Kinetics of Nitrification in Soil: Growth of the Nitrifiers1

A. DOUGLAS MCLAREN-

ABSTRACTNitrification leads to growth of nitrifiers but also includes

both maintenance and waste metabolism. The relative propor-tions of the three kinds of metabolism have yet to be measured,but waste and maintenance together are known to predominate.Rates of nitrification may be taken as proportional to growth ofnitrifiers provided populations are small compared to the carry-ing capacity of the environment and provided that substrateconcentrations are high enough to yield maximum specificgrowth rates. This applies to growth and metabolism in bothbatch culture and column perfusion studies.

Equations have been derived which describe populations ofnitrifying organisms as functions of depth of columns being per-fused with nutrients (ammonium or nitrite) and nutrient concen-tration profiles in an idealized soil. Typically, growth of nitriteoxidizers will lag behind ammonium oxidizers behind the flowfront during perfusion of short soil columns with ammonium.

Additional Key Words for Indexing: microbial ecology, dy-namics of nitrification, soil perfusion.

equation of the form (McLaren, 1970):

PREVIOUSLY we have described nitrification in soil interms of time and depth of penetration, for a uniform

rate of penetration of nutrient solution, in terms of an

d(S)dt

= A-dm~dt~

+ am +k"/3m(S)

Km + (S) [1]

where (S) is the substrate concentration, m is biomass, t istime, and Km is a saturation constant. A, a, and ft areproportionality constants and k" is a specific rate con-stant. The amount of enzyme per unit of in involved inwaste metabolism is given by /?. The first term in theequation is the Monod growth rate and the second is thePirt term for maintenance. The third represents wasteoxidation with energy loss through "leakage" from the cellmass (Hempfling and Vishniac, 1967; Stouthamer, 1969)and leading either to heat production or synthesis of ex-tracellular polymers. The rate of downward movement ofthe solution in cm/day within the soil is given by dX/dt =tk0 where k0 is the rate of penetration into the soil and eis an expansion factor (representing the factor of increaseof flow rate within the soil and given by 1/cc waterper cc soil).

On the assumption that the first two terms in equation[1] are small (5-15%) compared to the third quantity,which seems to be the case (Alexander, 1965; Stouthamer,1969), concentration gradients have been calculated forboth steady state and nonsteady-state conditions in soilcolumns (McLaren, 1969a). In order to perform integra-tions of equation [1] the simple logistic equation of Ver-hulst was invoked (in Pielou, 1969). It was assumed that

92 SOIL SCI. SOC. AMER. PROC., VOL. 35, 1971

available particle surface area (or nutrient in some cases)limits growth:

dm/dt = yin(\ — m/mmax) , [2]

where y is the growth rate constant and mmax is the maxi-mum population (saturation level) of a given nitrifier thatcan be achieved in 1 cc of perfused soil. Nitrification hasbeen found to follow an equation of the form of [2] in theperfusion apparatus of Lees (Lees and Quastel, 1946),which might suggest that a maximum population is reachedwhich depends on the total amount of ammonium added toa virgin soil system at the beginning. However, the popula-tion is maximal, as described by equation [2] as well, sinceon reperfusion with a fresh addition of ammonium the rateis constant and equals the maximum rate achieved duringthe first, enrichment period of growth and metabolism.Evidently a maximum population, mmax, was achieved longbefore the ammonium was exhausted.

Two additional questions can now be discussed in termsof the model chosen: (i) Under what conditions if anycan rates of nitrification be measures of population growthrates, and (ii) how does the population of nitrifiers developin a perfused column and what is the population distribu-tion with respect to depth X in the column.

THE MODEL

As long as m is much less than mmax, we may write equation[2] as

dm/dt = y

and by substitution, equation [1] as

= My + a. +dt Km + (S) 'I-

[2a]

[3]

At this point we must recall that 7 is a function of (S) and isusually taken as

7 =+

[4]

for which 7^, is the maximum specific growth rate for large(S) and Kg and 7^ are characteristic constants of the species.If indeed (S) is large in comparison with both Kg and Km [itis not known whether or not Kg equals Km (McLaren, 1970)]we may write

d(S)dt

= (Ay + a + k"/3)m = cm [3a]

where c is a sum of constants. For example, for ammonium,—d(NH4

+)/rff = cm0 exp (yo^O, with m0 the initial popula-tion of Nitrosomonas, and our equation integrates to

(NH4+)0 - (NH4+)

cm0 [exp(yo=iO - (N0a-) [5a]

for the initial oxidation of ammonium (with m « mmax).This initial period during which the approximation of equation[2a] is adequate lasts, for a ratio of mmax/m0 (equal to 100),for a time span of from about one to three generation times(compare equation [2]). Clearly a plot of log (NH,^ oxidized)versus time during which organisms are exposed to substratewill provide an approximate value of 7001 (Stojanovic and Alex-ander, 1958; Macura and Kunc, 1965; Seifert, 1969). Unfortu-•nately for this plot, analysis for residual ammonium is rarelysuitable owing to the adsorption of some of the ammonium byion exchange with soil, but one can analyze for nitrite on thebasis of the conservation equation (NO2~) ~ (NH4+)0 —(NH4

+) provided nearly all the ammonium oxide appears asnitrite, and generally it does (Alexander, 1961).

