Dynamics of nitrification in a continuous flow system

Soil Bid.

Biochem.

Vol.

5, pp. 531-543.

DYNAMICS

Pergamon

Press 1973. Printed in Great Britain

OF NITRIFICATION IN A CONTINUOUS FLOW SYSTEM M. J. BAZINand P. T. SAUNDERS

Department

of Microbiology

and

Department of Mathematics, Queen Road, London, W.8.. England (Accepted

26 October

Elizabeth

College,

Campden

Hill

1972)

Summary-Ammonium sulphate was added at constant rates to a column contammg a mixture of glass beads and marble chips inoculated with the nitrifying bacteria, Nitrosomonas ewopaea and Nitrobacter ugilis. Changing the rate of NH4’ addition caused nitrite and nitrate concentrations in the outflow to stabilize at characteristic steady state values after brief transition periods. The steady state concentrations were inversely related to the Row rate.The transition phaseswere characterized byovershoots in the concentrations of nitrite and nitrate when the flow was reduced and a smooth monotonic change to the new steady state values when it was increased. Three mathematical models were developed based on several simplifying assumptions and tested with the data. The best fit was obtained when it was assumed that growth of the organisms could be described by a modification of the VerhulstPearl logistic equation.

INTRODUCTION

experimental simulation of a soil column by means of a continuous flow system was suggested by Macura and Malek (1958). This technique differs from those employed previously, namely batch culture and percolation methods, in that it represents the soil as an open ecological system. As pointed out by Brock (1966) this is a more realistic approximation of the situation in nature where there is an input and output of nutrient and biomass. In addition, the consideration of a dynamic system as being open enables a wider variety of experimental and theoretical procedures to be employed than may be used for stationary states. The introduction of the chemostat as an alternative to batch culture, for example, has greatly increased our capacity to study the physiology of microbial growth. The purpose of this investigation was to develop a basis for studying quantitatively the consecutive oxidation of ammonium to nitrite and nitrate by two species of nitrifying bacteria in an open ecosystem. Throughout simplifying techniques and assumptions have been made in an attempt to gain insight into the underlying processes which control the system. The apparatus was designed to eliminate undefined environmental effects as far as possible and experimental parameters were chosen to facilitate the construction and testing of descriptive mathematical models.

THE

MATERIALS

Nitrosonzorm

AND

europaea,

METHODS

kindly supplied by Dr. N. Walker, Rothamsted Station, Harpenden, England, was maintained routinely in shake flasks using Skinner and Walker (1961). Nitrobacter agilis was cultured in a similar fashion described by Smith and Hoare (1968) and was provided by Dr. A. J. Smith, Agricultural Biochemistry, University College of Wales. For experimental purposes the bacteria were grown together in a manner 531

