotechnol. Bioengng 34,971. Ramkrishna, D., 1983, in Foundations of BiochemicalEngineering: Kinetics and Thermodynamics in Biological Systems (Edited by H. W. Blanch, E. T. Papoutsakis and G. N. Stephanopoulos), pp. 161-178. American Chemical Society, Washington, DC. Ramkrishna, D., Kompala, D. S. and Tsao, Cl. T., 1984a, in Frontiers in Chemical Reaction Engineering (Edited by L. K. Doraiswamy and M. A. Mashelkar), Vol. 1, pp. 241-260. Wiley Eastern Limited, New Delhi. Tessier, C., 1936, Ann. Physial. Physiochem. Biol. 12, 527. Verhulst, F., 1990, Nonlinear Difirential Equations and Dynamical Systems. Springer, Berlin, Heidelberg.
INTRODUcTION In biochemical reactions, a living microorganism such as a type of bacteria or a cell. is fed with a substrate S. As a consequence, the cell grows and the number of cells increases. The possible mechanism to express these observations was first postulated by Michaelis and Menten (1913) and based on their mechanism, a kinetic equation was first proposed by Monod (1949), which has a certain resemblance with the so-called Langmuir monolayer adsorption kinetics (Langmuir, 1916). Other single-substrate models have also been proposed elsewhere (Blackman, 1905; Tessier, 1936; Moser, 1958). These models may give a better fit to experimental data, but are not frequently used because of the mathematical complexities which one frequently would prefer to avoid. Microorganisms usually take up a number of different nutrients such as carbon, nitrogen, and mineral substances from the environment to grow. Under certain conditions, it has become apparent that the growth rate of an organism may be simultaneously limited by two or more substrates (Panikov, 1979; Court and Pirt, 1981; Harder and Dijhuizen, 1983). Bader (1978,1982) classified the double-substrate limitedgrowth into two categories; namely, noninteractive and interactive modes. The former implies that the growth rate of the organism can only be limited by one substrate at a time. The latter is based on the assumption that if two substrates are present in less than saturation concentrations (addition of more substrate would increase its growth rate), then both must affect the overall growth rate of the organism. Pavlou and Fredrickson (1989) extended a cybernetic modeling concept, developed by Ramkrishna and his coworkers (Ramkrishna, 1983; Ramkrishna et al., 1984; Kampala et al., 1984, 1986 Hhujati et al., 1985) to situations where nutrients (substrates) are not necessarily substitutable (Baltzis and Fredrickson, 1988). In their study of Lactobacilfus cnsei, McGee et al. (1972) proposed a double-substrate limited interactive growth model by simply multiplying two Monad single-substrate limited models together to give
cc _=_ Pm
Sl
S2
(K, + S,)(K2 + S,)
(1)
where p,,, is the model parameter showing the maximum specific growth rate, S1 and Sz are concentrations, and K, and K, are constants, respectively, for substrate 1 and 2. The quantities Si and Sz are what Pavlou and Fredrickson (1989) call complementary. Inhibition by substrate is less common but for some microbial systems the growth rate decreases at high substrate concentrations even though Michaelis-Menten kinetics are obeyed at low substrate concentrations. In this note, the characteristics of a double-substrate interactive model with inhibition effect(s) are presented for a continuously stirred tank bioreactor(CSTR). The model is formulated by extending McGee’s double-substrate limited interactive growth model with substrate inhibition effects us-
ing Andrews’ substrate (Andrew, l%R), which
inhibition gives
for each substrate
s,s2
P
-=
P,
model
Wi
+
SI + S:/Ki.
1W2
+ S2 + S,2/&,2)
(2)
In eq. (Z), KiV1 and &.z are inhibition coefIicients for substrate 1 and 2, respectively, and, when both become infinite, the substrate inhibition effects drop out and eq. (2) reduces to McGee’ formula, eq. (1).
ANALYTICAL FORMULATION A CSTR is fed with two limiting substrate concentrations S,i and SF2 with the dilution rate D. Then governing equations on the cell and two limiting substrates are
$ =(p dS, -= dt dsz = dt
where X is concentrations growth rate cells by the steady state, reduce to
D)X
fX + D(SF, -
S,)
I
gx +D(Sp2 -
S,)
the concentration of oells, S, and Sz are the of the limiting substrate 1 and 2, p is a specific of cells, and Yi and Yz are the yield factors of two limiting substrates 1 and 2, respectively. At the governing equations in dimensionless forms (6)
(4-d)y=O 6 -5y+d(a,-a)=0
- $J The dimensionless defined as
+ d(&
variables
a = SIlKI,
(7 - B) = 0.
are for steady-state
B f S21Kz,
d = D/L, YI = KI K/X,, kt = KifK,.,>
Y = XIX,
(8) values and Pa*)
4 = P/P,
(loa, b)
YZ = Kz K/X,
(1 la, b)
k, = WKc,.
