A cell population structuring model to estimate recombinant strain growth in a closed system

for subsequent search of the mode to increase protein accumulation during protealysin

producer cultivation

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Biofabrication 4 (2012) 019601 (1pp) doi:10.1088/1758-5082/4/1/019601

Erratum: A cell population structuringmodel to estimate recombinant straingrowth in a closed system for subsequentsearch of the mode to increase proteinaccumulation during protealysinproducer cultivationS P Klykov et al 2011 Biofabrication 3 045006

S P Klykov1,3, V V Kurakov1, V B Vilkov1, I V Demidyuk2,T Yu Gromova2 and D A Skladnev1

1 PHARM-REGION, Ltd, c.1, b.10, Baryshikha street, Moscow, 125222, Russia2 Institute of Molecular Genetics, Russian Academy of Sciences, Kurchatov Sq. 2, Moscow 123182,Russia

E-mail: smlk03@mail.ru, pharm-region@mail.ru, duk@img.ras.ru, microb52@mail.ru andskladda@gmail.com

Received 25 November 2011Published 23 February 2012Online at stacks.iop.org/BF/4/019601

There was an error in the published version of figure 4. Thecorrect figure is shown below.

3 Author to whom any correspondence should be addressed.

Figure 4. Dependence of enzymeactivity on growth time.Growth time, hours, is the obscissa. Activity P, Units ml−1, isthe ordinate axis. Experimental activity in Control 2005.

Model calculation of the activity in Control 2005.Experimental activity in Control 2010. Model calculation

of the activity in Control 2010. Predicted activity inExperiment 1 according to Control 2010 results. It is assumedthat specific rate of product destruction is equal to the specificdestruction rate in Control 2010. Experimental activity inExperiment 1. Predicted activity in Experiment 1 accordingto Control 2010 results. It is assumed that product destructiondoes not occur.

1758-5082/12/019601+01$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

IOP PUBLISHING BIOFABRICATION

Biofabrication 3 (2011) 045006 (12pp) doi:10.1088/1758-5082/3/4/045006

A cell population structuring model toestimate recombinant strain growth in aclosed system for subsequent search of themode to increase protein accumulationduring protealysin producer cultivationS P Klykov1,3, V V Kurakov1, V B Vilkov1, I V Demidyuk2,T Yu Gromova2 and D A Skladnev1

1 PHARM-REGION, Ltd, c.1, b.10, Baryshikha street, Moscow, 125222, Russia2 Institute of Molecular Genetics, Russian Academy of Sciences, Kurchatov Sq. 2, Moscow 123182,Russia

E-mail: smlk03@mail.ru, pharm-region@mail.ru, duk@img.ras.ru, microb52@mail.ru andskladda@gmail.com

Received 14 December 2010Accepted for publication 12 September 2011Published 25 October 2011Online at stacks.iop.org/BF/3/045006

AbstractIn this paper we have proposed a new structured population growth model, further developinga model previously proposed by the authors. Based on this model, optimal growthcharacteristics of the recombinant strain Escherichia coli BL-21 (DE3) [pProPlnHis6] weredetermined, which allowed us to increase the output of metalloproteinase by 300%. We haveexperimentally demonstrated the applicability of the new model to cell cultures with implantedplasmids and the potential practical use for an output increase of a wide variety of biosynthesisprocesses.

(Some figures in this article are in colour only in the electronic version)

Notation

LGP logarithmic growth phase;GIP growth inhibition phase;S substrate concentration, g l−1;X biomass concentration, OD units or

g l−1;OD unit of an optical density;τ time, hour;P products-metalloproteinase, units

P ml−1;dP/dτ absolute rate of product synthesis, units

of P(ml h)−1;Q = −dS/dτ absolute rate of substrate consumption;QO2 oxygen mass exchange rate, mmole

O2/(volume units per time units);

3 Author to whom any correspondence should be addressed.

J stochiometric factor of energy substrate(S) oxidation, Joule of S/mmoleO2;

μ specific growth rate of biomass X,(hour)−1;

q = Q/X orq = (1/X) ∗dP/dτ specific rate of substrate utilization

or product synthesis, units of S(unitsof X h)−1 or units of P(ml units ofX h)−1;

a trophic coefficient, amount of energysubstrate consumed for the synthesis ofa biomass unit, Joule of S/Joule of X orunits of S/units of X;

f amount of energy substrate accumulatedin biomass X during cultivation on asynthetic medium, Joule of S/Joule ofX or units of S/units of X;

1758-5082/11/045006+12$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK

Biofabrication 3 (2011) 045006 S P Klykov et al

m energy maintenance coefficient, therate of substrate consumption formaintaining viability of one biomassunit per a unit of time, Joule of S(Jouleof X h)−1 or units of S(units of X h)−1;

A = m/a (1) Parameter describes a delay of thebiomass growth rate; (2) specific rate ofaccumulation of stable cells, h−1;

Xp maximum biomass concentration, whenall the energy generated duringcultivation is consumed for cell viabilitymaintenance;

XLim biomass concentration in the end ofexponential growth phase and beginningof growth inhibition phase;

Xst concentration of the biomass of zero agecells (stable), the content of resting cells,OD units or g l−1;

Xdiv = X − Xst concentration of proliferation biomass,OD units or g l−1;

τLim time of exponential growth phasetermination, hour;

XLimst concentration of stable cells at the end

of exponential growth phase, OD unitsor g l−1;

R ratio of Xst to biomass X, relativecontent of stable cells in the biomass,synchronization degree, part of 1;

Xl initial biomass concentration in LGPcorresponding the beginning of popu-lation structuring, OD units or g l−1;

Xfinal final biomass concentration, at whichR = 1 (when energy consumption islimited);

krdiv, kr

st, ksdiv, ks

st constants of metabolite and substratebiochemical reaction rates, [units of P(or S)/(ml∗ units of biomass per onehour)];

PLim metabolite concentration or ac-tivity at the end of LGP andbeginning of GIP, units ofP ml−1;

P0 metabolite concentration or activity inLGP, when biomass structuring occursat X = Xl , units of P ml−1.

