BIOTECHNOLOGY AND BIOENGINEERING, VOL. XVIII, PAGES 239-252 (1976)
Protozoan Feeding and Bacterial Wall Growth ANTONIO BONOMI* and A. G. FREDRICKSON, Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 66466
Summary Monod‘s model is often assumed to describe the kinetics of feeding of a protozoan population on a bacterial population in a chemostat. An earlier study (J. L. Jost e t al., J. Bacteriol., 113, 84 (1973)) of the feeding of Tetrahymena pyriformis on either Escherichia coli or Azotobacter vinelandii found that this model correctly predicted the occurrence of sustained oscillations of population densities but made predictions of minimum bacterial population densities that were much smaller than those observed. The earlier study removed the discrepancy between the model and data by replacing Monod’s model with a different model. I t is shown in the present study that the discrepancy can be explained equally as well if Monod’s model for the feeding relation is retained and if, in addition, growth of bacteria on the chemostat walls is allowed for in the model equations.
INTRODUCTION have dealt with the “feeding,” A number of recent “grazing,” or (‘predation’’of a population of protozoans on a population of bacteria in a chemostat. The simplest model of such a system that seems to have any possibility of being successful is that in which one assumes that the reproduction rate of the protozoa is limited by the density of the bacteria, that the reproduction rate of the bacteria is limited by the concentration of a single chemical substance dissolved in the incoming feed liquid, and that Monod’s model (with biomass concentrations replaced by population densities) describes reproduction and consumption of the “substrate” by both populations. The equations which result from this model are
*Permanent address: Department of Chemical Engineering, Escola PolitBcnica, University of S&oPaulo C. P. 8174, SLo Paulo, Brazil. 239
@ 1976 by John Wiley & Sons, Inc.
240
BONOMI AND FREDRICKSON
ds - 1 _ - -(sf e dt
- s)
1 psb Y K + s
--
(1c)
where p , b are population densities of protozoa and bacteria, respectively, s, sf are concentrations of the rate-limiting substrate for bacterial growth in the chemostat and the feed to the chemostat, respectively, 0 is the holding time (reciprocal dilution rate) of the chemostat, v, p are maximum specific reproduction rates of the protozoa and bacteria, respectively, X , Y are yield coefficients for reproduction of the protozoa and bacteria, respectively, and L, K are Michaelis constants of Monod’s model for reproduction of the protozoa and bacteria, respectively. These equations were first published by Drake et al.’ and the Bungay9 and analyses of them were published by Canale3s4and Jost et al.5 Under most operating conditions (i.e., conditions of e and sf), eq. (1) predicts that the system will exhibit a state of sustained oscillations of all concentration quantities rather than a steady state of coexistence or a steady state where one or both of the populations becomes extinct. Tsuchiya et al. actually observed such oscillations with the system Dictyostelium discoideum (a cellular slime mold) and Escherichia coli and they demonstrated that eq. (1) gave a good description of their observations. However, Canale et al.’ could not fit their data on the system Tetrahymena pyriformis-Aerobacter aerogenes using that equation. They suggested that death of the populations and the reuse of carbohydrates released by metabolic processes or by death and lysis were important in their system and so rendered its behavior more complex than that predicted by eq. (1). Van den Endes studied the system T. pyriformis-Klebsiella aerogenes in a “continuous culture” in a single run at an unspecified holding time; the system exhibited a series of damped oscillations of concentrations in approaching what appeared to be a steady state. Jost et al.9 studied the systems T. pyriformis-E. coli and T. pyriformis-Azotobacter vinelandii in chemostat cultures. The constants V , p , L, K , X , and Y of eq. (1)were determined by independent experiments and used to predict the results of chemostat experiments. The equation predicted that sustained oscillations of concentrations should have occurred at the operating conditions used and this was observed experimentally; however, the minimum bacterial densities predicted by the model vlrere exceedingly small. For instance, if
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
241
eq. (1) is integrated numerically for a holding time of 20 hr and a feed substrate concentration of 0.52 g/liter, using the same numerical values of the model parameters v, p , L, K,X ,and Y that are subsequently used to construct Figure 1 in this study, one finds that the minimum bacterial density is about 7 x liter-’ so that in a 100 ml chemostat (the size used by Jost et al.9) only 7 X bacteria are predicted to be present at the time the bacterial density achieves its minimum. Equation (1) predicts that the system recovers from this minimum, but of course that would hardly be possible. If the stochastic, discrete version of eq. (1)had been used, it would surely have predicted that the protozoa would consume all of the bacteria and that the consequent nongrowing protozoan cells would then be washed from the chemostat. Experimentally, however, extinction of the populations was never observed. The data obtained by Jost et al.9 suggested to them that their “predators” became less effective at low densities of the “prey;” i.e., that the predation rate fell off more rapidly than the first power of the prey density as the latter became small, as predicted by Monod’s model. Hence, they used a model which amounted to replacing the factor vbp/(L b) which appears in eq. (1) by the factor vb2p/ ( L , b) (L 2+ b ) , where L1 and Lz are new constants to be determined. This “multiple saturation” model also predicted sustained oscillations of concentrations under certain operating conditions, but the minimum bacterial densities predicted were now much higher and in fact were of the order of magnitude of those actually observed. Moreover, analysis of the new model led to the prediction that the dynamical behavior of the system could be changed from a state of sustained oscillations of concentrations to a steady state (approached via damped oscillations) by keeping the same concentration of glucose in the feed but progressively increasing the holding time.5 Experim e n t ~showed ~ that the system did indeed exhibit this change as the holding time was increased. No such prediction concerning the effect of increased holding time is made by eq. (1) and so the experimental results were regarded as strong evidence against Monod’s model and in favor of the multiple saturation model for feeding of the ciliates on bacteria. There is, however, at least one other possibility for explaining the apparent failure of RiIonod’s model found by Jost et al.9 Van den Endes has pointed out that even when the liquid in a culture vessel is well stirred, the environment therein is still not spatially homogeneous because of the unavoidable presence of the vessel walls. If bacteria can attach themselves to the walls and grow there, and if such attached
+
+
BONOMI AND FREDRICKSON
242
cells are not captured by the protozoans, then bacteria that slough off the vessel walls will continually reinoculate the liquid portion of the culture. Clearly, the effects of reinoculation from the wall will be greatest when the bacterial population in the liquid is least and it will prevent the bacterial density from falling as low as it otherwise would. Hence, the present study will deal with the effects of wall growth of the bacteria on the dynamics of the protozoan-bacterial system propagated in a chemostat.
