analysis of a class of monod-like functions that link specific growth

Transcription

analysis of a class of monod-like functions that link specific growth
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 provide a link between specific growth rate (x0 (t) /x (t)) and delayed growth response (DGR) in a class of delay differential
equation models for microbial growth in batch and continuous culture. The connection between specific growth rate and
delayed growth response, which has not been considered in previous studies of these models, is then employed in studying a
model of continuous culture competition between two species
of microorganisms that are “Monod–equivalent” in that they
have the same maximal specific growth rate (µm ) and half–
saturation constant (Kh ). It is shown that the species with the
smaller DGR is a superior competitor if the dilution rate of the
chemostat is high but that the species with the larger DGR is
a 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 of
the French biologist Jacques Monod (1910–1976) who employed such
functions in his pioneering studies on quantitative aspects of microbial
growth [6]. The Monod model for a microorganism being grown in
batch culture and whose growth rate is limited only by the availability
of a single essential nutrient (substrate) is
(2)
s0 (t) =
−Y −1 µm s (t)
x (t) ,
Kh + s (t)
(3)
x0 (t) =
µm s (t)
x (t) ,
Kh + s (t)
t≥0
t≥0
c
Copyright Applied
Mathematics Institute, University of Alberta.
69
70
SEAN ELLERMEYER
where s (t) and x (t) are, respectively, the concentrations of the substrate
and the microorganism at time t. The derivation of the Monod model
is 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, defined
as x0 (t) /x (t), should be constant for all t ≥ 0. Furthermore, the
specific growth rate depends on s according to a function of the
form (1). Thus there exist a > 0 and b > 0 such that if s ∈ [0, ∞)
is fixed, then
as
x0 (t)
=
x (t)
b+s
for all t ≥ 0.
(M2) The rate of substrate consumption at any time t is proportional to
the rate of biomass formation at time t. Thus, there exists c > 0
such that
s0 (t) = −cx0 (t) for all t ≥ 0.
Model (2)–(3) is produced from hypotheses (M1) and (M2) by interpreting the modelling parameters a, b, and c in terms of the following
observable parameters:
•
•
•
µm , called the maximal specific growth rate, and defined to be the
constant specific growth rate that results when the substrate concentration is maintained in excess for all time
Kh , called the half–saturation constant, and defined to be the substrate concentration at which the specific growth rate is half–maximal
Y , called the yield constant, and defined to be the ratio of biomass
produced per unit mass of substrate consumed.
By formally setting s = ∞ (interpreted to mean that substrate is
available in abundance), we obtain from hypothesis (M1) and the definition of µm that a = µm . Likewise, by setting s = Kh , we obtain
µm Kh / (b + Kh ) = µh ≡ µm /2, and hence b = Kh , from (M1) and the
definition of Kh . Since hypothesis (M2) implies that
1
x (t) − x (0)
=
s (0) − s (t)
c
for all t > 0,
we obtain c = Y −1 from the definition of Y .
The functions
Y −1 µm s
p (s) =
Kh + s
A CLASS OF MONOD-LIKE FUNCTIONS
and
µ (s) =
71
µm s
Kh + s
are called, respectively, the per capita substrate uptake function (or the
functional response) and the specific growth rate function for the Monod
model. Both p and µ are Monod functions and they are simply related
to each other via µ = Y p. We make this observation in order to contrast
it with the more complicated relationship that exists between µ and p
when a delayed growth response is taken into account in the modelling
process.
A model for single substrate dependent batch culture growth that
takes a delayed growth response into account was formulated and analyzed in [2]. This model, which we will refer to as the DGR model, takes
the form
(4)
s0 (t) =
−Y −1 µm exp (µm τ ) s (t)
x (t) ,
(2 exp (µh τ ) − 1) Kh + s (t)
(5)
x0 (t) =
µm exp (µm τ ) s (t − τ )
x (t − τ ) ,
(2 exp (µh τ ) − 1) Kh + s (t − τ )
t≥0
t≥τ
where s (t) and x (t) are as defined for the Monod model and τ > 0 is
a fixed microbe–substrate specific amount of time that is assumed to
elapse between the consumption of substrate and the biomass formation
that results from this consumption. A unique positive solution of system
(4)–(5) is determined for all t ≥ τ by specifying an initial microbe concentration, x (t) = x0 (t) > 0, t ∈ [0, τ ], along with an initial substrate
concentration, s (0) = s0 ≥ 0.