1. Introduction

Studies on microorganism growth and biosynthesis ofmetabolites have given rise to a great number ofmathematical models describing the dynamics of biomassgrowth and nutrient substrate consumption. Many models ofmicroorganism growth are based on J Monod’s growth modelor its variants [1, 2]. The basic idea of these models consists inthe restriction or limitation of the specific growth rate of a cellpopulation either by concentrations of a limiting substrate orof a biosynthetic end-product. However, limited applicationareas and numerous exceptions are the usual drawbacks ofsuch models that necessitate the construction of a new model.

The basic disadvantage of the J Monod model described in [2]is an ambiguity about the physical sense of its parameters (oreven absence of this sense).

The works cited above also describe the so-called logisticcurves characterizing all phases of microorganism growth.However, the model data have no wide practical application indesigning biotechnological production technologies yet.

The Volterra model considers a variant of the logistic curvedescribing the phase when all nutrients are completely utilizedand the cell population growth reaches the stationary phaseand/or the phase of cell destruction [2].

The fact that the models mentioned do not consider thefactors influencing cell growth, for instance oxygen mass-exchange, is a common disadvantage [2].

A number of mathematical models are also available todescribe product biosynthesis. Among these, the Leudeking–Piret model is one of the most well known [3].

One of the characteristic features of the Leudeking–Piretmodel is that factors either related or not related to cellgrowth contribute to the kinetics of metabolite production.For the cases when the studied substance is a final product ofmetabolism related to energy consumption, the first member ofthe Leudeking–Piret equation describes biomass cell growth,while the second one characterizes the amount of energyconsumed for cell viability maintenance. However, if thetarget products are not related to biomass energy metabolism,the Leudeking–Piret equation is difficult to interpret.

Time dependence of metabolite concentration in thebatch process may be rather complicated. In the growthmedium several substances, which then are exposed to furthertransformations, can be accumulated. In some cases, acomplicated kinetics of metabolite production may indicatemodifications in the mechanism of cell metabolism under theinfluence of growth condition changes. The kinetics of acetate,butyrate, acetone and butanol production by Clostridiumacetobutilicum [2] serves as a good example of these processes.

The production of metabolites can also be accompaniedby their chemical transformations in the growth medium. Suchcomplicated processes, thus, should require the inclusion ofthese reactions, if they are well studied, into a mathematicalmodel. Undoubtedly, this would make the mathematicaldescription more sophisticated.

Currently, the models of biomass growth, substrateconsumption and metabolite synthesis are, as a rule,subdivided into two groups: structured and unstructuredmodels [2].

The structured model implies the availability of more thanone component to describe the structure of a cell populationand its viability. The unstructured model suggests that thecell population is homogeneous and only one component, forexample, biomass X, is used for its characterization.

Since unstructured models are rather simple, they are oftenused for research.

In all of the above-mentioned models of cell growth,energy consumption and metabolite synthesis are calculatedby different equations describing the amounts of consumedsubstrates and synthesized products. From our point of view,this approach is not correct, since both substrates and products

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Biofabrication 3 (2011) 045006 S P Klykov et al

are substances, which undergo various transformations duringfermentation, and thus should be described by one and thesame equation.

Free access of oxygen to cells during cultivation playsthe key role in aerobic processes. Since oxygen deficiencyslows cell growth down resulting in the reduction of aerobicmicroorganism biomass output and synthesis of the majorityof metabolites, it is important to provide oxygen inflow sothat the concentration of the dissolved oxygen in the substratewould not be lower than 10% of a complete oxygen saturationlevel.

In [4], a biotechnological method providing optimumconditions for cell growth has been described. The methodsuggests that during cultivation microorganism culturesreceive additional feeding from a balanced nutrient mediumconcentrate. The inflow rate and the amount of feedingconcentrate exclude both the lack of nutritious substrates, andgrowth inhibition owing to high concentration of nutrient com-ponents. In this case, oxygen mass exchange rate in the cultureremains constant and oxygen concentration tends to zero.

The Pirt–Marr equation,

Q = −dS/dτ = adX/dτ + mX, (1)

proposed for the calculation of the energy substrateconsumption rate, has been modified for the analysis of energyconsumed by cells for growth and viability maintenance. Thiswas done by expressing Q through the oxygen mass exchangerate as follows:

Q = JQO2 . (2)

This yields the initial equation (3) for the unstructuredmathematical model of cell growth limited by oxygenconsumption. All the equation constituents are expressed inenergy units:

JQO2 = adX/dt + mX. (3)

The solution of the basic equation (3) allows for theformulation of fundamental laws of culture growth underconditions of a limited oxygen supply rate. Unlike earlierexisting conceptions, it is shown that a linear decrease inabsolute growth rate of the biomass and hyperbolic reduction inthe specific growth rate is a function of biomass concentration,and that the energy substrate consumption rate specified byoxygen mass exchange rate is constant. Methods for definingthe parameters of the unstructured model proposed includedgrowth efficiency and energy substrate consumption (m, a andA= m/a), which were not previously used in any practical wayto estimate periodic culture growth. Parameter A describes adelay of the total biomass growth rate.

Studies on the effect of Salmonella culture growth rateon cell survival under adverse external influences [5] showedthat during GIP, if there is a lack of dissolved oxygen, theaccumulation of stable cells occurs at a constant specificrate equal to that of the growth delay (A = m/a). Theshare of stable cells within the population is obviously equalto that of nonproliferating cells, which consume energyonly for viability maintenance. Methods for the definitionof parameters of the structured model describing substrateconsumption and metabolite biosynthesis on the basis of

preliminary calculated parameters of the unstructured modelwere designed.

Thus, the proposed structured model assumes that withina growing population there are two groups of cells essentiallydiffering in their physiology. Group I represents newlygenerated (young) cells and group II contains cells beingin a state of active proliferation. Although the cells ofgroup I are often called ‘resting’ cells [1], in our opinion theseare the cells of ‘zero age’ [2], i.e. the cells being in phase G1

or in phase V as designated for eukaryotes and prokaryotes,respectively. Group I cells exhibit minimal physiologicalfunctions, and for each cell these functions are constant. Acharacteristic feature of the cells is that they consume energysubstrates only for their viability maintenance. In [5] thesecells are called ‘stable’.