MODIFICATION OF THE MODEL TO ACCOUNT FOR WALL GROWTH OF THE BACTERIA We assume that the feeding of the protozoa on bacteria in the liquid is described adequately by Monod's model, so th a t eq. (1) will be valid in the case where the internal surface-to-volume ratio ( A I V ) of the chemostat is negligible. Our task is then to add terms to correct for a nonnegligible surface-to-volume ratio. I n order to do this, we assume that the model of wall growth proposed by Topiwala and Hamer'o is valid for the bacteria. The elements of this model are that 1) there is an equilibrium surface density of bacteria (b') which is achieved very quickly and which is maintained by a balance of the reproduction rate and slough-off rate and 2) the specific growth rate of bacteria on the wall is the same as the specific growth rate of bacteria in the liquid. Equilibrium densities b' which have been observed"J2 for growth on glass surfaces range from 106 to 108 cm-2 and these correspond to something less than a monolayer of closely packed cells on the wall. Hence, wall growth need not be visible in order to be present. The modified version of eq. (1) is then
_ dP - - - p 1+ - dt
8
db -
1- b
dt
~
e
VbP L + b
- ~ xvbp- + ~ (2b) (b + $ b') Y K+s
+ K" ( +6 S
+7 A b')
- 1
ds 1 = - ( s / - s) - - _ _ ps dt 0
When b' is regarded as constant, as in the model of Topiwala and Hamer, eq. (2) will admit two steady states: coexistence of bacteria and protozoans ( b , p > 0; 0 < s < sf) and washout of the protozoans
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
243
( b > 0 ; p = 0 ; 0 < s < s,). The stability of these steady states with respect t o small perturbations is discussed below.
CHOICE OF MODEL PARAMETERS The objective of the present calculations is to see if by appropriate choice of model parameters it is possible to make eq. (2) correct the apparent failure of Monod’s model found by Jost et al.9for (say) the Tetrahymena-Azotobacter system. The parameters v, p, K , X , and Y are not subject to choice since they are common to the multiple saturation model and to the present model. The values of these constants reported by Jost13 for the Tetrahymena-Azotobacter system grown in glucose-minimal salts medium a t 28°C are: v = 0.20 hr-l; g/liter; X = 1.6 X and Y = p = 0.23 hr-’; K = 1.2 X 2.1 X 10” g-l. These values will be used in all subsequent calculations. The parameters L and (A/V)b’ are subject to choice, but their orders of magnitude are known. Thus, A / V of the 100 ml chemostats used by Jost was about 1.5 X lo3 cm2/liter and b’ should be in the range of lo6 - lo*cm-2 so that (A/V)b’ should be within a n order of magnitude or so of 1.5 X 1Olo liter1. Similarly, the value of L used must be such as to produce a plot of protozoan specific reproduction rate versus bacterial density that agrees with that yielded by the multiple saturation model at all except the lowest bacterial densities. Hence, an estimate of L was found by fitting (via least squares) Monod’s model to “data” generated by the multiple saturation model with L1 = 1.3 X lo9 liter-’, Lz = 1.7 X lo9 liter-’ (these were the values used by JostI3 for feeding of Tetrahymena on Azotobacter). The result was that L should be approximately 7 x 109 liter-’. In order t o arrive a t values of L and (A/V)b’ that “best” fitted the data of Jost et ~ t l . ,the ~ following procedure was used. Base values of L = 6.9 X lo9liter-’ and (A/V)b’ = 2.4 x lo9liter-’ were chosen and eq. (2) was integrated numerically for operating conditions (0 in hr; s/ in g glucose/liter) of (5.9; 0.48), (9.6; 0.52), (20; 0.52), and (40; 0.52). These conditions correspond, except for differences in initial conditions and slight differences in sf values, to actual experiments performed by Jost; the results of three of them are given as Figures 4-6 in the study by Jost et al.9 Values of L and (A/V)b’ were varied systematically around the base case and the four integrations described above were repeated for each pair of values of the adjustable parameters and the nature of the solutions of eq. (2) was noted. I n general, it was found that increasing (A/V)b’ and
BONOMI AND FRED RICKSON
244
decreasing L had stabilizing effects on system dynamics; that is, large values of (A/V)b’ and small values of L would make the coexistence steady state a stable focus or even a stable node for most values of 8 whereas the reverse combination of values of ( A / V ) b ’ and L would make that steady state an unstable focus for most values of 8. The problem was then to choose a pair of values of the adjustable parameters such that the predicted effect of holding time on system dynamics matched the experimental results reported by Jost et al.9 in the “best” way. 8.5.9 h Sf-0.48 g/l
Protozoans
2 0.2
0.0
0
10
20
30
Time, days 101
0
5
10 T h e . days ibl
Fig. 1. (Continued)
15
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
245
The values of the adjustable parameters that produced the “best” match between prediction and observation were L = 6.9 X log liter’ and (A/V)b’ = 4.8 x lo8 liter-’. Figure 1 shows the results of numerical integrations of eq. (2) for different values of 6 and s/ using the foregoing “best” values of L and ( A / V ) b ’ ;these results are to be compared with the experimental data of Figures 4-6 in
sf= 0.52 g/l Protozoans
v)
0.2 0.I
0
5
10 Tlrne.doys ICI
-02
-
15
246
BONOMI AND FREDRICKSON
the study by Jost et al? Comparison will show that the following relations are both predicted and observed. 1) The system exhibits sustained oscillations a t short holding times b u t it exhibits damped oscillations at long holding times and the change from sustained to damped oscillations occurs at a holding time somewhat greater than 23 hr. 2) The period of the oscillations (sustained or damped) is reduced by almost an order of magnitude upon moving from 0 = 5.9 hr to e = 40 hr. 3) The peak population densities of the protozoa are not much affected by holding time whereas those of the bacteria are reduced by about two orders of magnitude upon moving from e = 5.9 hr to e = 40 hr. 4) The sugar concentration achieves a high-nearly equal to that in the feed-and nearly steady level at the longer holding times used. Hence, the “best” values of L and ( A / V ) b ’ are capable of making the Nonod-Topiwala and Hamer model fit data in a semiquantitative fashion. However, the “best” values are not fixed very closely by the available data, for the agreements between predictions and data listed above are also obtained if L and ( A / V ) b ’ are varied somewhat-say by a factor of two or so-from the reported “best” values. With ( A / V ) b ’ = 4.8 X lo8 liter-’ and A / V = 1.5 X lo3 cm2/liter (see above), the “best” value of b’ is 3.2 X lo5 cm-2. Values of b‘ which have been used with the Topiwala and Hamer model of wall growth range from lo6to los cm-2 so that the present “best” value is reasonable. Comparison of Figure 1 with the data of Jost et al.9 will also show that agreement between prediction and observation is only semiquantitative as the curves of Figure 1 cannot be superimposed closely on the curves of Figures 4-6 in the study by Jost et aL9Discrepancies are probably due primarily to 1) differing initial conditions between experiments and numerical integrations of the model, 2) failure of the model to account for changes in model parameters as the physiological states of the populations change (which they do), and 3) the random nature of population-changing events a t the very low densities which sometimes occurred in the experiments. AIodels that are much more complicated than those of Rlonod-Topiwala and Hamer or the multiple saturation model would be required to account for the latter two effects. Jost13 integrated eq. (1)-but with the term v b p / ( L b ) replaced by the multiple saturation term v b 2 p / ( L 1 b) (L2 b)-for the same pairs of values of e and s/ used i; Figure 1 in the present study. The curves that he generated are virtually identical with the curves shown in Figure 1 in the present study. Hence, on the basis of the
+
+
+
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
247
data presented by Jost et ~ t l .the , ~ multiple saturation model with no wall growth and Monod’s model with wall growth are equally likely explanations of their observations.