The DGR model is derived from two hypotheses:
(H1) Substrate consumption occurs according to a Monod process (just
as in the Monod model). Thus there exist a > 0 and b > 0 such
that
−as (t)
s0 (t) =
x (t) for all t ≥ 0.
b + s (t)
(H2) Biomass formation at time t occurs at a rate proportional to the
rate of substrate consumption at time t − τ . Thus there exists
c > 0 such that x0 (t) = −cs0 (t − τ ) for all t ≥ τ .
Interpretations of the modelling parameters a, b, and c are obtained
from hypotheses (H1) and (H2) and the definitions of µm , Kh , and Y as
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SEAN ELLERMEYER
follows: First, we note that (H1) and (H2) imply that
x0 (t) =
cas (t − τ )
x (t − τ ) , t ≥ τ .
b + s (t − τ )
Setting s (t) = s (constant) for all t ≥ 0 in the above equation produces
the linear delay differential equation
(6)
x0 (t) =
cas
x (t − τ ) ,
b+s
t ≥ τ.
It was proved in Appendix A of [2] that for any given A ≥ 0, every
positive solution of the equation x0 (t) = Ax (t − τ ) satisfies
lim
t→∞
x0 (t)
=r
x (t)
where r is the unique solution of r exp (rτ ) = A. Thus every positive
solution of equation (6) satisfies
x0 (t)
= µ (s)
t→∞ x (t)
lim
where µ (s) is the unique solution of
µ (s) exp (µ (s) τ ) =
cas
.
b+s
By formally setting s = ∞, we obtain
µm exp (µm τ ) = ca.
Likewise, by setting s = Kh , we obtain
µh exp (µh τ ) =
caKh
.
b + Kh
Thus a = c−1 µm exp (µm τ ) and b = (2 exp (µh τ ) − 1) Kh . In addition,
since hypothesis (H2) implies that
x (t) − x (τ )
= c,
s (0) − s (t − τ )
we obtain c = Y .
t ≥ τ,
A CLASS OF MONOD-LIKE FUNCTIONS
73
Based on the fact that all positive solutions of equation (5) with fixed
s ∈ [0, ∞) satisfy limt→∞ x0 (t) /x (t) = µ (s) where µ (s) is defined by
(7)
µ (s) exp (µ (s) τ ) =
µm exp (µm τ ) s
,
(2 exp (µh τ ) − 1) Kh + s
we find it appropriate to define the specific growth rate function for the
DGR model to be the function µ : [0, ∞) → [0, µm ) defined implicitly
for each s ∈ [0, ∞) by equation (7). If τ = 0, then µ is a Monod
function (and the DGR model is identical to the Monod model), but µ
is not a Monod function if τ > 0. The per capita substrate consumption
function, p, in the DGR model is related to µ according to µ exp (µτ ) =
Y p.
The fundamental difference in the properties of solutions of the Monod
model and solutions of the DGR model can be observed by considering
the situation of excess substrate (s (0) >> 0). In this situation, the
Monod model predicts that the culture will begin growing at its maximal specific growth rate (µm ) immediately after inoculation, but the
DGR model predicts that the culture will only achieve specific growth
rate µm after some time has passed. The latter scenario is more in accordance with observations that are typically made in actual experiments,
where an initial “lag phase” is usually observed to precede a phase of
approximately constant exponential growth [3].
The main objective of our present work is to analyze the dependence
of the specific growth rate function, µ, on the response time, τ , for given
values of µm > 0 and Kh > 0. With the Monod model as a basis, µm and
Kh completely determine the specific growth rate of a batch culture and
also determine which of two or more competing species (with differing
µm and Kh ) will persist in continuous culture while driving the other
species to extinction [4]. We will show in Section 2 that, for fixed µm and
Kh , increasing τ has a “flattening” effect on µ. Thus, a microorganism
with large τ is one that requires a large substrate concentration in order
to achieve maximal specific growth rate but, on the other hand, is able
to maintain an approximately constant (but less than maximal) growth
rate over a wide range of substrate concentrations. A result of this, as
we will show in Section 3, is that if two microbial species with identical
µm and Kh , but with different τ , compete for a shared substrate in
continuous culture, then the species with smaller τ has a competitive
advantage if the removal rate is high but the species with the larger τ
has a competitive advantage if the removal rate is low.