All these discussions give good theoretical basis forthe dynamics of the structure formation of microbialpopulations limited by the lack of oxygen in terms of energyconsumption and cell viability maintenance. An analysisof the experimental and literature data made it possible topropose a new structured model of cell population growth[6–8]. On the basis of the model, consumption of substratesused for cell construction and synthesis of metabolites in thecultures consisting of two groups of cells differing in energyconsumption was described.

The performed experiments and analysis of the literaturedata have proved that the proposed model is adequate forthe description of a wide range of microorganisms (fromobligate aerobes up to obligate anaerobes), synthesis of variousproducts (antibiotics, organic acids, alcohol, poly-β-butyricacid, polysaccharides, etc) and consumption of substrates(glucose, nitrogen and phosphorus) [6, 7]. The present papershows the feasibility of the structured and unstructured modelsfor estimation of the growth of recombinant microorganismstrains and expression of foreign proteins in them.

An analytical solution of constitutive equations (1)–(3)for the unstructured model of biomass growth is shown in[4]. Equations (1)–(6) presented in table 1 describe changesin the consumption of energy substrate S, growth of biomassX, absolute growth rate of a total biomass dX/dτ and specificgrowth rate μ.

In [8], it was shown that quantity of nonproliferating zeroage cells consuming energy only for viability maintenancechanges in direct proportion to the change of energyconsumption for viability maintenance, i.e. adXst = dSst (3a).

On the other hand, apparently dSst/dτ = mXst (3b) sincecells of zero age (stable cells) consume an energy substrateonly for viability maintenance. Superposition of the last twoequations gives equation (28), table 1.

It is also obvious that if m → 0, then, according toequation (1), the rate of energy substrate consumptionQ is proportional to the absolute biomass growth ratedX/dτ , and exponential cell population growth is observed.For GIP, m �= 0.

Successful research in the field of recombinant productsperformed during the last 30 years has led to an increase oftheir manufacturing. Taking into account all of the above-mentioned, we have found it interesting to use the proposed

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Biofabrication 3 (2011) 045006 S P Klykov et al

Table 1. Basic equations for GIP.

Function Equation ##

1 2 3Q = − dS

dτ= f (X) Q = a dX

dτ+ mX (1)

Q = f (QO2) Q = JQO2 (2)Q = f (X) JQO2 = adX/dτ+mX = mXp (3)dSst = f (dXst) dSst = adXst (3a)dSst/dτ = f (Xst) dSst/dτ = mXst (3b)dX

dτ= f (X) dX

dτ= A(Xp − X) (4)

μ = dX

dτ× 1

X= f (X) μ = A

(Xp

X− 1

)(5)

X = f (τ) X = Xp − (Xp − XLim) exp[−A(τ − τLim)] (6)dXst

dτ× 1

XstdXst

dτ× 1

Xst = m

a= A (7)

Xst = f (τ) Xst = XstLim exp[A(τ − τLim)] (8)

R = f (μ) R = K

A2X2p

×(

μ

A+ A

μ+ 2

)(9)

R = f (X) R = K

A2(XpX−X2)(10)

K K = dX

dτ× dXst

dτ= A2 Xst

Lim × (Xp − XLim) (11)

τfinal τfinal = 1A

ln Xp−XLimXp−Xfinal + τLim (12)

τfinal τfinal = 1A

ln XfinalXst

Lim+ τLim (13)

Xfinal Xfinal = Xp

2 + 12

√X2

p − 4XstLim

(Xp − XLim) (14)

Xfinal Xfinal = Xp+ 1A

√A2X2

p−4K

2 (15)

μfinal μfinal = A(

Xp

Xfinal − 1)

(16)dS

dτ= f (Xst,Xdiv) construction substrate consumption rate − dS

dτ= kst

s Xst + kdivs Xdiv (17)

dP

dτ= f (Xst, Xdiv)product synthesis rate dP

dτ= kst

pXst + kdivp Xdiv (18)

P = f (X)

product concentration P = PLim +(

kdivp

A

)[Xp ln Xp−XLim

Xp−X− (X − XLim)

(1 +

XstLim

Xp−X

)](19)

P = f (X)

product concentrationP = PLim +

(kdivp

A

){Xp ln Xp−XLim

Xp−X− (X − XLim)

}+

+ 1A(kst − kdiv)Xst

LimX−XLimXp−X

(20)

S = f (X)–construction substrate concentrationS = SLim −

(kdivs

A

){Xp ln Xp−XLim

Xp−X− (X − XLim)

}−

−(

1A

)(kst

s − kdivs )Xst

LimX−XLimXp−X

(21)

q = f(R) q = kdiv +(kst − kdiv

)R (22)

XstLim Xst

Lim = 2(X2

Lim

/X2

p

)(Xp − XLim) (23)

XstLim Xst

Lim = (Xp − Xfinal)Xfinal/(Xp − XLim) (24)

models for the description and intensification of the growth ofgenetically modified strains and biosynthesis of recombinantmetabolites.

For testing the proposed model, E. coli strain BL-21(DE3) [pProPlnHis6] [9], a producer of protealysin(metalloproteinase belonging to the family of thermolysins)[10–12] was used. Enzymes of this group can serve as potentialdrugs; in addition, they have a considerable innovativepotential for biotechnological application [13]. Currently,recombinant producers [14–19] are used for thermolysin-likeproteases. In the proposed paper, an attempt to increasethe efficiency of metalloproteinase expression by recombinantE. coli strain BL-21 (DE3) [pProPlnHis6] owing to the increaseof oxygen mass exchange rate in a reactor followed by aconsequent correction of nutrient medium composition hasbeen made. The correction of medium composition is requireddue to a significant increase of the rates of all metabolic

processes in the reactor. In this case, additional nutrients arerequired. This is also stipulated by the increase of energy usedfor maintaining the producer viability during biosynthesis ofsubstances genetically foreign for a cell host.

