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Bistability in the chemostat
Ecological Modelling, 43 (1988) 287-301 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
287
BISTABILITY IN THE CHEMOSTAT R. KREIKENBOHM and E. BOHL
Fakultiit fiir Mathematik, Universitiit Konstanz, Postfach 5560, D-7750 Konstanz (Federal Republic of Germany) (Accepted 25 May 1988) ABSTRACT Kreikenbohm, R. and Bohl, E., 1988. Bistability in the chemostat. Ecol. Modelling, 43: 287-301. A general model for the interaction of two bacterial populations with interfering metabolic activities is provided. We discuss the dynamics of the system. As a result we obtain bistability under continuous-flow conditions if the growth rate of the first organism in the food chain depends on the substrate of the second, and if the growth rate of the second organism depends on the substrate of the first. During the anaerobic degradation of toxic substances, such an arrangement consisting of two organisms and two substrates can be found. In this case, we suggest explicit growth-rate expressions on the basis of experimental results and thermodynamic considerations. Using these expressions we can explain the disfunctioning of an anoxic digestor after a sudden increase of an environmental pollutant. INTRODUCTION In m i c r o b i a l ecosystems, the d e g r a d a t i o n of o r g a n i c m a t t e r p r o c e e d s in a f o o d chain consisting of two or m o r e m i c r o o r g a n i s m s with d i f f e r e n t metabolic activities. T h e simplest of these a l i g n m e n t s is the c o m m e n s a l i s t i c c o c u l t u r e with o n l y two different species. This system can b e r e p r e s e n t e d b y the following r e a c t i o n scheme: X
Y=
U
V
i n this f o o d chain Y a n d V d e n o t e the i n t e r a c t i n g m i c r o b i a l p o p u l a t i o n s , X is the substrate o f the first organism, a n d U stands for a p r o d u c t that at the s a m e time serves as the sole substrate o f the s e c o n d species. A theoretical analysis o f this c o m m e n s a l i s t i c system u n d e r c o n t i n u o u s - c u l t u r e c o n d i t i o n s has b e e n p e r f o r m e d b y Reilly (1974). Stable coexistence of Lactobacillus plantarum Y a n d Propionibacterium shermanii V has b e e n o b s e r v e d in a m i x e d c o n t i n u o u s culture growing o n glucose X a n d lactate U, respectively ( L e e et al., 1976). A t a n y given dilution rate there was o n l y o n e stable s t e a d y state a n d n o oscillations occurred. M o r e c o m p l i c a t e d m i x e d cultures consisting o f two d i f f e r e n t m i c r o b i a l p o p u l a t i o n s have b e e n investigated e x p e r i m e n t a l l y ( Y e o h et al., 1968; M i u r a 0304-3800/88/$03.50
© 1988 Elsevier Science Publishers B.V.
288
et al., 1978) and have been analysed theoretically (Miura et al., 1980). In these systems two or even more interactions between both species coexist. Furthermore, oscillatory phenomena were found under continuous-culture conditions. In methanogenic anaerobic environments the degradation of fatty acids, alcohols, aromatic compounds and amino acids is carried out by bacteria belonging to two different metabolic groups that are combined in syntrophic associations (Zehnder, 1978; Bryant, 1979). The members of the first group are Ha-producing acetogenic bacteria Y that convert the substrates X mentioned above to hydrogen U, acetate and CO 2. For thermodynamic reasons the release of hydrogen is limited to very low partial pressures (McInerney and Bryant, 1980). Therefore, the extent to which the substrate X is degraded by the organism Y depends on the efficiency of the removal of the product U by the methanogenic bacteria V of the second group. Here, a generalized reaction sequence can be given in the following form: X
~
U
~
A dynamic model of this type of interaction has been examined theoretically (Kreikenbohm and Bohl, 1986) and it has been used for the determination of the growth parameters of an isovalerate degrading strain cultivated in a defined coculture with Methanospirillum hungatei (Bohl and Kreikenbohm, 1987). In this contribution we present a model for the degradation of organic compounds that are considered to be toxic to either member of a syntrophic association. For example, the degradation of toluene, phenol, catechol, long-chain fatty acids, and unsaturated hydrocarbons such as isoprene can be summarized in the following way:
X,
~
U
,~
Here, the consumption of the molecule X by the organism Y shows substrate inhibition at high concentrations. Furthermore, the second organism in the food chain is also inhibited by molecule X because of a negative influence on the functional activities of the microbial cells. Finally, the production of the intermediate U is thermodynamically limited as discussed above. For example, the anaerobic degradation of phenolics by an undefined microbial community from domestic sludge has been examined (Fedorak
289
and Hrudey, 1984) under batch culture conditions. As a result, the authors have shown that the phenol-degrading bacteria are inhibited by their own substrate at a concentration of about 5 mM. Their metabolic partners (methanogens) stopped methane production when phenol was present at about 20 m M. Furthermore, evidence has been provided that, under anaerobic conditions, the phenol-degrading microorganisms are H2-producing acetogenic bacteria (Szewzyk et al., 1986). Thus, these authors worked with a food chain as presented by our third reaction scheme. With little modification this scheme may also be valid for the decontamination of pesticides by a microbial food chain. Our model allows us to understand the difficulties in the enrichment studies on organisms that degrade toxic compounds. Furthermore, we can explain the disfunctioning of an anoxic digestor from the sewage plants after suddently increasing the input of an environmental pollutant. This increase in the concentration of a toxic substance may be due to an accident: from a source with a very high concentration the pollutant enters the stream of waste water. Thereby, neither the volume of the waste water nor its flow rate is altered significantly. Furthermore, other parameters like temperature or pH value are also unchanged. As a result, the microbial community is disturbed so deeply that the whole degradation process stops at all the different stages. In this contribution, we model the situation described above as a syntrophic coculture of two organisms in the chemostat. We assume that the thermodynamics of hydrogen release and the inhibition by the inflowing substrate are the dominant factors. Using the appropriate growth rate expressions, our model shows hysteretic phenomena. Therefore, it allows us to understand that, at the same dilution rate, the degradation of the substrate is either nearly complete or restricted to a few percent. DYNAMICAL SYSTEM OF MIXED CONTINUOUS CULTURES
Continuous culture experiments of mixed microbial populations with metabolic reactions obeying any of the given schemes can be described by the following dynamical system:
t l(X, u) y
f¢--
Y1
)~' = ( ~ I ( X , fi
+ D(XR-- X )
u)--O)y
t,,(x, u)
(lb)
u)
y
~/1
v - Du
3'2 > 0 ,
(lc)
Y2
t)= (/,2(x, u ) - D ) v 3'1 > 0 ,
(la)
D>0,
(ld) x R>0
(le)
