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Structured kinetic model for Xanthomonas campestris growth
Enzyme and Microbial Technology 34 (2004) 583–594
Structured kinetic model for Xanthomonas campestris growth F. Garcia-Ochoa∗ , V.E. Santos, A. Alcon Departamento de Ingenieria Quimica, Facultad CC. Quimicas, Universidad Complutense, Madrid 28040, Spain Received 8 December 2002; accepted 8 January 2004
Abstract A structured kinetic model for Xanthomonas campestris growth is proposed. The formulation of the model is made according to those steps employed for kinetic modeling of reaction networks. First, simplified reaction schemes are proposed, which are formed by four or five reactions using lumping and pseudosteady-state assumption for different compounds. Several expressions for the kinetic equations have been considered, checking their validity with the evolution of experimental data taking into account the different reactions in the scheme of reactions assumed. Afterwards, parameter values are calculated by regression to experimental data, using simple and multiple non-linear techniques. Then, model discrimination is carried out using both statistical and physical restrictions. The model finally proposed has four key compounds: ammonium, RNA, DNA and intracellular proteins. The analysis of the intracellular compounds (RNA, DNA and proteins) has been carried out using an experimental technique based in flow cytometry. Three experiments (under the same operational conditions but with different initial nitrogen concentrations in the medium) have been fitted to the model to obtain the values of the parameters, showing a prediction very close to the experimental data. The model is able to predict different system evolutions for different nitrogen initial concentrations. © 2004 Elsevier Inc. All rights reserved. Keywords: Cell model; Xanthomonas campestris; Kinetic modeling; Intracellular compounds; Flow cytometry
1. Introduction Xanthan gum is the polysaccharide produced by Xanthomonas campestris, an obligatory aerobic Gram-negative bacterium. This gum is employed as stabilizer, emulsifier and suspending agent due to its special rheological properties [1] in many industries such as food (E-415), pharmaceutical, cosmetics, etc. [2]. Xanthan gum production rate is greatly influenced by several variables, such as temperature, oxygen mass transfer rate and the nutrients employed in the medium. Previous works have shown that there are optimum medium composition [3] and conditions [4] to carry out this process: 28 ◦ C, 257 ppm of ammonium and enough available dissolved oxygen in the broth without an excessive hydrodynamic stress for the cells. To describe the influence of these variables and to predict the evolution of the system if any perturbation should occur different kinetic models have been proposed. First of all unstructured kinetic models [4–9] considering growth, consumption of substrates (carbon and nitrogen sources) and xanthan production itself in a set of differential equations ∗
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were proposed. These models are not able to involve the influences of the variables commented on; therefore, more complex kinetic models had to be proposed. Temperature and oxygen mass transfer influences have been considered using metabolic kinetic models [10,11]; these models take into account a simplified reaction scheme using lumping of metabolic pathways into the carbon metabolism, yielding global reactions as a sum of the reactions involved in the pathway considered such as ‘glucose total catabolism’, ‘maintenance energy’, ‘oxidative phosphorylation’ and ‘xanthan production’. However, since growth is regarded as one lumped process, the influence of the nitrogen source is not considered in this kind of models, and therefore an optimal value for the nitrogen source concentration can only be found by experimental observations [3]. The proposal of a set of lumped reactions describing nitrogen metabolism is the way to involve the influence of this nutrient in the process description; therefore, a structured kinetic model to describe growth had to be proposed. The proposal of a structured kinetic model involves a wide knowledge of the microbial metabolism, therefore, all such models are proposed for microorganisms such as Escherichia coli, Bacillus subtilis and Sacharomyces cerevisiae [12–14]. These models present a very high number of parameters (between 70 and 200) usually estimated from
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Nomenclature Cj C DNA f(CK ) F FCM INT INT-O
ki kI Ki KMRN N NR r ri rs rsi R RNA RNAI
Rj SKM SSR t ts
concentration of component j (g/l) key compounds concentration vector nitrogen bases forming deoxyribonucleic acid concentration function in the kinetic expression Fischer’s F statistical parameter flow cytometry lumped species involved in transformation of RNA bases into DNA bases oxidized lumped species involved in transformation of RNA bases into DNA bases kinetic constant of reaction i inhibition constant (gIPR/h gA) saturation constant of reaction i (gN/l) Kinetic Modeling of Reaction Networks number of key compounds in the system number of reactions considered in the scheme reaction rates vector reaction rate of the reaction i (g/l h) integral of the reaction rates vector integral of the reaction rate of the reaction i (g/l) production rates vector nitrogen bases forming ribonucleic acid lumped species to be reduced to obtain nitrogen bases forming deoxyribonucleic acid production rate of the compound j (gj/l h) structured kinetic models square sum of residuals time (h) Student’s t statistical parameter
Greek letters νij stoichiometric coefficient of the compound j in the reaction matrix of the stoichiometric coefficients Subscripts 0 initial value A ammonium Ac critical ammonium concentration aaNFB non-forming bases aminoacids DNA nitrogen bases forming DNA EPR extracellular proteins IPR intracellular proteins RNA nitrogen bases forming RNA RNAI intermediary RNA nitrogen bases to form DNA nitrogen bases X biomass