(1% b)
The quantity X0 is a scaling factor usually taking the maximum number attainable in the process. Then the dimensionless specific growth rate, eq. (2), becomes
The washout steady-state solution to the above three simultaneously coupled equations, where y = 0, c+ = a, and
2169
2170
Shorter Communications
BP = 8, is a trivial one which is a washout. solution, eq. (6) becomes
For the nontrivial
d=d
(14)
and from eqs (7) and (S), we obtain
(BP-B) -@F- N By rearranging
Yl
z
UW
eqs (15a) and (7), we obtain
(15’3 and
Equation (13) in the a-p phase plane with constant d (= 4) is the growth contour curves and eq. (15b) is the stoichiometric line with the slope of yl/y2 through the feed point (c+, BP). The multiplicity is determined by the number of intersection point(s) between these two curves from eqs (13) and (15b). The criterion of a local stability is that the eigenvalues of the Jacobian matrix evaluated at the respective steady state have negative real parts (Verhulst, 1990). The elements of the Jacobian matrix are
Substrate inhibition for one substrate only When only one of the substrates d( causes an inhibition effect, then k2 = 0. The growth contour curves for three dilution rates, 0.45,0.50, and 0.55, with k, = 0.1 are shown in Fig. 2(a). This figure shows that the minimum of the U-shape curves moves upward and the width decreases as the dilution rate increases. At the limiting dilution rate, dli, = 0.6125, the two vertical portions in the U-shape curve merge to form a line. If the inhibition effect is due to the substrate 2 or /I (k, = 0), the U-shape profiles in the phase diagram will show exactly the same profiles except for interchanging the coordinates a with 8. The growth contour curve, d, = 0.45, and the stoichiometric line (y, = yr = 0.8; CQ= 9, BF = 12) intersect at point A (0~= 1.6, p = 4), which is the only stable steady state for this set of feed substrate concentration. When the dilution rate increases from dl = 0.45 to dz = 0.50, the stoichiometric line intersects the growth contour curve dl twice at B (a = 2.15, /? = 5.20) and C (a = 6.50, fi = 9.50). Thus, there are two steady states in which B is stable and C is unstable steady state, respectively. The bifurcation diagram for the case is presented in Fig. 2(b) showing the hysteresis effect. The state with the lowest substrate concentration is the desired state between the two possible steady states. Thus, B is the desired stable steady state.
wherefis are the left-hand side of eqs (6)<8) and zjs are the variables, y, a, and p, respectively. 6.00
-
a
k,=k,=O
RESULTS
Numerical simulations for the following three cases: (1) without substrate inhibition for both substrates, (2) with substrate inhibition for one substrate only, and (3) with substrate inhibition for both substrates, are performed to obtain the growth contour curves in the c+_Bphase plane and the stoichiometric line as a function of dilution rate, d. Both substrates without substrate inhibition For the case k, and k2 are zero, eq. (13) reduces to the case of Bader (1978, 1982) given in eq. (1). The growth contour curves in the m-p plane are hyperbola as shown in Fig. l(a) for three dilution rates, d = 0.24,0.34, and 0.54, in which the curves move upwards as the dilution rate increases. Depending upon the initial feed concentration, (a,, BF), and the yield coefficient ratio, yl/yz. there are two possible classes of steady states: (i) stat& which are formed by an intersection of a stoichiometric line and a contour curve as shown at the points A and B, respectively, on the d, and d2 contour lines, and (ii) a state in which the contour curve is always above the feed point (aF, /$), where a stoichiometric line and a contour curve never intersect with each other as shown for the d3 contour line. The former case is the nontrivial steady state. As the dilution rate increases, the contour curve moves upbards, while the stoichiometric line remains unchanged. It shows that the steady state moves from state A to state 3, resulting in a shift of the nontrivial steady state along the stoichiometry line as the dilution rate increases. The latter case is the trivial steady state, which is the washout condition. The concentration profiles of y, a, and fi with a feed concentration given in Fig. 1 (a) as a function of dilution rate are presented in Fig. l(b). As shown in the figure, there is only one steady state and it is a stable one. The dilution rate at y = 0 is the washout steady state. The Poincark-Bendixson theorem (Verhulst, 1990) states that the washout steady state has to be stable, since it is impossible to have a limit cycle around the washout point (the cell cannot be created after the cell is gone!). Figure l(b) is also known as a bifurcation diagram, in which solid lines stand for stable steady region and dotted lines represent unstable steady region.
z
6.00
-
4.00
-
b 2 z z
0.00 ,,.,.,....,......... 0.00 2.00
(a)
4.60
Concentration
(b)
Dilution
of
= . . . . . . . . .d, . . . . .0.26 . ..I 6dO 6
Substrate.
Rate.
3
c I-I
d I-I
Fig. 1. The case of both substrates without substrate inhibition: (a) growth contour curves in the substrate concentrations phase diagram; and (b) bifurcation diagram for concentrations of cell and substrates as a function of dilution rate.