290 where D is the dilution rate, and y, v denote the population densities of the interacting species Y, V. In the culture vessel the substrates X, U are present at the concentrations x, u, whereas x R is the concentration of the substrate X in the reservoir. As indicated, the specific growth rates/_h(x, u), ~t2(x, u) depend on both substrates X and U. For the metabolism of the first organism we assume that the degradation of one molecule of substrate X is directly accompanied by the extrusion of one molecule of product U. A n y delay due to internal storage formation is excluded, so that the molecules U are immediately available as substrate to the organism V. Finally, equations (la) and (lc) imply that consumption and production are strictly growth-dependent processes. For the first organism Y in the third reaction scheme we choose the following growth-rate expression:
/ /~l(x, u)
=
x /~m~,l
--
u
gl
x 2
if
X-- K11u > 0
(2)
K 2 + x + K 3 u + 'K--4
0
otherwise
with K i > 0 (i = 1, 2, 3, 4). Here, /Lm~,l denotes the m a x i m u m specific growth rate of the organisms Y. The expression (2) ensures that these bacteria stop growing if the following expression fails: (3)
u < Klx
K a can be interpreted via the thermodynamics involved in the reaction from X to U (Kreikenbohm and Bohl, 1986); K 2 is a Michaelis-Menten-type constant, and g 3 a c t s as an inhibition constant related to the negative influence of the substrate U on its own production; K 4 is a constant that models the inhibition of the organism Y by its own substrate X. If K 4 ~ ~ , the growth rate expression (2) reduces to the form used for the degradation of nontoxic substrates by the Hz-producing acetogenic bacteria (Kreikenbohm and Bohl, 1986). If K 1 ~ ~ and K 3 = 0, we have the simplest substrate inhibition model analysed by Edwards (1970). Finally, if K 1 ~ ~ , K 3 = 0 , K 4 ~ oO, the growth-rate expression (2) reduces to the relation introdtrced by M o n o d (1942). For the second population V of the food chain in the third reaction scheme we use the following growth-rate expression: ~2(x, u ) -
~max2~/ K6 Ks+u Kr+x
(4)
291 with Ks, K 6 > 0. Here, /.tmax2 gives the maximum specific growth rate of the organisms V; K 5 is a Michaelis-Menten-type constant, and K 6 is an inhibition constant. Expression (4) has been introduced by Pirt (1975) to model the influence of a non-competative inhibitor that interacts with the microbial cell. For example, the negative influence of unsaturated long-chain fatty acids on methanogenic bacteria can be interpreted this way (Prins et al., 1972). This sort of inhibitory action leads to a decrease in the actual growth rate without affecting the specifity of the degradation process. Therefore, the affinity constant K 5 is unchanged with regard to the Monod relation. ANALYSIS OF SYSTEM (1) For the discussion of (1) it is important to note that we do not need the explicit representations of the growth-rate expressions (2) and (4). It is rather enough to consider qualitative assumptions on /.q(x, u),/~2(x, u) which we summarize as: ~i>0,
D>__0, / t ~ L i p ( R 2 ) ,
XR>0,
/tl(0, u ) = 0 ,
#2(x,0)=0
for
x>0,
#l(x,u)>0,
~2(x,u)>0,
~l(x, 0 ) > 0
i=1,2
(5a)
u>__0, /~2(0, X R ) > 0
(5b)
for
x>0,
u>0
(5c)
Expression (5b) says that the growth rate of a population vanishes in the absence of food. The Lipschitz-class appearing in (5a) contains, in particular, all functions which are continuously differentiable with respect to both variables. Our function (2) is in that Lipschitz-class but not continuously differentiable. For a more detailed discussion of the steady states we need some further technical assumptions, which we mention later. Let (x(t), y(t), u(t), v(t)) be a solution of (1) with initial values: x(0)>__0,
y(0)>0,
u(0)>0,
v(0)>0
(6)
Then this solution exists for all times t > 0 and all components of the solution remain non-negative for any t > 0. Furthermore, the functions:
wl(t)=x(t)+
")tll y ( t ) - x
R
(7a)
w2(t ) = x(t) + u(t) + V2' o ( t ) - x R
(7b)
satisfy:
wj(t) = wj(0) e x p ( - D t )
for
t > 0,
j = 1, 2
To prove these statements we only need (5).
(8)
292 In this section we describe the stationary solutions of our system (1). The steady-state system reads: (9a)
--~l(X, U) ~ l l y + D ( x R - x) = 0 (/11(x,
u)-D)
(#2(x, u ) - D )
y-0,
(9b)
v=0
~l(X, U) ~/lly--~[L2(X , U) ~21U--Ou = 0
(9c)
It is easily seen that any solution of (9) is of the form: ( x , v , ( x R - x ) , u, v ~ ( x . -
(10)
x - u))
To ensure non-negativity of all components in (10) we need: 0 <_~X'~ 0 ~_~U,
(11)
X'+-U~_~XR
The two parameters x, u satisfy one of the following conditions: X=XR,
(12a)
U=0
Ill(x, XR--X )=D, ~,,(x, u) = D,
U=XR--X
(12b)
(12c)
~q(x, u) = D
Accordingly, we have three solution branches of the steady-state system (9) which m a y be described as: X = XR, ,
y = 0,
u = 0,
Izl(X, X R - - X ) = D
,
v= 0
y='Yl(XR--X),
~l(X, u ) = O ,
y=y(XR--X
~ 2 ( x , u) = D ,
.
= ~2(x.-
(13a) U=XR--X ,
V=0
)
(13b) (13C)
x - u)
To see this, just combine (10) and (12). For a stability analysis of the various steady states we consider the dilution rate D to be our control parameter. Then we may characterize the three branches (13) in a (D, x)-plane. In the case of (13a), the branch is just the straight line parallel to the D-axis on the level x R (see Fig. 1). It is easily shown that this branch consists of stable stationary points if D > D2, and it is unstable if D < D x where: 3 2 := ~I(XR, 0) > 0
(14)
Next, we consider (13b). We must plot the graph of:
gl(X):=ld,l(X, XR--X),
X E [0, XR]
(I5)
defined by the first equation in (13b). By (5b) this branch joins the origin with the point B = ( D 2, XR) in Fig. 1. The point B is a bifurcation point
293
since it lies on the two branches (13a) and (13b). To analyse the stability of (13b) we must discuss the function:
XR--X),
[0, XR]
(16)
By (5) we have h ( 0 ) < 0, h ( X R ) > 0. We are now going to introduce the additional assumptions: /~,x(x, u ) > 0 , /~2u(x, u ) > 0 (lVa) ~tlu(X, u ) < 0 , ~tzx(X, u) < 0 (17b) at all steady states on the branch (13b). Expression (17a) means that, at the steady states, both organisms are supported by their respective food, and (17b) means that, at these steady states, the first organism is inhibited by its product and the second organism is inhibited by the food of the first organism. With this assumption, h defined in (16) is strictly m o n o t o n e increasing on the interval [0, xR]. Hence, there exists a unique zero 2 (0, xR). By definition we have: /~,(Y, x R - 2 ) - t ~ 2 ( 2 ,
x R - ~) = 0
(18a)
D, :-/~,(Y, x R - Y) (lab) It is easily proved that the branch (13b) is stable if x ~ (Y, xR) and unstable if x ~ (0, Y). This is indicated in Fig. 1 where we plot the unstable parts as a broken line and the stable parts as a solid line. The change in stability occurs at the point A in Fig. 1. Its corresponding solution (13b) reads:
(.~, ]tl(X R - ~), x R - .~, 0) (19) We finally turn to the branch (13c). It is defined by the solution set of the two-dimensional system (12c) under the constraints: O
X B (13a) X!~I.......................................... ...-..........~
DI
0 2
0
Fig. 1. Bifurcation diagram representing the stable ( ) and the unstable ( . . . . . . of the solution branches (13a), (13b), (13c) indicating no hysteresis phenomenon.