OPERATING DIAGRAM FOR THE SYSTEM The stability of the two steady states allowed by eq. (2) may be established in the usual way. Stability depends on operating conditions (el sf),of course, and this dependence is shown on the operating diagram, Figure 2. The model parameters chosen to construct this diagram are those listed above, the “best” values of L and ( A / V ) b ’ being used. Denote the steady state where b > 0 and p = 0 as steady state 1 and denote the steady state where b, p > 0 as steady state 2. Then in region I1 of the operating diagram, steady state 1 is a stable node whereas steady state 2 does not exist. In regions (111-V) of the operating diagram, steady state 1 is unstable, a saddle point, and steady state 2 exists and is meaningful in the sense that b, p > 0; 0 < s < sf. In region 111,steady state 2 is a stable node, in region IV it is a stable focus, and in region V it is an unstable focus. Region V is therefore the region where sustained oscillations of concentrations are expected and Figure l a and b confirm their existence under these
0 0.0
I
I
0.I
0.2 Sf
I
I
“0.4
0.5
A
* g/1
Fig. 2. Operating diagram for eq. (2) using model parameters listed for Fig. 1. The various regions of the diagram are explained in the text.
248
BONOMI AND FREDRICKSON
conditions. The stabilizing effect of wall growth of the bacteria is manifested as the (upper) horizontal asymptote of the boundary between regions IV and V. In the absence of wall growth, Monod’s model predicts a vertical asymptote for that boundary (see Fig. 2 in the study by Jost et a1.5). On the other hand, the operating diagram for the system without wall growth but with the multiple saturation model replacing Monod’s model for feeding of the protozoa also possesses a horizontal asymptote for the (upper) boundary between regions IV and V of the operating diagram (see Fig. 3 in the study by Jost et al.5). No region I is shown in Figure 2 since this corresponds to the nonexistent state of extinction of both populations; such a state is possible if there is no wall growth and the two operating diagrams in the study by Jost et al.5 do have regions labeled I.
EXTINCTION IN THE ABSENCE OF WALL GROWTH It was stated above that if the feeding of the protozoans follows Monod’s model and there is no wall growth of the bacteria, then the bacteria, and so the protozoa, should become extinct in a laboratorysize chemostat operated with a holding time of 20 hr and a feed glucose concentration of 0.52 g/liter. Some ecologists have inferred from statements of this type that predators and prey cannot coexist in a habitat that does not provide places-such as the walls of a chemostat-or times where or when the prey are immune from the attacks of the predators. The model described here indicates th a t this view is somewhat extreme, because the model does not predict extinction in the absence of wall growth under all operating conditions The question of extinction can be attacked rigorously only from a probabilistic viewpoint. Bartlett14 has carried out the necessary calculations for the discrete, stochastic analog of the Lotka-Volterra predator-prey equations, and in principle the same thing could be done for the discrete, stochastic version of eq. (1). However, extinction probabilities can be found from such models only by extensive calculations of the Monte Carlo type and so a simpler though nonrigorous approach is indicated. One possibility is as follows. Under operating conditions where eq. (1) predicts that the coexistence steady state is an unstable focus (i.e., in region V of the operating diagram shown in Fig. 2 in the study by Jost e t al.5), numerical integration of the equation shows that their solutions are stable limit cycles. The number of organisms present at a time of minimum population density depends not only on the density a t that time but also on the total volume of culture in the apparatus. If
PROTOZOAN FEEDING A N D BACTERIAL WALL GROWTH
249
this number is of the order of 1 or smaller, we can regard extinction as virtually certain whereas if it is much greater than 1, we can regard extinction as highly unlikely. Minimum densities of the bacterial and protozoan populations predicted by eq. (1) for operating conditions such that the coexistence steady state is an unstable focus have been estimated by integrating eq. (1) numerically for initial conditions p = 2 X lo6 liter-l, b = 1 x lo9 liter+, and s = 0.1 g/liter; the model parameters were the same as those given above. Values of p and b at the second minimum achieved after startup are reasonable estimates of minimum densities in the limit cycles and these are the values given in Table I. Inspection of Table I shows that, for a given volume of apparatus extinction of the bacteria (and so also of the protozoa) becomes more likely as both holding time and feed substrate concentration are increased. The table also shows that under most operating conditions, extinction, if it occurs, would involve both populations, first the bacteria and then the ciliates. For a 10 hr holding time, however, extinction of the ciliates alone might be observed. The most interesting feature of the table is the effect of enrichment (increase of s,). One sees that at all holding times, enrichment increases the probability of extinction of both populations. Other models of predation also predict this effect of enrichment.15J6 The values given in Table I are for a particular set of model parameters. If other model parameters were used, the values in Table I would of course be changed. One expects, however, that the qualitative pattern established by the parameters used would not be significantly changed.
