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CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 11, Number 1, Spring 2003

ANALYSIS OF A CLASS OF MONOD-LIKE

FUNCTIONS THAT LINK SPECIFIC GROWTH

RATE TO DELAYED GROWTH RESPONSE

SEAN ELLERMEYER

ABSTRACT. We analyze a family of functions that pro-vide a link between specific growth rate (x (t) /x (t)) and de-layed growth response (DGR) in a class of delay differentialequation models for microbial growth in batch and continu-

ous culture. The connection between specific growth rate anddelayed growth response, which has not been considered in pre-vious studies of these models, is then employed in studying amodel of continuous culture competition between two speciesof microorganisms that are Monodequivalent in that theyhave the same maximal specific growth rate (m) and halfsaturation constant (Kh). It is shown that the species with thesmaller DGR is a superior competitor if the dilution rate of thechemostat is high but that the species with the larger DGR isa better competitor if the dilution rate is low.

1 Introduction A function of the form

(1) f (s) =as

b + s,

for given a > 0 and b > 0, is called a Monod function in honor ofthe French biologist Jacques Monod (19101976) who employed suchfunctions in his pioneering studies on quantitative aspects of microbialgrowth [6]. The Monod model for a microorganism being grown inbatch culture and whose growth rate is limited only by the availabilityof a single essential nutrient (substrate) is

s (t) =Y 1ms (t)

Kh + s (t)x (t) , t 0(2)

x (t) =ms (t)

Kh + s (t)x (t) , t 0(3)

Copyright cApplied Mathematics Institute, University of Alberta.

69

70 SEAN ELLERMEYER

where s (t) and x (t) are, respectively, the concentrations of the substrateand the microorganism at time t. The derivation of the Monod modelis based on two simple hypotheses:

(M1) If the substrate concentration can be maintained constant (s (t) =s for all t 0), then the specific growth rate of the culture, definedas x (t) /x (t), should be constant for all t 0. Furthermore, thespecific growth rate depends on s according to a function of theform (1). Thus there exist a > 0 and b > 0 such that if s [0,)is fixed, then

x (t)

x (t)=

as

b + sfor all t 0.

(M2) The rate of substrate consumption at any time t is proportional tothe rate of biomass formation at time t. Thus, there exists c > 0such that

s (t) = cx (t) for all t 0.

Model (2)(3) is produced from hypotheses (M1) and (M2) by inter-preting the modelling parameters a, b, and c in terms of the followingobservable parameters:

m, called the maximal specific growth rate, and defined to be theconstant specific growth rate that results when the substrate concen-tration is maintained in excess for all time

Kh, called the halfsaturation constant, and defined to be the sub-strate concentration at which the specific growth rate is halfmaximal

Y , called the yield constant, and defined to be the ratio of biomassproduced per unit mass of substrate consumed.

By formally setting s = (interpreted to mean that substrate isavailable in abundance), we obtain from hypothesis (M1) and the def-inition of m that a = m. Likewise, by setting s = Kh, we obtainmKh/ (b + Kh) = h m/2, and hence b = Kh, from (M1) and thedefinition of Kh. Since hypothesis (M2) implies that

x (t) x (0)

s (0) s (t)=

1

cfor all t > 0,

we obtain c = Y 1 from the definition of Y .The functions

p (s) =Y 1ms

Kh + s

A CLASS OF MONOD-LIKE FUNCTIONS 71

and

(s) =ms

Kh + s

are called, respectively, the per capita substrate uptake function (or thefunctional response) and the specific growth rate function for the Monodmodel. Both p and are Monod functions and they are simply relatedto each other via = Y p. We make this observation in order to contrastit with the more complicated relationship that exists between and pwhen a delayed growth response is taken into account in the modellingprocess.

A model for single substrate dependent batch culture growth thattakes a delayed growth response into account was formulated and ana-lyzed in [2]. This model, which we will refer to as the DGR model, takesthe form

s (t) =Y 1m exp (m) s (t)

(2 exp (h) 1) Kh + s (t)x (t) , t 0(4)

x (t) =m exp (m) s (t )

(2 exp (h) 1) Kh + s (t )x (t ) , t (5)

where s (t) and x (t) are as defined for the Monod model and > 0 isa fixed microbesubstrate specific amount of time that is assumed toelapse between the consumption of substrate and the biomass formationthat results from this consumption. A unique positive solution of system(4)(5) is determined for all t by specifying an initial microbe con-centration, x (t) = x0 (t) > 0, t [0, ], along with an initial substrateconcentration, s (0) = s0 0.

The DGR model is derived from two hypotheses:

(H1) Substrate consumption occurs according to a Monod process (justas in the Monod model). Thus there exist a > 0 and b > 0 suchthat

s (t) =as (t)

b + s (t)x (t) for all t 0.

(H2) Biomass formation at time t occurs at a rate proportional to therate of substrate consumption at time t . Thus there existsc > 0 such that x (t) = cs (t ) for all t .