) parts
294 (12c). The corresponding point on the branch (13c) is exactly (19). Hence, it is a bifurcation point on both branches (13b, c) (see point A in Fig. 1). Furthermore, (5b, c) yields that x = 0, u = 0 is the only solution of (12c) for D = 0. The corresponding point on the branch (13c) is: (0, "YlXR, 0, )'2XR)
(21)
We easily verify that (21) is not a point on the branch (13b) so that the origin in Fig. 1 is not a bifurcation point. Only the projections of the two branches (13b, c) drawn in Fig. 1 coincide in the origin. For a further discussion of the branch (13c) we assume that, for any x E (0, XR), there is a unique solution (u(x), D(x)) of the system (12c). T h e n the corresponding point on the branch (13c) is:
(x, "YI(XR-- X), U(X), ~'2(XR-- X-- U(X)))
(22)
with the control parameter D(x) for 0 < x < x R. If we now assume (17) for (x, u(x)), x ~ (0, XR), then the function u(x) is strictly m o n o t o n e increasing. Since u(0) = 0, u(~) = x R - ~, we have:
x+u(x)
for
x ~ [0, ~1
x + u(x) > x R
for
x ~ ( Y , XR]
(23)
In view of (20) we are therefore only interested in the steady states (22) with x ~ (0, ~], where Y is defined by (18a). In the projection of Fig. 1, this part of the branch is characterized by the function D(x), the graph of which joins the origin and the point A. We can now prove that all parts on this graph with: D'(x) >0
(24)
are stable, and all points with: D'(x) <0
(25)
are unstable steady states on the branch (13c). Since an inequality (24) prevails in Fig. 1, the branch is indicated as a solid line (a completely stable branch). There are of course other possibilities, for example, that given in Fig. 2. Here, the system shows a hysteresis p h e n o m e n o n . In this respect, the following representation:
D'(x) = [ ( # i x / * 2 , - g2xlJ'lu)(~2u -
~1u)-1]( x'
U(X))
(26)
for the derivative of D is of interest. In view of (24), (25), it shows that the sign of D ' is completely determined by the sign of the quantity: (~.glx/X2, - ~,~2xglu)(x, u(x))
(27)
295
i
7
.............-" bl .........................(al ...............
x o
S
g
~3
-It)
OOO
005
OIO
01s
o20
025
0 30
c~ o~ o7
0itufi0n Rote 0 [I/h] Fig. 2. Bifurcation diagram of the solution branches on the basis of the rate expressions (2),
(4). By (17) the first product of this difference is always positive so that (24) (and therefore Fig. 1) holds if the last product in (27) vanishes in the interval (0, ~). This means that a hysteresis phenomenon necessarily requires that the function /L~ depends on u and the function /~2 depends on x. In particular, it is necessary that the growth rate of the second organism explicitely depends on the substrate x which is the food of the first organism. If the second is unaffected by the food of its partner, the system is unable to undergo hysteresis loops. NUMERICAL EXPERIMENTS
In this section, we examine the behaviour of our system (1) in the region of bistability between D 1 and D 3 in Fig. 2. This figure is drawn on the basis of the model parameters x R = 5 mM, ~maxl = 0 . 2 h - 1 , It 1 = 10 g/mol, K~ = 0.1, K 2 = 1 0 ~tM, K 3=0.1, K 4 = 5 mM, /Zr,ax2=0.1 h -1, "/2=6 g/mol, K 5 = 5 ~tM, K 6 = 5 mM. The solid trace of the three branches (a), (b), (c) depicted in Fig. 2 gives the steady state concentration of the substrate X, which should be attained by any evolution started in its vicinity. As stated before, at any given dilution rate between D~ and D 3 there are two stable steady states. In Table 1, the two steady-state values of the substrate concentrations x, u and the population densities y , u are given at three different dilution rates D (D1, D3). A glance at Table l a and lb shows the differences: in Table l a (third branch) a cooperation between the two organisms Y and V is obtained; in contrast, population V in Table lb (second branch) is no longer in the chemostat and hence substrate X remains almost unconsumed.
296 TABLE 1 D e p e n d e n c e of substrate c o n c e n t r a t i o n s x, u a n d bacterial p o p u l a t i o n densities y, v o n dilution rate D (a) D, 3rd b r a n c h D ( h - 1)
x (~M)
y (mg/1)
u (~M)
v (mg/1)
0.051 0.064 0.077
76.059 149.443 470.241
49.2394 48.5056 45.2976
5.3682 9.6683 26.7293
29.511 29.045 27.018
(b) D, 2nd b r a n c h D ( h - 1)
x (mM)
y (mg/1)
u (~M)
v (mg/1)
0.051 0.064 0.077
4.762 4.822 4.884
2.382 1.782 1.159
238.24 178.19 115.86
0 0 0
TABLE 2 Values of the c o n c e n t r a t i o n change necessary to move the whole system from the third to the second b r a n c h D ( h - 1)
x (m M )
y (mg/1)
u (m M )
v (mg/l)
0.051 0.064 0.077
284 11.5 1.5
- 49.233 - 46.1 - 26.1
26100 34.8 2.1
- 29 - 18.9 - 3.2
Negative values m e a n that this c o m p o n e n t has to be diminished. TABLE 3 D e p e n d e n c e of substrate c o n c e n t r a t i o n s x, u a n d bacterial p o p u l a t i o n densities y, v o n dilution rate D without substrate inhibition of X o n Y (a) D, 3rd b r a n c h D ( h - 1)
x (I~M)
y (mg/1)
u ( ~tM )
v (mg/1)
0.052 0.064 0.077
79.372 147.156 416.891
49.2063 48.5284 45.831
5.5989 9.6557 25.157
29.49 29.059 27.348
(b) D, 2rid b r a n c h D ( h - 1)
x (m M )
y (mg/1)
u (/t M )
v (mg/1)
0.052 0.064 0.077
4.657 4.683 4.712
3.434 3.171 2.883
343.43 317.1 288.28
0 0 0
297 TABLE 4 V a l u e s of c o n c e n t r a t i o n c h a n g e n e c e s s a r y to m o v e t h e w h o l e s y s t e m f r o m t h e t h i r d to t h e s e c o n d b r a n c h in t h e c a s e w i t h o u t s u b s t r a t e i n h i b i t i o n o f X o n Y D ( h - 1)
x (m M)
y (rag/l)
u (m m )
v (mg/1)
0.052 0.064 0.077
260 13 2.1
- 49.2062 - 47.8 - 34
9.3.106 110 3.4
- 29 - 20 - 4.8
N e g a t i v e v a l u e s m e a n t h a t this c o m p o n e n t h a s to b e d i m i n i s h e d .
F r o m the ecological point of view, a reactor is operating well only if it always remains on the third branch. Therefore, we have determined the magnitude of perturbation that forces the components of our system to evolve towards the steady-state values of the second branch. As indicated b y Table lb, the system on this branch is nearly unable to degrade the toxic substance X. The numerical experiments have been performed at fixed dilution rates (see the first column in Table 2). We start the evolution with three components equal to the steady state values. The remaining component has been varied until the evolution goes to the second branch. The results are given in Table 2. It clearly shows the sensitivity of any c o m p o n e n t depending on the dilution rate between the critical parameters D 1 and D 3. For comparison, we also examined our system without substrate inhibition of X on Y. The result is given in Tables 3a, 3b and 4. In this case, the region of bistability goes from D 1 = 0.05105 h -1 to D 3 = 0.07843 h - 1 (see
7t •
(o . I Z : - ~ : . ~ : : . : ~
..............
= = .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
e0
72
g
0
. . . .
0.00
I
. . . .
0.05
t o,
I
. . . .
0.I0
I
. . . .
015
t c6 Dilution Rate
I
020
. . . .
I
0.25
.
.
.
.
.
030
t [l/h]
Fig. 3. D i a g r a m o f t h e s o l u t i o n b r a n c h e s o b t a i n e d w i t h o u t s u b s t r a t e i n h i b i t i o n o f X o n Y.
298 50
-z
~0 ~
3°
g ~
20
..............................,..... \
/.:
1\\,\.~. .
'"... ..........................-."'/
f ..........