literature data. As commented above, these kinetic models are needed to explain the influence of some variables, but the models to be employed must be simpler than the models previously quoted if the parameters are going to be calculated by fitting to experimental data. Therefore, they have to include an appropriate number of key compounds, no more than four or five, and parameters. The aim of this work is the proposal of a simple biochemical structured kinetic model for Xanthomonas campestris growth involving the influence of initial nitrogen concentration. The model is applied to experimental data using the methodology employed in kinetic modeling of reaction networks. The experimental data for intracellular compounds are obtained using flow cytometry by means of a previous reported experimental procedure [15]. 1.1. Kinetic Model proposal As commented above, there is no precedent in literature for obtaining the parameter values of simplified structured models by statistical fitting of experimental data. Therefore, the development of a methodology to carry out this complex task is needed. This problem is quite similar to the Kinetic Modeling of Reaction Networks (KMRN), in which the kinetic analysis needs much more information than in the case of single reactions, because both stoichiometry and thermodynamic are more complex and the combination of reaction schemes and kinetic equations yields a higher number of possible kinetic models and a model discrimination procedure must be carried out [16]. In the KMRN two main different ways have been followed to study these complex systems: (1) to study the scheme reactions one by one to reduce the problem to a sum of single reactions -but when the system complexity is high this is no longer applicable- and, (2) to propose a reaction scheme, usually very simplified, using lumping of similar compounds, to assume kinetic equations (as simple as possible, potential or hyperbolic equations) and to obtain the parameter values; afterwards, model predictions have to be compared to experimental data and any of the model assumptions changed until a good agreement between experimental data and model predictions is attained. Discrimination can be carried out using statistical and physical criteria, as applied in KMRN [16]. The proposal of structured kinetic models (SKM) can be carried out using the same methodology developed for KMRN. The main steps needed to attain a SKM are the following: simplified metabolic pathway proposal, kinetic equations proposal and model discrimination. The formulation of a simplified metabolic pathway has to be preceded by the determination of the chemical species to be involved in the system. A stoichiometric study of the system has to be carried out to obtain the following information: number of key compounds, number of independent relationships between the chemical species in the system and
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to know how to obtain the changes in the system composition knowing the changes in key compound concentrations. This information is essential for both the determination of the analytical method to know the changes in the system composition and for proposing a simplified reaction scheme, which considers the main reactions in the metabolism of the microorganism studied. Therefore, a previous knowledge of the metabolic pathways involved in the metabolism of the microorganism studied is needed. The simplified metabolic pathway proposal has to be preceded by a rigorous analysis of the experimental data, to observe the tendencies in the evolutions of the different compounds analyzed. The information obtained by means of the stoichiometric study has to be considered to propose a scheme of reactions. The simplification of the metabolic pathway proposed has to be carried out using lumping to reduce the number of compounds to be considered. When lumping is employed it is assumed that similar compounds, which are involved in similar reactions, can be joined together. The application of this simplification has been generally performed using good sense: molecule reactivity, molecule size, etc. [16]. The equations joining the measured magnitudes (production rates or concentrations) with the magnitudes to be determined as operational conditions function (reaction rates) are the following: Rj =
NR
νij × ri
(j = 1, . . . , N)
(1)
i=1
The integration of the Eq. (1) yields the following expression, to be used with integral data (Cj versus t): Cj − Cj0 =
t NR νij ri dt i=1
(j = 1, . . . , N)
(2)
0
The proposal of kinetic equations can be performed by different methods. Differential methods are based on the application of Eq. (1) to the N key compounds of the system and integral methods are based on the application of Eq. (2) to the N key compounds. Usually the form of the functions ri have to be assumed as different functions of the species concentrations of the system, which are different for the several reactions taking place in the reaction scheme assumed. The set of equations obtained for the N key compounds by means of Eqs. (1) and (2) can be expressed by means of a matrix form, as follows: R=×r
(3)
C = × rs
(4)
Garc´ıa-Ochoa and Romero [16] proposed the use of an equation set in terms of reaction rates of the key compounds instead of the equation set usually employed in terms of productions rates (Eqs. (3) and (4)). This set of equations
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of reaction rates can be expressed in matrix form both in differential (Eq. (5)) and integral form (Eq. (6)): r = −1 × R
(5)
rs = −1 × C
(6)
This method can be employed only when the stoichiometric matrix, , is square, that is to say, when the number of key compounds is equal to the number of reactions in the scheme assumed. This method gives several advantages both from the calculation point of view (the mathematical methods needed are simpler than in the production rates method, and the calculation of each parameter value is isolated) and from the conceptual point of view (the complex reaction network can be treated as a sum of single reactions). The application of Eq. (6) to the N key compounds of the system yields a set of independent equations. Once the kinetic equation forms of each reaction in the model are assumed, a multiple-response fit of the experimental data to the proposed model has to be employed to obtain the values of the parameters as a set. Afterwards parameter values are calculated, the model reproducibility has to be proved; when the fit is not acceptable another model has to be checked. Combinations between different kinetic equations for the same reaction scheme yield the different models. Usually more than one model can describe the system; therefore model discrimination criteria have to be established. The criteria for model discrimination can be grouped together into two main types [16]: statistical and physical criteria. Statistical criteria are based on the comparison of statistical parameters, such as the Student’s t or the Fischer’s F that are tabulated for different confidence levels and they are functions of the number of experimental data employed in the fitting and the number of parameters to be calculated. Another parameter to compare the goodness of the model fitting is the residual sum of squares; which must be as low as possible, but the values obtained under experimental error have to be considered as the same value. Physical criteria are imposed criteria such as the following: the values of all the parameters obtained have to be greater than zero, and all the parameters of the model have to show a logical tendency versus the variables studied, such as medium composition, in this case.
2. Materials and methods 2.1. Microorganism The microorganism employed was the bacterium Xanthomonas campestris NRRL B-1459 supplied by the Northern Regional Research Laboratory of the U.S. Department of Agriculture (Peoria, IL).