Interpretations of the modelling parameters a, b, and c are obtainedfrom hypotheses (H1) and (H2) and the definitions of m, Kh, and Y as

72 SEAN ELLERMEYER

follows: First, we note that (H1) and (H2) imply that

x (t) =cas (t )

b + s (t )x (t ) , t .

Setting s (t) = s (constant) for all t 0 in the above equation producesthe linear delay differential equation

(6) x (t) =cas

b + sx (t ) , t .

It was proved in Appendix A of [2] that for any given A 0, everypositive solution of the equation x (t) = Ax (t ) satisfies

limt

x (t)

x (t)= r

where r is the unique solution of r exp (r) = A. Thus every positivesolution of equation (6) satisfies

limt

x (t)

x (t)= (s)

where (s) is the unique solution of

(s) exp ( (s) ) =cas

b + s.

By formally setting s = , we obtain

m exp (m) = ca.

Likewise, by setting s = Kh, we obtain

h exp (h) =caKh

b + Kh.

Thus a = c1m exp (m) and b = (2 exp (h) 1)Kh. In addition,since hypothesis (H2) implies that

x (t) x ()

s (0) s (t )= c, t ,

we obtain c = Y .

A CLASS OF MONOD-LIKE FUNCTIONS 73

Based on the fact that all positive solutions of equation (5) with fixeds [0,) satisfy limt x

(t) /x (t) = (s) where (s) is defined by

(7) (s) exp ( (s) ) =m exp (m) s

(2 exp (h) 1) Kh + s,

we find it appropriate to define the specific growth rate function for theDGR model to be the function : [0,) [0, m) defined implicitlyfor each s [0,) by equation (7). If = 0, then is a Monodfunction (and the DGR model is identical to the Monod model), but is not a Monod function if > 0. The per capita substrate consumptionfunction, p, in the DGR model is related to according to exp () =Y p.

The fundamental difference in the properties of solutions of the Monodmodel and solutions of the DGR model can be observed by consideringthe situation of excess substrate (s (0) >> 0). In this situation, theMonod model predicts that the culture will begin growing at its max-imal specific growth rate (m) immediately after inoculation, but theDGR model predicts that the culture will only achieve specific growthrate m after some time has passed. The latter scenario is more in accor-dance with observations that are typically made in actual experiments,where an initial lag phase is usually observed to precede a phase ofapproximately constant exponential growth [3].

The main objective of our present work is to analyze the dependenceof the specific growth rate function, , on the response time, , for givenvalues of m > 0 and Kh > 0. With the Monod model as a basis, m andKh completely determine the specific growth rate of a batch culture andalso determine which of two or more competing species (with differingm and Kh) will persist in continuous culture while driving the otherspecies to extinction [4]. We will show in Section 2 that, for fixed m andKh, increasing has a flattening effect on . Thus, a microorganismwith large is one that requires a large substrate concentration in orderto achieve maximal specific growth rate but, on the other hand, is ableto maintain an approximately constant (but less than maximal) growthrate over a wide range of substrate concentrations. A result of this, aswe will show in Section 3, is that if two microbial species with identicalm and Kh, but with different , compete for a shared substrate incontinuous culture, then the species with smaller has a competitiveadvantage if the removal rate is high but the species with the larger has a competitive advantage if the removal rate is low.

74 SEAN ELLERMEYER

2 Properties of the family of specific growth rate functions

For fixed m > 0, Kh > 0, and 0, the specific growth rate function = (s) associated with the DGR model (4,5) is defined by

(8) exp () =m exp (m) s

(2 exp (h) 1) Kh + s

where h = m/2. Assuming m > 0 and Kh > 0 to be fixed, we willconsider the properties of the family of functions { (, )}0. It willbe shown that each function (, ) has qualitative properties similarto those of a Monod function and that lim (, ) = h uniformlyon any compact interval [s1, s2] (0,). The latter fact implies thatamong all microorganisms with given m and Kh, those with large have specific growth rates that remain close to h over a wide range ofsubstrate concentrations.

For each fixed 0, it is easily seen from the definition (8) that (, 0) = 0, (, Kh) = h, and lims (, s) = m. Also, since

s=

+ 1

(

1

s

1

(2 exp (h) 1) Kh + s

)

> 0 for s (0,)

and

2

s2<

2

(2 exp (h) 1)Kh + s

s< 0 for s (0,) ,

it can be seen that (, ) is monotone increasing and concave down on[0,). Hence, the functions (, ) have the same qualitative character-istics as a Monod function (1).

Next, we wish to show that if 0 < s < Kh, then (, s) is a monotoneincreasing function of , and that if s > Kh, then (, s) is a monotonedecreasing function of . We also wish to show that lim (, s) = hfor each fixed s > 0. In order to prove these facts, we introduce anotherfamily {y (, )}0 where y is defined for each 0 and s 0 by

(9) y = y (, s) =mKh exp (h)

(2 exp (h) 1) Kh + s.