.I..t./"'//
01 o
. . . . . ~o
--S-'~'~ ~oo
i~o
.
~oo
.
.
.
~o
.
300
Time [h] Fig. 4. E v o l u t i o n at the dilution rate D = 0 . 0 6 4 h -1 with x ( 0 ) = l l (------), x; ( . . . . . . ), y; ( ), u; ( . . . . . ), v.
mM
as initial value:
Fig. 3). The steady-state values of the second and the third branch are only slightly different from those with inhibition of X on Y (see Tables la, lb). To demonstrate the differences in the outcome of an experiment in the region of bistability, we start our system ((1), (2), (4)) at D -- 0.064 h -1 with two different sets of initial values. In both cases, y(0), u(0), v(0) are the steady-state values taken from Table la. In the first experiment, x(0) is set at 11 m M (see Fig. 4) whereas, in the second case, x(0) = 12 m M has been chosen (see Fig. 5). In the first experiment, all components attain 99% of the steady-state values of the third branch within 200 h, while the other experiment shows that, after 600 h, all components of our system are fairly close to the steady-state values of the second branch. Next, we have tried to stop the system on its way towards the steady state on the second branch. After 200 h of the experiment in Fig. 5, we regulated the dilution rate D to a value which allows the system to return to the third so
E g ._
L
30
g
o
zo
~
~o 0
100
200
300
GO0
500
600
Time [h] Fig. 5. E v o l u t i o n at the dilution rate D = 0 . 0 6 4 h -1 w i t h x ( 0 ) = 1 2 m M
as initial value.
299
507
[ ...............
+5
i
+"+ ++o +++
i
i
-
/, .....
~ + +s I-....................
=8 ~ zo ,\
y..'"i,
.................................................................................................. //
~s
~
+i~ IOL.s--~" ..... " ..............:_~iT:;i o
;-----i--T--__ 0
100
200
300
s~O0
.
SO0
.
.
.
.
600
.
.
700
.
. . . . ,,
800
Tim~[h~
900
1000
Fig. 6. Effect of a change in the dilution rate from D = 0.064 h -1 to D = 0.058 h -1 after 200 h of evolution.
+l 50-
25 '"..
<~'-' .................. ~+~-m'si\"---_ ........................... ~o
~,
1o-~
"-..._.
0 0
100
200
300
6.00
SO0
600
700
800
9110
1000
Time[h] Fig. 7. Effect of a change in the dilution rate from D = 0.064 h -1 to D = 0.059 h -1 after 200 h of evolution.
TABLE 5 Values to which the dilution rate D must be reduced to move the system back towards the third branch after several hours T of disfunctioning at D = 0.064 h - t T (h)
D (h -1)
100 200 300 400 500 600
0.062 0.058 0.053 0.051 0.050 0.050
300 branch. Figure 6 shows that a change from D = 0.064 h -1 to D = 0.058 h -a brings the system back to the third branch while a change to D = 0.059 h-1 does not (Fig. 7). Note Table 5, which shows the values to which D must be reduced after T hours of disfunctioning at the rate D = 0.064 h -~ to guarantee a recovery of the system in the above-mentioned sense. DISCUSSION To attain bistability in a microbial food chain with two members cultivated under continuous-flow conditions, it is necessary that the growth rate /~1 of the first organism Y is negatively influenced by the substrate U of the second organism V and that the growth r a t e / & of this species is diminished b y the addition of the substrate X of the first one. On the other hand, in the special case of phenol-degrading organisms Y, experiments have shown that these bacteria are also inhibited by c o m p o u n d X (Fedorak and Hrudey, 1984). Theoretically, this additional substrate inhibition is accompanied b y a shortage of the stable part of the second branch. This branch is characterized by a very insufficient degradation of the inflowing compound. Only the third branch ensures that the degradation of an incoming toxic substance is almost complete. In semicontinuous culture experiments with undefined enrichment populations the dominant role of toxic substrates (phenol, m- and p-cresol) has been exemplified (Fedorak and Hrudey, 1986). In these experiments, diluted phenolic coal-conversion waste-water, which is a fairly complex medium, has been used. After repetitive addition of substrate to a batch culture, the toxic c o m p o u n d s have always accumulated and only a short period of complete removal has been observed. But no experiment has been performed that shows how stability can be regained. Finally, it will be useful to examine a microbial food chain with two members fed with one substrate and exchanging another one under defined continuous-culture conditions. REFERENCES Bohl, E. and Kreikenbohm, R., 1987. Syntrophic cocultures in nature and in model systems. In: J. Warnatz and W. J~iger (Editors), Modelling of Chemical Reactor Systems. Springer, Berlin, pp. 181-193. Bryant, M.P., 1979. Microbial methane production - theoretical aspects. J. Anim. Sci., 48: 193-201. Edwards, V.H., 1970. The influence of high substrate concentrations on microbial kinetics. Biotechnol. Bioeng., 12: 679-712. Fedorak, P.M. and Hrudey, S.E., 1984. The effects of phenol and some alkyl phenolics on batch anaerobic methanogenesis. Water Res., 18: 361-367.
301 Fedorak, P.M. and Hrudey, S.E., 1986. Anaerobic treatment of phenolic coal conversion wastewater in semicontinuous cultures. Water Res., 20: 113-122. Kreikenbohm, R. and Bohl, E., 1986. A mathematical model of syntrophic cocultures in the chemostat. FEMS Microbiol. Ecol., 38: 131-140. Lee, I.H., Fredrickson, A.G. and Tsuchiya, H.M., 1976. Dynamics of mixed cultures of Lactobacillus plantarum and Propionibacteriurn shermanii. Biotechnol. Bioeng., 18: 513-526. McInerney, M.J. and Bryant, M.P., 1980. Syntrophic associations of Hz-utilizing methanogenic bacteria and Hz-producing alcohol and fatty acid-degrading bacteria in anaerobic degradation of organic matter. In: G. Gottschalk, N. Pfennig and H. Werner (Editors), Anaerobes and Anaerobic Infections. Fischer, Stuttgart, pp. 117-126. Miura, Y., Sigiura, K., Yoh, M., Tanaka, H., Okazaki, M. and Komemushi, S., 1978. Mixed culture of Mycotorula japonica and Pseudomonas oleovorans on two hydrocarbons. J. Ferment. Technol., 56: 339-344. Miura, Y., Tanaka, H. and Okazaki, M., 1980. Stability analysis of commensal and mutual relations with competitive assimilation in continuous mixed culture. Biotechnol. Bioeng., 22: 929-946. Monod, J., 1942. Recherches sur la Croissance des Cultures Bactrriennes. Hermann, Paris, 211 pp. Pirt, S.J., 1975. Principles of Microbe and Cell Cultivation. Blackwell, Oxford, 274 pp. Prins, R.A., Van Nevel, C.J. and Demeyer, D.I., 1972. Pure culture studies of inhibitors for methanogenic bacteria. Antonie van Leeuwenhoek J. Microbiol. Serol., 38: 281-287. Reilly, P.J., 1974. Stability of commensalistic systems. Biotechnol. Bioeng., 16: 1373-1392. Szewzyk, U., Szewzyk, R. and Schink, B., 1985. Methanogenic degradation of hydroquinone and catechol via reductive dehydroxylation to phenol. FEMS Microbiol. Ecol., 31: 79-87. Yeoh, H.T., Bungay, H.R. and Krieg, N.R., 1968. A microbial interaction involving combined mutualism and inhibition. Can. J. Microbiol., 14: 491-492. Zehnder, A.J.B., 1978. Ecology of methane formation. In: R. Mitchell (Editor), Water Pollution Microbiology, 2. Wiley, New York, pp. 349-376.
