Original article Biomath 2 (2013), 1312301, 1–9

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Cell Growth Models Using Reaction Schemes:
Batch Cultivation

Svetoslav Markov
Institute of Mathematics and Informatics

Bulgarian Academy of Sciences
Sofia, Bulgaria

smarkov@bio.bas.bg

Received: 27 October 2013, accepted: 30 December 2013, published: 18 January 2014

Abstract—Simple structured mathematical models of
bacterial cell growth are proposed. The models involve
fractions of bacterial cells related to their physiological
phases. Reaction schemes involving the biomass of the cell
fractions, the substrate and the product are proposed in
analogy to reaction schemes in enzyme kinetics. Apply-
ing the mass action law these reaction schemes lead to
dynamical models represented by systems of ODE’s. All
parameters of the models are rate constants with clear
biological or biochemical meaning. The proposed models
generalize classical bacterial growth models and offer more
flexible tools for modelling and control of biotechnological
processes. In this paper the study is focused on batch
cultivation models. We formulate a hypothesis that cell
growth models can be entirely based on reaction schemes.

I. INTRODUCTION
Bacterial cell growth involves reaction steps such as

cell phase transition, reproduction and mortality. We
propose reaction schemes combining such reaction steps.
We then apply the mass action law to formulate simple
dynamical models of bacterial growth which account for
the physiological states of bacterial cells. The cells are
classified into fractions (compartments) related to their
physiological states.

Classical bacterial cell growth models make use of
the assumption that all cells are in the same physi-
ological state at a given moment t. Therefore these
models involve a single variable representing the biomass
(population density) of the organisms. It has been recog-
nized since long that most classical cell growth models

describe adequately real processes under favorable envi-
ronmental conditions, whenever organisms are steadily
at log (exponential) phase when the cells actively divide
and grow at a maximal rate. However, the environment
in bio-reactors may be perturbed by various factors and
the cells change their physiological state.

Under perturbed conditions classical cell growth mod-
els may fail to reflect adequately the dynamics of the
bio-processes. Modifications of these models in various
directions have been proposed in the literature. A well-
studied direction is to allow some of the parameters in
the model, such as the “specific growth rate”, to depend
on the nutrient substrate and on other quantities such as
temperature, pH etc. [3]. Another modelling direction—
not so well explored—is based on the assumption that
the bacterial cells are not simultaneously in the same
physiological state (as assumed in classical models).
Such models are named “structured” [9].

A. Structured cell growth models

An important feature of bacteria is their physiological
state/phase (lag, log, stationary, dead). The physiolog-
ical state depends on the environmental conditions for
the cell population. We assume that not all cells are
simultaneously in the same state. For example, the log
(exponential) phase is the time period when the cells
reproduce themselves. It is realistic to assume that not
all cells are in log state at a given moment t. Assume
that the environmental conditions have been unfavorable
and then have rapidly improved so that the cells start

Citation: Svetoslav Markov, Cell Growth Models Using Reaction Schemes: Batch Cultivation, Biomath 2
(2013), 1312301, http://dx.doi.org/10.11145/j.biomath.2013.12.301

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S Markov, Cell Growth Models Using Reaction Schemes: Batch Cultivation

changing their physiological state from lag to log phase.
Some cells that are in better physiological condition
than others and will start to reproduce earlier. There
could exist cells that are not able to reproduce at all
(genetically damaged, defect cells), but still viable (in
lag phase). We assume that certain fractions of cells
may react differently under similar conditions changing
dynamically their physiological states (say, from lag
to log phase and conversely). Thus organisms can be
classified into two (or more) fractions corresponding
to their physiological states and degree of adaptation
ability. Mathematically this means to assign different
phase variables for the biomass (population density) of
the respective fractions of organisms and to study the
dynamics of these fractions as separate cell populations.