Thus, the objective of the proposed paper is to demonstratenovel structured and unstructured models of recombinantstrain growth and metabolite biosynthesis with the purposeto intensify cultivation processes and to increase the targetproduct output.

2. Materials and methods

2.1. Escherichia coli strain BL-21 (DE3) [pProPlnHis6] aproducer of protealysin

This was previously constructed on the base of E. coli strainBL-21 (DE3) widely used for recombinant protein production[9]. This strain exhibits lysogenic properties with respect to

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Biofabrication 3 (2011) 045006 S P Klykov et al

prophage λDE3 bearing prophage T7 RNA pol gene underthe control of promotor lacUV5, IPTG, galactose and glucosebeing the inducers and repressors, respectively.

2.1.1. Total proteolytic activity. TPA was determined fromthe ability of the strain to hydrolyse of azocasein [10]. For thispurpose 50 μl of the enzyme solution were added to 100 μlof 1% azocasein solution in 20 mM Tris-HCl (pH 8,0). Themixture was incubated at 37 ◦S for 15 min. The reaction wasceased by adding 200 μl of 10% trichloracetic acid solution.The sediment was separated by centrifuging at 9500 g for5 min. 50 μl of 5 M NaOH were added to 250 μl of thesupernatant. Light absorption was measured at 450 nm on aplate photometer.

The amount of the enzyme increasing the light absorptionby 1 OD units per 1 min was considered as 1 unit of the activity.Enzyme activity was determined after the destruction of thebiomass and metalloprotease formation. For this purpose,the cells were centrifuged and then resuspended in 20 mMTris-HCl, pH 8,0. The cells suspension at a concentrationof approximately 30 g l−1 was exposed to ultrasonication for1.5–2 min 5–7 times and incubated at 4 ◦S for 24–48 h.

2.2. Cultivation of Escherichia coli BL-21 (DE3)[pProPlnHis6]

A standard scheme of culture preparation was used: producer-collection culture on Petri dishes-inoculation flasks-reactor.

The cultivation was performed in a reactor with a workingvolume of 5–5.5 l, in a liquid medium, containing (g l−1):tripton—15; yeast extract—7.5; sodium chloride—1.0;potassium di-phosphate—5.0; potassium monophosphate—2.5; glucose-3.0; ampicillin—0.1; antifoam—1.0 ml, atpH 7,0 ± 0,2. At this stage 2 control cultivations (Control 2005and Control 2010) were performed. Experimental cultivation(experiment 1) was conducted at high oxygen inflow rateunder stirring and broken glucose feeding according to pHand pO2 detector signals. Flasks were inoculated with a smallamount of inoculate taken with a microbiological loop fromthe surface of the bacterium layer in Petri dishes. Each ofthe five flasks containing 0.2 l of the inoculation material wasused to inoculate the reactor containing 4–4.5 l of the nutrientmedium with all the necessary additives.

Stirrer rotations and the amount of oxygen consumed wereadjusted automatically according to the signal of pO2 detector.

Prior to inoculation, a small amount (2 ml l−1) of 50%glucose solution was poured into the reactor. Before the initialamount of glucose had been consumed, pH was maintainedat 7,0 with 15% NaOH solution according to the call of thepH detector. After the consumption of the initial amount ofglucose, it was fed in low doses through a peristaltic pumpaccording to the pH detector signal. The rate of pumping wasadjusted to provide glucose feeding (dry weight) within therange from 1 to 3 g (l h)−1. Alkaline feeding was stopped toavoid glucose overdose. IPTG (0.01 g l−1) was introducedinto Control 2005 and Control 2010 reactors after microbialsuspension had reached optical density equal to 2.5 and2 OD units, respectively. The inductor was placed intoexperiment 1 reactor at OD units equal to 7 OD units.

2.3. Description of a model for the estimation of Escherichiacoli strain BL-21 (DE3) [pProPlnHis6] growth andmetalloproteinase biosynthesis

The basic equations of the proposed model are presented intable 1.

2.3.1. Unstructured model equations. Equations (1)–(6),except for (3a) and (3b), deduced in [5] represent basicequations of the unstructured model of microorganism growth.

The presented equations are used for

• analysis of oxygen mass exchange in the cultural medium,kd ;

• analysis of physiological producer constants, m, a, A,μmax, τLim, XLim, Xp.

Oxygen mass exchange can be measured by other well-known methods and selected for cultivation. If this parameteris unknown, it can be calculated using equations (2) and (3) ofthe unstructured model.

Parameters of the unstructured model τLim, XLim, Xp

depend on the selected oxygen mass exchange. All theparameters depend on nutrient medium composition andcultivation conditions.

Then all the parameters of the unstructured model areused for calculating the parameters of the structured modeland biosynthesis constants.

Experimental biomass concentration parameters are usedfor the estimation of biomass growth according to theunstructured model. Sampling is performed at equal timeintervals �τ = τ i+1 − τ i = const, at which biomassconcentration changes, �Xτ = Xτ+�τ − Xτ .

According to equation (3) for GIP the following isdeduced:

Xτ+�τ = Xτ + �X (25)

Xτ + �X = Xp − (Xp − Xτ)∗ exp(−A∗�τ), (26)

or

Xτ+�τ = Xp − (Xp − Xτ)∗ exp(−A∗�τ) (27)

�X = Xp − Xτ − (Xp − Xτ)∗ exp(−A∗�τ) (28)

�X = (Xp − Xτ)(1 − exp(−A∗�τ)) (29)

at �τ = const (30) equation (29) represents a linear regressionfunction of Xτ

�X = (1 − exp(−A∗�τ))Xp − (1 − exp(−A∗�τ))Xτ .

(31)

In the point of intersection with the ordinate axis, at X = 0,equation (31) is converted as follows:

�X = (1 − exp(−A∗�τ))Xp = �0X, (32)

where �0X is the point where the regression line crosses theordinate axis (31).