287
BISTABILITY IN THE CHEMOSTAT R. KREIKENBOHM and E. BOHL
Fakultiit fiir Mathematik, Universitiit Konstanz, Postfach 5560, D-7750 Konstanz (Federal Republic of Germany) (Accepted 25 May 1988) ABSTRACT Kreikenbohm, R. and Bohl, E., 1988. Bistability in the chemostat. Ecol. Modelling, 43: 287-301. A general model for the interaction of two bacterial populations with interfering metabolic activities is provided. We discuss the dynamics of the system. As a result we obtain bistability under continuous-flow conditions if the growth rate of the first organism in the food chain depends on the substrate of the second, and if the growth rate of the second organism depends on the substrate of the first. During the anaerobic degradation of toxic substances, such an arrangement consisting of two organisms and two substrates can be found. In this case, we suggest explicit growth-rate expressions on the basis of experimental results and thermodynamic considerations. Using these expressions we can explain the disfunctioning of an anoxic digestor after a sudden increase of an environmental pollutant. INTRODUCTION In m i c r o b i a l ecosystems, the d e g r a d a t i o n of o r g a n i c m a t t e r p r o c e e d s in a f o o d chain consisting of two or m o r e m i c r o o r g a n i s m s with d i f f e r e n t metabolic activities. T h e simplest of these a l i g n m e n t s is the c o m m e n s a l i s t i c c o c u l t u r e with o n l y two different species. This system can b e r e p r e s e n t e d b y the following r e a c t i o n scheme: X
Y=
U
V
i n this f o o d chain Y a n d V d e n o t e the i n t e r a c t i n g m i c r o b i a l p o p u l a t i o n s , X is the substrate o f the first organism, a n d U stands for a p r o d u c t that at the s a m e time serves as the sole substrate o f the s e c o n d species. A theoretical analysis o f this c o m m e n s a l i s t i c system u n d e r c o n t i n u o u s - c u l t u r e c o n d i t i o n s has b e e n p e r f o r m e d b y Reilly (1974). Stable coexistence of Lactobacillus plantarum Y a n d Propionibacterium shermanii V has b e e n o b s e r v e d in a m i x e d c o n t i n u o u s culture growing o n glucose X a n d lactate U, respectively ( L e e et al., 1976). A t a n y given dilution rate there was o n l y o n e stable s t e a d y state a n d n o oscillations occurred. M o r e c o m p l i c a t e d m i x e d cultures consisting o f two d i f f e r e n t m i c r o b i a l p o p u l a t i o n s have b e e n investigated e x p e r i m e n t a l l y ( Y e o h et al., 1968; M i u r a 0304-3800/88/$03.50
© 1988 Elsevier Science Publishers B.V.
288
et al., 1978) and have been analysed theoretically (Miura et al., 1980). In these systems two or even more interactions between both species coexist. Furthermore, oscillatory phenomena were found under continuous-culture conditions. In methanogenic anaerobic environments the degradation of fatty acids, alcohols, aromatic compounds and amino acids is carried out by bacteria belonging to two different metabolic groups that are combined in syntrophic associations (Zehnder, 1978; Bryant, 1979). The members of the first group are Ha-producing acetogenic bacteria Y that convert the substrates X mentioned above to hydrogen U, acetate and CO 2. For thermodynamic reasons the release of hydrogen is limited to very low partial pressures (McInerney and Bryant, 1980). Therefore, the extent to which the substrate X is degraded by the organism Y depends on the efficiency of the removal of the product U by the methanogenic bacteria V of the second group. Here, a generalized reaction sequence can be given in the following form: X
~
U
~
A dynamic model of this type of interaction has been examined theoretically (Kreikenbohm and Bohl, 1986) and it has been used for the determination of the growth parameters of an isovalerate degrading strain cultivated in a defined coculture with Methanospirillum hungatei (Bohl and Kreikenbohm, 1987). In this contribution we present a model for the degradation of organic compounds that are considered to be toxic to either member of a syntrophic association. For example, the degradation of toluene, phenol, catechol, long-chain fatty acids, and unsaturated hydrocarbons such as isoprene can be summarized in the following way:
X,
~
U
,~
Here, the consumption of the molecule X by the organism Y shows substrate inhibition at high concentrations. Furthermore, the second organism in the food chain is also inhibited by molecule X because of a negative influence on the functional activities of the microbial cells. Finally, the production of the intermediate U is thermodynamically limited as discussed above. For example, the anaerobic degradation of phenolics by an undefined microbial community from domestic sludge has been examined (Fedorak
289
and Hrudey, 1984) under batch culture conditions. As a result, the authors have shown that the phenol-degrading bacteria are inhibited by their own substrate at a concentration of about 5 mM. Their metabolic partners (methanogens) stopped methane production when phenol was present at about 20 m M. Furthermore, evidence has been provided that, under anaerobic conditions, the phenol-degrading microorganisms are H2-producing acetogenic bacteria (Szewzyk et al., 1986). Thus, these authors worked with a food chain as presented by our third reaction scheme. With little modification this scheme may also be valid for the decontamination of pesticides by a microbial food chain. Our model allows us to understand the difficulties in the enrichment studies on organisms that degrade toxic compounds. Furthermore, we can explain the disfunctioning of an anoxic digestor from the sewage plants after suddently increasing the input of an environmental pollutant. This increase in the concentration of a toxic substance may be due to an accident: from a source with a very high concentration the pollutant enters the stream of waste water. Thereby, neither the volume of the waste water nor its flow rate is altered significantly. Furthermore, other parameters like temperature or pH value are also unchanged. As a result, the microbial community is disturbed so deeply that the whole degradation process stops at all the different stages. In this contribution, we model the situation described above as a syntrophic coculture of two organisms in the chemostat. We assume that the thermodynamics of hydrogen release and the inhibition by the inflowing substrate are the dominant factors. Using the appropriate growth rate expressions, our model shows hysteretic phenomena. Therefore, it allows us to understand that, at the same dilution rate, the degradation of the substrate is either nearly complete or restricted to a few percent. DYNAMICAL SYSTEM OF MIXED CONTINUOUS CULTURES
Continuous culture experiments of mixed microbial populations with metabolic reactions obeying any of the given schemes can be described by the following dynamical system:
t l(X, u) y
f¢--
Y1
)~' = ( ~ I ( X , fi
+ D(XR-- X )
u)--O)y
t,,(x, u)
(lb)
u)
y
~/1
v - Du
3'2 > 0 ,
(lc)
Y2
t)= (/,2(x, u ) - D ) v 3'1 > 0 ,
(la)
D>0,
(ld) x R>0
(le)