Structured models inspired by models of phytoplank-
ton [2] have been proposed and used in [4], [8], [9].
We accept the idea of structured models in so far that
cell population is subdivided into fractions. In addition
our modelling approach is tightly related to enzyme
kinetics. This contributes to more clarity in the biological
interpretation of the partaking terms in the models. Our
approach aims to assign clear biological meaning to all
parameters involved. In particular, no parameters depend
on other quantities, that is, only numeric parameters are
used. We propose reaction schemes for the transition
dynamics of the cell fractions and the substrate/products
involved. We believe that such reaction schemes will
be helpful in understanding the biological mechanisms
of the cell growth dynamics and, in particular, in un-
derstanding the biological nature of the physiological
states of the fractions involved—for instance, to what
extend the phases are related to the cell cycle or to
cell communications. Cell fractions are assumed to be
strongly related to the individual cell cycle in the works
[4], [8], [9]. Our biological interpretations are closer
to those in [13]; there an interesting structured three-
phase mathematical model stressing on the enzymatic
competence of the cells is proposed and discussed.

B. Relation to enzyme kinetics

Once postulating that cell growth is tightly related to
enzyme activity of the cells, then it is natural to make use
of basic enzyme kinetics in the modelling of cell growth.
From the perspective of cell growth dynamics bacteria
can be viewed as complexes of enzymes processing
nutrient substrates. Thus cells are similar to enzymes
in that both process specific substrates. The only big
difference between cells and enzyme complexes is the

reproduction property that does not exist in the world of
enzymes (viewed as chemical substances).

We recall some familiar knowledge from enzyme
kinetics. The Michaelis-Menten differential equation de-
scribing the uptake of a substrate by a specific enzyme

ds/dt = −const s/(Km + s)

will be briefly referred as the MM-model. The MM-
model is used to represent Monod specific growth rate
function µ(s) = const s/(Km + s) which is an integral
part of various classical cell growth models [11]. For
the construction of our models we make use of the
familiar reaction scheme between an enzyme E with a
single active site and a substrate S, forming an enzyme-
substrate complex C, which then yields product P :

S + E
k1−→←−
k−1

C
k2−→ P + E.

We shall call this scheme Henri-Michaelis-Menten
(HMM) reaction scheme in tribute to the prominent
scholars Victor Henri, Leonor Michaelis and Maud
Menten [5], [6], [10], cf. also [14],[16]. The HMM-
scheme leads to the familiar enzyme kinetics model
involving four phase variables: the concentrations of the
substrate S, the product P and the concentrations of
the two forms of the enzyme—free E and bound C.
Denoting the concentrations: s = [S], e = [E], c =
[C], p = [P ], and applying the mass action law we
obtain a system of four ODE’s:

ds/dt = −k1es + k−1c, de/dt = −k1es + (k−1 + k2)c,
dc/dt = k1es− (k−1 + k2)c, dp/dt = k2c,

to be further called the HMM-model.

In contrast to the MM-model the HMM-model de-
scribes not only the dynamics of the concentration of the
substrate S, but also the dynamics of the concentrations
of the two fractions of the enzyme—free E and bound
C plus the product P . Knowing that the solution of the
MM-model is an approximation of the solution for the
substrate in the HMM-model [12], it is natural to come
to the idea to use the exact HMM-model—instead of
the approximate MM-model—when constructing a cell
growth model.

The idea of using the HMM-mechanism for modelling
cell growth requires to introduce phase variables cor-
responding to certain fractions of the cell population
which we relate to cell phases. Our modelling approach
to bacterial growth has been explained in some detail

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S Markov, Cell Growth Models Using Reaction Schemes: Batch Cultivation

in [1], where some initial variants of structured two-
phase models are proposed. Here we revisit some of
these models by basing them more tightly on the HMM
reaction scheme, resp. HMM-model.

Let us briefly sketch the idea of our modelling ap-
proach. Recall that in a familiar classical cell growth
model the utilization of the substrate s by the cell
biomass z is described by: ds/dt = − δµz, where
µ is Monod specific growth rate function µ(s) =
const s/(K + s) [11]. If the cell biomass z is nearly
constant—as is usually the desired case under a contin-
uous cultivation process—then this model coincides with
the MM-model for the substrate uptake by corresponding
enzymes. Several remarks can be made. First, the con-
nection between the two models is obvious as substrate
nutrients are utilized by the cell via corresponding en-
zyme complexes in the cell cytoplasm; this explains why
classical Monod type models are so successful and pop-
ular. Second, recall that the MM-model is an approxima-
tion of the accurate HMM-model, hence classical models
are likely to be approximations of certain more accurate
models involving additional phase variable(s). Third, we
know that the above mentioned approximation is a good
one only if the ratio enzyme/substrate is small, which is
rarely the case when realistic cell growth is considered
[15]. Fourth, passing from the MM-model to the more
general HMM-model we get free of rational functions in
the right hand side of the model such as s/(K +s) (and
have only simple polynomial expressions), however, for
the expense of introducing additional phase variable(s).
Having in mind these remarks one can ask what would be
the natural way to upgrade classical cell growth models
in order to make them more adequate.