From this it follows that

1 − �X/Xp = exp(−A∗�τ). (33)

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Biofabrication 3 (2011) 045006 S P Klykov et al

Expression (33) can be presented in the following form:

A = −[ln(1 − �0X/Xp)]/�τ. (34)

Similarly to formula (34) for A of logarithmic growth phase(LGP), equations for μmax can be deduced:

Xτ+�τ = Xτ + �X = Xτ exp(μmax∗�τ), (35)

from which

�X = [exp(μmax∗�τ) − 1]∗Xτ , (36)

and

μmax = [Ln(1 + M)]/�τ, (37)

where M = [exp(μmax∗�τ ) − 1] is the inclination angle

tangent of the straight line �X = f (X) for LGP.In the work given, μmax was calculated by a standard

technique [8]. In this case, dependence of LGP naturallogarithms, X, on time τ was built and the tangent of theinclination angle of the obtained straight line was determined.These values were compared to the results obtained withequation (37).

For the analysis of biomass growth, the followingparameters are determined first: biomass concentration,XLim, and time, τLim, corresponding to LGP terminationand beginning of GIP; hypothetical maximum biomassconcentration, Xp, when all energy transformed by the systemis consumed for biomass viability maintenance; A = m/a isthe specific growth delay rate of the biomass during GIP.

2.3.2. Equations for the structured model. Equation (7)was theoretically proved in [6–8]. This equation indicatesthat in any population limited in energy consumption, Xst,a specific rate of the accumulation of nonproliferating cellsconsumed energy only for viability maintenance is constantand equal to A. This is due to the fact that the energy consumedfor cell viability maintenance causes a total reduction in thegrowth rate of proliferating cells, Xdiv, and, as a consequence,of the population growth rate as a whole. In this case,nonproliferating cell accumulation is directly proportional toconcentration Xst at any instant time.

Equation (8) represents an integrated form of equation (7).This equation shows exponential character of Xst accumulationin time.

Equations (9)–(16) were deduced in [5–8].Equations (9) and (10) represent a share of

nonproliferating cells in the population. In [8], it was shownthat R = Xst/X also describes the degree of synchronization ofzero age cultures. All physiological functions of zero age cellsare minimal, and cell resistance to adverse external influencesis maximal owing to temporarily inhibited metabolism. It isshown [5, 7, 8] that if growth of the biomass is limited only byenergy inflow, all proliferating cells are converted into stableones after which further proliferation ceases. Thus, the wholeamount of energy is consumed for cell viability maintenance.It is obvious that equality R = 1 is true for this case.

Physical meaning of parameter K in equation (11) consistsin the fact that with growth limitation strengthening (increasethe duration of growth phase), the effect from one and the samework of biochemical mechanisms underlying cell processes

continuously decreases, i.e. the number of proliferating cellsreduces over time. At the same time, there is an acceleratedincrease in the quantity of stable cells consumed energy onlyfor viability maintenance (see equations (7) and (8)). Thelatter statement should probably be understood to reflect theincrease of stable cell numbers due not only to the terminationof early cell proliferation cycles, but also to the progressivefailure of stable cells to proliferate; otherwise, the cells couldstart proliferation again in the absence of limits.

Equations (12)–(16) describe the time of culture growthtermination and specific growth rate at the moment, when thediscussed unstructured model of cell growth and biosynthesisduring GIP, as it is presented in table 1, does not work. In thiscase, other equations are required (see below).

Equations (17), (18) and (19)–(21) were considered in[6–8]. The physiological processes occurring in proliferatingand stable cells differ greatly and are diametrically opposed.Therefore, subdivision of population cells into proliferating(Xdiv) and stable (Xst) ones makes it possible to useequations (17), (18), and table 1 to describe the consumptionrate of substrates utilized for cell construction and metabolitesynthesis:

dP(or − S)/dτ = kdivP,SX

div + kstP,SX

st ·We assume that metabolites are synthesized only byproliferating cells. Nonproliferating cells, as a rule, destroythese products. Therefore, signs of the constants for metabolitesynthesis and degradation are opposite. The same should bestated for substrates utilized for cell construction.

If stable cells do not influence synthesis of metabolites(and substrate utilization), i.e. kst

P,S = 0, then the synthesis iscarried out by proliferating cells and can be described by theintegrated equation (19). If both proliferating cells and zeroage cells participate in the synthesis, then the accumulationof metabolites is described by equation (20). Similarlyequation (21) is proposed for the consumption of substratesused for cell construction. An analysis of equations (17)–(21)shows that metabolite synthesis proceeds with rate constants,whose signs are opposite to each other if the influence of thestable cells on the product output is not equal to 0. This meansthat groups of proliferating and stable cells behave differentlyduring product biosynthesis: the former group synthesizesmetabolites, the latter destroys them.

It was found out that if cell growth process is limitednot only by energy consumption but also by any substrateparticipating in cell construction, then the increase of biomassyield may suddenly stop. Studies on the dynamics ofbiomass accumulation limited by substrates utilized for cellconstruction [8] showed that this parameter can be presentedas

Rfinal = kdiv/(kdiv − kst). (38)

The required final value of Rfinal is expressed by an identicalequation, if it is necessary to terminate the biosynthesis processin order to avoid the destruction of metabolite products byactive cells when kst

P �= 0. The situation, when maximumof biomass accumulation does not coincide with that ofmetabolite accumulation, i.e. the latter occurs earlier than the

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Biofabrication 3 (2011) 045006 S P Klykov et al

former, is actually rather common. For this case, equation (15)would have the following appearance:

Xfinal =Xp + 1

A

√A2X2

p − (1/Rfinal)∗4K

2. (39)

Equations (12)–(14) and (16) in this case are transformedaccording to (39).

In the studies of growing cultures, especially, ofPseudomonas and Yersinia pestis, by cytorefractometricmethods [20], we have repeatedly noticed that a certain numberof nonproliferating cells are always present in the exponentialgrowth phase. As the cell growth reaches GIP, the number ofsuch cells increases significantly.

Equations (12)–(14) and (16) then can be transformedaccording to (39).