290 where D is the dilution rate, and y, v denote the population densities of the interacting species Y, V. In the culture vessel the substrates X, U are present at the concentrations x, u, whereas x R is the concentration of the substrate X in the reservoir. As indicated, the specific growth rates/_h(x, u), ~t2(x, u) depend on both substrates X and U. For the metabolism of the first organism we assume that the degradation of one molecule of substrate X is directly accompanied by the extrusion of one molecule of product U. A n y delay due to internal storage formation is excluded, so that the molecules U are immediately available as substrate to the organism V. Finally, equations (la) and (lc) imply that consumption and production are strictly growth-dependent processes. For the first organism Y in the third reaction scheme we choose the following growth-rate expression:
/ /~l(x, u)
=
x /~m~,l
--
u
gl
x 2
if
X-- K11u > 0
(2)
K 2 + x + K 3 u + 'K--4
0
otherwise
with K i > 0 (i = 1, 2, 3, 4). Here, /Lm~,l denotes the m a x i m u m specific growth rate of the organisms Y. The expression (2) ensures that these bacteria stop growing if the following expression fails: (3)
u < Klx
K a can be interpreted via the thermodynamics involved in the reaction from X to U (Kreikenbohm and Bohl, 1986); K 2 is a Michaelis-Menten-type constant, and g 3 a c t s as an inhibition constant related to the negative influence of the substrate U on its own production; K 4 is a constant that models the inhibition of the organism Y by its own substrate X. If K 4 ~ ~ , the growth rate expression (2) reduces to the form used for the degradation of nontoxic substrates by the Hz-producing acetogenic bacteria (Kreikenbohm and Bohl, 1986). If K 1 ~ ~ and K 3 = 0, we have the simplest substrate inhibition model analysed by Edwards (1970). Finally, if K 1 ~ ~ , K 3 = 0 , K 4 ~ oO, the growth-rate expression (2) reduces to the relation introdtrced by M o n o d (1942). For the second population V of the food chain in the third reaction scheme we use the following growth-rate expression: ~2(x, u ) -
~max2~/ K6 Ks+u Kr+x
(4)
291 with Ks, K 6 > 0. Here, /.tmax2 gives the maximum specific growth rate of the organisms V; K 5 is a Michaelis-Menten-type constant, and K 6 is an inhibition constant. Expression (4) has been introduced by Pirt (1975) to model the influence of a non-competative inhibitor that interacts with the microbial cell. For example, the negative influence of unsaturated long-chain fatty acids on methanogenic bacteria can be interpreted this way (Prins et al., 1972). This sort of inhibitory action leads to a decrease in the actual growth rate without affecting the specifity of the degradation process. Therefore, the affinity constant K 5 is unchanged with regard to the Monod relation. ANALYSIS OF SYSTEM (1) For the discussion of (1) it is important to note that we do not need the explicit representations of the growth-rate expressions (2) and (4). It is rather enough to consider qualitative assumptions on /.q(x, u),/~2(x, u) which we summarize as: ~i>0,
D>__0, / t ~ L i p ( R 2 ) ,
XR>0,
/tl(0, u ) = 0 ,
#2(x,0)=0
for
x>0,
#l(x,u)>0,
~2(x,u)>0,
~l(x, 0 ) > 0
i=1,2
(5a)
u>__0, /~2(0, X R ) > 0
(5b)
for
x>0,
u>0
(5c)
Expression (5b) says that the growth rate of a population vanishes in the absence of food. The Lipschitz-class appearing in (5a) contains, in particular, all functions which are continuously differentiable with respect to both variables. Our function (2) is in that Lipschitz-class but not continuously differentiable. For a more detailed discussion of the steady states we need some further technical assumptions, which we mention later. Let (x(t), y(t), u(t), v(t)) be a solution of (1) with initial values: x(0)>__0,
y(0)>0,
u(0)>0,
v(0)>0
(6)
Then this solution exists for all times t > 0 and all components of the solution remain non-negative for any t > 0. Furthermore, the functions:
wl(t)=x(t)+
")tll y ( t ) - x
R
(7a)
w2(t ) = x(t) + u(t) + V2' o ( t ) - x R
(7b)
satisfy:
wj(t) = wj(0) e x p ( - D t )
for
t > 0,
j = 1, 2
To prove these statements we only need (5).
(8)
292 In this section we describe the stationary solutions of our system (1). The steady-state system reads: (9a)
--~l(X, U) ~ l l y + D ( x R - x) = 0 (/11(x,
u)-D)
(#2(x, u ) - D )
y-0,
(9b)
v=0
~l(X, U) ~/lly--~[L2(X , U) ~21U--Ou = 0
(9c)
It is easily seen that any solution of (9) is of the form: ( x , v , ( x R - x ) , u, v ~ ( x . -
(10)
x - u))
To ensure non-negativity of all components in (10) we need: 0 <_~X'~ 0 ~_~U,
(11)
X'+-U~_~XR
The two parameters x, u satisfy one of the following conditions: X=XR,
(12a)
U=0
Ill(x, XR--X )=D, ~,,(x, u) = D,
U=XR--X
(12b)
(12c)
~q(x, u) = D
Accordingly, we have three solution branches of the steady-state system (9) which m a y be described as: X = XR, ,
y = 0,
u = 0,
Izl(X, X R - - X ) = D
,
v= 0
y='Yl(XR--X),
~l(X, u ) = O ,
y=y(XR--X
~ 2 ( x , u) = D ,
.
= ~2(x.-
(13a) U=XR--X ,
V=0
)
(13b) (13C)
x - u)
To see this, just combine (10) and (12). For a stability analysis of the various steady states we consider the dilution rate D to be our control parameter. Then we may characterize the three branches (13) in a (D, x)-plane. In the case of (13a), the branch is just the straight line parallel to the D-axis on the level x R (see Fig. 1). It is easily shown that this branch consists of stable stationary points if D > D2, and it is unstable if D < D x where: 3 2 := ~I(XR, 0) > 0
(14)
Next, we consider (13b). We must plot the graph of:
gl(X):=ld,l(X, XR--X),
X E [0, XR]
(I5)
defined by the first equation in (13b). By (5b) this branch joins the origin with the point B = ( D 2, XR) in Fig. 1. The point B is a bifurcation point
293
since it lies on the two branches (13a) and (13b). To analyse the stability of (13b) we must discuss the function:
XR--X),
[0, XR]
(16)
By (5) we have h ( 0 ) < 0, h ( X R ) > 0. We are now going to introduce the additional assumptions: /~,x(x, u ) > 0 , /~2u(x, u ) > 0 (lVa) ~tlu(X, u ) < 0 , ~tzx(X, u) < 0 (17b) at all steady states on the branch (13b). Expression (17a) means that, at the steady states, both organisms are supported by their respective food, and (17b) means that, at these steady states, the first organism is inhibited by its product and the second organism is inhibited by the food of the first organism. With this assumption, h defined in (16) is strictly m o n o t o n e increasing on the interval [0, xR]. Hence, there exists a unique zero 2 (0, xR). By definition we have: /~,(Y, x R - 2 ) - t ~ 2 ( 2 ,
x R - ~) = 0
(18a)
D, :-/~,(Y, x R - Y) (lab) It is easily proved that the branch (13b) is stable if x ~ (Y, xR) and unstable if x ~ (0, Y). This is indicated in Fig. 1 where we plot the unstable parts as a broken line and the stable parts as a solid line. The change in stability occurs at the point A in Fig. 1. Its corresponding solution (13b) reads:
(.~, ]tl(X R - ~), x R - .~, 0) (19) We finally turn to the branch (13c). It is defined by the solution set of the two-dimensional system (12c) under the constraints: O
X B (13a) X!~I.......................................... ...-..........~
DI
0 2
0
Fig. 1. Bifurcation diagram representing the stable ( ) and the unstable ( . . . . . . of the solution branches (13a), (13b), (13c) indicating no hysteresis phenomenon.