Experimental the medium of in the medium Department of similar to that

532

M. J. BAZIN

AND

P. T. SAUNDERS

described by Macura (1961). The apparatus consisted of a glass tube of 1.5 cm internal diameter filled to a height of 31.5 cm with O-18 mm diameter glass beads. In order to minimize the inhibitory effect of acid production by the bacteria approximately 30 marble chips were interspersed among the glass beads. The base of the tube was plugged with glass wool to hold the beads in position and terminated by a Pasteur pipette through which the outflow solution passed. The entire tube was covered with aluminium foil. An inoculum for the tube was prepared by centrifuging batch cultures of each of the nitrifiers, mixing the pellets, washing twice and suspending in ammonium-free medium of the following composition per litre: K,HPOL, I.0 g; NaCI, 0.5 g; MgS0,.7H,O, 0.5 g; ‘iron solution’ of Smith and Hoare (1968), 1.0 ml; and trace salts solution of Smith and Hoare (1968), I.0 ml. The suspension was then thoroughly mixed with the glass beads and marble chips in the column. The above medium containing 106 pg/ml NH, +-N as (NH,),SO, was added continuously at constant rates to the top of the column by means of a Watson-Marlow MHRE flow inducer. The column was also aerated from the top with filtered air at a rate just sufficient to provide perceptible positive pressure at the outflow. The experiment was carried out at room temperature (approx. 22°C). The effluent solution was dispensed in 5.0 ml volumes to a fraction collector. Initially precautions were taken to immobilize bacterial activity in the samples by adding a solution of glacial acetic acid and sodium acetate to each of the collecting tubes. This was found later to be unnecessary as the effects of any nitrification that continued to occur in the tubes were negligible compared to the efficiency of the estimation techniques. These precautionary measures were therefore discontinued. The output solution was assayed for nitrite using Griess-llsovay reagent and for nitrate by the phenoldisulphonic acid method described by Snell and Snell (1949). A simplifying assumption that a negligible amount of ammonium nitrogen failed to metabolize to nitrite or nitrate is made throughout this paper. Thus the concentration of ammonium in the effluent solution was calculated by subtracting the total nitrite plus nitrate from the original ammonium concentration in the input medium. After inoculation, medium was added to the column at a constant rate for about 20 days, while the stability of the system was tested. After this the flow rate was changed periodically and the response of the system measured in terms of nitrite and nitrate production. At the termination of the experiment the pore space or void volume, that is the volume in the column occupied by the solution, was determined gravimetrically by drying out the column. It was found to be 13.71 ml. Solutions of mathematical models simulating the dynamic behaviour of the system were approximated using numerical techniques on a CDC 6600 digital computer. RESULTS

After inoculation and after each change of flow rate a transition period occurred during which the concentration of nitrite and nitrate in the effluent either increased or decreased. Following the transition phases these concentrations stabilized. Their mean values as a function of flow rate, together with relevant statistics, are given in Table I. Of significance are the estimates of the slopes (,B) of the nitrite data with time as the independent variable. These were calculated by linear regression and tested by a r-test described by Chatfield (1970). In all but two cases /3 was found not to differ significantly from zero (P < 0.05). This provides strong evidence that the concentration of nitrite was independent of time; true steady state conditions prevailed.

NITRIFICATION TABLE

1.00 1.41 233 3.53 5.48 7.50

1.

31.12 24.68 14.48 10.54 8.85 7.25

STEADYSTATE

3.04 0.46 1.01 0.94 1.01 0.94

DYNAMICS

CONCENTRATIONSOF

-0.08 -0.04 -0.12 -0.08 -0.02 -0.22

533

NITRITE,NITRATEAND

M2

s3

63

8 32 10 18 13 6

26.43 14.84 8.03 4.74 4.28 2.26

1.27 0.23 1.10 1.09 0.82 0.62

6 = standard deviation; F = flow rate (ml/h); 3, = nitrite &g/ml N02+-N); regression line; M = number of data points; s 3 = nitrate @g/ml NO3--N); NH,+-N).

AMMONIUM

M3

5 8 5 8 5 4

s,

48.45 66.48 83.49 90.72 92.87 96.49

/? = slope of the nitrite s, = ammonium(~g/ml

The reproducibility of the steady states was tested by returning to the initial flow rate of 3.53 ml/h after an interval during which two other flow rates were employed. The mean nitrite concentration on the first occasion was found to be IO.54 pg/ml NO,-N (6 = O-94) and on the second 10.94 pg/ml N02-N (6 = 1.04), and did not significantly differ (P < 0.01). The transient behaviour of the system depended on whether the flow rate was increased or decreased. In general, lowering the flow rate increased the concentrations of nitrite and nitrate in the outflow solution while increasing the flow rate decreased these concentrations. In Fig. 1 a shift in flow from 3.73 to 1.41 ml/h is illustrated. Immediately after the shift there was a short lag and then the concentration of ammonium fell as the nitrite and nitrate concentrations rose. The rate of increase in nitrite was greater than that of nitrate. Of particular interest was the overshoot of the two anions and the undershoot of the cation before steady state.

f=3,53

ml/h

f=l,41

ml/h

100

z

70

."i _ 60 E m e c z

50 40

Fraction

No.

FIG. 1. Transient behaviour of the column after a change in flow rate (f) from 3.53 to 1.41 ml/h. NH,+-N (0); N02--N (0); N03--N (x).