DISCUSSION Jost et al. 5.9 reconciled a discrepancy between the experiment and a theory of feeding by discarding Monod’s model; the rationalization for their multiple saturation model was based on the existence of a life cycle of development of individual ciliates. The present calculations show that the discrepancy can also be removed by retaining Monod’s model for the feeding relation and allowing for the effectsof wall growth of the bacteria. Hence, we believe that as matters now stand, the explanation proposed in this study is as likely as the explanation proposed by Jost et a1.5,9 Moreover, there is an additional bit of data available which lends support to the wall growth explanation. Tsuchiya et a1.6 found that Monod’s model without wall growth of the bacteria would explain their data on the Dictyostelium
-
x x 1.67 x 1.66 x
7.38 1.81
2.85 1.23 7.03 1.42
104 104
2.99 1.68 8.30 X lo2 1.73 104
107 105
1.73 1.00
x x x
x x
10-3 103
103
103 103
2.61 X lo6 1.16 x 103
0.3
107 106
x x
1.38 X lo6 1.65 x 104
4.02 X lo8 2.29 X lo6
3.97 5.35
1.27 X lo8 2.56 x 104
0.2
109 105
x x
0.1
x x
10-4 10 2.58 X 1.01 x 102
1.37 7.43
6.03 6.51
4.45 4.20
2.41
7.08
x
x
x
10-14
10-9
10-4
5.18 X 102 1.44
4.02 x 104 4.36 X 10 1.33 5.14 X 10
0.5
0.4
*The model parameters used were the same as those used to construct Figs. 1 and 2, except that here ( A / V ) b ’ = 0. Densities are given in cells/liter. The upper figures are the bacterial densities whereas the lower figures are the protozoan densities.
25
20
15
10
(hr)
(g/liter)
\-
TABLE I Minimum Densities of the Bacterial and Protozoan Populations Predicted by Eq. (1) Under Operating Conditions Such that the Coexistr ence Steady State is an Unstable Focusa
tn
g U 2 m
8 5
W 0
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
251
discoideum-E. coli system. The mode of feeding of Dictyostelium amoebae differs from that of Tetrahymena cells. The latter create currents of water toward their (‘mouths’’ by the beating of their cilia and capture bacterial cells suspended in the water; the former engulf bacterial cells that they encounter in their wanderings. It seems unlikely that Tetrahymena cells could create currents strong enough to remove bacterial cells attached to a wall, but the normal situation with Dictyostelium is for the amoebae to glide over solid surfaces and engulf bacteria found lying on it. Thus, growth on the wall may provide immunity for bacteria from the feeding of ciliates, but not from the feeding of amoebae. Clearly, the element of conjecture here is large. Nevertheless, the probable importance of wall growth in microbial feeding relations has been established. What is needed now are experiments-such as testing the dynamical effectsof varying the internal surface-to-volume ratios of chemostats and measurements of bacterial surface densitiesto establish or disprove its actual significance. Obviously, care must be taken to eliminate or correct for the type of complications found by Canale et al.’ or by van den Ende.s
Nomenclature A b b’
internal area of chemostat wetted by culture population density of bacteria in liquid population density of bacteria on walls Michaelis constant for reproduction of bacteria K Michaelis constant for reproduction of protozoa L L1,LZ constants in multiple saturation model population density of protozoa in liquid P concentration of substrate for bacterial growth in liquid in chemostat s concentration of substrate for bacterial growth in liquid feed to chemostat Sf time t volume of liquid culture in chemostat V yield coefficient for reproduction of protozoa X Y yield coefficient for reproduction of bacteria Greek Letters holding time of chemostat maximum specific reproduction rate of bacteria fi maximum specific reproduction rate of protozoa Y
e
One of the authors (AB) acknowledges the support provided by fellowships from the 3M Company and from the Funda@o de Amparo B Pesquisa do Estado de SSo Paulo (FAPESP), SSo Paulo, Brazil. The authors thank the University of Minnesota Computer Center for a gift of computer time.
252
BONOMI AND FREDRICKSON
References 1. J. F. Drake, J. L. Jost, A. G. Fredrickson, and H. M. Tsuchiya, in Bioregenerative Systems, NASA Special Publication 165, U. S. Government Printing Office, Washington, D. C., 1968, pp. 87-94. 2. H. R. Bungay, I11 and M. L. Bungay, Adv. Appl. Microbiol., 10, 269 (1968). 3. R. P. Canale, Biotechnol. Bioeng., 11, 887 (1969). 4. R. P. Canale, Biotechnol. Bioeng., 12, 353 (1970). 5. J. L. Jost, J. F. Drake, H. M. Tsuchiya, and A. G. Fredrickson, J. Theor. Biol., 41, 461 (1973). 6. H. M. Tsuchiya, J. F. Drake, J. L. Jost, and A. G. Fredrickson, J. Bacteriol., 110, 1147 (1972). 7. R. P. Canale, T. D. Lustig, P. M. Kehrberger, and J. E. Salo, Biotechnol. Bioeng., 15, 707 (1973). 8. P. van den Ende, Science, 181, 562 (1973). 9. J. L. Jost, J. F. Drake, A. G. Fredrickson, and H. M. Tsuchiya, J. Bacteriol., 113, 834 (1973). 10. H. H. Topiwala and G. Hamer, Biotechnol. Bioeng., 13, 919 (1971). 11. D. H. Larsen and R. L. Dimmick, J. Bacteriol., 88, 1380 (1964). 12. R. J. Munson and B. A. Bridges, J. Gen. Microbiol., 37, 411 (1964). 13. J. L. Jost, Ph.D. Thesis, University of Minnesota, Minneapolis, Minnesota, 1972. 14. M. S. Bartlett, Biometrika, 44, 27 (1957). 15. M. L. Rosenzweig, Science, 171, 385 (1971). 16. A. G. Fredrickson, J. L. Jost, H. M. Tsuchiya, and Ping-Hwa Hsu, J . Theor. Biol., 38, 487 (1973).