) parts
294 (12c). The corresponding point on the branch (13c) is exactly (19). Hence, it is a bifurcation point on both branches (13b, c) (see point A in Fig. 1). Furthermore, (5b, c) yields that x = 0, u = 0 is the only solution of (12c) for D = 0. The corresponding point on the branch (13c) is: (0, "YlXR, 0, )'2XR)
(21)
We easily verify that (21) is not a point on the branch (13b) so that the origin in Fig. 1 is not a bifurcation point. Only the projections of the two branches (13b, c) drawn in Fig. 1 coincide in the origin. For a further discussion of the branch (13c) we assume that, for any x E (0, XR), there is a unique solution (u(x), D(x)) of the system (12c). T h e n the corresponding point on the branch (13c) is:
(x, "YI(XR-- X), U(X), ~'2(XR-- X-- U(X)))
(22)
with the control parameter D(x) for 0 < x < x R. If we now assume (17) for (x, u(x)), x ~ (0, XR), then the function u(x) is strictly m o n o t o n e increasing. Since u(0) = 0, u(~) = x R - ~, we have:
x+u(x)
for
x ~ [0, ~1
x + u(x) > x R
for
x ~ ( Y , XR]
(23)
In view of (20) we are therefore only interested in the steady states (22) with x ~ (0, ~], where Y is defined by (18a). In the projection of Fig. 1, this part of the branch is characterized by the function D(x), the graph of which joins the origin and the point A. We can now prove that all parts on this graph with: D'(x) >0
(24)
are stable, and all points with: D'(x) <0
(25)
are unstable steady states on the branch (13c). Since an inequality (24) prevails in Fig. 1, the branch is indicated as a solid line (a completely stable branch). There are of course other possibilities, for example, that given in Fig. 2. Here, the system shows a hysteresis p h e n o m e n o n . In this respect, the following representation:
D'(x) = [ ( # i x / * 2 , - g2xlJ'lu)(~2u -
~1u)-1]( x'
U(X))
(26)
for the derivative of D is of interest. In view of (24), (25), it shows that the sign of D ' is completely determined by the sign of the quantity: (~.glx/X2, - ~,~2xglu)(x, u(x))
(27)
295
i
7
.............-" bl .........................(al ...............
x o
S
g
~3
-It)
OOO
005
OIO
01s
o20
025
0 30
c~ o~ o7
0itufi0n Rote 0 [I/h] Fig. 2. Bifurcation diagram of the solution branches on the basis of the rate expressions (2),
(4). By (17) the first product of this difference is always positive so that (24) (and therefore Fig. 1) holds if the last product in (27) vanishes in the interval (0, ~). This means that a hysteresis phenomenon necessarily requires that the function /L~ depends on u and the function /~2 depends on x. In particular, it is necessary that the growth rate of the second organism explicitely depends on the substrate x which is the food of the first organism. If the second is unaffected by the food of its partner, the system is unable to undergo hysteresis loops. NUMERICAL EXPERIMENTS
In this section, we examine the behaviour of our system (1) in the region of bistability between D 1 and D 3 in Fig. 2. This figure is drawn on the basis of the model parameters x R = 5 mM, ~maxl = 0 . 2 h - 1 , It 1 = 10 g/mol, K~ = 0.1, K 2 = 1 0 ~tM, K 3=0.1, K 4 = 5 mM, /Zr,ax2=0.1 h -1, "/2=6 g/mol, K 5 = 5 ~tM, K 6 = 5 mM. The solid trace of the three branches (a), (b), (c) depicted in Fig. 2 gives the steady state concentration of the substrate X, which should be attained by any evolution started in its vicinity. As stated before, at any given dilution rate between D~ and D 3 there are two stable steady states. In Table 1, the two steady-state values of the substrate concentrations x, u and the population densities y , u are given at three different dilution rates D (D1, D3). A glance at Table l a and lb shows the differences: in Table l a (third branch) a cooperation between the two organisms Y and V is obtained; in contrast, population V in Table lb (second branch) is no longer in the chemostat and hence substrate X remains almost unconsumed.
296 TABLE 1 D e p e n d e n c e of substrate c o n c e n t r a t i o n s x, u a n d bacterial p o p u l a t i o n densities y, v o n dilution rate D (a) D, 3rd b r a n c h D ( h - 1)
x (~M)
y (mg/1)
u (~M)
v (mg/1)
0.051 0.064 0.077
76.059 149.443 470.241
49.2394 48.5056 45.2976
5.3682 9.6683 26.7293
29.511 29.045 27.018
(b) D, 2nd b r a n c h D ( h - 1)
x (mM)
y (mg/1)
u (~M)
v (mg/1)
0.051 0.064 0.077
4.762 4.822 4.884
2.382 1.782 1.159
238.24 178.19 115.86
0 0 0
TABLE 2 Values of the c o n c e n t r a t i o n change necessary to move the whole system from the third to the second b r a n c h D ( h - 1)
x (m M )
y (mg/1)
u (m M )
v (mg/l)
0.051 0.064 0.077
284 11.5 1.5
- 49.233 - 46.1 - 26.1
26100 34.8 2.1
- 29 - 18.9 - 3.2
Negative values m e a n that this c o m p o n e n t has to be diminished. TABLE 3 D e p e n d e n c e of substrate c o n c e n t r a t i o n s x, u a n d bacterial p o p u l a t i o n densities y, v o n dilution rate D without substrate inhibition of X o n Y (a) D, 3rd b r a n c h D ( h - 1)
x (I~M)
y (mg/1)
u ( ~tM )
v (mg/1)
0.052 0.064 0.077
79.372 147.156 416.891
49.2063 48.5284 45.831
5.5989 9.6557 25.157
29.49 29.059 27.348
(b) D, 2rid b r a n c h D ( h - 1)
x (m M )
y (mg/1)
u (/t M )
v (mg/1)
0.052 0.064 0.077
4.657 4.683 4.712
3.434 3.171 2.883
343.43 317.1 288.28
0 0 0
297 TABLE 4 V a l u e s of c o n c e n t r a t i o n c h a n g e n e c e s s a r y to m o v e t h e w h o l e s y s t e m f r o m t h e t h i r d to t h e s e c o n d b r a n c h in t h e c a s e w i t h o u t s u b s t r a t e i n h i b i t i o n o f X o n Y D ( h - 1)
x (m M)
y (rag/l)
u (m m )
v (mg/1)
0.052 0.064 0.077
260 13 2.1
- 49.2062 - 47.8 - 34
9.3.106 110 3.4
- 29 - 20 - 4.8
N e g a t i v e v a l u e s m e a n t h a t this c o m p o n e n t h a s to b e d i m i n i s h e d .
F r o m the ecological point of view, a reactor is operating well only if it always remains on the third branch. Therefore, we have determined the magnitude of perturbation that forces the components of our system to evolve towards the steady-state values of the second branch. As indicated b y Table lb, the system on this branch is nearly unable to degrade the toxic substance X. The numerical experiments have been performed at fixed dilution rates (see the first column in Table 2). We start the evolution with three components equal to the steady state values. The remaining component has been varied until the evolution goes to the second branch. The results are given in Table 2. It clearly shows the sensitivity of any c o m p o n e n t depending on the dilution rate between the critical parameters D 1 and D 3. For comparison, we also examined our system without substrate inhibition of X on Y. The result is given in Tables 3a, 3b and 4. In this case, the region of bistability goes from D 1 = 0.05105 h -1 to D 3 = 0.07843 h - 1 (see
7t •
(o . I Z : - ~ : . ~ : : . : ~
..............
= = .
.
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.
.
e0
72
g
0
. . . .
0.00
I
. . . .
0.05
t o,
I
. . . .
0.I0
I
. . . .
015
t c6 Dilution Rate
I
020
. . . .
I
0.25
.
.
.
.
.
030
t [l/h]
Fig. 3. D i a g r a m o f t h e s o l u t i o n b r a n c h e s o b t a i n e d w i t h o u t s u b s t r a t e i n h i b i t i o n o f X o n Y.
298 50
-z
~0 ~
3°
g ~
20
..............................,..... \
/.:
1\\,\.~. .
'"... ..........................-."'/
f ..........
.I..t./"'//
01 o
. . . . . ~o
--S-'~'~ ~oo
i~o
.