534

M. J. BAZIN AND P. T. SAUNDERS

In Fig. 2 an increase in the flow rate is illustrated. Again there appears to be a lag phase before the change in the concentration of the ions and the rate of change of nitrite was greater than that of nitrate. The ammonium concentration increased while nitrite and nitrate decreased smoothly to new steady state values with no overshoot and undershoot comparable to those observed when the flow rate was decreased.

I I f=I.41

100

f=5,48ml/h

ml/h

l-

90

-

80

-

200

I

I 210

220

Fraction

230

I 240

No.

FIG. 2. Effects of a shift-up in flow rate from 1.41 to 5.48 ml/h. NH++-N

NO3--N

(a);

NO,--N

(0);

(x).

DISCUSSION Macura and Kunc (1963) obtained steady state conditions when they supplied a column of ammonof soil with 17 pg/ml or 35 pg/ml NH4 +-N. However, with a higher concentration ium their system did not stabilize even after over 25 days of culture. In this respect their results differ from those reported here where a series of steady states were observed at an input concentration of 106 pg/ml NH 4+-N. Macura and Kunc offer as an explanation of their results the suggestion that the factor limiting the growth of the bacteria is the degree of absorption of ammonium on to soil particles rather than the concentration of ammonium entering the column. Lees and Quastel (1946) have shown that nitrifiers preferentially oxidize ammonium in the absorbed state and it is possible that differences in ability to absorb ammonium between soil and glass beads might explain the two results. Theoretical aspects of nitrification in a continuous flow system have been considered by McLaren (1969a, b; 1970; 1971). McLaren (1969a) has proposed a model which describes the concentration of ammonium, nitrite and nitrate in a column as a function of distance down the column. The oxidation of ammonium is considered to be a pure exponential function’of depth. Consider McLaren’s equation (4) :

NITRIFICATION

DYNAMICS

535

(NH,+) = (NH,+)O exp (--k;X) where X is distance down the column, (NH,+), the column and k; is defined by:

is the concentration of ammonium entering

K; = kJako. Here kl is a rate constant for the ammonium oxidation reaction, E is an expansion factor constant for any given system and k, is the rate of flow through the system. Substituting for K, in the original equation: (NH,+) = (NH,+),

exp (-k,X/eko).

As McLaren points out, for any given column depth the concentration of ammonium is timeinvariant and the equation represents a steady state situation. Application of this equation to the steady state data described there is therefore possible if we consider (NH,+) to be the output concentration of ammonium and X to be a constant representing the total height of the column. Letting a = k,X/e and taking logarithms both sides: In (NH,“)

= In (NH,+),

- a/k,.

In a later publication McLaren (1970) extended his analysis to specifically include the case where ammonium concentration is high compared to the saturation constant of iVitro.somonas and this model is more directly applicable to the experimental conditions employed in our study. McLaren’s steady state equations can be reduced to the following form: In (NH,+) = ci - c2/ko - (NH,+)/c, where cl, c2, cd are constants. This relationship differs from the last only in the final term which will have the effect of reducing the ordinate values of a plot of the data at high flow rates where (NH,‘) is comparatively large. At lower flow rates the effect of the last term decreases. A plot of In (NH,+) against the reciprocal of flow rate is shown in Fig. 3. The relationship is distinctly nonlinear and approximates the form predicted by McLaren’s (1970) model.

T

100

+I*

l*

90 $ *O z

l

.

T 70i

.

!-

+$ 60 z 01 5O+ 0 =; ," 40-

l

P ;i 30 0

I

I

0.2

04

I/

FIG. 3. Steady state ammonium

concentration

1 O-6

I 0.8

I 0.1

fiowraie

plotted against the reciprocal

of flow rate.