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2.2. Experimental set-up and procedure Experiments were carried out in a commercial 1.5-l fermentor (B. Braun Model Biostat M). Inoculum was built up as described in a previous work [4]. The production medium employed was optimized elsewhere [3], using 40 g/l of sucrose as carbon source. The production medium without the carbon source was sterilized into the vessel. The carbon source was separately sterilized and introduced afterwards into the vessel. The pH electrode was calibrated before fermentor sterilization. For the calibration of the dissolved oxygen electrode, after polarizing at least 6 h, using oxygen desorption with nitrogen and oxygen absorption with air. Afterwards, inoculum was introduced into the vessel through a membrane septum. Temperature was controlled by means of a PID controller, using another controller for stirrer speed. At several times during the process, 10 ml of culture broth were withdrawn aseptically from the fermentor for analysis. 2.3. Analytical methods Biomass was determined as OD at 540 nm with a Spectrophotometer Shimadzu (Model UV 1603). The nitrogen source was analyzed employing a selective ammonia electrode (Orion, model 95–12). Flow cytometry has been the analytical technique employed for intracellular compounds quantification. The samples from the bioreactor were centrifuged at 4000 rpm for 30 min and the pellet was resuspended in a buffer Tris 0.1 M, NaCl 0.1 M and pH 7.4. The cells were washed three times and resuspended in the same buffer. Afterwards, cells were centrifuged at 4000 rpm for 30 min and the pellet was resuspended in 50 ml of the fixative solution (glutaraldehyde 3% in Tris buffer). Cells were incubated at 4 ◦ C for 4 h. After washing with Tris buffer, the samples were diluted to obtain a concentration between 200 and 2000 cells/l. Afterwards cells were stained for protein analysis with a fluorescein isothiocyanate 0.1 mg/ml solution and excess dye was removed by washing and rinsing in Tris buffer 0.1 M, NaCl 0.1 M, pH 7.4. For nucleic acids analysis samples were incubated at 37 ◦ C for 40 min with deoxyribonuclease solution (1 mg/ml) for RNA and with ribonuclease solution (1 mg/ml) for DNA. After incubation the enzymes were removed by centrifugation and pellet was resuspended in Iodide propidium 0.1 mg/ml solution as stain for nucleic acids and incubated for 10 min at 4 ◦ C. The fluorescence intensity data mean of FCM can be extrapolated using the corresponding calibration curve obtained in a previous work [15] and the results can be expressed finally as compound concentration in broth (g/l). 2.4. Experiments Three experiments were performed at 28 ◦ C employing different ammonium initial concentrations: 130, 257 and 475 ppm. Biomass, ammonium, DNA, RNA and intracellular protein evolution were measured. Extracellular proteins
concentration can be calculated by means of a nitrogen mass balance in the system. 2.5. Parameter calculation methods Parameter values have always been obtained by means of non-linear regression techniques [17]. This kind of regression technique was applied both in single and multiresponse way. The integral method has been employed for obtaining the parameter values for the kinetic model considered in this work. For this purpose, it was necessary to perform the integration of the equations of the models by means of numerical techniques employing a fourth-order Runge-Kutta algorithm coupled with the Marquardt algorithm [17] for multiresponse fit and Simpson algorithm for simple response regression when the ‘reaction rates method’ was applied. For statistical meaning of the fits the Fisher’s F and Student’ t-test were used. Moreover, the confidence interval of the parameters and the sum of squares of residuals referred to the data number have been calculated. 3. Results and discussion The proposal of a structured kinetic model for Xanthomonas campestris growth has been carried out following the method described above, according to the following steps. 3.1. Simplified metabolic pathway proposal First of all, a simplified scheme of reactions must be assumed, taking into account that hundreds of reactions inside the cell can be lumped together into pseudochemical reactions. The structured kinetic model proposed breaks the cell into three different lumped groups of macromolecules: DNA, RNA and proteins which represent natural grouping of biomass components. It is assumed that the composition of the cell membranes is time invariant and has no influence on metabolism. Other compounds necessary for growth such as carbon source or phosphate are assumed to be available in excess and consequently cellular growth rate is not controlled by them. In this work, it is proposed a reaction scheme of the nitrogen source metabolism into the cell (because nitrogen is assumed to be the limiting substrate in this process). The simplification of the metabolism has been carried out as in the metabolic model previously commented on [11]: the reactions forming a pathway have been assumed to obtain only one global reaction. The compounds involved in the pathway have been lumped into general compounds. Therefore, it is considered that all the nitrogen is obtained from NH4 + and all the hydrocarbons present in the pathway are assumed to be glucose. The ammonium metabolism is branched in two directions: (a) non-forming bases aminoacids synthesis; (b) forming bases aminoacids synthesis, as can be seen in Fig. 1. The total protein (intracellular and extracellular) protein has been assumed to be
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587
INTRACELLULAR PROTEIN (IPR)
r1
Non-forming bases amminoacids (aaNFB)
EXTRACELLULAR PROTEIN (EPR)
NH4+
r3
RNA
r4
RNAI
r2 Forming bases amminoacids (aaFB)
r5
DNA
Fig. 1. Simplified metabolic pathway assumed.
formed by non-forming bases aminoacids and the nucleic acids (DNA and RNA) by forming bases aminoacids synthesis. There are some main assumptions in this procedure: • It is considered that there is a transformation of some molecules of RNA bases to DNA bases. This consideration has been made due to two main assumptions: (a) the deoxyribonucleotides are obtained from ribonucleotides by means of an oxido-reduction reaction; (b) the uracile molecule (RNA specific component) is transformed into a timine one (DNA specific component). These RNA bases have been named as RNAI . • In the reduction of RNAI into DNA (transformation of ribonucleotides into deoxyribonucleotides), there is a lost of an oxygen atom for each RNAI base; therefore, an intermediary enzymatic pathway is involved and grouped in an hypothetical compound called ‘INT’ that yields and oxidised compound, named as ‘INT-O’. The molecular formulas assumed for the intracellular compounds considered (RNA, DNA and proteins) have
been obtained as average of the molecular formulas of the different components—aminoacids or bases—assumed as forming those molecules. Because X. campestris is able to synthesize all the 20 aminoacids, proteins (both, intracellular and extracellular) have been considered formed by non-forming bases aminoacids, with an average molecular formula taking into account 18 non-forming bases aminoacids, yielding: C5.6 H10.1 O2.3 N1.4 S0.1 . On the other hand, forming-bases aminoacids (glycine and aspartic acid), which average molecular formula considered has been C3 H5.5 O3 N, are the reactant considered to take part in the synthesis of RNA. Therefore, RNA has been considered as bases (adenine, guanine, cytosine and uracile) obtained from the forming-bases aminoacids, its average molecular formula being: C9.5 H11.75 O12.75 N3.75 P3 . In a similar way, DNA has been considered as formed by adenine, guanine, timine and cytosine, yielding the following average molecular formula C9.75 H12.5 O11.75 N3.75 P3 . All these assumptions yield a reaction scheme formed by five lumped reactions (Eqs. (7)–(10))
• Non-forming bases aminoacids (r1 ): 0.933C6 H12 O6 + 1.4NH4 + + 0.216NAD+ + 0.1CoASH + ATP ↓ C5.6 H10.1 O2.3 N1.4 S0.1 + 0.216(NADH, H+ ) + 0.1CoA + 3.29H2 O + (ADP + Pi )
(7)
• Forming bases aminoacids (r2 ): 0.5C6 H12 O6 + NH4 + + 2.25NAD+ → C3 H5.5 O3 N + 2.25(NADH, H+ )
(8)
• RNA bases synthesis (r3 and r4 ): 1.08C6 H12 O6 + C3 H5.5 O3 N + 2.74NH4 + + 3.5ATP + 4.83NAD+ + 0.25GTP + 3C10 H12 O13 N5 P3 ↓ C9.5 H11.75 O12.75 N3.75 P3 +5.73H2 O+3.5(ADP+Pi )+4.83(NADH+H+ )+0.25(GDP+Pi )+3C10 H12 O10 N5 P2
(9)
• DNA synthesis (r5 ): 0.041C6 H12 O6 + 0.258(NADPH, H+ ) + C9.5 H11.75 O12.75 N3.75 P3 + 1.24INT ↓ C9.75 H12.5 O11.75 N3.75 P3 + 0.258NADP+ + 1.24INT − O.