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SEAN ELLERMEYER
2 Properties of the family of specific growth rate functions
For fixed µm > 0, Kh > 0, and τ ≥ 0, the specific growth rate function
µ = µ (s) associated with the DGR model (4,5) is defined by
(8)
µ exp (µτ ) =
µm exp (µm τ ) s
(2 exp (µh τ ) − 1) Kh + s
where µh = µm /2. Assuming µm > 0 and Kh > 0 to be fixed, we will
consider the properties of the family of functions {µ (τ, ·)}τ ≥0 . It will
be shown that each function µ (τ, ·) has qualitative properties similar
to those of a Monod function and that limτ →∞ µ (τ, ·) = µh uniformly
on any compact interval [s1 , s2 ] ⊆ (0, ∞). The latter fact implies that
among all microorganisms with given µm and Kh , those with large τ
have specific growth rates that remain close to µh over a wide range of
substrate concentrations.
For each fixed τ ≥ 0, it is easily seen from the definition (8) that
µ (τ, 0) = 0, µ (τ, Kh ) = µh , and lims→∞ µ (τ, s) = µm . Also, since
µ
∂µ
=
∂s
µτ + 1
1
1
−
s (2 exp (µh ) τ − 1) Kh + s
> 0 for s ∈ (0, ∞)
and
∂2µ
2
∂µ
<−
·
< 0 for s ∈ (0, ∞) ,
∂s2
(2 exp (µh τ ) − 1) Kh + s ∂s
it can be seen that µ (τ, ·) is monotone increasing and concave down on
[0, ∞). Hence, the functions µ (τ, ·) have the same qualitative characteristics as a Monod function (1).
Next, we wish to show that if 0 < s < Kh , then µ (·, s) is a monotone
increasing function of τ , and that if s > Kh , then µ (·, s) is a monotone
decreasing function of τ . We also wish to show that limτ →∞ µ (τ, s) = µh
for each fixed s > 0. In order to prove these facts, we introduce another
family {y (τ, ·)}τ ≥0 where y is defined for each τ ≥ 0 and s ≥ 0 by
(9)
y = y (τ, s) =
µm Kh exp (µh τ )
.
(2 exp (µh τ ) − 1) Kh + s
The key properties of this family are
(10)
y=
Kh
µ exp ((µ − µh ) τ ) , τ ≥ 0, s > 0
s
A CLASS OF MONOD-LIKE FUNCTIONS
75
and
∂y
= y (µh − y) .
∂τ
(11)
In addition, it can be seen that
µ
∂µ
=
(µm − µ − y) .
∂τ
µτ + 1
(12)
If 0 < s < Kh , then µ < µh (since µ is a monotone increasing function
of s) and, by definition (9), we have y > µh . Thus y > µ and
Kh
µ exp ((µ − µh ) τ ) > µ
s
by property (10). Therefore
1
µ > µh − ln
τ
Kh
s
and we conclude that limτ →∞ µ = µh . By similar reasoning, it can be
shown that if s > Kh , then
µh < µ < µ h +
1
ln
τ
s
Kh
and hence that limτ →∞ µ = µh in this case as well.
To prove that µ converges to µh in monotone fashion, we first observe
that
µ (0, s) + y (0, s) = µm for all s ≥ 0.
If 0 < s < Kh , then y > µh and ∂y/∂τ < 0 for all τ ≥ 0 by property
(11). We claim that, in addition, it must also be the case that µ (τ, s) +
y (τ, s) < µm for all τ > 0. If this were not the case, then there would
exist some τ > 0 such that µ (τ, s) + y (τ, s) ≥ µm and ∂ (µ + y) /∂τ ≥ 0.
However, this would yield the contradiction
0≤
∂ (µ + y)
µ (µm − µ − y) + (µτ + 1) y (µh − y)
=
< 0.