536

M. J. BAZIN AND P. T. SAUNDERS

In their final forms the models of McLaren (1969b, 1971) purport to describe the dynamic behaviour of ammonium, nitrite, nitrate and microbial biomass as a function of depth. We have considered the system with respect to time only by making the simplifying assumption that the bacteria in the column were at steady state with respect to distance down the column. To a close approximation this is probably valid with respect to Nitrosomonas in the case under study where relatively high concentrations of ammonium were present. However, a flow rate dependent gradient of nitrite down the column probably occurred limiting the growth of Nitrobacter and thereby changing the concentration of this organism in the system. If it is further assumed that the rates of nitrite and nitrate production are proportiona to the biomass concentrations of Nitrosomonas and ~itrobacter, that the amount of nitrogen utilized for the formation of bacterial protoplasm is negligible, that the only products of ammonium metabolism are nitrite and nitrate and that the effect of hydrodynamic dispersion is small, then the following set of equations may be used to evaluate the steady state behaviour of the system: Change in ammonium = input of -washout -loss due to growth of concentration ammonium Nitrosomonas. Change in Nitrosomonas = growth-growth limitation factor. -washout -loss due to growth of Change in nitrite = production by Nitrobacter. Nitrosomonas Change in Nitrobacter = growth-growth limi~tion factor. Change in nitrate = production by Nitrobacter-washout.

(1’) (2’) (3’) (4’) (5’)

These equations may be written in differential form: dS,ldt dmddt dS,/dt dm,/dt dSJdt where:

S, Sr S, S3 m, m2 ~1 X k, k, D

= = = = = = = = = = =

= = = = =

DS, - DS1 - pm, pm1 -fi(md k,ml - DS, - Am, Am, -&(m2> k2m, - D&

(1) (2) (3) (4) (5)

input concentration of ammonium in pg/ml NH,+-N output concentration of ammonium in pg/ml NH4+-N output concentration of nitrite in pgjrnl N02--N output concentration of nitrate in pg/ml N03--N density of Nitrosomonas in &ml N density of Nitrobacter in pg/ml N specific growth rate of Nitrosomonas specific growth rate of Nitrobacter reaction constant for nitrite formation reaction constant for nitrate formation rate of washout -j,/V, wherej, is the flow rate and Vis the void volume.

These equations are analogous to those developed by McLaren except that here we choose time as the independent variabIe and consider them to represent the situation where the system is at steady state with respect to distance down the column. The specific growth rates p and Amay be substituted by the saturation function described by Novick (I 955) : I” = PJJG h = &J,I(L

+ S,> + S,)

(6) (7)

NITRIFICATION

DYNAMICS

537

where pn, and X, are the maximum specific growth rates and K and L are the saturation constants for Nitrosomonas and Nitrobacter respectively. Equation (6) can be simplified in the context of the experimental conditions considered here. According to Knowles, Downing and Barrett (1965) the saturation constant, K, is in the order of l-2 pg ml NH,+-N. Compared to this value the concentration of ammonium in the column was always high so :

K + S1 2 S,. Substituting

this expression

in to (6) yields: CL+ Pm*

(8)

A similar approximation is not valid for equation (7) as the concentration of nitrite was not large enough. As it was initially assumed that all the nitrogen in the column was conserved as ammonium, nitrite or nitrate it follows that: s, -

s, = s, + s,.

(9)

The functions fi and fi described the way in which microbial growth rate in the column is limited. These are unknown but by using arbitrary functions and testing them against the data it is possible to eliminate unsuitable choices and perhaps gain some insight into the mechanism of inhibition. Three expressions were tried in an attempt to quantitatively simulate the nitrification process : _h = Wm fi = k,‘m12

fl= k,“Dm12

.L = k,Dm,

(10)

fi = k41m22 f2= k4”Dm22.

(11) (12)

Equations (7-9) were substituted into (l-5) and then each of the expressions in (10-12) were used in turn ro replace the functions fiand f2. Thus three sets of equations were derived each differing only by the way in which growth limitation was expressed. In the terminology of Williams (1971) each model differed from the other by a single anacalyptic assumption, i.e. each model contained only one assumption to be tested by the experimental data. Equations (l-5) after (6), (7) and (10) are substituted in become: dS,/dt dm,ldt dS,ldt dm,ldt dS,ldt

= = = = =

D (S,, - S,) - pmml pmml - k,Dml k,m, - DS, - X,,&m,/(L + S,) A,S2m2/(L -I- S,) - k,Dm, k2m2 - DS,.