(10)
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
Compound concentration (g/L)
588
Key
Compound
Curve
A RNA DNA IPR X
1 2 3 4 5
1.00
0.75
5
0.50
4 0.25
3
1
2
0.00 0
10
20
30
40
50
60
t (h) Fig. 2. Experimental results (points) and reproduction obtained using the Kinetic Model 5 for the experiment carried out using an initial ammonium concentration of 130 ppm.
Afterwards, a stoichiometric study of the X. campestris growth involving nine species (NH4 + , cofactors, ATP, non forming bases aminoacids, forming bases aminoacids, glucose, RNA, RNAI and DNA) has been performed showing that there are five key compounds: NH4 + , non forming bases aminoacids (intracellular and extracellular proteins), RNA, RNAI and DNA were chosen to be analysed. Therefore, the five stoichiometric relationships considered (Eqs. (7)–(10)) among these nine species are independent. As indicated above, flow cytometry has been the analysis technique employed for intracellular compounds quantification (DNA, RNA and intracellular protein). Afterwards, the extracellular protein concentration has been obtained by means of a nitrogen balance.
In Figs. 2–4 the tendencies of the experimental data points obtained in this work can be observed during time fermentation. As can be seen in all of them, the intracellular protein concentration is always higher than nucleic acids concentration, and DNA concentration is always higher than RNA concentration. Theoretical biomass (considered as a sum of DNA, RNA and intracellular proteins) is close to experimental biomass values. Moreover, no maximum in intracellular compounds evolution is observed, therefore, pseudosteady state assumptions can be considered for both forming bases aminoacids and RNAI species. These assumptions yield two simplified reaction schemes, which are going to be considered in this work (see Table 1):
1.75 Compoound concentration (g/L)
Key 1.50 1.25
Compound
Curve
A RNA DNA IPR X
1 2 3 4 5
5
1.00
4 0.75 0.50
3
1
0.25
2
0.00 0
10
20
30
40
50
60
t (h) Fig. 3. Experimental results (points) and reproduction obtained using the Kinetic Model 5 for the experiment carried out using an initial ammonium concentration of 257 ppm.
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
589
Compound concentration (g/L)
1.75 Key
Compound
Curve
A RNA DNA IPR X
1 2 3 4 5
1.50 1.25 1.00
Key
Curve without Inhibition term with Inhibition term
5
0.75
4 0.50
1
3
0.25
2
0.00 0
10
20
30
40
50
t (h) Fig. 4. Experimental results (points) and reproduction obtained using the Kinetic Model 5 for the experiment carried out using an initial ammonium concentration of 457 ppm: (a) without inhibition term and (b) with inhibition term.
• Reaction Scheme 1 (RS-1): pseudosteady state is assumed for both intermediary compounds. The scheme is formed by three reactions (r1 , r3 and r5 ). • Reaction Scheme 2 (RS-2): pseudosteady state is assumed only for RNAI . The reaction scheme involves four reactions (r1 , r2 , r3 and r5 ). 3.2. Kinetic equations assumption The following step in the kinetic modeling is the proposal of kinetic equations for the different reactions involved in the two reaction schemes considered. Because there is no information in literature about the kinetic expressions for the reactions considered in Xanthomonas campestris growth,
different kinetic equations have been checked using the ‘reaction rates method’ [16], that allows to test different expressions to find the best one in each case. The application of this method yields both the best kinetic equations for each reaction and the initial values of the kinetic parameters to be employed in a multiple response non-linear fitting. The application of this method needs the knowledge of the kinetic model expressed as production rates of the key compounds. The application of the pseudosteady state assumption to the scheme RS-1 yields the following relationships between the reaction rates of the scheme considered: dCRNAI = RRNAI = 0 ⇒ RRNAI = r4 − r5 ⇒ r4 = r5 dt (11)
Table 1 Kinetic Models proposed Reaction scheme RS-1
aaNFB
r1 r3
NH4 +
RNA
r5 DNA RS-2
r1
aaNFB
NH4 +
RNA r3 r2 aaFB
r5 DNA
Kinetic equation
Reference
r 1 , r3 , r5 = k i C X C A r1 , r3 , r5 = ki CX CA /(Ki + CA ) r1 = k i C X C A r3 , r5 = ki CX CA /(Ki + CA ) r1 = ki CX CA /(Ki + CA ) r 3 , r5 = k i C X C A
Model 1 Model 2 Model 3
r 1 , r3 , r5 = k i C X C A r2 = k i C A r1 , r3 , r5 = ki CX CA /(Ki + CA ) r2 = k i CX CA r1 = ki CX CA /(Ki + CA ) r 3 , r5 = k i C X C A r2 = k i C A r1 = k i C X C A r3 , r5 = ki CX CA /(Ki + CA ) r2 = ki CA
Model 4 Model 5 Model 6 Model 7
Model 8
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F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
dCaaNFB = RaaNFB = 0 ⇒ RaaNFB dt = r2 − r3 − r4 ⇒ r2 = r3 + r4 = r3 + r5