Accepted for Publication November 4, 1975
Protozoan Feeding and Bacterial Wall Growth ANTONIO BONOMI* and A. G. FREDRICKSON, Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 66466
Summary Monod‘s model is often assumed to describe the kinetics of feeding of a protozoan population on a bacterial population in a chemostat. An earlier study (J. L. Jost e t al., J. Bacteriol., 113, 84 (1973)) of the feeding of Tetrahymena pyriformis on either Escherichia coli or Azotobacter vinelandii found that this model correctly predicted the occurrence of sustained oscillations of population densities but made predictions of minimum bacterial population densities that were much smaller than those observed. The earlier study removed the discrepancy between the model and data by replacing Monod’s model with a different model. I t is shown in the present study that the discrepancy can be explained equally as well if Monod’s model for the feeding relation is retained and if, in addition, growth of bacteria on the chemostat walls is allowed for in the model equations.
INTRODUCTION have dealt with the “feeding,” A number of recent “grazing,” or (‘predation’’of a population of protozoans on a population of bacteria in a chemostat. The simplest model of such a system that seems to have any possibility of being successful is that in which one assumes that the reproduction rate of the protozoa is limited by the density of the bacteria, that the reproduction rate of the bacteria is limited by the concentration of a single chemical substance dissolved in the incoming feed liquid, and that Monod’s model (with biomass concentrations replaced by population densities) describes reproduction and consumption of the “substrate” by both populations. The equations which result from this model are
*Permanent address: Department of Chemical Engineering, Escola PolitBcnica, University of S&oPaulo C. P. 8174, SLo Paulo, Brazil. 239
@ 1976 by John Wiley & Sons, Inc.
240
BONOMI AND FREDRICKSON
ds - 1 _ - -(sf e dt
- s)
1 psb Y K + s
--
(1c)
where p , b are population densities of protozoa and bacteria, respectively, s, sf are concentrations of the rate-limiting substrate for bacterial growth in the chemostat and the feed to the chemostat, respectively, 0 is the holding time (reciprocal dilution rate) of the chemostat, v, p are maximum specific reproduction rates of the protozoa and bacteria, respectively, X , Y are yield coefficients for reproduction of the protozoa and bacteria, respectively, and L, K are Michaelis constants of Monod’s model for reproduction of the protozoa and bacteria, respectively. These equations were first published by Drake et al.’ and the Bungay9 and analyses of them were published by Canale3s4and Jost et al.5 Under most operating conditions (i.e., conditions of e and sf), eq. (1) predicts that the system will exhibit a state of sustained oscillations of all concentration quantities rather than a steady state of coexistence or a steady state where one or both of the populations becomes extinct. Tsuchiya et al. actually observed such oscillations with the system Dictyostelium discoideum (a cellular slime mold) and Escherichia coli and they demonstrated that eq. (1) gave a good description of their observations. However, Canale et al.’ could not fit their data on the system Tetrahymena pyriformis-Aerobacter aerogenes using that equation. They suggested that death of the populations and the reuse of carbohydrates released by metabolic processes or by death and lysis were important in their system and so rendered its behavior more complex than that predicted by eq. (1). Van den Endes studied the system T. pyriformis-Klebsiella aerogenes in a “continuous culture” in a single run at an unspecified holding time; the system exhibited a series of damped oscillations of concentrations in approaching what appeared to be a steady state. Jost et al.9 studied the systems T. pyriformis-E. coli and T. pyriformis-Azotobacter vinelandii in chemostat cultures. The constants V , p , L, K , X , and Y of eq. (1)were determined by independent experiments and used to predict the results of chemostat experiments. The equation predicted that sustained oscillations of concentrations should have occurred at the operating conditions used and this was observed experimentally; however, the minimum bacterial densities predicted by the model vlrere exceedingly small. For instance, if
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
241
eq. (1) is integrated numerically for a holding time of 20 hr and a feed substrate concentration of 0.52 g/liter, using the same numerical values of the model parameters v, p , L, K,X ,and Y that are subsequently used to construct Figure 1 in this study, one finds that the minimum bacterial density is about 7 x liter-’ so that in a 100 ml chemostat (the size used by Jost et al.9) only 7 X bacteria are predicted to be present at the time the bacterial density achieves its minimum. Equation (1) predicts that the system recovers from this minimum, but of course that would hardly be possible. If the stochastic, discrete version of eq. (1)had been used, it would surely have predicted that the protozoa would consume all of the bacteria and that the consequent nongrowing protozoan cells would then be washed from the chemostat. Experimentally, however, extinction of the populations was never observed. The data obtained by Jost et al.9 suggested to them that their “predators” became less effective at low densities of the “prey;” i.e., that the predation rate fell off more rapidly than the first power of the prey density as the latter became small, as predicted by Monod’s model. Hence, they used a model which amounted to replacing the factor vbp/(L b) which appears in eq. (1) by the factor vb2p/ ( L , b) (L 2+ b ) , where L1 and Lz are new constants to be determined. This “multiple saturation” model also predicted sustained oscillations of concentrations under certain operating conditions, but the minimum bacterial densities predicted were now much higher and in fact were of the order of magnitude of those actually observed. Moreover, analysis of the new model led to the prediction that the dynamical behavior of the system could be changed from a state of sustained oscillations of concentrations to a steady state (approached via damped oscillations) by keeping the same concentration of glucose in the feed but progressively increasing the holding time.5 Experim e n t ~showed ~ that the system did indeed exhibit this change as the holding time was increased. No such prediction concerning the effect of increased holding time is made by eq. (1) and so the experimental results were regarded as strong evidence against Monod’s model and in favor of the multiple saturation model for feeding of the ciliates on bacteria. There is, however, at least one other possibility for explaining the apparent failure of RiIonod’s model found by Jost et al.9 Van den Endes has pointed out that even when the liquid in a culture vessel is well stirred, the environment therein is still not spatially homogeneous because of the unavoidable presence of the vessel walls. If bacteria can attach themselves to the walls and grow there, and if such attached
+
+
BONOMI AND FREDRICKSON
242
cells are not captured by the protozoans, then bacteria that slough off the vessel walls will continually reinoculate the liquid portion of the culture. Clearly, the effects of reinoculation from the wall will be greatest when the bacterial population in the liquid is least and it will prevent the bacterial density from falling as low as it otherwise would. Hence, the present study will deal with the effects of wall growth of the bacteria on the dynamics of the protozoan-bacterial system propagated in a chemostat.