On the other hand, at low and intermediate values of ( S ) ,the growth rate constant 7 will vary with (5), which in turnis a function of time.

Equation [5a] is for a batch culture. In a column, recalling thatas the ammonium front moves forward, for a total travel timeT = XT/ek0, organisms behind the front will be stimulated tomultiply, and at any distance Xt, the time since the solutionreached that distance is (T — t) = (XT — Xt)/ek0. Thus, forthe column we have

+)/dt = cm0 exp [y (XT - Xt)/ek0] .

Letting XT = L, the length of the column, and integrating, weobtain

(NH4+) = (NH4+)0

{exp [y^L/fko] - exp [7xi (L - X ) / f k 0 ] } [5b]

and

(NH4+)0 - (NH4+) = [5c]

for the concentration of product at the exit after a flow timeL/ek0. These equations will hold as long as equation [2a] isapplicable and a plot of log (NO2-) [or log {(N<V) + (NO3-)}if Nitrobacter are present] versus flow time can again give avalue for 7oo! cf. equation [5a] (Macura and Kunc, 1965).

Traditionally short soil columns have been reperfused repeat-edly (Quastel and Scholefield, 1951) and it is necessary to showwhether YOOI can be obtained in this way (Stojanovic and Alex-ander, 1958). After many cycles of reperfusion the column willbecome enriched with respect to nitrifiers; according to equa-tion [2], dm/dt = 0, and it is important to note that under cer-tain conditions, such as at low concentrations of ammonium,the substrate concentration can decrease as an exponentialfunction of time (McLaren, 1969). This decrease can clearlynot be used to measure y or yoo.

During the first few cycles, exponential growth may obtain.Thus, after the first pore volume of substrate has reachedX = L the population is no longer wj0 throughout the column;the population distribution may then be expressed as m =m0 exp [y(L — X ) / e k 0 ] . After a second flowtime L/ek0 [re-membering (NH4

+)0 is relatively high and not significantlydecreased by a few reperfusions] the population distributionbecomes m = W0 exp [y(L — X)/ek0] exp [yL/ek0 and forn cycles m = m0 exp (ynL/ek^) exp ( — yX/ek0).

Let the volume of perfusate in the soil column at any timebe v and the total volume of solution in the apparatus be V.The appearance of nitrite during reperfusion will thus have anoverall concentration in the reservoir of

MCLAREN: KINETICS OF NITRIFICATION IN SOIL: GROWTH OF NITRIFIERS 93

(NO2-)nv/K

= ( —- exp (ynL/ek0) exp (—y X/fk0) dxJ <^

[5d]

The situation for Nitrobacter (mz~) is less clear; since theinitial concentration of nitrite (NO2~) may be zero, equation[4] must be invoked from the very beginning of the ammoniumflow for the growth of this species. Thus, for Nitrobacter

and again our solution is of the form

[exp (y T) - 1]

where the total time of reperfusion is T. Similar equationsshould apply if a soil is perfused with nitrite and a growth rateconstant y»2 for Nitrobacter can be determined.

For an enriched soil equation [1] becomes

d(S)dt

= <*m k"/3mmax(S)Km + (S)

[la]

Y,, (NO2-)m2[6]

and a solution of this equation requires a knowledge of (NO2-)as a function of X. Furthermore, the relationship among m2,(NO2-) and X, corresponding to equation [1] will generallyhave the form (McLaren, 1969a)

dt+

(NH4+

and a way must eventually be found to distinguish betweenwaste and maintenance metabolism. By means of incorporationof 14CO2 into the biomass followed by slowly removing ammo-nium, possibly one could find the minimum concentration ofammonium flowing through the column required to just main-tain a healthy population of cells and below which cell deathmight be accompanied by a marked increase in rate of librationof 14C-containing products of necrosis. This would give a valuefor a as the rate of oxidation of substrate per unit of biomassjust sufficient to prevent catabolism of cells.

Another useful datum might come from nitrification dataon a soil subjected to enough ionizing radiation to kill cellsbut not enough to destroy enzyme activity (McLaren, 1969b);then an oxidation study with a variation in substrate concen-tration could provide data from which k"p and Km could becalculated. Integration of equation [la] for low substrate con-centrations suggests that there is a time ts at which S reacheszero, and knowing this time plus these constants, a value for acould be calculated.