(la) (24 (34 (44 (54

As we consider our models only to represent steady state conditions we can solve these equations by setting the differentials equal to zero. When this is done the dependent variables m, and m, can be eliminated by substitution and the following linear relationship derived:

Df,/(,?,

+ 3,) = k,/k,

- k4D2 S3/k2 (3, + &).

In this and subsequent equations the bar above a value signifies a steady state value. Similarly when (6), (7) and (11) are substituted into equations (l-5) the following tions result: dS,ldt dm,ldt

= D (SO - S,) - pm ml = pmml - k,m12

(104 equa-

(lb) (2b)

538

M. J. BAZIN

Equating

AND

P. T. SAUNDERS

dS,/dt

= k,m,

dm,ldt

= A,S2m21(L + S,) -

dS,ldt

= k,

these equations

DS, - h,S,m,/(L

-

m2

+ 5’2)

(3b) (4b)

k, m22

(5b)

D&.

-

to zero and eliminating

m, and m, results in :

The growth limiting functions in (12) can also be substituted state relationship in this case is:

into (l-5)

and the steady

SZ + S, = p,,21k3 D2.

(12a)

The equations (lOa-12a) represent the growth limiting functions of (10-12) respectively. They are all linear relationships which can be represented by the equation for a straight line : J’

=

ax + b.

As the equations contain only constants or measured steady state data the appropriate values for y and x can be calculated and plotted. If the resulting graph is not a straight line then a poor fit to the data is implied and the model under consideration can be discarded. Thus for equation (lOa) :

and

These are plotted in Fig. 4. The relationship can be interpreted as linear so at this stage we cannot dismiss the functions in (10) as possible ways of representing growth limitation.

OI

80 D2s,/(s,

FIG. 4. Transformed

ship suggests

+ !?.,) ~10~

steady state data plotted according to equation (lOa). The linear relationthat the relationship in (10) might be suitable growth-limiting expressions.

NITRIFICATION

DYNAMICS

539

For equation (11 a) : and This relationship is plotted in Fig. 5. Again a linear interpretation cannot be discounted so the functions in (11) must also be considered. On the other hand Fig. 6 shows the transformed data of (12a) where: y = s, -/- 3, and x = 1/D2. Here there is a distinctly nonlinear relationship and so the functions in (12) may be eliminated from consideration. 60 t

/’ 50 -

/* / 0

40 -

l/

WY + 30 v?

0

0' 0 /'

20 -

l'

A

r'

IO 0'

op

5

IO

I5

I/D

Steady state nitrite and nitrate plotted against the reciprocal of the dilution rate. AS implied by equation (lla) a straight line suggests that equations (11) might describe the way in which microbial growth is limited.

FIG. 5.

I 50

I 100

I 150

I 200

I/D2 FIG. 6. Here the transformed

steady state data of equation (12a) are plotted. The deviation from linearity implies that equations (12) are unsuitable growth-limiting expressions.

540

M. J. BAZIN

AND

P. T. SAUNDERS

It is now necessary to distinguish between (IO) and (11) as suitable choices for growthlimiting expressions, i.e. to distinguish between the models described by equations (la-5a) and (1 b-5b). In order to do this estimates of the constants in these equations must be made. From the data of Knowles et al. (1965) we assigned pm = 1.8 h-l, A,= 2.0 h-l and L = 2.0 pg/ml NOz--N. Equations (1a-5a) can be reduced to the relationship in (10a) at steady state and the linear plot shown in Fig. 4 derived. The slope of this line is -k,/k, and the intercept is k,/k3. Thus an estimate of the ratios of these constants can be made by linear regression of the transformed data. In order or obtain values for k,‘-k4’ in equations (1 b-5b) a further steady state transformation was required. By simple.algebraic rearrangement of the steady state relationships the following equations can be derived:

k,’ p,/k3’

Ds2

kz’

= P,,,~/D CS2 + S,) = l/k,’ + [Xms2/(L + S2)](l/k1’k4’ = D& kq’ (L + $)/Am s,.