~oo
.
.
.
~o
.
300
Time [h] Fig. 4. E v o l u t i o n at the dilution rate D = 0 . 0 6 4 h -1 with x ( 0 ) = l l (------), x; ( . . . . . . ), y; ( ), u; ( . . . . . ), v.
mM
as initial value:
Fig. 3). The steady-state values of the second and the third branch are only slightly different from those with inhibition of X on Y (see Tables la, lb). To demonstrate the differences in the outcome of an experiment in the region of bistability, we start our system ((1), (2), (4)) at D -- 0.064 h -1 with two different sets of initial values. In both cases, y(0), u(0), v(0) are the steady-state values taken from Table la. In the first experiment, x(0) is set at 11 m M (see Fig. 4) whereas, in the second case, x(0) = 12 m M has been chosen (see Fig. 5). In the first experiment, all components attain 99% of the steady-state values of the third branch within 200 h, while the other experiment shows that, after 600 h, all components of our system are fairly close to the steady-state values of the second branch. Next, we have tried to stop the system on its way towards the steady state on the second branch. After 200 h of the experiment in Fig. 5, we regulated the dilution rate D to a value which allows the system to return to the third so
E g ._
L
30
g
o
zo
~
~o 0
100
200
300
GO0
500
600
Time [h] Fig. 5. E v o l u t i o n at the dilution rate D = 0 . 0 6 4 h -1 w i t h x ( 0 ) = 1 2 m M
as initial value.
299
507
[ ...............
+5
i
+"+ ++o +++
i
i
-
/, .....
~ + +s I-....................
=8 ~ zo ,\
y..'"i,
.................................................................................................. //
~s
~
+i~ IOL.s--~" ..... " ..............:_~iT:;i o
;-----i--T--__ 0
100
200
300
s~O0
.
SO0
.
.
.
.
600
.
.
700
.
. . . . ,,
800
Tim~[h~
900
1000
Fig. 6. Effect of a change in the dilution rate from D = 0.064 h -1 to D = 0.058 h -1 after 200 h of evolution.
+l 50-
25 '"..
<~'-' .................. ~+~-m'si\"---_ ........................... ~o
~,
1o-~
"-..._.
0 0
100
200
300
6.00
SO0
600
700
800
9110
1000
Time[h] Fig. 7. Effect of a change in the dilution rate from D = 0.064 h -1 to D = 0.059 h -1 after 200 h of evolution.
TABLE 5 Values to which the dilution rate D must be reduced to move the system back towards the third branch after several hours T of disfunctioning at D = 0.064 h - t T (h)
D (h -1)
100 200 300 400 500 600
0.062 0.058 0.053 0.051 0.050 0.050
300 branch. Figure 6 shows that a change from D = 0.064 h -1 to D = 0.058 h -a brings the system back to the third branch while a change to D = 0.059 h-1 does not (Fig. 7). Note Table 5, which shows the values to which D must be reduced after T hours of disfunctioning at the rate D = 0.064 h -~ to guarantee a recovery of the system in the above-mentioned sense. DISCUSSION To attain bistability in a microbial food chain with two members cultivated under continuous-flow conditions, it is necessary that the growth rate /~1 of the first organism Y is negatively influenced by the substrate U of the second organism V and that the growth r a t e / & of this species is diminished b y the addition of the substrate X of the first one. On the other hand, in the special case of phenol-degrading organisms Y, experiments have shown that these bacteria are also inhibited by c o m p o u n d X (Fedorak and Hrudey, 1984). Theoretically, this additional substrate inhibition is accompanied b y a shortage of the stable part of the second branch. This branch is characterized by a very insufficient degradation of the inflowing compound. Only the third branch ensures that the degradation of an incoming toxic substance is almost complete. In semicontinuous culture experiments with undefined enrichment populations the dominant role of toxic substrates (phenol, m- and p-cresol) has been exemplified (Fedorak and Hrudey, 1986). In these experiments, diluted phenolic coal-conversion waste-water, which is a fairly complex medium, has been used. After repetitive addition of substrate to a batch culture, the toxic c o m p o u n d s have always accumulated and only a short period of complete removal has been observed. But no experiment has been performed that shows how stability can be regained. Finally, it will be useful to examine a microbial food chain with two members fed with one substrate and exchanging another one under defined continuous-culture conditions. REFERENCES Bohl, E. and Kreikenbohm, R., 1987. Syntrophic cocultures in nature and in model systems. In: J. Warnatz and W. J~iger (Editors), Modelling of Chemical Reactor Systems. Springer, Berlin, pp. 181-193. Bryant, M.P., 1979. Microbial methane production - theoretical aspects. J. Anim. Sci., 48: 193-201. Edwards, V.H., 1970. The influence of high substrate concentrations on microbial kinetics. Biotechnol. Bioeng., 12: 679-712. Fedorak, P.M. and Hrudey, S.E., 1984. The effects of phenol and some alkyl phenolics on batch anaerobic methanogenesis. Water Res., 18: 361-367.
301 Fedorak, P.M. and Hrudey, S.E., 1986. Anaerobic treatment of phenolic coal conversion wastewater in semicontinuous cultures. Water Res., 20: 113-122. Kreikenbohm, R. and Bohl, E., 1986. A mathematical model of syntrophic cocultures in the chemostat. FEMS Microbiol. Ecol., 38: 131-140. Lee, I.H., Fredrickson, A.G. and Tsuchiya, H.M., 1976. Dynamics of mixed cultures of Lactobacillus plantarum and Propionibacteriurn shermanii. Biotechnol. Bioeng., 18: 513-526. McInerney, M.J. and Bryant, M.P., 1980. Syntrophic associations of Hz-utilizing methanogenic bacteria and Hz-producing alcohol and fatty acid-degrading bacteria in anaerobic degradation of organic matter. In: G. Gottschalk, N. Pfennig and H. Werner (Editors), Anaerobes and Anaerobic Infections. Fischer, Stuttgart, pp. 117-126. Miura, Y., Sigiura, K., Yoh, M., Tanaka, H., Okazaki, M. and Komemushi, S., 1978. Mixed culture of Mycotorula japonica and Pseudomonas oleovorans on two hydrocarbons. J. Ferment. Technol., 56: 339-344. Miura, Y., Tanaka, H. and Okazaki, M., 1980. Stability analysis of commensal and mutual relations with competitive assimilation in continuous mixed culture. Biotechnol. Bioeng., 22: 929-946. Monod, J., 1942. Recherches sur la Croissance des Cultures Bactrriennes. Hermann, Paris, 211 pp. Pirt, S.J., 1975. Principles of Microbe and Cell Cultivation. Blackwell, Oxford, 274 pp. Prins, R.A., Van Nevel, C.J. and Demeyer, D.I., 1972. Pure culture studies of inhibitors for methanogenic bacteria. Antonie van Leeuwenhoek J. Microbiol. Serol., 38: 281-287. Reilly, P.J., 1974. Stability of commensalistic systems. Biotechnol. Bioeng., 16: 1373-1392. Szewzyk, U., Szewzyk, R. and Schink, B., 1985. Methanogenic degradation of hydroquinone and catechol via reductive dehydroxylation to phenol. FEMS Microbiol. Ecol., 31: 79-87. Yeoh, H.T., Bungay, H.R. and Krieg, N.R., 1968. A microbial interaction involving combined mutualism and inhibition. Can. J. Microbiol., 14: 491-492. Zehnder, A.J.B., 1978. Ecology of methane formation. In: R. Mitchell (Editor), Water Pollution Microbiology, 2. Wiley, New York, pp. 349-376.