In the given work, biosynthesis parameters weredetermined for cells in GIP. The calculated values wereused to estimate the conformity of the data obtained for thecells in GIP. As is well known, biosynthesis of metabolitesoften occurs during GIP. In [8], the factors affectingcell population structuring during LGP were investigated.Equations describing metabolite accumulation in this growthphase are as follows:

Xst = 2Xl(1 − Xl/X), (40)

R = 2Xl(1/X − Xl/X2) (41)

and

P = P0 + (kdiv/μmax)(X − Xl)

+ 2[(kst − kdiv)/μmax]Xl[ln(X/Xl) + Xl/X − 1] (42)

where

Xl = X2Lim/Xp (43)

is the initial biomass concentration in LGP, corresponding tothe beginning of cell population structuring.

2.4. Technique for the estimation of bacterium growth andmetabolite biosynthesis

Parameters Xtheor for LGP were calculated accordingto standard exponential equation (1, 2): Xtheor =X0

∗ exp[μmax∗τ ],where μmax and X0 were determined from

LnXexperiment = f (τ ) calculated from the data presented infigure 1. μmax was also calculated from equation (37) andcompared with the parameter calculated according to theabove equations. For this purpose, �X for the cells in LGPwas estimated according to experimental data Xexperiment fromwhich corresponding values �X were calculated for each ofthe specified sampling interval. For each previous value ofthe experimental biomass concentration change, �X, i.e. �Xτ

= X(τ+�τ) − Xτ ., dependence �Xτ = f (Xτ ) was then builtaccording to equation (36) and μmax was determined from(37) (see figure 2).

Parameters Xtheor for GIP were calculated on the base ofexperimental data Xexperiment (figure 1), according to whichvalues �X for sampling within specified time �τ , wereobtained as a difference between previous and subsequent X,i.e. �Xτ = X(τ+�τ) − Xτ . Then, dependence �Xτ = f (Xτ )was drawn, and corresponding constants �0X, Xp, A were

Table 2. Estimated model parameters.

Control Control ExperimentParameter 2005 2010 2010

X0,[OD units] 0,20 0,65 1,1μmax,[1,2], [h−1] 0,740 0,550 0,649μmax,(37),[h−1] 0,821 0,438 0,622A,[h−1] 0,330 0,381 0,395XLim,[OD units] 0,75 4 7τLim,[h] 4,5 4 3Xp ,[OD units] 4,33 13,33 24,5�τ ,[h] 1 1 1�0X,[OD units] 1,224 4,210 7,90XLim

St,[OD units] 0,21 1,68 2,86K,[(OD units)2 h−2] 0,0840 2,2792 7,7942PLim,[units ml−1] 0,04 0,7 1,225kdiv,[units (ml OD units h)−1] 0,0535 0,200 0,200kst,[units (ml OD units h)−1] ∼0 −0,164 −0,045Xl ,[OD units] 0,13 1,20 2,0P0,[units ml−1] 0 0,22 0,20Rfinal = RforPmax (38) 1 0,549 0,826

determined. �0X is an intersection of the regression line (31)and ordinate axis; Xp is an intersection of the regression line(31) and abscissa axis; A is calculated from formula (34) (seefigure 2). The boundary point of two growth phases (XLim,τLim) is determined from the intersection of straight lines (31)and (36) (see figure 2).

Biosynthesis parameters are calculated fromequations (10) and (22). For this purpose Xst

Lim is pre-liminarily estimated by equation (23). Then the valuesof the biosynthesis specific rate are calculated. (qP )τ isestimated from [�P/�τ ]τ = [P(τ i+�τ) − Pτ ]/�τ and (qp)τ= (1/Xτ )[�P/�τ ]τ . On the basis of Xp, XLim, A, τLim

previously estimated from equation (8) dependence Xtheor =f (τ ) is deduced (figure 1). Xst

Lim calculated above makes itpossible to obtain dependence R = f 1(τ ) = f 2(X). Then thestraight line (qP )τ = kdiv

P +(kstP − kdiv

P

)Rτ (figure 3), at R =

0, cuts off a line segment equal to kPdiv on the ordinate axis.

Inclination angle tangent of this line is equal to kstP − kdiv

P ,from which kst

P can be calculated.The integrated accumulation of metabolite Ptheor is

estimated according to equation (19) or (20): if kstP = 0, then

equation (19) is used, if kstP �= 0, equations (20) and (21) are

suitable.Experimental results on E. coli strain BL-21 (DE3)

[pProPlnHis6], X, metalloproteinase P synthesis and thecorresponding calculated parameters Xtheor and Ptheor forLGP were compared with the model values. The latterwere obtained from the calculated parameters for GIP,Xp, XLim, Xst

Lim, A, kdivP , kst

P , and from μmax determinedpreliminary for LGP. Equation (40) for calculating thenumber of stable cells was used. For the description ofmetabolite accumulation during LGP, equation (42), whereP0 is the product concentration at the moment, when biomassstructuring supposed to occur during LGP, is used.

Parameters P0 and PLim were calculated from the resultsof cell cultivation in fermentations Control 2005 and Control2010 and are presented in table 2.

As was noted above, often there is a situation whenbiomass growth stops growing while metabolite biosynthesis

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Biofabrication 3 (2011) 045006 S P Klykov et al

Figure 1. Control 2005 cultivation. Experimental biomass, X, and biomass calculated according to unstructured model Xtheor, stable(nonproliferating) cells, Xst, proliferating cells, Xdiv. Growth time, hours, is the abscissa axis. X, OD units, is the ordinate axis.

X, Xtheor, Xst, Xdiv.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

1,2

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

Figure 2. Control 2005 cultivation. Determination of unstructured model parameters, A, μmax, τLim, XLim, Xp , �0X. Parameter values arepresented in table 2. X, OD units, is the abscissa axis. �X, OD units/hour, is the ordinate axis. Experimental data for LGP,

Linearization for GIP, Linearization for LGP, Calculation data for GIP. �XLGP = 1,273X; �XGIP = −0,2814X + 1,224.

continues for some time. It was found that the absoluterate of product synthesis can be described also by equation(18). However, if no changes of R occur, then X, Xst and Xdiv

remain constant. The integrated accumulation of biosynthesismetabolites in this case can be expressed by the followingequation:

P = PfinalGIP + kdiv ∗ XdivfinalGIP ∗ (τ − τfinalGIP)

+ kst ∗ XstfinalGIP ∗ (τ − τfinalGIP), (44)

where PfinalGIP is the final concentration of the product, whichis determined from equation (20), table 1, Xdiv

finalGIP and XstfinalGIP

are the corresponding final concentrations that can either becalculated using equations (6), (8), (38) and (39) or determinedexperimentally.