∂τ
µτ + 1
Since µ (τ, s) + y (τ, s) < µm for all τ > 0, equation (12) implies that
∂µ/∂τ > 0 for all τ > 0 and hence that µ (·, s) is a monotone increasing
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SEAN ELLERMEYER
FIGURE 1: Graphs of typical µ1 = µ (τ1 , ·) and µ2 = µ (τ2 , ·) with
identical µm and Kh and τ1 < τ2 . These graphs were generated with
Maple using µm = 2, Kh = 1, τ1 = 2, and τ2 = 10.
function of τ . By similar reasoning, it can be shown that if s > Kh ,
then µ (·, s) is a monotone decreasing function of τ .
Since µ is a monotone increasing function of s for each fixed τ and a
monotone (increasing if 0 < s < Kh and decreasing if s > Kh ) function
of τ for each fixed s, we observe that µ (τ, s1 ) ≤ µ (T, s) ≤ µ (τ, s2 ) for
all τ ≥ 0, T ≥ τ , and s ∈ [s1 , s2 ] where 0 < s1 < µh < s2 . This
implies that limτ →∞ µ (τ, ·) = µh uniformly on any compact interval
[s1 , s2 ] ⊆ (0, ∞).
Two typical specific growth rate functions, µ1 (τ1 , ·) and µ2 (τ2 , ·) with
common µm and Kh and τ1 < τ2 , are illustrated in Figure 1. This figure
illustrates the essential difference that arises in using the DGR model
rather than the Monod model in modelling microbial growth. With the
Monod model as a basis, two microbial species with the same µm and
Kh have the same specific growth rates at any substrate concentration.
However, with the DGR model as a basis, two species with identical
µm and Kh , but with different τ , have different specific growth rates at
every substrate concentration other than s = 0, s = Kh , and s = ∞. At
substrate concentrations s > Kh , the species with the smaller τ has the
larger specific growth rate. At substrate concentrations 0 < s < Kh , the
situation is reversed. An important additional observation that we wish
to make is that it is not necessary to formulate the specific growth rate
function (8) in terms of the half–saturation constant, Kh . We have done
so 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
α ∈ (0, 1), define Kα (which we might call the α–saturation constant)
according to µ (Kα ) = µα ≡ αµm , and consider the class of specific
growth rate functions {µ (τ, ·)}τ ≥0 defined according to
µ exp (µτ ) =
(α−1
µm exp (µm τ ) s
.
exp ((1 − α) µm τ ) − 1) Kα + s
This more general approach allows us to make comparisons between two
species with the same maximal specific growth rate and α–saturation
constant but different response times. In this case, an analysis similar
to the one we have provided in the case α = 1/2 shows that the species
with the smaller response time has a greater specific growth rate at
substrate concentrations s > Kα but has a lesser specific growth rate at
substrate concentrations 0 < s < Kα .
3 Competition in continuous culture Continuous culture of
microorganisms is a process in which a well–stirred culture vessel is
continuously supplied at a constant rate with fresh growth medium containing the growth–limiting substrate at a fixed concentration sf , while
the contents of the culture vessel are simultaneously allowed to flow out
of the vessel at the same rate. If F is the common input and output flow
rate (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 culture
is called a chemostat. In contrast to batch culture, a chemostat allows
for the maintenance of a constant substrate concentration in the culture
vessel. For a thorough exposition of the basic mathematical theory of
the chemostat, the reader should consult [7].
We now consider the scenario of two Monod–equivalent species of
microorganisms competing for the same growth–limiting substrate in
continuous culture. By “Monod–equivalent”, we mean that each species
has the same maximal specific growth rate, µm , and the same half–
saturation constant, Kh . However, we assume that each species has a
different response time. It will be seen that the DGR model (modified to
include two species in continuous culture) predicts that the species with
the smaller response time has a competitive advantage if the dilution
rate is high but that the species with the larger response time has a
competitive advantage if the dilution rate is low.
The concentrations of the two species, which we will call species 1 and
species 2, will be denoted by x1 and x2 and their respective response
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SEAN ELLERMEYER
times will be denoted by τ1 and τ2 , with the assumption that τ1 < τ2 .