To answer this question we propose suitable reaction
schemes for the cell growth dynamics based on the
HMM-model. This paper focuses on cell growth mod-
els related to batch cultivation. Continuous cultivation
models are proposed and studied in a forthcoming work.

C. Model assumptions

There may be various biological interpretations of the
phases depending on the particular situations (bacterial
species, nutrient substrate, environmental conditions etc).
The following interpretations seem plausible in certain
conditions.

We start with the following abstractions. Enzyme com-
plexes are considered as the most important components
of the cells (from the perspective of their production
abilities). Apart of enzymes bacterial cells contain other

substances such as specific proteins, nucleotides, water,
minerals etc. For the purposes of this study we shall
consider these substances as playing auxiliary role in the
metabolic processes. We assume that the total biomass of
the bacteria is proportional to the total mass of enzymes
in the bacteria cells. Hence, in cell growth dynam-
ics bacteria can be considered as organized complexes
of enzymes that perform specific metabolic functions
generalizing the functions of particular enzymes. As
already mentioned the functions of specific enzyme-
substrate pairs are modelled by enzyme kinetics systems
of ODE’s, e. g. the HMM-model. Considered as sets
of enzymes, cells can also be modelled by the enzyme
kinetics systems of ODE’s.

The total microbial biomass increases for the ex-
pense of the utilized nutrient substances and the product
substances produced by the bacteria. From an abstract
perspective, proteins reproduce themselves—via the ri-
bosome mechanisms. Part of the newly built products
“come into life” as newborn cell components. For sim-
plicity we may assume that one of the newly formed
daughter cells coincides with the mother cell (which
corresponds to reproduction by cellular budding).

Classical bacterial growth models typically make use
of a single variable for the biomass concentration. The
use of a single variable for the biomass does not permit
to distinguish cells of different physiological states at a
given moment t. In this work we distinguish between
two main phases of bacterial cells:

— cells in lag phase denoted as X-cells;
— cells in log phase denoted as Y -cells.

Let us recall some characteristics of these two phases.
Lag phase X-cells do not reproduce and their metabolic
activity is limited. X-cells appear as dormant being
stressed by environmental perturbations, nutrient limita-
tion etc. The moment when X-cells start to be metabol-
ically active by consuming nutrient substrates and form-
ing products P , then we consider these cells as log phase
Y -cells, that is X-cells transform into Y -cells.

Y -cells are metabolically active; they consume nutri-
ent substrate S from the environment to form product
P . The latter can be viewed as the set of all biochemical
substances (like proteins, nucleotides, etc) needed for
the cell functioning, including reproduction. We assume
that Y -cells are in reproduction state at the current
moment. Reproduction can be viewed as part of the
production process; since the newly formed substances
P are component part of the newborn cells we assume
that this part of P transforms into new cells.

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S Markov, Cell Growth Models Using Reaction Schemes: Batch Cultivation

After reproduction (by binary fission) the newborn
daughter cells are considered as Y-cells [7]. Thus newly
built metabolic products become live Y -cells. The ulti-
mate goal of the cell activities is cell reproduction. Log
phase Y -cells are those that have reached this goal, in
contrast to lag phase X-cells that are in an apparently
dormant phase.

Both fractions of X- and Y -cells are assumed to
be homogeneously distributed (stirred) in the volume,
so that we can work with their biomass concentrations
(population densities) x,y, respectively. The same refers
to the substances S and P which are also considered
well stirred.

In agreement with the above assumptions we assign
the following biological/biochemical meanings to the
variables used in the proposed model:

s = s(t) is the concentration of all nutrient substrates
S in the bioreactor (fermenter) at moment t that are used
for the bacterial growth;

x = x(t) is the biomass (concentration or population)
of X-cells being in lag phase at time t; these bacteria
are not metabolically active. X-cells do not consume S
and do not produce P , neither they reproduce at time t.
(Nevertheless, x is not constant, due to transition of the
phases.)

y = y(t) is the biomass concentration of Y -cells in log
phase at time t; Y -cells utilize product S and produce
substances P . Y -cells are in the state of reproduction at
time t. The substances metabolized by the Y -cells are
part of the cell products denoted P .

z = z(t) = x + y is the total biomass concentration
that is the sum of the two fractions x = x(t) and y =
y(t).

p = p(t) is the concentration of all product substances
P metabolized by the Y -cells including those excreted
and those build up for growth and reproduction. Products
P are partly included in the newborn daughter cells.