(12)
Therefore, the production rates of the key compounds according to this scheme are the following: dCA = −1.4r1 − r2 − 2.74r3 − 2.74r4 dt = −1.4r1 − 3.74r3 − 3.74r5
(13)
dCRNA = r3 dt dCDNA = r5 dt
(14) (15)
The application of Eq. (6) to the N key compounds of both simplified schemes yields a set of independent equations. The reaction rates method has been applied in integral and differential form for the two reaction schemes considered. The expressions obtained for the reaction rates of reactions 1, 3 and 5 are the same in both cases, for scheme RS-1 and for scheme RS-2. These kinetic equations in integral form are the following: t rS1 = (18) r1 dt = (CIPR − CIPR0 ) + CEPR
0
t
rS3 =
0 t
r3 dt = CRNA − CRNA0
(19)
r5 dt = CDNA − CDNA0
(20)
dCIPR dCAPR = r1 − (16) dt dt Assuming pseudosteady state only for RNAI lumping species in the scheme RS-2, the same relationship between the reaction rates expressed by Eq. (11) is obtained. The kinetic model expressed as production rates of the key compounds is now formed by Eqs. (14)–(16) together with the following instead of Eq. (13):
rS5 =
dCA = −1.4r1 − r2 − 2.74r3 − 2.74r4 dt = −1.4r1 − r2 − 2.74r3 − 2.74r5
Three kinds of kinetic equations have been checked for each reaction: (17)
0.016
0
Nevertheless, for scheme RS-2 it is necessary to include also the reaction 2: t rS2 = r2 dt = −(2.74(CRNA − CRNA0 ) 0
+ 2.74(CDNA − CDNA0 ) + 1.4(CIPR − CIPR0 ))
First order power-law equation :
ri = kCA
(21)
(22)
0.040 1
0.012 3
0.030 2
0.008 2
0.020
0.004
0.010
1 0.000 0
10
20
30 t (h)
(a)
40
50
60
0.000
0
10
20
0.016 1
0.020
3
40
Curve Kinetic equation [22] 1 2 [23] 3 [24]
0.012 0.015
2
3
0.008
30
(b)
2
0.010
0.004 0.005
1 0.000 0
(c)
10
20
30 t (h)
40
50
60
0.000
(d)
0
10
20
30
40
50
60
t (h)
Fig. 5. Results obtained when the ‘reaction rates’ method is applied in differential form to the experiment carried out using 257 ppm of initial ammonium concentration employing different kinetic expressions for the four reactions involved in the reaction schemes assumed (RS-1 and RS-2): (a) r1 ; (b) r2 ; (c) r3 and (d) r5 .
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594 0.4
3
591
1
0.20 2 0.3
1
0.15
2
0.2
0.10 Curve Kinetic equation 1 [22] 2 [23] 3 [24]
0.1
0.05 0.00
0.0 0
10
20
30
40
50
60
t (h)
(a)
0
10
20
(b)
30
40
0.35
0.30
1
0.20
2
0.30
2
1
0.25
0.15
0.20
0.10
0.15
0.05
0.10
0.0
60
3
3
0.25
50
t (h)
0.0 0
10
20
(c)
30 t (h)
40
50
0
60
(d)
10
20
30 t (h)
40
50
60
Fig. 6. Results obtained when the ‘reaction rates’ method is applied in integral form to the experiment carried out using 257 ppm of initial ammonium concentration employing different kinetic expressions for the four reactions involved in the reaction schemes assumed (RS-1 and RS-2): (a) r1 ; (b) r2 ; (c) r3 and (d) r5 .
Second order power-law equation : Hyperbolic equation :
ri = kCX
ri = kCX CA
CA CA + K i
(23) (24)
As commented above, the reaction rates method can be applied both in integral and differential form. As an example, Figs. 5 and 6 show the results obtained when the ‘reaction rates’ method is applied to the experimental data using an initial ammonium concentration of 257 ppm. Fig. 5 shows the results obtained when this calculation method is applied in differential form: solid squares are the sum of derivatives of experimental values in each case and the tendencies presented by the kinetic equations proved (Eqs. (22)–(24)), are presented as solid lines. In Fig. 5 can be observed that three of these reaction rates (r1 , r3 and r5 ) show an autocatalytic behavior that can be explained only by means of kinetic equations involving biomass concentration (Eqs. (23) and (24)). On the other hand, the reaction number 2 does not show such tendency; therefore, only Eq. (22) must be more adequate for kinetic modeling of this reaction.
Fig. 6 shows the results obtained when the ‘reaction rates’ method is applied in integral form; in this case the reactions 1, 3 and 5 show a sigmoid tendency (which can be obtained for functions such as those of Eqs. (23) and (24)); nevertheless, reaction 2 presents a tendency to saturation (only achieved from functions such as those of Eq. (22)). The same tendencies have been observed when this method has been applied to all the experiments and the same conclusions than those from Figs. 5 and 6 can be reached. 3.3. Kinetic Models discrimination The combinations between the different kinetic equations previously obtained (for reactions 1–3 and 5) and the different reaction schemes assumed (RS-1 and RS-2) yield eight different models, as can be seen in Table 1. Experimental data have been fitted to all these models using a nonlinear multiple-response algorithm joined to a subroutine for integration, as indicated above.