(13) (14) (15)

DS2)

Estimates of kS’ and k,’ were made from (13) and (15) by averaging the transformed steady state data and linear regression was employed to calculate kl’ and kq’ from (14) where the slope is l/kl’k4’ and the intercept, IlkI’. This procedure yielded values of k,’ = 2.52, k2’ = O-59, kJ’ = 0.62 and kq’ = 0.68. Solutions to the two sets of equations (la-5a) and (1 b-5b) were approximated by the fourth order Runge-Kutta method on a digital computer. The initial biomass concentrations were unknown but both set equal to I.0 pg/ml N which seems a reasonable estimate. The systems were tested for several flow rates and results compared with those obtained

0 170 -

2-9

z

-

2.8

E \ s

0.257

--I-

-2.7

Nitrobacter

-

2-4

-23 -

D 2

2-2

z? " iG

200

Time. FIG. 7. Numerical

solution

h

of equations (lb-5b) with k,’ = 2.52, k,’ = 0.59, k,’ = 0.62 and the simulated results, the open circles nitrate data and the closed circles nitrite data.

kq’ = 0.68. The solid lines represent

NITRIFTCATION

DYNAMICS

541

experimentally. Several values for the constants k, to k4 conforming to the ratios k,/k, = 5.3 and k,/k, = 0.045 were used in equations (la-5a) but in all cases the results obtained deviated from the experimental data in excess of 15 orders of magnitude. On the other hand equations (lb-5b) gave a much better fit to the data. In Fig. 7 the simulated results for 6 flow rates in the order that they were employed in the experiment are presented. For comparison the data is also shown. The steady states achieved by simulation compared to those obtained experimentally are given in Table 2 and shown fair agreement. As can be predicted by equation (2b) the concentration of Nitrosomonas is constant after an initial growth period and is not dependent on the dilution rate of the system. On the other hand the concentration of Nitrobacter changes from one steady state to another although the magnitude of the changes is small. The transient behaviour of the system when the flow rate is increased is approximated fairly well by the model. Any difference in the time taken for the transitions is not a serious fault as it merely represents a scaling factor on the time axis. The rate of decrease in nitrite concentration appears to be greater than that of nitrate and the transition from one steady state to another is a smooth monotone curve. On the other hand a step-down in flow rate shows significant differences between theoretical and experimental results. Although the differences in the rate of change of the ions is similar to that found experimentally the model does not predict the overshoots which occurred when the flow rate of the experimental system was decreased.

TABLE 2. COMPARISONOFSTEADYSTATE

Dilution rate W’) 0.073 0.103 0.170 0.257 0.400 0.547

DATA

WITH THAT PREDICTED BY THEMODEL

Steady state nitrite data h/ml NO2 --N)

Simulated steady state nitrite

Steady state nitrate data b/ml NO,--N)

Simulated steady state nitrate

31.20 24.68 14.48 10.54 8.85 7.25

29.43 22.72 15.74 11.72 8.59 6.89

26.43 14.84 8.03 4.74 4.28 2.26

22.19 15.43 9.03 5.74 3.51 2.45

Simulated Simulated steady state steady state Nitrosomonas Nitrobacter concentration concentration 2.89 2.89 2.89 2.89 2.89 2.89

2.75 2.70 2.61 2.51 2.38 2.28

The model defined by equations (1 b-5b) thus represents the experimental results to a fair approximation. It must be emphasized, however, that the experimental basis on which the model was developed, chiefly that the concentration of ammonium was saturating, will invalidate application to situations where ammonium is limiting. In addition, the simplifying procedures employed, both experimentally and theoretically, do not allow direct comparison to the process of nitrification in nature. However, one of the chief aims of theoretical model construction is not to obtain a precise mathematical description of a process but rather to gain insight into the system under study. In the case at hand the model represents the experimental findings fairly closely except for one important difference. This is the occurrence of an overshoot phenomenon when a step down in flow rate is made. It illustrates the importance of transient behaviour of microbial systems. The response of dyanmic systems to environmental perturbations is one of the most widely used techniques in systems analysis because transient phenomena characterize systems much more precisely than steady state data. Unfortunately such methods are seldom applied to microbial ecosystems. Here the