The above model assumptions, abstractions and in-
terpretations suggest direct relations to enzyme kinetics.
Thus X-cells can be related to free enzymes, as they
are not involved in production or reproduction at a given
time moment t; Y -cells can be related to bound enzymes,
as their enzyme complexes are actively engaged with
production and reproduction at time t.

Our biological assumptions are close to the ones
discussed in [13], [4], [8], [9]. Our proposed models are

also based on biochemical arguments related to enzyme
kinetics. However, in contrast to the cited works, we
make use of the Henri-Michaelis-Menten mechanism
involving the concentrations of substrate, enzymes (free
and bound) and product (and not just of the single
Michaelis-Menten differential equation for the substrate).

D. Reaction steps of the cell growth process

In the course of model construction we formulate
certain possible reaction steps of the cell growth process,
e. g.:

- cell growth reduces or decreases owing to limitation
of nutrient substrate; cells stop product formation and
stop reproducing themselves;

- cell growth increases due to abundance and utiliza-
tion of nutrient substrate;

- cell population rapidly increases due to reproduction
(by binary fission);

- transition of substrate into product via cell metabolic
enzymes;

- transition of metabolized product into living cells;
- excretion of waste products of cell metabolism;
- death and disintegration of living cells;
- cell growth inhibition due to excess of substrate.

Some of the above reaction steps are related to the
transition of cells from one phase to another. Especially
important are the transitions from lag to log phase and
vice versa. The last two steps concern the stationary
and death phases. These two phases are also significant,
but in this work we try to ignore them for the sake
of simplicity, mainly focusing on the reaction steps
involving lag-log transitions.

II. BATCH CULTIVATION: REACTION SCHEMES

A. The lag–log cell phase transition

We shall borrow the basic lag–log cell phase transition
reaction scheme from Henri-Michaelis-Menten enzyme
kinetic [12], [14], identifying free enzymes with X-
cells (bacteria in lag phase) and bound enzymes with
Y -cells (log phase bacteria). Following the familiar re-
action scheme for the substrate-enzyme kinetics when
the enzymes possess a single active site we have:

S + X
k1−→←−
k−1

Y
k2−→ P + X, (RSP)

wherein k1,k−1,k2 are rate constants.

Passing from a model describing a specific substrate-
enzyme reaction to a model describing a substrate-cell
activity is a jump from molecular level to cell population
level. A possible justification of such a jump is that:

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S Markov, Cell Growth Models Using Reaction Schemes: Batch Cultivation

i) cells can be viewed as (organized) complexes of
enzymes, ignoring thus other components of the cells; ii)
nutrient substrates can be restricted to a few (sometimes
a single) limiting substrate(s); iii) the cell enzymes
process the nutrient substrate S to produce new complex
substances (like proteins, via the ribosome mechanism)
partly included in product P in the right hand-side of
reaction scheme (RSP); iv) X-cells that become engaged
in metabolic/reproductive processes transform (change
their status)from lag into log phase Y -cells.

The product P in the scheme (RSP) can be viewed
as (part of) the highly organized substances needed to
complete the enzymatic competence of the cell to be
able to reproduce. We can think of the Y -cells as having
reached the highest inner protein-nucleotide organization
needed to reproduce. We shall thus assume that the Y -
cells are those engaged in production and reproduction
at the current moment t, in contrast to X-cells that do
not (re)produce at moment t.

In the scheme (RSP) the Y -cells correspond to bound
enzymes C = E-S producing P . A formal analogy
between the S-E pairs and the S-X pairs is that an
S-E pair is an enzyme in a special temporal state
(bound enzyme, engaged in a product formation process)
and similarly a S-X pair is a cell in a temporal state
(occupied in a (re)production process).