MODIFICATION OF THE MODEL TO ACCOUNT FOR WALL GROWTH OF THE BACTERIA We assume that the feeding of the protozoa on bacteria in the liquid is described adequately by Monod's model, so th a t eq. (1) will be valid in the case where the internal surface-to-volume ratio ( A I V ) of the chemostat is negligible. Our task is then to add terms to correct for a nonnegligible surface-to-volume ratio. I n order to do this, we assume that the model of wall growth proposed by Topiwala and Hamer'o is valid for the bacteria. The elements of this model are that 1) there is an equilibrium surface density of bacteria (b') which is achieved very quickly and which is maintained by a balance of the reproduction rate and slough-off rate and 2) the specific growth rate of bacteria on the wall is the same as the specific growth rate of bacteria in the liquid. Equilibrium densities b' which have been observed"J2 for growth on glass surfaces range from 106 to 108 cm-2 and these correspond to something less than a monolayer of closely packed cells on the wall. Hence, wall growth need not be visible in order to be present. The modified version of eq. (1) is then
_ dP - - - p 1+ - dt
8
db -
1- b
dt
~
e
VbP L + b
- ~ xvbp- + ~ (2b) (b + $ b') Y K+s
+ K" ( +6 S
+7 A b')
- 1
ds 1 = - ( s / - s) - - _ _ ps dt 0
When b' is regarded as constant, as in the model of Topiwala and Hamer, eq. (2) will admit two steady states: coexistence of bacteria and protozoans ( b , p > 0; 0 < s < sf) and washout of the protozoans
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
243
( b > 0 ; p = 0 ; 0 < s < s,). The stability of these steady states with respect t o small perturbations is discussed below.
CHOICE OF MODEL PARAMETERS The objective of the present calculations is to see if by appropriate choice of model parameters it is possible to make eq. (2) correct the apparent failure of Monod’s model found by Jost et al.9for (say) the Tetrahymena-Azotobacter system. The parameters v, p, K , X , and Y are not subject to choice since they are common to the multiple saturation model and to the present model. The values of these constants reported by Jost13 for the Tetrahymena-Azotobacter system grown in glucose-minimal salts medium a t 28°C are: v = 0.20 hr-l; g/liter; X = 1.6 X and Y = p = 0.23 hr-’; K = 1.2 X 2.1 X 10” g-l. These values will be used in all subsequent calculations. The parameters L and (A/V)b’ are subject to choice, but their orders of magnitude are known. Thus, A / V of the 100 ml chemostats used by Jost was about 1.5 X lo3 cm2/liter and b’ should be in the range of lo6 - lo*cm-2 so that (A/V)b’ should be within a n order of magnitude or so of 1.5 X 1Olo liter1. Similarly, the value of L used must be such as to produce a plot of protozoan specific reproduction rate versus bacterial density that agrees with that yielded by the multiple saturation model at all except the lowest bacterial densities. Hence, an estimate of L was found by fitting (via least squares) Monod’s model to “data” generated by the multiple saturation model with L1 = 1.3 X lo9 liter-’, Lz = 1.7 X lo9 liter-’ (these were the values used by JostI3 for feeding of Tetrahymena on Azotobacter). The result was that L should be approximately 7 x 109 liter-’. In order t o arrive a t values of L and (A/V)b’ that “best” fitted the data of Jost et ~ t l . ,the ~ following procedure was used. Base values of L = 6.9 X lo9liter-’ and (A/V)b’ = 2.4 x lo9liter-’ were chosen and eq. (2) was integrated numerically for operating conditions (0 in hr; s/ in g glucose/liter) of (5.9; 0.48), (9.6; 0.52), (20; 0.52), and (40; 0.52). These conditions correspond, except for differences in initial conditions and slight differences in sf values, to actual experiments performed by Jost; the results of three of them are given as Figures 4-6 in the study by Jost et al.9 Values of L and (A/V)b’ were varied systematically around the base case and the four integrations described above were repeated for each pair of values of the adjustable parameters and the nature of the solutions of eq. (2) was noted. I n general, it was found that increasing (A/V)b’ and
BONOMI AND FRED RICKSON
244
decreasing L had stabilizing effects on system dynamics; that is, large values of (A/V)b’ and small values of L would make the coexistence steady state a stable focus or even a stable node for most values of 8 whereas the reverse combination of values of ( A / V ) b ’ and L would make that steady state an unstable focus for most values of 8. The problem was then to choose a pair of values of the adjustable parameters such that the predicted effect of holding time on system dynamics matched the experimental results reported by Jost et al.9 in the “best” way. 8.5.9 h Sf-0.48 g/l
Protozoans
2 0.2
0.0
0
10
20
30
Time, days 101
0
5
10 T h e . days ibl
Fig. 1. (Continued)
15
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
245
The values of the adjustable parameters that produced the “best” match between prediction and observation were L = 6.9 X log liter’ and (A/V)b’ = 4.8 x lo8 liter-’. Figure 1 shows the results of numerical integrations of eq. (2) for different values of 6 and s/ using the foregoing “best” values of L and ( A / V ) b ’ ;these results are to be compared with the experimental data of Figures 4-6 in
sf= 0.52 g/l Protozoans
v)
0.2 0.I
0
5
10 Tlrne.doys ICI
-02
-
15
246
BONOMI AND FREDRICKSON
the study by Jost et al? Comparison will show that the following relations are both predicted and observed. 1) The system exhibits sustained oscillations a t short holding times b u t it exhibits damped oscillations at long holding times and the change from sustained to damped oscillations occurs at a holding time somewhat greater than 23 hr. 2) The period of the oscillations (sustained or damped) is reduced by almost an order of magnitude upon moving from 0 = 5.9 hr to e = 40 hr. 3) The peak population densities of the protozoa are not much affected by holding time whereas those of the bacteria are reduced by about two orders of magnitude upon moving from e = 5.9 hr to e = 40 hr. 4) The sugar concentration achieves a high-nearly equal to that in the feed-and nearly steady level at the longer holding times used. Hence, the “best” values of L and ( A / V ) b ’ are capable of making the Nonod-Topiwala and Hamer model fit data in a semiquantitative fashion. However, the “best” values are not fixed very closely by the available data, for the agreements between predictions and data listed above are also obtained if L and ( A / V ) b ’ are varied somewhat-say by a factor of two or so-from the reported “best” values. With ( A / V ) b ’ = 4.8 X lo8 liter-’ and A / V = 1.5 X lo3 cm2/liter (see above), the “best” value of b’ is 3.2 X lo5 cm-2. Values of b‘ which have been used with the Topiwala and Hamer model of wall growth range from lo6to los cm-2 so that the present “best” value is reasonable. Comparison of Figure 1 with the data of Jost et al.9 will also show that agreement between prediction and observation is only semiquantitative as the curves of Figure 1 cannot be superimposed closely on the curves of Figures 4-6 in the study by Jost et aL9Discrepancies are probably due primarily to 1) differing initial conditions between experiments and numerical integrations of the model, 2) failure of the model to account for changes in model parameters as the physiological states of the populations change (which they do), and 3) the random nature of population-changing events a t the very low densities which sometimes occurred in the experiments. AIodels that are much more complicated than those of Rlonod-Topiwala and Hamer or the multiple saturation model would be required to account for the latter two effects. Jost13 integrated eq. (1)-but with the term v b p / ( L b ) replaced by the multiple saturation term v b 2 p / ( L 1 b) (L2 b)-for the same pairs of values of e and s/ used i; Figure 1 in the present study. The curves that he generated are virtually identical with the curves shown in Figure 1 in the present study. Hence, on the basis of the
+
+
+
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
247
data presented by Jost et ~ t l .the , ~ multiple saturation model with no wall growth and Monod’s model with wall growth are equally likely explanations of their observations.