The key properties of this family are

(10) y =Khs

exp (( h) ) , 0, s > 0

A CLASS OF MONOD-LIKE FUNCTIONS 75

and

(11)y

= y (h y) .

In addition, it can be seen that

(12)

=

+ 1(m y) .

If 0 < s < Kh, then < h (since is a monotone increasing functionof s) and, by definition (9), we have y > h. Thus y > and

Khs

exp (( h) ) >

by property (10). Therefore

> h 1

ln

(

Khs

)

and we conclude that lim = h. By similar reasoning, it can beshown that if s > Kh, then

h < < h +1

ln

(

s

Kh

)

and hence that lim = h in this case as well.To prove that converges to h in monotone fashion, we first observe

that (0, s) + y (0, s) = m for all s 0.

If 0 < s < Kh, then y > h and y/ < 0 for all 0 by property(11). We claim that, in addition, it must also be the case that (, s) +y (, s) < m for all > 0. If this were not the case, then there wouldexist some > 0 such that (, s)+y (, s) m and ( + y) / 0.However, this would yield the contradiction

0 ( + y)

=

(m y) + ( + 1) y (h y)

+ 1< 0.

Since (, s) + y (, s) < m for all > 0, equation (12) implies that/ > 0 for all > 0 and hence that (, s) is a monotone increasing

76 SEAN ELLERMEYER

FIGURE 1: Graphs of typical 1 = (1, ) and 2 = (2, ) withidentical m and Kh and 1 < 2. These graphs were generated withMaple using m = 2, Kh = 1, 1 = 2, and 2 = 10.

function of . By similar reasoning, it can be shown that if s > Kh,then (, s) is a monotone decreasing function of .

Since is a monotone increasing function of s for each fixed and amonotone (increasing if 0 < s < Kh and decreasing if s > Kh) functionof for each fixed s, we observe that (, s1) (T, s) (, s2) forall 0, T , and s [s1, s2] where 0 < s1 < h < s2. Thisimplies that lim (, ) = h uniformly on any compact interval[s1, s2] (0,).

Two typical specific growth rate functions, 1 (1, ) and 2 (2, ) withcommon m and Kh and 1 < 2, are illustrated in Figure 1. This figureillustrates the essential difference that arises in using the DGR modelrather than the Monod model in modelling microbial growth. With theMonod model as a basis, two microbial species with the same m andKh have the same specific growth rates at any substrate concentration.However, with the DGR model as a basis, two species with identicalm and Kh, but with different , have different specific growth rates atevery substrate concentration other than s = 0, s = Kh, and s = . Atsubstrate concentrations s > Kh, the species with the smaller has thelarger specific growth rate. At substrate concentrations 0 < s < Kh, thesituation is reversed. An important additional observation that we wishto make is that it is not necessary to formulate the specific growth ratefunction (8) in terms of the halfsaturation constant, Kh. We have doneso here in order to be able to draw comparisons between the DGR model

A CLASS OF MONOD-LIKE FUNCTIONS 77

and the Monod model. To be more general, we could choose arbitrary (0, 1), define K (which we might call the saturation constant)according to (K) = m, and consider the class of specificgrowth rate functions { (, )}0 defined according to

exp () =m exp (m) s

(1 exp ((1 ) m) 1)K + s.

This more general approach allows us to make comparisons between twospecies with the same maximal specific growth rate and saturationconstant but different response times. In this case, an analysis similarto the one we have provided in the case = 1/2 shows that the specieswith the smaller response time has a greater specific growth rate atsubstrate concentrations s > K but has a lesser specific growth rate atsubstrate concentrations 0 < s < K.

3 Competition in continuous culture Continuous culture ofmicroorganisms is a process in which a wellstirred culture vessel iscontinuously supplied at a constant rate with fresh growth medium con-taining the growthlimiting substrate at a fixed concentration sf , whilethe contents of the culture vessel are simultaneously allowed to flow outof the vessel at the same rate. If F is the common input and output flowrate (volume/time) and V is the (constant) volume of the culture vessel,then D = F/V is called the specific removal rate of the microorganism.The laboratory apparatus that is used in performing continuous cultureis called a chemostat. In contrast to batch culture, a chemostat allowsfor the maintenance of a constant substrate concentration in the culturevessel. For a thorough exposition of the basic mathematical theory ofthe chemostat, the reader should consult [7].

We now consider the scenario of two Monodequivalent species ofmicroorganisms competing for the same growthlimiting substrate incontinuous culture. By Monodequivalent, we mean that each specieshas the same maximal specific growth rate, m, and the same halfsaturation constant, Kh. However, we assume that each species has adifferent response time. It will be seen that the DGR model (modified toinclude two species in continuous culture) predicts that the specie...