The (RSP) reaction scheme represents three reaction
steps in the cell growth process, involving the two
X −→←− Y transitions plus the S −→ P transformation.
The scheme shows that Y -cells are entirely dependant on
the available nutrient substrate S. Indeed, the presence
of S stimulates (initiates) the transition of lag phase X-
cells into log phase Y -cells. On the other side, substrate
limitation leads to a decrease (up to disappearance) of
Y -cells, resp. of product formation (including reproduc-
tion). The reverse transition X ←− Y can be interpreted
as the case when X-cells absorb a certain amount of
nutrient substrate but then do not process it (and excrete
back the substrate). This reaction step seems to have
minor impact on the total cell growth process.

Applying the mass action law to reaction scheme
(RSP) leads to the following system of ordinary differ-
ential equations:

ds/dt = −k1xs + k−1y
dx/dt = −k1xs + k−1 y + k2y
dy/dt = k1xs−k−1 y −k2y
dp/dt = k2y

(1)

familiar from enzyme kinetic textbooks [12]. Appropri-
ate initial conditions corresponding to a batch cultivation
process are: s(0) = s0, x(0) = x0, y(0) = y0, p(0) =
0.

The terms with k−1 in the right hand-side of (1) can
be suppressed as k−1 is expected to be small, then we
obtain a simpler model:

ds/dt = −k1xs
dx/dt = −k1xs + k2y
dy/dt = k1xs−k2y
dp/dt = k2y

(2)

corresponding to the reaction scheme:

S + X
k1−→ Y k2−→ P + X, (RSRs)

Systems (1), (2) modelize a batch mode bioreactor
under the assumption that cells do not reproduce nor
die. The basic models (1), (2) take into account substrate
limitation in so far that only the log phase biomass
declines due to substrate depletion; however the total
biomass concentration z = x + y = const remains
constant in spite of substrate depletion. As in enzyme
kinetics here we see again the conservation law for
the substrate ds/dt + dy/dt + dp/dt = 0 as well. A
realistic cell growth model should include the above
mentioned additional reaction steps, such as reproduction
and mortality in order to encompass the complete cell
growth process.

B. Reproduction

Reproduction can be viewed as the most important
part of the total “production” process—during reproduc-
tion a mother cell produces two daughter cells, thus one
more cell appears as result of this process.

Reproduction: a simple reaction scheme. Y -cells
are engaged in formation (biosynthesis) of products P ,
which are important components of newborn cells. Thus
we may view at reproduction as part of the production
process. We assume that: a) cell growth is mainly due
to reproduction—hence to Y -cells; b) Y -cells utilize
product P to reproduce; c) newborn cells are Y -cells,
that is newborn cells do reproduce (by binary fission) [7].
Following these assumptions a simple reaction scheme
for the growth-reproduction process is:

P + Y
k3−→ 2Y. (RSG)

From a “mechanistic” point of view reaction scheme
(RSG) can be interpreted as follows: the mother Y -cell
transforms into one of the daughter cells, while the other

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daughter cell is built by means of product P components.
In other words, a certain part of product P “comes into
life” as a newborn cell. Putting reaction schemes (RSP)
and (RSG) together:

S + X
k1−→←−
k−1

Y
k2−→ P + X,

P + Y
k3−→ 2Y, (RSP–RSG)

and applying the mass action law, we obtain the model:

ds/dt = −k1xs + k−1y
dx/dt = −k1xs + k−1 y + k2y
dy/dt = k1xs−k−1 y −k2y + k3py
dp/dt = k2y −k3py

(3)

with initial conditions (corresponding to a batch cultiva-
tion process): s(0) = s0, x(0) = x0, y(0) = y0, p(0) =
0.

Respectively, assuming k−1 = 0 we obtain the simpler
model:

ds/dt = −k1xs
dx/dt = −k1xs + k2y
dy/dt = k1xs−k2y + k3py
dp/dt = k2y −k3py

(4)

induced by the reaction scheme:

S + X
k1−→ Y k2−→ P + X,

P + Y
k3−→ 2Y. (RSPs)–(RSG)

C. Comparison to classical models

We next compare model (3) with the classical model

ds/dt = −δµ(s)z
dz/dt = δµ(s)z
µ(s) = s/(Km + s).

(5)

Proposition. Classical model (5) is a special case of
model (3) under the assumption that the biomass x and
the product p are (nearly) constant during the process.

