Abstract
In the present work, a generalization of the classical model of Monod accounting the influence of both delayed and instant mortalities on the dynamics of the micro-organism population is proposed. The model was analysed and compared with respect to its quality and applicability for simulation of the cultivation process of micro-organisms. Existence of a unique global positive solution of the Cauchy problem for the proposed model is proved and explicit relations between the decay parameters and the nutrition substrate concentration are obtained. These mathematical results allow us to calculate the nutrient substrate concentration which guarantees that the biomass concentration is maximal for every specific type of taxonomic groups of micro-organisms (bacteria, yeasts).
Introduction
It is well known that the Monod-type microbial growth models describe adequately bioprocesses appearing in bioreactors and this explains why the Monod-type models are still actual from the theoretical as well as practical point of view.[Citation1] The classical model of Monod [Citation2] of aerobic periodic cultivation of micro-organisms (bacteria, yeasts)(1) where
, and
are the concentrations of micro-organisms and the substrate, respectively, has been studied in details by many authors.[Citation1–6] The function
is the intrinsic specific rate of micro-organism population growth and the parameter
is called economic coefficient (rate of yield). Note that
is designed to reflect the limiting influence of the substrate on the microbial growth. It is practically established that the model of microbial growth (1) with monotonously increasing functions of Monod type
or of Webb type
, where
(maximal specifically possible producing rate),
(constant of half saturation) and
(inhibition constant) are positive constants, adequately describes the dynamics of this process at certain favourable conditions permitting the micro-organisms actively to produce specific enzymes, which are necessary for assimilation and dissimilation of the nutrient substrates. Thus the micro-organisms reproduce themselves at the maximal possible rate
. However, that small or large amount of substrate may have an inhibiting (decreasing) effect on the specific rate of microbial growth. In order to reflect this phenomenon in model (1), Haldane [Citation7] and Andrews [Citation3] have suggested the unimodal functions
and
, respectively. These functions are also similar and special cases of the Webb [Citation8] function
which is unimodal too, when
. Besides,
if
. It means that the parameter
determines in a way the inhibitory phase of the population growth.[Citation6] The basic properties and the graphs of all the four functions are given in [Citation9] where the system
(2) introduced in [Citation10] is under consideration. Here,
is the specific rate of decay of the micro-organism population. The necessity of models of the kind (2) arises due to unfavourable conditions in the bioreactor. Theoretical and computational analysis of model (2) is fulfilled in [Citation10]. This includes establishing of explicit dependencies between
and
for all four above-mentioned functions as well as between
and
for the first three of them when
. In a previous work,[Citation11] we study the delay analogue of (2), namely
(3) with an initial condition
(4) where
,
:
,
and
is the specific rate of decay of the micro-organism population. The model (3) was proposed under the reasonable assumption that the individuals of every kind of population have their specific average lifetime
in the bioreactor, which implies that the population decay at the moment
is directly proportional to the micro-organism quantity at the moment
.
Materials and methods
Materials
Our mathematical model is applicable for all types of bioreactors for aerobic periodic cultivation of micro-organisms (bacteria, yeasts).
Methods
Since the time of V. Volterra, functional-differential equations (FDEs) are widely used to model biological processes. The transmission of control signals in biological systems is related to such long processes as birth, growth (development) and death. Because of this, the evolution of biological systems depends in an essential way on the whole previous history, and can be modelled in general only by FDEs. Moreover, using FDEs allows us to take into account various insecure factors such as finite lifetime and interaction time; inhomogeneity of the populations lifetime; finite acceptance time for external signals and finite time for elaborating counteractions; pollution effects, resulting in additional mortality with time delay; and spatial environmental heterogeneity. The importance of the aftereffects in population dynamics, and the new effects stipulated by it, determines the practical reason to create delay models which are used to control processes of microbiological growth of cells and production of a useful product. We consider one of them, describing the periodical aerobic reproduction of micro-organisms.
Results and discussion
Statement of the problem
From the biotechnological point of view, we presume that the micro-organism mortality is one of the most significant factors influencing successful micro-organism cultivation. Therefore, it is very important to create models which take into account more precisely the micro-organism mortality impact on the population dynamics. In the present paper, we follow this direction and study a combination of models (2) and (3) of the kind(5)
(6) where
,
:
,
,
are the specific rates of decay of the micro-organism population. The model (5) describes more precisely the impact of the microorganism mortality for different kinds of micro-organism populations in comparison with the models (2) and (3) taking into account not only the micro-organism mortality in the same moment, but also that in a previous moment. It means that the individuals of every kind of population have their own specific average lifetime
in the bioreactor. It is clear from the biological point of view that the micro-organism mortality in the moment
is caused in general by natural reasons, i.e. it is proportional to the quantity of those micro-organisms that have begun their lives in the moment
described by the term
included in the model (5). Thus, we take into account the influence of both (instant and delayed) micro-organism mortalities on the population dynamics. Under biological reasons, one may suppose
(a majority of the micro-organism population will die after the expiry of its average lifetime
), but in our exposition below we will not do that.