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Table 2 Kinetic parameter values and statistical parameters values obtained for Kinetic Models 5–8 based on RS-2 Model 5
CN0 = 257 ppm
CN0 = 475 ppm
Model 7
Model 8
Parameter
Value
F
SSR
Parameter
Value
F
SSR
Parameter
Value
F
SSR
Parameter
Value
F
SSR
k1
0.571 ± 0.021
986.2
0.75
k1 K1
0.151 ± 0.002 0.089 ± 0.001
1925
0.52
k1 K1
0.102 ± 0.003 0.182 ± 0.001
1125
0.96
k1 k3
0.801 ± 0.001 0.271 ± 0.002
1399
0.89
k3
0.141 ± 0.015
k3 K3
0.029 ± 0.002 0.039 ± 0.002
k3
0.080 ± 0.002
K3 k5
0.042 ± 0.003 0.054 ± 0.004
k5
0.195 ± 0.008
k5
0.060 ± 0.002
k5
0.119 ± 0.001
K5
0.079 ± 0.002
k2
0.039 ± 0.004
k5 k2
0.080 ± 0.002 0.057 ± 0.020
k2
0.038 ± 0.020
k2
0.035 ± 0.001
k1
0.571 ± 0.030
k1 K1 k3
0.100 ± 0.001 0.131 ± 0.020 0.203 ± 0.002
k1 K1 k3
0.100 ± 0.005 0.042 ± 0.003 0.302 ± 0.002
k1 k3 K3
0.300 ± 0.002 0.018 ± 0.002 0.050 ± 0.002
1989
0.095
k3
0.152 ± 0.030
K3 k5
0.048 ± 0.001 0.039 ± 0.003
k5
0.340 ± 0.002
k5 K5
0.039 ± 0.003 0.110 ± 0.001
k5 k2
0.199 ± 0.020 0.039 ± 0.020
K5 k2
0.112 ± 0.008 0.054 ± 0.002
k2
0.055 ± 0.002
k2
0.076 ± 0.002
k1
0.611 ± 0.030
k1 K1
0.130 ± 0.002 0.110 ± 0.002
k1 K1
0.100 ± 0.002 0.062 ± 0.001
k1 k3
0.531 ± 0.001 0.017 ± 0.002
611.2
0.93
k3
0.182 ± 0.002
k3 K3
0.026 ± 0.001 0.037 ± 0.001
k3
0.199 ± 0.005
K3 k5
0.057 ± 0.002 0.035 ± 0.002
k5
0.202 ± 0.010
k5
0.048 ± 0.002
k5
0.200 ± 0.001
K5
0.100 ± 0.003
k2
0.065 ± 0.003
K5 k2
0.099 ± 0.008 0.071 ± 0.005
k2
0.059 ± 0.002
k2
0.060 ± 0.001
F (95% confidence) = 2.49
2725
295.1
0.06
0.75
F (95% confidence) = 2.66
3825
185.3
0.005
0.95
F (95% confidence) = 2.85
2352
211.8
0.10
1.2
F (95% confidence) = 2.49
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CN0 = 130 ppm
Model 6
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The selection of the best kinetic model must be carried out employing discrimination criteria. The first criteria employed to discriminate the kinetic models have been statistical criteria: Student’s t-test, Fischer F test and minimum square sum of residuals (SSR) have been employed in this work. The kinetic models obtained from scheme RS-1 (Model 1 to Model 4) do not fulfill statistical criteria (in most experiments there is not algorithm convergence). On the other hand, all the models obtained from scheme RS-2 (Models 5–8)—see Table 2—fulfill statistical criteria: Student’s t and Fischer’s F are always higher than those ones tabulated for 95% of confidence level and the values of the SSR obtained are significantly lower in all cases. The second discrimination criteria employed to discern among all the models that fulfill the previous criteria have been physical ones. Firstly, all the parameters obtained must be positive. Secondly, parameter values must show a logical tendency with operational conditions: initial ammonium concentration, in this case; thus similar values of the parameters must be obtained when this condition is changed. In Table 2 it can be observed that the four models corresponding to the RS-2 yield positive parameter values for all their kinetic parameters. On the other hand, for Model 5 the parameter values are very similar for runs carried out with different initial ammonium concentration, while those values obtained applying Model 7 are different from one to another run, thus, Model 7 does not fulfill the criteria about tendency in parameter values. 3.4. Model validation The kinetic model selected in this work has finally been Model 5 and experimental data reproductions are shown in Figs. 2–4. In these figures it can be observed the good agreement between experimental data and model predictions. Nevertheless, the reproduction obtained for the experiment carried out using the highest ammonium initial concentration (450 ppm) could be improved. In Fig. 4 it can be observed that when such ammonium concentration is employed the microorganism does not consume the entire nitrogen source, but X. campestris grows less than in the experiment carried out using 257 ppm of initial ammonium concentration. This can be due to an inhibitory effect of ammonium on X. campestris growth; this effect has been taken into account in the Model 5 into the reaction production rate of intracellular proteins (Eq. (25)) assuming the inhibitory expression proposed by Wayman and Tseng [18], which consider a critical concentration of the inhibitory substrate (CAc ), in this work it was assumed to be 350 ppm. dCIPR dCEPR = r1 − − (kI (CA0 − CAc )) (25) dt dt A new non-linear multiple-response fit has been carried out to obtain the value of the inhibitory parameter considered (kI ). Fig. 4 shows the reproduction of the experimental data obtained by means of the Kinetic Model 5 including the
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inhibitory effect of ammonium showing a better reproduction of both biomass and intracellular proteins experimental data (dotted lines).
4. Conclusions A method has been described to determine a structured kinetic model for X. campestris growth according to the steps quoted above. The use of the ‘method of reaction rates’ allows forming an idea about the kinetic equations to be considered in the simplified reaction scheme previously assumed. The method applied in this work for model discrimination, based on statistical and physical criteria, has yielded a structured model able to describe X. campestris growth with a reduced number of equations and parameters. The final structured kinetic model obtained to describe Xanthomonas campestris growth is formed by the following set of five differential equations, with the parameter values and confidence intervals indicated: (0.584 ± 0.060)CA CX dCA = 18 −1.4 dt 136.9 (0.048 ± 0.020)CA (0.158 ± 0.026)CA CX − − 2.74 103.5 475.25 (0.199 ± 0.020)CA CX (26) − 2.74 463.0 dCRNA = (0.158 ± 0.026)CA CX dt dCDNA = (0.199 ± 0.020)CA CX dt dCIPR dCEPR = (0.584 ± 0.060)CA CX − dt dt ⇒ CA0 ≤ 0.350
(27) (28)
(29a)
dCIPR dCEPR = (0.584 ± 0.060)CA CX − dt dt − (0.080 ± 0.001)(CA0 −0.350) ⇒ CA0 ≥ 0.350 (29b) dCX dCRNA dCDNA dCIPR = + + (30) dt dt dt dt This model is able to fit very well data obtained in different runs carried out with different initial nitrogen concentrations, as shown in Figs. 2–4. Moreover, it is also able to predict different growth rates when that operational condition changes.