The specific growth rate and per capita substrate consumption function
for species i are, respectively, µi ≡ µ (τi , ·) as defined by (8) and pi =
Yi−1 µi exp (µi τi ). The DGR model for competition between the two
species can be written in terms of the µi as
(13)
s0 (t) = D (sf − s (t)) −
2
X
Yi−1 µi (s (t)) exp (µi (s (t)) τi ) xi (t)
i=1
(14)
0
x1 (t) = µ1 (s (t − τ1 ))
· exp ((µ1 (s (t − τ1 )) − D) τ1 ) x1 (t − τ1 ) − Dx1 (t)
(15)
x02 (t) = µ2 (s(t − τ2 ))
· exp ((µ2 (s (t − τ2 )) − D) τ2 ) x2 (t − τ2 ) − Dx2 (t) .
or, equivalently, in terms the the pi as
(16)
s0 (t) = D (sf − s (t)) −
2
X
pi (s (t)) xi (t)
i=1
0
(17)
x1 (t) = Y1 exp (−Dτ1 ) p1 (s (t − τ1 )) x1 (t − τ1 ) − Dx1 (t)
(18)
x2 (t) = Y2 exp (−Dτ2 ) p2 (s (t − τ2 )) x2 (t − τ2 ) − Dx2 (t) .
0
where sf and D are, respectively, the concentration of substrate in the
fresh medium and the specific removal rate (as defined above). Note
that the xi equations each contain factors of exp (−Dτi ). These factors
account for the substrate stored inside cells that is lost in the washout
without contributing to the formation of new biomass in the culture
vessel.
If the removal rate is sufficiently low (specifically, D < µm ), then there
exist unique substrate concentrations, λ1 and λ2 , such that µ1 (λ1 ) =
µ2 (λ2 ) = D. Since λi is the substrate concentration at which the specific
growth rate of species i is equal to its specific removal rate, λi is called the
break–even substrate concentration for species i. If, in addition, the input
concentration is sufficiently high (specifically, sf > λ1 and sf > λ2 ),
then system (13)–(15) and the equivalent system (16)–(18) have three
A CLASS OF MONOD-LIKE FUNCTIONS
79
equilibrium points: E0 = (sf , 0, 0), E1 = λ1 , Y1 e−Dτ1 (sf − λ1 ) , 0 , and
E2 = λ2 , 0, Y2 e−Dτ2 (sf − λ2 ) . The equilibrium point E0 corresponds
to both competitors being extinct in the culture vessel; whereas the
equilibria Ei , i = 1, 2, correspond to organism i being maintained at a
positive steady state concentration in the culture vessel with organism
j (j 6= i) extinct.
It was proved in [8] that if µm > D, and λi < λj < sf , then every
solution of system (16)–(18) with positive xi component converges to Ei
as t → ∞. Since we are assuming τ1 < τ2 , our analysis of the functions
µi carried out in Section 2 shows that λ1 < λ2 if D > µh and λ2 < λ1
if D < µh . Thus, species 1 persists and species 2 becomes extinct if
D > µh , but the situation is reversed if D < µh .
We conclude with some remarks concerning the two model formulations (13)–(15) and (16)–(18). In previous studies of DGR competition
models with constant delays [1,5,8], the focus has been on model (16)–
(18) with the assumption that p is a Monod function or, more generally,
that p is a C 1 monotone increasing function that is bounded above and
satisfies p (0) = 0. In terms of p, the break–even substrate concentration is determined by the condition p (λ) = DeDτ . However, since
the previous work [1,5,8] did not consider the specific growth rate function associated with p via the response time τ , it has heretofore been
unclear that the condition p (λ) = DeDτ should in fact define λ as the
break–even substrate concentration. The equivalent condition µ (λ) = D
clearly provides a much clearer intuitive interpretation of the break–even
concentration. In addition, the condition λi < λj that determines the
winner of a competition implies the condition µj (λi ) < µi (λi ) = D,
which gives the intuitively reasonable conclusion that, at equilibrium,
species i is growing at a rate equal to its removal rate and species j
is growing at a rate less than its removal rate. However, when translated in terms of functional responses, the condition λi < λj implies
pj (λi ) < Yj−1 Yi exp (D (τj − τi )) pi (λi ), and this condition is difficult
to reconcile with intuition. These considerations indicate that it should
be advantageous to take specific growth rate functions into account in
future work involving microbial growth and competition models with
delayed growth response.
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SEAN ELLERMEYER
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Department of Mathematics, Kennesaw State University, 1000 Chastain Road,
Box 1204, Kennesaw, GA 30144-5591
E-mail address: [email protected]

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