In our consideration, we make the following assumption: the material composition is uniform in the reactor and intracellular, while nonuniform space distribution is ignored.
Our basic purpose in this work is to give an explicit answer of the following two practical questions which play an important role in the aerobic periodic cultivation of micro-organisms:
For every choice of the function
, how to calculate practically the minimal concentration of the nutrient substrate
How to establish practically that the biomass concentration is maximal, i.e. the cultivation process enters in the stationary phase.
Let be an arbitrary interval. For every function
, we set by definition
if
. We shall denote by
the Euclidean norm of
, by
the set of all continuous vector functions
and by
the set of all vector functions
which are absolutely continuous on every closed subinterval of
. For the vector function
, we set by definition
if
and
. Further, we shall use the following definitions:
Definition 2.1.
Vector function ,
,
is said to be a solution of the initial value problem (IVP) (5)–(6) in the interval
if it satisfies EquationEquation (5)
(5) for almost all
and the initial condition (6) for
.
Definition 2.2.
A solution of the IVP (5)–(6) in the interval
is said to be positive (nonnegative) in the interval
,
if
for all
.
Definition 2.3.
We will say that the property P is ultimately fulfilled for some function if there exists a point
, such that for the function
the property P holds for each
.
The basic tasks concerning the model (5) to be solved in the present paper are the existence and uniqueness of a solution of the IVP (5)–(6), analysis of the dynamics of when the nutrient substrate diminishes on a finite or infinite period and studying the influence of the correlation between the parameter
and the values of the function
on the dynamics of
.
Main results
We denote by (H) the following conditions:
(H1).
(H2).
(H3). There exists
Lemma 1.
Let the conditions (H) be fulfilled.
Then there exists a function which is the unique solution of the IVP (5)–(6) in
.
Proof. Denote by the initial vector function and define for each
the function
(7)
where ,
for
.
Let us denote by the set of all vector functions
, such that
,
and the restriction
is a continuous vector function. We will denote by
the set
equipped with the metric function
,
(see [Citation12] Chapter 3, Subs. 2.4), where
for all
and if
, then
. We set
for
and
for
.
From the conditions (H), it follows that if the function is defined by (7), then the map
is continuous for any
. Moreover, for every fixed
, the function
is continuous in every function
. Let
be an arbitrary point,
and consider the neighbourhood
. Then, there exists a constant
such that the inequality
holds for every two points
. Since
,
is uniformly continuous in
,
and
is uniformly continuous in
, then there exists a point
such that the IVP (5)–(6) has a unique solution on the interval
. The function
defined by equality (7) is sublinear: that is, for each point
, the inequality holds. Therefore,
(see [Citation12] Chapter 3, Subs. 2.2–2.4).
Theorem 2.
Let the conditions (H) be fulfilled.
Then for every solution of the IVP (5)–(6) for which there exists a point
such that
and
for
, then
for each
.
Proof. Assume there exists a point such that
and
for
. Then, it follows from (5) that
and consequently
is either an inflection point or a point of local minimum for
. Since
, then there exists
such that
for
. From the first equation of (5) it follows that
cannot be neither an inflection point nor a point of local minimum for
.
Corollary 3.
Let the conditions (H) be fulfilled.
Then for every solution of the IVP (5)–(6) for which there exists a point
such that
and
for
there exists a point
such that
,
and
for
.
Proof. Assume there does not exist a point such that
and
. Then, we obtain that
for
, which contradicts the conclusion of Theorem 2.
Theorem 4.
Let the conditions (H) be fulfilled.
Then for every solution of the IVP (5)–(6) there exists a point such that the solution
is positive and
is decreasing for
.
Proof. Let the function be a solution, existing according to Lemma 1, in the interval
. From conditions (H) it follows that there exists a point
such that
is a positive solution in the interval
. It is easy to see that from (5) it follows
for
. Now assume there exists a point
such that
and
for
. From (5) it follows that
and from condition (H3) it follows that
is either an inflection point for
or
has minimum in the point
which is impossible because from (5) it follows that
can be neither an inflection point, nor a point of local minimum for
. Therefore,
for
. Then, there exists a point
such that
for
and Theorem 2 implies
for
. From conditions (H) and (5), it follows that
is decreasing for
.