Acknowledgments The Ministry of Science & Technology, “Programa de Procesos y Productos Quimicos del Plan Nacional de I+D”,
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has supported this work under contract with reference PPQ2001-1361-C02-01.
References [1] Margaritis A, Zajic JE. Biotechnology review: mixing, mass transfer and scale-up of polysaccharide fermentations. Biotechnol Bioeng 1978;20:939–1001. [2] Kennedy JF, Bradshaw IJ. Production, properties and applications of xanthan. Prog Indus Microbiol 1984;19:319–71. [3] Garc´ıa-Ochoa F, Santos VE, Fritsch AP. Nutritional study of Xanthomonas campestris in xanthan gum production by factorial design of experiments. Enz Microbiol Technol 1992;14:991–6. [4] Garc´ıa-Ochoa F, Santos VE, Alcón A. Xanthan gum production: an unstructured kinetic model. Enz Microbiol Technol 1995;17:206–17. [5] Moraine RA, Rogovin P. Kinetics of polysaccharide B-1459 fermentation. Biotechnol Bioeng 1966;8:511–24. [6] Moraine RA, Rogovin P. Xanthan biopolymer production at increased concentration by pH control. Biotechnol Bioeng 1971;13:381– 91. [7] Weiss RM, Ollis DF. Extracellular microbial polysaccharides. I. Substrate, biomass and product kinetic equations for batch xanthan gum fermentation. Biotechnol Bioeng 1980;22:859–73. [8] Pinches A, Pallent LJ. Rate and yield relationships in the production of xanthan gum by batch fermentations using complex and chemically defined growth media. Biotechnol Bioeng 1986;28:1484–96.
[9] Quinlan AV. Kinetics of secondary metabolite synthesis in batch culture when two different substrates limit cell growth and metabolite production: xanthan synthesis by Xanthomonas campestris. Biochem Eng IV Ann NY Acad Sci 1986;469:259–69. [10] Pons A, Dussap CG, Gros JB. Modeling Xanthomonas campestris batch fermentation in a bubble column. Biotechnol Bioeng 1989;33:394–405. [11] Garc´ıa-Ochoa F, Santos VE, Alcón A. Metabolic structured kinetic model for xanthan production. Enz Microbiol Technol 1998;23:75– 82. [12] Domach MM, Leung SK, Cahm RE, Cocks GC, Shuler ML. Computer model for glucose-limited growth of a single cell of Escherichia coli B/r-A. Biotechnol Bioeng 1984;26:203–16. [13] Shuler ML. Invited review on the use of chemically structured models for bioreactors. Chem Eng Commun 1985;36:161–89. [14] Jeong JW, Snay J, Ataai MM. A mathematical model for examining growth and sporulation processes of Bacillus subtilis. Biotechnol Bioeng 1990;35:160–84. [15] Garc´ıa-Ochoa F, Santos VE, Alcón A. Intracellular compounds quantification by means of flow cytometry in bacteria: application of xanthan production by Xanthomonas campestris. Biotechnol Bioeng 1998;51:87–94. [16] Garc´ıa-Ochoa F, Romero A. Kinetic modeling of reaction networks. Trends Chem Eng 1993;1:286–305. [17] Marquardt AW. An algorithm for least-squares estimation of nonlinear parameters. J Soc Indus Appl Math 1963;11:431–41. [18] Wayman M, Tseng MC. Inhibition-threshold substrate concentrations. Biotechnol Bioeng 1976;18:383–7.
Structured kinetic model for Xanthomonas campestris growth F. Garcia-Ochoa∗ , V.E. Santos, A. Alcon Departamento de Ingenieria Quimica, Facultad CC. Quimicas, Universidad Complutense, Madrid 28040, Spain Received 8 December 2002; accepted 8 January 2004
Abstract A structured kinetic model for Xanthomonas campestris growth is proposed. The formulation of the model is made according to those steps employed for kinetic modeling of reaction networks. First, simplified reaction schemes are proposed, which are formed by four or five reactions using lumping and pseudosteady-state assumption for different compounds. Several expressions for the kinetic equations have been considered, checking their validity with the evolution of experimental data taking into account the different reactions in the scheme of reactions assumed. Afterwards, parameter values are calculated by regression to experimental data, using simple and multiple non-linear techniques. Then, model discrimination is carried out using both statistical and physical restrictions. The model finally proposed has four key compounds: ammonium, RNA, DNA and intracellular proteins. The analysis of the intracellular compounds (RNA, DNA and proteins) has been carried out using an experimental technique based in flow cytometry. Three experiments (under the same operational conditions but with different initial nitrogen concentrations in the medium) have been fitted to the model to obtain the values of the parameters, showing a prediction very close to the experimental data. The model is able to predict different system evolutions for different nitrogen initial concentrations. © 2004 Elsevier Inc. All rights reserved. Keywords: Cell model; Xanthomonas campestris; Kinetic modeling; Intracellular compounds; Flow cytometry
1. Introduction Xanthan gum is the polysaccharide produced by Xanthomonas campestris, an obligatory aerobic Gram-negative bacterium. This gum is employed as stabilizer, emulsifier and suspending agent due to its special rheological properties [1] in many industries such as food (E-415), pharmaceutical, cosmetics, etc. [2]. Xanthan gum production rate is greatly influenced by several variables, such as temperature, oxygen mass transfer rate and the nutrients employed in the medium. Previous works have shown that there are optimum medium composition [3] and conditions [4] to carry out this process: 28 ◦ C, 257 ppm of ammonium and enough available dissolved oxygen in the broth without an excessive hydrodynamic stress for the cells. To describe the influence of these variables and to predict the evolution of the system if any perturbation should occur different kinetic models have been proposed. First of all unstructured kinetic models [4–9] considering growth, consumption of substrates (carbon and nitrogen sources) and xanthan production itself in a set of differential equations ∗
Corresponding author. Tel.: +34-91-3944176; fax: +34-91-3944171. E-mail address: [email protected] (F. Garcia-Ochoa).