General solutions for the second question are difficult andonly an outline of procedures with examples are now presented.

The simplest case with ammonium is to keep (NH^) high.The integral of equation [2] for Nitrosomonas (OTI) is

- [ A 2dm2

dt£2",82m2(N02-)K

with dt = dX/fk0.

[7]

From data tabulated previously (McLaren, 1970), a concen-tration of 100 ppm ammonium nitrogen is well above Kg forNitrosomonas (Kai X 1.1 ppm), but for Nitrobacter Kg issomewhere between 2 and 23 ppm nitrite-nitrogen for viablecells. Kmz is perhaps somewhat higher for subcellular enzymesystems. Thus, at low values of X, (NO2~) will be low andlow compared with K92 and Km2, and one will have to deter-mine the unknown quantities in equations [6] and [7] eventuallyfor a complete solution.

If we ignore fixation of nitrogen by the biomass as before,integration of equation [7], even without the two A dm/dtterms, is most cumbersome unless 7: = y2, which is generallynot the case.

Obviously there are two extreme cases for which equation[6] can be easily integrated and the results are of interest,because of the general conclusions possible. At small (NO2~),we have

exp - [2b]

where yj = yooj. This is an equation representing restrictedgrowth. Implicitly equation [2b] carries the notion that pro-vided 7j is constant a built-in "clock" or growth pressure ismanifest such that m± is independent of (NH4+) and dependentonly on the time for which Nitrosomonas is in contact with sub-strate. Substituting (L — X)/ek0 for the time that Nitrosomonashas had to grow in a column of total length L, at some depthin the column X, after the ammonium front has passed X andreached L, we have (the uncertain concentration gradient atthis front has been acknowledged, McLaren, 1969a)

1 + M' exp (yiA7e*0)[2c]

where

~xi ~ m0l\

m~0 —— J exp = M

= y'CC2(NO2-)m2(l -

and at large (NO2-)

dm2/dt =

[6a]

[6b]

The solution of equation [6b] is similar to equation [2b] and,if Yooj = 7oo2, as may sometimes be the situation (Knowleset al. 1965), the population of Nitrobacter would parallel thatof Nitrosomonas (see Fig. 1 below). Since, however, growth ofNitrobacter depends on (NO2~) and this may be arbitrarilysmall, we can not expect such a superimposition of growth andinstead expect growth to Nitrobacter to lag behind that ofNitrosomonas.

If the time of growth is short and (NO3-) is therefore small,we can write

dm2/dt = y' (NO2~)w2 [6c]

and

(NOS-) s (NH4+)0 - (NH4+) .

In order to find (NO2~) we write

94 SOIL SCI. SOC. AMER. PROC., VOL. 35, 1971

expy.r t [8]

for large ammonium concentrations [(NH4+)0 » Kgl] and

short growth times, by inserting the integral of equation [2a]into equation [3a]. The rate of consumption of nutrient duringgrowth at X is given by

Clm0 (L - X)/f] [8a]

where / = ek0. To find the concentration as a function of Xthis equation is integrated, differentiated with respect to X,and again integrated to give

(N02-) = (NH 4+) 0 - (NH4

+)

e x p [ y x ( L - . [8b]

Substitution of equation [8b] into equation [6cj and integrat-ing we find

In

(L - X)e x p ( y _ L / f )

exp[y. (L - X)/f\\. [6dl

As already indicated, i.e., for short growth times and large(NH4

+)0 , the growth of Nitrosomonas is given by

mi = m0 exp [y (L — X)/f] [2a—1]i xi

an equation for free or unrestricted growth.For large growth times, i.e., with both L and L — X large,

the concentration of ammonium is found by substituting equa-tion [2c] in equation [3a] and integrating. The result is, forrestricted growth.

(NH4+) = (NH4+)0 - Clmm

-[f"T

log1 + Mexp(-yML/7)

-yx (L - X)/f]n

}\' [8c]

[Since (NH4+)g (NH4+)0, inspection of equation [8c] suggeststhat when A" = L = Llnax all ammonium is oxidized. This dis-tance is a function of (NH4

+)0, ct, 7ooj, and 7??mnx1 and thusdepends on the organism and the carrying capacity of the soildefined as mmax.] Under these conditions, clearly (NO2~) =(NH4+)0 — (NH4

+) — (NO3-), provided we neglect thesmall amount of nitrogen taken up as biomass, and the rate ofchange of d(NO2~)/d/ is given by equation [7], i.e., by

Thus, a general solution of equation [6] would require a knowl-edge of the rate of nitrate production. As soon as the concen-tration of (NC>2~) materially exceeds K02, however, dmz/dtwill be given by equation [6b].

x 1.0

0.5 -

20 40 60

D e p t h X , c

80 100

Fig. 1—Population distribution of Nitrosomonas with depth,according to equation [2c], with L = 100 cm, fco = 3-85cm/day, m»iaji/moi = 10 and mol =; 105 cells per cc soil.