Let us denote by
Theorem 5.
Let the conditions (H) be fulfilled.
Then, for every solution of the IVP (5)–(6) we have
.
Proof. Assume that . Then, from condition (H3) it follows that
. Since
and
for
, then from Theorem 4 it follows that the inequalities
are obviously true for
and therefore,
and
is strictly increasing in the same interval. From (1.5) and Theorem 4 it follows that the inequalities
hold. Assume there exists a point such that
and denote
. Then we have
and
. Condition (H3), Theorem 2 and the assumption
imply the inequalities
and therefore the following inequality
(8) holds for
. Considering that
and
, from inequalities (8) it follows
which contradicts
. Therefore,
for each
and the first equation of (5) implies that
. Consequently, the function
is negative and decreasing for all sufficiently large
which contradicts the assumption that
.
Theorem 6.
Let the following conditions be fulfilled:
Conditions (H) are fulfilled.
Then for any positive solution of the IVP (5)–(6) in
, the following equalities
and
are valid.
Proof. Let be a positive solution of the IVP (5)–(6). Then, Theorem 5 implies the inequalities
. From (5) it follows that
is a positive decreasing function for each
and consequently
. Since in virtue of Theorem 5 we have that
, then there exists a unique point
such that
(apparently
). Therefore, either
for each
which is impossible, or there exists a point
such that
. If we denote by
, then we have
. Suppose that there exists a point
such that
and denote
. Then, evidently,
and
for
.
Consider the case . Then, for each
for which
it follows from (5) that
and since
, therefore we have
which is a contradiction. In the case when
and
, the following inequalities
(9)
hold. From (9) it follows that which is impossible.
Consequently, and let
. Then,
and from (5) for which
, we obtain
In the case when , the inequalities
hold and therefore
which is impossible. Then, we ultimately have that
and therefore
. Let us assume that
. Since
, ultimately then it follows from (5) that
and consequently, we have
which is a contradiction. Therefore,
.
Discussion
The proposed model (5) is analysed and compared from mathematical point of view with respect to its quality and applicability for simulation of the cultivation process of micro-organisms. This analysis includes proof of existence of a unique global positive solution of the Cauchy problem (5)–(6) (Lemma 1). The biological sense of Theorem 2 and Corollary 3 is that if the concentration of the nutrient substrate in some finite moment becomes equal to zero, then it is impossible to have live micro-organisms after this moment. Moreover, Theorem 6 implies that if the nutrition substrate vanishes ( ), then the concentration of the live micro-organisms vanishes too
. Thus, the model (5) describes the real process more adequately in comparison to the classical model (2) where the nutrition substrate can vanish (
), but the concentration of the live micro-organisms stay positive (
) and even increases, which is impossible from the biological point of view.
The biological sense of Theorem 5 and the second equation from the model (5) is that the inequality(10) implies increasing reproductive rate of the micro-organism population (at least for period
) and there exists a moment
such that the biomass concentration is maximal and
(11)
The importance of these relations for the practice is that they give explicit (computational) answers to the questions (1) and (2).
The inequality (10) allows for every specific choice of the function practically to calculate the minimal concentration of the nutrient substrate
which is necessary for an increasing bacterial concentration at least for period
.
From the equality (11), we can calculate the critical concentration of the nutrition substrate which guarantees that the biomass concentration is maximal. It is enough to measure periodically the nutrient substrate concentration until it reaches the critical level
. Solve EquationEquation (11)
(11) with respect to s:
Let us explain this when, for example ,. It follows from (10) that if
, then we will have increasing bacterial concentration at least for period
and when the nutrient substrate concentration
reaches, according to (11), the critical level
, then the biomass concentration is maximal.
Conclusion
In this work, a model of aerobic periodic cultivation of micro-organisms (bacteria, yeasts) is introduced and studied. It is more precise comparing to previous models.
Our model allows us to calculate practically the minimal initial concentration of the nutrient substrate, which is necessary to start an increasing micro-organism reproduction for a given period, for every specific rate of micro-organism population growth. Moreover, it allows with a simple measuring of the concentration of the nutrient substrate to establish when the biomass concentration is maximal, i.e. the cultivation process enters in the stationary phase.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
This work was partially supported by Plovdiv University NPD [grant number NI13 FMI-002].
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