OPERATING DIAGRAM FOR THE SYSTEM The stability of the two steady states allowed by eq. (2) may be established in the usual way. Stability depends on operating conditions (el sf),of course, and this dependence is shown on the operating diagram, Figure 2. The model parameters chosen to construct this diagram are those listed above, the “best” values of L and ( A / V ) b ’ being used. Denote the steady state where b > 0 and p = 0 as steady state 1 and denote the steady state where b, p > 0 as steady state 2. Then in region I1 of the operating diagram, steady state 1 is a stable node whereas steady state 2 does not exist. In regions (111-V) of the operating diagram, steady state 1 is unstable, a saddle point, and steady state 2 exists and is meaningful in the sense that b, p > 0; 0 < s < sf. In region 111,steady state 2 is a stable node, in region IV it is a stable focus, and in region V it is an unstable focus. Region V is therefore the region where sustained oscillations of concentrations are expected and Figure l a and b confirm their existence under these
0 0.0
I
I
0.I
0.2 Sf
I
I
“0.4
0.5
A
* g/1
Fig. 2. Operating diagram for eq. (2) using model parameters listed for Fig. 1. The various regions of the diagram are explained in the text.
248
BONOMI AND FREDRICKSON
conditions. The stabilizing effect of wall growth of the bacteria is manifested as the (upper) horizontal asymptote of the boundary between regions IV and V. In the absence of wall growth, Monod’s model predicts a vertical asymptote for that boundary (see Fig. 2 in the study by Jost et a1.5). On the other hand, the operating diagram for the system without wall growth but with the multiple saturation model replacing Monod’s model for feeding of the protozoa also possesses a horizontal asymptote for the (upper) boundary between regions IV and V of the operating diagram (see Fig. 3 in the study by Jost et al.5). No region I is shown in Figure 2 since this corresponds to the nonexistent state of extinction of both populations; such a state is possible if there is no wall growth and the two operating diagrams in the study by Jost et al.5 do have regions labeled I.
EXTINCTION IN THE ABSENCE OF WALL GROWTH It was stated above that if the feeding of the protozoans follows Monod’s model and there is no wall growth of the bacteria, then the bacteria, and so the protozoa, should become extinct in a laboratorysize chemostat operated with a holding time of 20 hr and a feed glucose concentration of 0.52 g/liter. Some ecologists have inferred from statements of this type that predators and prey cannot coexist in a habitat that does not provide places-such as the walls of a chemostat-or times where or when the prey are immune from the attacks of the predators. The model described here indicates th a t this view is somewhat extreme, because the model does not predict extinction in the absence of wall growth under all operating conditions The question of extinction can be attacked rigorously only from a probabilistic viewpoint. Bartlett14 has carried out the necessary calculations for the discrete, stochastic analog of the Lotka-Volterra predator-prey equations, and in principle the same thing could be done for the discrete, stochastic version of eq. (1). However, extinction probabilities can be found from such models only by extensive calculations of the Monte Carlo type and so a simpler though nonrigorous approach is indicated. One possibility is as follows. Under operating conditions where eq. (1) predicts that the coexistence steady state is an unstable focus (i.e., in region V of the operating diagram shown in Fig. 2 in the study by Jost e t al.5), numerical integration of the equation shows that their solutions are stable limit cycles. The number of organisms present at a time of minimum population density depends not only on the density a t that time but also on the total volume of culture in the apparatus. If
PROTOZOAN FEEDING A N D BACTERIAL WALL GROWTH
249
this number is of the order of 1 or smaller, we can regard extinction as virtually certain whereas if it is much greater than 1, we can regard extinction as highly unlikely. Minimum densities of the bacterial and protozoan populations predicted by eq. (1) for operating conditions such that the coexistence steady state is an unstable focus have been estimated by integrating eq. (1) numerically for initial conditions p = 2 X lo6 liter-l, b = 1 x lo9 liter+, and s = 0.1 g/liter; the model parameters were the same as those given above. Values of p and b at the second minimum achieved after startup are reasonable estimates of minimum densities in the limit cycles and these are the values given in Table I. Inspection of Table I shows that, for a given volume of apparatus extinction of the bacteria (and so also of the protozoa) becomes more likely as both holding time and feed substrate concentration are increased. The table also shows that under most operating conditions, extinction, if it occurs, would involve both populations, first the bacteria and then the ciliates. For a 10 hr holding time, however, extinction of the ciliates alone might be observed. The most interesting feature of the table is the effect of enrichment (increase of s,). One sees that at all holding times, enrichment increases the probability of extinction of both populations. Other models of predation also predict this effect of enrichment.15J6 The values given in Table I are for a particular set of model parameters. If other model parameters were used, the values in Table I would of course be changed. One expects, however, that the qualitative pattern established by the parameters used would not be significantly changed.