Shorter
Communications
-
I J
5e
5000.00
2 B
2.50 I
I1
105000.00
& drm
=
5
0.6125
03 = 2.00
2 d
1.50
1 b .-s
40.00
P T
II
1 ~~~
1.00
8 E 0
0.50
__
0.00 c
0.00
2.00
4.00
Concentration
@I
0.11
e
50.00 1
2171
6. 0
8.00
of Substrate.
10.00
12.00
)
2.w
4.00
s.00
Concentration
(a)
I.00
of
lo.00
Sub&rote.
0
12.00
zz
P
,2,m :___-_-__________-I
----_---__-___-__
10.00
IT----+ \\ ,
r \
-
,
II
/J
1
’ ‘I , 1
v
4.00
14
-t-t
=H
76.00 ,
6.00
12.00
I
I
0.00
0.20
(W
0.40 Dilution
Rote,
0.60
0.00
0
Fig. 2. Tbe case of substrate inhibition for one substrate only (k, = 0): (a) growth contour curves in the substrate concentrations phase diagram; and (b) bifurcation diagram for concentrations of cell and substrates as a function of dilution rate.
A range
of controlling
dilution
rates may be defined
d(y = 0) < d < d(bd/ba
= 0).
t
(b)
d 1-1
as (17)
For this example, it is obtained as: 0.46 < d < 0.53. Within narrowly defined region as given in eq. (17). there are three possible steady states which may exist at a given dilution rate. Two out of three are stable steady states, which include the washout steady state. Proper initial conditions are necessary to ensure that the system will attain the desired steady state (Liu and Hsu, 1991). a
Substrate inhibirion for both substrates The contour curves of eq. (13) for four dilution rates, 0.11, 0.13, 0.15, and 0.16, with kL = 0.1 and k2 = 2.0 in the a-@ phase plane are a family of closed loops, as shown in Fig. 3(a). When the dilution rate increases, the closed-loop contour curve reduces its size and finally approaches a point, where the dilution rate is the limiting dilution rate. The characteristic number of steady states depends on the stoichiometric line and the feed concentration (ag, pp). For the case where the stoichiometric line M with a low feed concentration (aF = 1.89, pF = 0.41) and slope yI/yz = 0.18/1.4, in the region where the contour curves resemble the
Dilution
Rote.
d I-1
Fig. 3. The case of substrate inhibition for both substrates only (k, = 0.1 and k2 = 2.0): (a) growth contour curves in the substrate concentrations phase diagram; and (b) bifurcation diagram for concentrations of cell and substrates as a function of dilution rate.
case of no inhibition, the concentration profiles of y, u, and B as a function of dilution rate will be the same as that for no inhibition as shown in Fig. l(a). As the feed concentration increases to (aF = 10, BP = LO), the stoichiometric line with the same slope becomes the H-line and intersects with the contour curve d, twice at B (a = 5.18, /3 = 0.36) and C (a = 8.65, p = 0.80) points. The bifurcation diagram, Fig. 3(b), with a high feed concentration as given in Fig. 3(a), resembles the case of single inhibition. The characteristics are exactly the same as the case of single inhibition. The range of controlling dilution rate is 0.12 < d -=c0.14. For generalization, the limiting dilution rate as a function of inhibition coefficient kl with three values of kl, 0.0, 0.1, and 0.5, is presented in Fig. 4. It shows that the lower the inhibition coefficients the higher the limiting dilution rate. DISCUSSION The numerical
AND CONCLUSION
values used in the simulation are typical cases to exemplify the effect of double substrates. The higher the feed concentration (CL=.pr) the wider the choice of clilution rate is, regardless to whether inhibition e&et(s) is present with the substrate. However, the effect of feed concentration on the “attainability” of the optimal stable steady state
2172
Shorter Communications 1.20
kp = a.00
,
1.00
Inhibition
2.60
Coefficient,
3.00
0.5 4.
k,
Fig. 4. The limiting dilution rate as a function of inhibition coefficient, kl, with parameter kz = 0, 0.1 and 0.5.
has to be taken into consideration, which has been discussed by Liu and Hsu (1991). The main effect of substrate inhibition is to generate multiple steady states showing bifurcation phenomena. The characteristics are to limit the concentration of substrate above which the cells cannot grow and to reduce the operable dilution rate. These are shown in the contour curves in the concentration phase diagram, Figs 2(a) and 3(a) for one substrate inhibition only and for both substrates, respectively, With a double inhibition effect, the contour curve becomes enclosed. Further increase in the dilution rate, causes the enclosed area to reduce its size and finally approach a point. With one or two substrate inhibition, the cell concentration profile no longer monotonically decreases with dilution rate and has alimiting dilution rate as shown in Figs 2 and 3 for the two cases and eq. (17). Acknowledgement-The authors gratefully acknowledge the support of the National Science Council (NSC SO-0402EOO2-23), Republic of China. HWAI-SHEN LIU SHENG-SHIH WANG HSIEN-WEN HSU’ Department of Chemical Engineering National Taiwan University Taipei, Taiwan, ROC
‘Present address: Department of Chemical Engineering, University of Tennessee, Knoxville, TN 37996-2200, U.S.A.
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