3. Results and discussion

The possibility to increase the output of the target productsynthesized by E. coli strain BL-21 (DE3) [pProPlnHis6] isdetermined by the presence of the following genes:

(1) ompT responsible for the inhibition of protease activityfor product output increase.

(2) lon responsible for the inhibition of intracellular ATP-dependent protease for product output increase.

(3) lacUV5 being a promoter repressed by glucose.

Gene 1 and 2 determine the selection of nitrogen componentof the nutrient medium, acid casein hydrolysate. Amino acids

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Biofabrication 3 (2011) 045006 S P Klykov et al

-0,10

-0,05

0,00

0,05

0,10

0,15

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

Figure 3. Control 2005 cultivation. Determination of parameters kdiv and kst according to the structured model. Parameter values arepresented in table 2. R, portion 1, is the abscissa axis. q, units P/(ml∗ OD∗hour), is the ordinate axis. Experimental data for GIP,

Linearization for LGP. q = –0,0476R + 0.0535.

are the source of nitrogen in the caseine hydrolysate. lacUV5determines the mode of glucose feeding of the producer withinduced product expression. It is efficient to introduce glucoseinto the nutrient medium at small doses (1–3 g (1 h)−1) in aform of 50% solution according to the pH detector signal. Suchmode of glucose feeding allows for the producer to remain ina stated of glucose limitation. In this state, glucose does notinduce lac-operon repression.

Experimental results obtained during the cultivationof producer X, metalloproteinase P biosynthesis and thecorresponding parameters Xtheor and Ptheor are presentedin figures 1–4. Parameters of producer biomass andmetalloproteinase changes calculated according to the modelare also depicted in the figures.

The parameters of growth and biosynthesis shown intable 2 were calculated from experimental data X (figure 1)and P (figure 4), according to the technique described above.

A mode of calculating the parameters presented in table 2is shown in figures 2 and 3, the Control 2005 experiment beingtaken as an example. Parameters of the other processes werecalculated by the similar mode.

The producer growth estimation was performed from alimited number of points (from 4 to 6). This provides thelimited accuracy of the related analyses possible for thesecases. For LGP, calculation of μmax by two well-known modesshowed practically similar values (table 2). However, even atthe accuracy given, the description of growth and biosynthesisboth during GIP and LGP has appeared comprehensible tosum up the results allowing for forecasting metalloproteinasebiosynthesis processes. Nutrient medium composition andcultivation conditions have been respectively corrected.

Enzyme activity in the intermediate samples of all threeexperiments is depicted in figure 4. The area of gray color in

the center of figure 4 represents an expectation value of theenzyme activity for experiment 1 performed at high oxygenmass exchange rate and increased content of nutrients.

The upper curve limiting this area is built using a productsynthesis constant corresponding to the Control 2010 processon the assumption that product degradation does not take place.The rest of the parameters correspond to the characteristicsof the experiment 1 process. The lower curve differs fromthe upper one because degradation constant kst equal to thevalue determined for the Control 2010 process is taken intoconsideration.

The enzyme activity values in the control experiments,Control 2005 and Control 2010, were obtained whencultivation was performed at a low oxygen exchange rate,and the decreased content of nutrients is depicted in figure 4.These values were used as the basis for modeling experiment1 processes.

As is seen from figure 4, the data for experiment1 satisfactorily fall into the area marked in gray. Thiscorrespondence has been forecasted. Results of mathematicaldescription of three points of experiment 1 were obtainedby a trial and error method using the constants of productsynthesis and destruction (see table 2) at the minimized root-mean-square deviation. Definition by the method described insection 2.4 is unsuitable for the given case, since quantity ofpoints for the analysis (three points) is not enough. In orderto receive a comprehensible result by the method proposed, aminimal quantity of the points should be no less than 4.

For the Control 2005 process, product destructionconstant is equal to 0. For the Control 2010 process, productdestruction constant, kst, makes about half of the productsynthesis constant, kdiv. This means that more intensive growth

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Biofabrication 3 (2011) 045006 S P Klykov et al

Figure 4. Dependence of enzyme activity on the growth time. Growth time, hours, is the abscissa. Activity P, units ml−1, is the ordinateaxis. Experimental activity in Control 2005. Model calculation of the acvitity in Control 2005. Experimental activity in Control2010. Model calculation of the activity in Control 2010. Predicted activity in Experiment 1 according to Control 2010 results.It is assumed that specific rate of product destruction is equal to the specific destruction rate in Control 2010. Experimental activity inExperiment 1. Predicted activity in Experiment 1 according to Control 2010 results. It is assumed that product destruction does notoccur.

(a) (b) (c)

Figure 5. (a) The change of decimal logarithm of the concentration of mammalian cell HUVEC-population corresponding to theexperimental and model data (according to the proposed model); (b) experimental data on the quantity of cells of different age cycle;(c) model data (according to the proposed model) on the quantity of cells of different age cycle. Reproduced by permission [25],coordinated by Dr J Tyson.

of the producer in Control 2010 has not been provided withthe required additional amount of nutrients that has causeddegradation of target protein being a potential source of carbonand nitrogen for cells.

During experiment 1, a successful attempt to preventthe undesirable degradation of the protein by increasing thecontent of nutrients in the initial medium and duly addition ofglucose as an energy source has been undertaken. Thus, theproduct destruction constant, kst, was lowered approximatelyfour-fold.

Knowing the amount of energy inflow to the system,which can be calculated from equations (1)–(3), using thedynamics of glucose solution inflow expressed in terms ofheat of combustion and Xp expressed in the units of biomass

heat combustion, it is possible to calculate m(3) and a valueseasily.