Proof. We sum up the second and the third equations
in (3) to obtain a relation for the total biomass z = x+y:

dz/dt = dx/dt + dy/dt = k3py.

Integrating in the interval [0, t] we have

z = x + y = x0 + y0 + k3I, I(t) =
∫ t
0
p(τ)y(τ)dτ.

Let us assume that the biomass x is at a steady state:

dx/dt = −k1xs + k−1 y + k2y = 0.

From (3) this implies ds/dt = −k2y. Substituting x
by z −y we have

−k1xs+k−1 y+k2y = −k1(z−y)s+k−1 y+k2y = 0,

which gives

y = sz/(Km + s), Km = (k−1 + k2)/k1.

Substituting the above expression for y in ds/dt =
−k2y we have

ds/dt = −k2sz/(Km + s),

which gives the first equation of (5): ds/dt = −δµ(s)z
with δ = k2.

To obtain the second equation of (5) we substitute
y = sz/(Km + s) in the last equation of (3): dp/dt =
k2y −k3py to obtain (using dz/dt = k3py):

dp/dt = k2zs/(Km + s)−dz/dt

or

dz/dt = k2zs/(Km + s)−dp/dt.

Assuming a constant production of p, i. e. approxi-
mately dp/dt = 0, we arrive at the classical model (5)
for the substrate consumption and the biomass dynamics
in a batch reactor. �

Remark. Note that if the biomass is nearly constant, i.
e. z = z0, then the ODE for the consumption of s in (5)
turns into the Michaelis-Menten ODE of the substrate
uptake in a chemostat: ds/dt = −k2z0s/(Km + s).

Proposition 1 shows that the basic Monod model (5) is
an approximation of model (3) under the assumption of
(nearly) constant biomass and product. In other words,
model (3) generalizes model (5).

D. Mortality

To model the process of cell mortality, we first should
decide which cell fractions are most susceptible of dying.
One possibility is to introduce a separate cell fraction, as
done in [13]. In this work we shall try to keep our model
mathematically simple and decide against introducing a
new cell fraction, resp. a new phase variable. From the
two fractions X and Y we have to choose which one is
more likely to die. In this work we accept that the X-cell
fraction is the one that is more likely to be affected (but
we leave this decision open for future examination). We
thus accept here that only X-cells die and disintegrate.
In addition, we assume that disintegrated cells transform
partially to substrate S and partially to product P :
X

ks−→ S, X
kp−→ P. (RSD)

Putting all three reaction schemes (RSP), (RSG) and
(RSD) together we have:

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S + X
k1−→←−
k−1

Y
k2−→ P + X,

P + Y
k3−→ 2Y, X ks−→ S, X

kp−→ P.
wherein k1,k−1,k2,k3 and ks,kp are rate constants.

Applying the mass action law the above global scheme
leads to the dynamical model

ds/dt = −k1xs + k−1y + ksx
dx/dt = −k1xs + k−1 y + k2y − (ks + kp)x
dy/dt = k1xs−k−1 y −k2y + k3py
dp/dt = k2y −k3py + kpx

(6)

wherein the parameters k1,k−1,k2,k3 and ks,kp are
rate constants.

Biological meaning of the terms in the models. The
reaction schemes make the biological interpretation of
the terms in model (6) almost obvious. Here they are:

k1xs represents the consumption of S by bacteria X
and the transition of bacteria X into bacteria Y ;
k−1y describes an amount of substrate concentration

temporal;y stored by the cells but not further processed;
k2y describes the amount of product concentration

formed by Y -cells and the transition of Y -cells into X-
cells;
k3py describes the increase of the cell population due

to reproduction and the decrease of P due to its transition
to newly formed Y -cells (reaction scheme (RSG));
ksx describes the decay of bacteria X and the disinte-

grated part of the dead cells transforming into substrate
S—reaction scheme (RSD);
kpx describes the decay of bacteria X and the disin-

tegrated part of the dead cells transforming into product
P —reaction scheme (RSD).

The parameters have the meaning of specific rate con-
stants as follows: k1 is the substrate utilization rate, k−1
is the substrate non-utilization rate, k2 is the production
rate, k3 is the reproduction rate, ks and kp are death
rates.

Whenever appropriate we may also use specific reac-
tion steps describing waste product excretion, e. g.