542

M. J. BAZIN

AND

P. T. SAUNDERS

overshoot phenomena points to an omission in the descriptive equations that have been proposed. In general, at least for linear systems, an overshoot response of the type recorded requires a second order term. As Denbigh, Hicks and Page (1948) have pointed out equations of the form (1 bb5b) are unlikely to provide such a response. Despite these shortcomings the results do allow us to draw several important conclusions. In the first place from an experimental point of view the system described provides a simple and rapid method for studying the dynamics of nitrification. The use of glass beads instead of soil eliminates several unknown parameters which limit our ability to interpret results analytically. The addition of marble chips to the column allows stable steady states to be achieved and repeatable experimental conditions to be obtained. From such a starting point it should be possible to investigate the dynamic effect of not only flow rate perturbations but other environmental parameters such as temperature and nutrient concentration. From a theoretical standpoint the fit of the data indicates the possibility that the concentration of Nitrosomorlas is fairly constant throughout the column when saturating concentrations of ammonium are present and that the Nitrobacter concentration does not fluctuate greatly. The factors limiting microbial growth are basically Verhulst-Pearl logistic equations (Pielou, 1969) which in essence state that the specific growth rate of an organism is a decreasing linear function ofpopulation density. Thus in the column the growth of Nitrosomona and Nitrobacter is density dependent. Ardakani. Rehbock, and McLaren (1972) have also shown experimentally that the population of nitrifiers reaches a maximum according to the logistic equation and this type of growth limitation was predicted by McLaren ( 1969b, I97 1). Finally, the overshoot phenomena in the experimental results and their absence in the simulation suggests the possibility that for a more complete mathematical description of the system second order terms or distance as well as time dependent variables might be required. In conclusion we would like to stress the relevance in this study of the combination of experimental and theoretical considerations. The experimental procedures were designed so that appropriate simplifying assumptions for the models could be made. On the other hand theoretical aspects had to be modified to conform to experimental feasibility. In this respect, for example, flow rate was made the theoretical parameter rather than length of the column as the transient and steady state effects of changing the flow rate are much easier to experimentally determine. For problems of the nature considered here we believe such an approach is rewarding. Purely theoretical considerations too often lead to results which are difficult if not impossible to test experimentally. Alternatively. experiments which are not carried out to test a hypothesis, in this case structured in the form of a mathematical model, are not likely to lead to an understanding of the underlying processes in quantitative terms. AcknoM~le~~ements-This research was made possible by a Natural We thank Miss REKHA SHAH for her technical assistance.

Environment

Research

Council

grant.

REFERENCES ARDAKANI M.-S., REHBOC~ J. T. and MCLAREN A. D. (1973) PIVC. Soil Sri. Sue. Am. In press. BRICK T. D. (1966) Principles of Microbial Ecology. Prentice-Hall, Englewood Cliffs. CHATFIELD C. (1970) S/atistics for Techno/o,g?~. Penguin, Hardmondsworth. DENBIGH K. G., HICKS M. and PAGE F. M. (1948) The kinetics of open reaction systems. Tram Furuday Sot. 44, 479494. KNOWLES G., DOWNING A. L. and BARRETT M. J. (1965) Determination of kinetic constants for nitrifying bacteria in mixed cultures, with aid of an electronic computer. J. grn. hfiuobiol. 38, 363-276. LEES H. and QUASTEL J. H. (1946) Biochemistry of nitrification in the soil--I. Kinetics of and the effects of poisons on soil nitrification as studied by soil perfusion technique. Biochem. /. 40, 815-823. MACURA J. (1961) Continuous flow method in soil microbiology-I. &Jpardtus. Folia Microbial., Praha 6, 328-334.

NITRIFICATION

DYNAMICS

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