DISCUSSION Jost et al. 5.9 reconciled a discrepancy between the experiment and a theory of feeding by discarding Monod’s model; the rationalization for their multiple saturation model was based on the existence of a life cycle of development of individual ciliates. The present calculations show that the discrepancy can also be removed by retaining Monod’s model for the feeding relation and allowing for the effectsof wall growth of the bacteria. Hence, we believe that as matters now stand, the explanation proposed in this study is as likely as the explanation proposed by Jost et a1.5,9 Moreover, there is an additional bit of data available which lends support to the wall growth explanation. Tsuchiya et a1.6 found that Monod’s model without wall growth of the bacteria would explain their data on the Dictyostelium
-
x x 1.67 x 1.66 x
7.38 1.81
2.85 1.23 7.03 1.42
104 104
2.99 1.68 8.30 X lo2 1.73 104
107 105
1.73 1.00
x x x
x x
10-3 103
103
103 103
2.61 X lo6 1.16 x 103
0.3
107 106
x x
1.38 X lo6 1.65 x 104
4.02 X lo8 2.29 X lo6
3.97 5.35
1.27 X lo8 2.56 x 104
0.2
109 105
x x
0.1
x x
10-4 10 2.58 X 1.01 x 102
1.37 7.43
6.03 6.51
4.45 4.20
2.41
7.08
x
x
x
10-14
10-9
10-4
5.18 X 102 1.44
4.02 x 104 4.36 X 10 1.33 5.14 X 10
0.5
0.4
*The model parameters used were the same as those used to construct Figs. 1 and 2, except that here ( A / V ) b ’ = 0. Densities are given in cells/liter. The upper figures are the bacterial densities whereas the lower figures are the protozoan densities.
25
20
15
10
(hr)
(g/liter)
\-
TABLE I Minimum Densities of the Bacterial and Protozoan Populations Predicted by Eq. (1) Under Operating Conditions Such that the Coexistr ence Steady State is an Unstable Focusa
tn
g U 2 m
8 5
W 0
PROTOZOAN FEEDING AND BACTERIAL WALL GROWTH
251
discoideum-E. coli system. The mode of feeding of Dictyostelium amoebae differs from that of Tetrahymena cells. The latter create currents of water toward their (‘mouths’’ by the beating of their cilia and capture bacterial cells suspended in the water; the former engulf bacterial cells that they encounter in their wanderings. It seems unlikely that Tetrahymena cells could create currents strong enough to remove bacterial cells attached to a wall, but the normal situation with Dictyostelium is for the amoebae to glide over solid surfaces and engulf bacteria found lying on it. Thus, growth on the wall may provide immunity for bacteria from the feeding of ciliates, but not from the feeding of amoebae. Clearly, the element of conjecture here is large. Nevertheless, the probable importance of wall growth in microbial feeding relations has been established. What is needed now are experiments-such as testing the dynamical effectsof varying the internal surface-to-volume ratios of chemostats and measurements of bacterial surface densitiesto establish or disprove its actual significance. Obviously, care must be taken to eliminate or correct for the type of complications found by Canale et al.’ or by van den Ende.s
Nomenclature A b b’
internal area of chemostat wetted by culture population density of bacteria in liquid population density of bacteria on walls Michaelis constant for reproduction of bacteria K Michaelis constant for reproduction of protozoa L L1,LZ constants in multiple saturation model population density of protozoa in liquid P concentration of substrate for bacterial growth in liquid in chemostat s concentration of substrate for bacterial growth in liquid feed to chemostat Sf time t volume of liquid culture in chemostat V yield coefficient for reproduction of protozoa X Y yield coefficient for reproduction of bacteria Greek Letters holding time of chemostat maximum specific reproduction rate of bacteria fi maximum specific reproduction rate of protozoa Y
e
One of the authors (AB) acknowledges the support provided by fellowships from the 3M Company and from the Funda@o de Amparo B Pesquisa do Estado de SSo Paulo (FAPESP), SSo Paulo, Brazil. The authors thank the University of Minnesota Computer Center for a gift of computer time.
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BONOMI AND FREDRICKSON
References 1. J. F. Drake, J. L. Jost, A. G. Fredrickson, and H. M. Tsuchiya, in Bioregenerative Systems, NASA Special Publication 165, U. S. Government Printing Office, Washington, D. C., 1968, pp. 87-94. 2. H. R. Bungay, I11 and M. L. Bungay, Adv. Appl. Microbiol., 10, 269 (1968). 3. R. P. Canale, Biotechnol. Bioeng., 11, 887 (1969). 4. R. P. Canale, Biotechnol. Bioeng., 12, 353 (1970). 5. J. L. Jost, J. F. Drake, H. M. Tsuchiya, and A. G. Fredrickson, J. Theor. Biol., 41, 461 (1973). 6. H. M. Tsuchiya, J. F. Drake, J. L. Jost, and A. G. Fredrickson, J. Bacteriol., 110, 1147 (1972). 7. R. P. Canale, T. D. Lustig, P. M. Kehrberger, and J. E. Salo, Biotechnol. Bioeng., 15, 707 (1973). 8. P. van den Ende, Science, 181, 562 (1973). 9. J. L. Jost, J. F. Drake, A. G. Fredrickson, and H. M. Tsuchiya, J. Bacteriol., 113, 834 (1973). 10. H. H. Topiwala and G. Hamer, Biotechnol. Bioeng., 13, 919 (1971). 11. D. H. Larsen and R. L. Dimmick, J. Bacteriol., 88, 1380 (1964). 12. R. J. Munson and B. A. Bridges, J. Gen. Microbiol., 37, 411 (1964). 13. J. L. Jost, Ph.D. Thesis, University of Minnesota, Minneapolis, Minnesota, 1972. 14. M. S. Bartlett, Biometrika, 44, 27 (1957). 15. M. L. Rosenzweig, Science, 171, 385 (1971). 16. A. G. Fredrickson, J. L. Jost, H. M. Tsuchiya, and Ping-Hwa Hsu, J . Theor. Biol., 38, 487 (1973).
Accepted for Publication November 4, 1975