DISCUSSION

In order to evaluate the various equations we need toadopt values for various constants as found in the litera-ture. The distribution of Nitrosomonas in a column oflength L = 100 cm calculated according to equation [2c]is plotted in Fig. 1 with y^ = 1.43/day, tk0 = 3.9 x3.85 cm/day, 'wmax//M0

= 10, and »imax = 106 per cc soil.[The constants are those chosen previously (McLaren,1969a).] From data of Knowles et al. (1965), we mayassume y^j = yjo2 = 1.43 per day for purpose of illustra-tion and take Cj as 7.15 X 1Q-9 mg N/cell per day. Equa-tion [8b] giving (NH4+) as a function of depth with L =20 cm is plotted in Fig. 2. This value of L is small sinceLmnx is found to be about 224 cm (see equation [8c] anda plot of (NH4+) versus X with L = 100 cm, Fig. 3).Note that nitrification is but a small fraction of the totalpossible (compare Fig. 3) and provides for a semiquanti-tative validity for growth of Nitrobacter as represented byequation [6d] (Fig. 4). Even so, with mmaxi/m0l of 10,a departure of nitrification with free growth from restrictedgrowth is apparent very early, i.e., at short penetrationdistances into a soil column with L = 20 cm and L/f =1.33 days.

100

99

ECL

°- 98

97

5 10 15

Distance X, cm

20

Fig. 2—Nitrification by Nitrosomonas. Plot of equation [8b]with simple exponential growth (F.G.) and of equation [8c]with restricted growth (R.G.) with L, the column length,as 20 cm, and (NH4+)0 = 100 ppm N (0.1 mg N/cc). Theinitial number of Nitrosomonas is taken as 105/cc soil.

M C L A R E N : KINETICS OF NITRIFICATION IN SOIL; GROWTH OF NITRIFIERS 95

100

80

60

40

20

I

1 .0

20 40 60

D i s t a n c e X ,

80 100

Fig. 3—Nitrification with limited growth of Nitrosomonas. Plotof equation [8c] with L = 100 cm and (NH4+)o — 100ppm. The maximum population governed by the carryingcapacity is assumed to be 10G cells per cc soil.

It may be noted (Fig. 4) that the growth of Nitrobacterbehind the nutrient front is small compared to that ol Nitro-somonas. Eventually, however, as may be seen in Fig. 3,the concentration of nitrite will exceed Kg.,, and growth ofNitrobacter will more nearly be represented by equation[6b]; provided L is great enough, the population will tendtoward /Hn,ax2, the carrying capacity with respect to Nitro-bacter and will compete for space with Nitrosomonas atsmall values of X.

Incidentally, equation [2J may be expanded to that givenby Volterra (in Ayala, 1970) for the rate of growth of aspecies (a) competing with a second species, and (b) forthe same limited resource, namely food or space. Forexample, for nitrite we can write

j , , I , '"a 4>>nb \dma/dt = yama 1 - ——— - ———\ '"max,, >«max0 }

where a is Nitrobacter and b is Nitrocystis if present. Thepopulation of Nitrobacter is ma and mb is the populationof Nitrocystis and /MmaXo and /nmaX[) are the maximum pop-ulations of each, respectively, that can be obtained if onlyone of these is present. Likewise, the growth of Nitrocystiscan be written as

dmb/dt = ybmb ( '-mb

mn

0ma

Wmaxh

where <f> measures the inhibitory effect of one individual ofspecies b on the growth of species a and O is the effect ofone individual of species a on the growth of species b. Theequations assume that each individual of b decreases therate of growth of a by <£/wimaxa and vice versa. It can beshown that the two species can not coexist indefinitely (ifthese equations are valid) if they compete for the samelimiting resource (Ayala, 1970). This is a statement ofthe principle of competitive exclusion and may explainstatements in the literature such as the following: Nitro-somonas and Nitrobacter are "indisputably the most fre-quently isolated and hence presumably the most abundant

0.5 -

D e p t h X , c m

Fig. 4—Population of Nitrosomonas with free growth (F.G.),according to equation [2a-l] and with limiting growth (R.G.)by equation [2c]. Population of Nitrobacter as a function ofX by equation [6d] (free growth). L = 20 cm. K92 = 23 ppm.

of the autotrophs," although "five other genera of nitrogenautotrophs are recognized" (Alexander, 1965). Both yaand yb are functions of their respective Kg values, by equa-tion [4]; however, at some unique (S) they may be numeri-cally the same.

It would be worthwhile to see if, in general, only singlespecies of nitrifiers are found to be oxidizing either am-monia or nitrite in any individual soil sample.