If the oxygen mass exchange rate has not beenpreliminarily determined by any other way, it is possible tocalculate this parameter using equations (2) and (3).

Equations (1)–(6) for the estimation of total biomassare used to be applied within the unstructured model of cellpopulation growth and substrate consumption. In our paper,we pioneered the use of equations (7) and (8) for the structuredmodel with the purpose of obtaining a combined solution ofequations (17), (18) and (22) (also shown for the first time), in aform of equations (19)–(21). The pioneer solution of equation(22) had been confirmed with international patent PCT [21].

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Biofabrication 3 (2011) 045006 S P Klykov et al

One of the reviewers’ questions is as follows: Why do weuse the biosynthesis of a recombinant strain to demonstrateour model? The point is that biosynthesis processes, whichare described in the above-stated international Patent and thePatent of the Russian Federation, are ordinary biosyntheticprocesses carried out by usual ‘not recombinant’ strains ofbacteria, yeasts and fungi. Within the last 30 years, the roleof recombinant strains has sharply increased in science andindustry. Indeed, the recombinant strains are mainly used now.This circumstance has determined the choice of the researchobject.

The experimental and model data were analyzed usingf -tests included in the Excel Program [22–24]. For thispurpose, experimental data for Control 2005 and Control2010 were divided by the corresponding data, calculatedaccording to the model for Control 2005 and Control 2010. Ifthe model adequately describes the experiment, the obtainedaverage value of the ratio of the experimental and model data(their quotients) will be randomly grouped near 1. Indeed,according to the f -test for two experiments carried out bya ‘conventional’ method (Control 2005, Control 2010), theratio of the average value is equal to 0.96 ± 0.42, confidenceprobability being 95%.

Thus, the Fisher criterions is equal to 1.644, 3.695 and2.248 for various combinations of the number sequences forthe analysis that is less than the critical criterion equal to 5.05,the discretion range number being equal to 5. Hence, themodel is adequate for the experimental data.

For the model and experimental data of experiment 1,a similar analysis was made using two-selective f -tests atvarious dispersions as well. For this purpose, experimentaldata for experiment 1 were divided by the corresponding datacalculated according to the model for experiment 1 and werecompared with the number sequence of the ratio of the averagevalues of the experiment/model for Control 2005 and Control2010. The Fisher criterion is equal to 7.37, that is less thanthe critical criterion equal to 19.3, the discretion range numberbeing equal to 2. The ratio of the average value is equal to0.99 ± 0.15, the confidence probability being 95%.

Thus, there is no reason to consider the observeddifference between the number sequences statisticallysignificant.

This provides evidence to the fact that both the valuesbelong to one and the same general data set and that the modelis adequate.

Experimental confirmation of the conclusions of ourtheory was published in February 2011 by the Americanresearchers Dr John J Tyson et al [25]; it is shown in the lastdrawing in this work and figure 5 of this paper. Figure 5 fullyconfirms the predictions obtained by our model for changes inthe proportion of non-dividing (resting) cells (see equation forR). It is clear from the graph, which is designated by letter ‘B’in the figure, with our proposed model describing the authors’data better than the model offered by the authors themselves.

The fundamental equation for the unstructured model isthe Marr–Perth equation (1) for the phase of slower growthin the integral form (6). It can be easily obtained if therelevant work is done with equations (1) and (4) under ourmethodology.

The basic equations for the non-structured model are thefollowing: (7) integral equation for X, expressed by equation(6), and equation (17) and/or (18).

All other equations are auxiliary, derived from the basicequations, and are applied depending on the objectives set bya researcher or an engineer.

Moreover, from equations (1), (4), (6), (7) a generalequation was obtained:

dnXdiv

d(Xst)n= K

A2∗ (−1)(n−1)n!

(Xst )(n+1)− C, (45)

where n are the whole numbers, order of the derivative of afunction. Moreover, C = 1 if n = 1 and C = 0 if n � 2.The factor ‘C’ has the following physical sense: in GIPeach stable cell can become dividing again 1 time only,i.e. C(n=1) = 1; each such subsequent transition for eachseparate line of cell is impossible, i.e. C(n=2, ... ) = 0. Thisexpression is a generalizing equation for any component ofthe biomass and for total biomass and, thus, unifies structuredand unstructured models. To describe the biomass growth forany proposed model, the major differential equation is (45). Todescribe the cycle of the main substances a major differentialequation is (17) and/or (18).

These equations describe all the known diversity of theprocesses with S-like growth curves and changes in theconcentrations of substances in closed systems, which is anentirely new and previously unknown fact. It is knownthat a physical law means a generalization of a numericalrelationship between the objects of the real physical worldthat is running under specified conditions for the class ofthe objects and does not follow from any of the previouslydiscovered laws. There is no reason not to admit the twodescribed equations for the GIP as laws for GIP.

The data obtained were used for the selection oftechniques to increase the protein expression by geneticallymodified microorganisms.

4. Conclusions

(1) Growth characteristics of the recombinant strain andmetalloproteinase biosynthesis were calculated on thebasis of structured and unstructured population models.The results obtained allowed for the selection ofcultivation conditions providing the increase of targetproduct output in order to perform one more series ofthe experiment to design new more intensive cultivationregimens;

(2) It was shown that the producer cultivated in thepresence of different nitrogen sources and at differentmodes of glucose feeding exhibits different growth andbiosynthetic properties. Also, it was demonstratedthat metalloproteinase shows different biosyntheticproperties during cultivation on nutrient media of differentcompositions.

(3) Preferable substrates and cultivation regimens wereselected to optimize growth properties of the producer andto increase the target product (metalloproteinase) output.

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Biofabrication 3 (2011) 045006 S P Klykov et al

Acknowledgments

This work was supported in part by the Russian Foundationfor Basic Research (project no 09-04-00734). The authorscordially thank Dr V V Derbyshev for valuable advice givenduring the designing of the unstructured model and Dr V ASamoilenko and Ms T V Yakshina for assistance in experimentperformance.

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