P + Y
δ−→ Q, (RSW)

implying corresponding terms δpy in the dynamical
system.

We finish this section with the following hypothesis:

Hypothesis. Cell growth models for batch cultivation
can be based on reaction schemes, involving reaction
steps such as (RSG), (RSP), (RSD), (RSW).

III. COMPUTER EXPERIMENTS

We present the results of two computer experiments
based on the above reaction schemes. For the computer
experiments real experimental data for the biomass and
product concentrations have been used. The graphics of
the solutions demonstrate a good fit to the experimental
data. In particular the lag phase data for the biomass are
fitted better than when classical models such as (5) are
used.

Computer experiment 1. The model is:

ds/dt = −k1xs + k−1y
dx/dt = −k1xs + k−1y + k2y
dy/dt = k1xs−k−1y −k2y + βpy
dp/dt = k2y −γpy

(7)

Fig. 1. Numerical solution of cell growth model (7) using Matlab

The values of the parameters used in model (7) are
as follows: k1 = 3.48691,k−1 = 0,k2 = 7,β =
41.5095,γ = 86.4424;s0 = 1.4,x0 = 0.00450748. The
graphs of the solutions are represented in Fig. 1.

Model (7) is close to model (6) but is without mortality
terms and there we have β 6= γ. To explain why in this
experiment β < γ we need to introduce a waste product
excretion step, as done in the next experiment.

Computer experiment 2. The model is as follows:

ds/dt = −k1xs + k−1y
dx/dt = −k1xs + k−1y + k2y −kdx
dy/dt = k1xs−k−1y −k2y + (α−δ)py
dp/dt = k2y − (α + δ)py + kdx

(8)

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S Markov, Cell Growth Models Using Reaction Schemes: Batch Cultivation

The reaction scheme leading to model (8) involves a
decay step (RSD) and a waste product excretion step
(RSW) as follows:

S + X
k1−→←−
k−1

Y
k2−→ P + X,

P + Y
α−→ 2Y,

P + Y
δ−→ Q,

X
kd−→ P,

Fig. 2. Numerical solution of cell growth model (8) using Matlab

wherein Q represents waste product. The values of the
parameters used in model (8) are as follows: k1 =
4.18191,k−1 = 0,k2 = 4.50413,α = 60.6611,δ =
21.3493,kd = 0.136806;s0 = 1.55196,x0 =
0.00490581. The graphs of the solutions fitting real
experimental data are visualized in Fig. 2. A very good
fit can be observed.

IV. CONCLUSION

In this work we formulate certain reaction steps in the
cell growth process and try to base our models entirely
on proposed reaction schemes. Our reaction schemes for
the phase transition of the cells make use of appropriate
reaction schemes from enzyme kinetics. Elements of the
Michaelis-Menten enzyme kinetics can be observed in
classical cell growth models using the Monod specific
growth rate function µ. In this paper instead of the
(approximate) Michaelis-Menten enzyme kinetics for the

substrate uptake we use the (exact) Henri-Michaelis-
Menten enzyme kinetics (HMM-reaction mechanism).

It has been noted in the literature that classical Monod
type models often fail to describe adequately bio-reactors
under perturbed conditions. It has been also recognized
that Monod models lack (do not fit well) the lag phase
of the cells. Our numerical experiments make us believe
that structured models provide more flexibility and can
be better fitted to real data. The use of reaction schemes
makes the construction of particular cell growth models
simple and instructive; it also contributes to understand-
ing the underlying biological mechanism. Our hypothesis
is that cell growth models can be entirely based on
reaction schemes.

V. ACKNOWLEDGEMENTS

. The author is indebted to Prof. V. Beschkov from
the Institute of Chemical Engineering, BAS, and Prof.
M. Kamburova from the Institute of Microbiology, BAS,
for providing real experimental data on microbial growth
and for useful discussions on the behavior of the mi-
croorganisms and the proposed reaction schemes.

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http://dx.doi.org/10.11145/j.biomath.2013.12.301

	Introduction
	Structured cell growth models
	Relation to enzyme kinetics
	Model assumptions
	Reaction steps of the cell growth process

	Batch cultivation: reaction schemes
	The lag–log cell phase transition
	Reproduction
	Comparison to classical models
	Mortality

	Computer experiments
	Conclusion
	Acknowledgements
	References