0141-0229/$ – see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.enzmictec.2004.01.005
were proposed. These models are not able to involve the influences of the variables commented on; therefore, more complex kinetic models had to be proposed. Temperature and oxygen mass transfer influences have been considered using metabolic kinetic models [10,11]; these models take into account a simplified reaction scheme using lumping of metabolic pathways into the carbon metabolism, yielding global reactions as a sum of the reactions involved in the pathway considered such as ‘glucose total catabolism’, ‘maintenance energy’, ‘oxidative phosphorylation’ and ‘xanthan production’. However, since growth is regarded as one lumped process, the influence of the nitrogen source is not considered in this kind of models, and therefore an optimal value for the nitrogen source concentration can only be found by experimental observations [3]. The proposal of a set of lumped reactions describing nitrogen metabolism is the way to involve the influence of this nutrient in the process description; therefore, a structured kinetic model to describe growth had to be proposed. The proposal of a structured kinetic model involves a wide knowledge of the microbial metabolism, therefore, all such models are proposed for microorganisms such as Escherichia coli, Bacillus subtilis and Sacharomyces cerevisiae [12–14]. These models present a very high number of parameters (between 70 and 200) usually estimated from
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Nomenclature Cj C DNA f(CK ) F FCM INT INT-O
ki kI Ki KMRN N NR r ri rs rsi R RNA RNAI
Rj SKM SSR t ts
concentration of component j (g/l) key compounds concentration vector nitrogen bases forming deoxyribonucleic acid concentration function in the kinetic expression Fischer’s F statistical parameter flow cytometry lumped species involved in transformation of RNA bases into DNA bases oxidized lumped species involved in transformation of RNA bases into DNA bases kinetic constant of reaction i inhibition constant (gIPR/h gA) saturation constant of reaction i (gN/l) Kinetic Modeling of Reaction Networks number of key compounds in the system number of reactions considered in the scheme reaction rates vector reaction rate of the reaction i (g/l h) integral of the reaction rates vector integral of the reaction rate of the reaction i (g/l) production rates vector nitrogen bases forming ribonucleic acid lumped species to be reduced to obtain nitrogen bases forming deoxyribonucleic acid production rate of the compound j (gj/l h) structured kinetic models square sum of residuals time (h) Student’s t statistical parameter
Greek letters νij stoichiometric coefficient of the compound j in the reaction matrix of the stoichiometric coefficients Subscripts 0 initial value A ammonium Ac critical ammonium concentration aaNFB non-forming bases aminoacids DNA nitrogen bases forming DNA EPR extracellular proteins IPR intracellular proteins RNA nitrogen bases forming RNA RNAI intermediary RNA nitrogen bases to form DNA nitrogen bases X biomass
literature data. As commented above, these kinetic models are needed to explain the influence of some variables, but the models to be employed must be simpler than the models previously quoted if the parameters are going to be calculated by fitting to experimental data. Therefore, they have to include an appropriate number of key compounds, no more than four or five, and parameters. The aim of this work is the proposal of a simple biochemical structured kinetic model for Xanthomonas campestris growth involving the influence of initial nitrogen concentration. The model is applied to experimental data using the methodology employed in kinetic modeling of reaction networks. The experimental data for intracellular compounds are obtained using flow cytometry by means of a previous reported experimental procedure [15]. 1.1. Kinetic Model proposal As commented above, there is no precedent in literature for obtaining the parameter values of simplified structured models by statistical fitting of experimental data. Therefore, the development of a methodology to carry out this complex task is needed. This problem is quite similar to the Kinetic Modeling of Reaction Networks (KMRN), in which the kinetic analysis needs much more information than in the case of single reactions, because both stoichiometry and thermodynamic are more complex and the combination of reaction schemes and kinetic equations yields a higher number of possible kinetic models and a model discrimination procedure must be carried out [16]. In the KMRN two main different ways have been followed to study these complex systems: (1) to study the scheme reactions one by one to reduce the problem to a sum of single reactions -but when the system complexity is high this is no longer applicable- and, (2) to propose a reaction scheme, usually very simplified, using lumping of similar compounds, to assume kinetic equations (as simple as possible, potential or hyperbolic equations) and to obtain the parameter values; afterwards, model predictions have to be compared to experimental data and any of the model assumptions changed until a good agreement between experimental data and model predictions is attained. Discrimination can be carried out using statistical and physical criteria, as applied in KMRN [16]. The proposal of structured kinetic models (SKM) can be carried out using the same methodology developed for KMRN. The main steps needed to attain a SKM are the following: simplified metabolic pathway proposal, kinetic equations proposal and model discrimination. The formulation of a simplified metabolic pathway has to be preceded by the determination of the chemical species to be involved in the system. A stoichiometric study of the system has to be carried out to obtain the following information: number of key compounds, number of independent relationships between the chemical species in the system and
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to know how to obtain the changes in the system composition knowing the changes in key compound concentrations. This information is essential for both the determination of the analytical method to know the changes in the system composition and for proposing a simplified reaction scheme, which considers the main reactions in the metabolism of the microorganism studied. Therefore, a previous knowledge of the metabolic pathways involved in the metabolism of the microorganism studied is needed. The simplified metabolic pathway proposal has to be preceded by a rigorous analysis of the experimental data, to observe the tendencies in the evolutions of the different compounds analyzed. The information obtained by means of the stoichiometric study has to be considered to propose a scheme of reactions. The simplification of the metabolic pathway proposed has to be carried out using lumping to reduce the number of compounds to be considered. When lumping is employed it is assumed that similar compounds, which are involved in similar reactions, can be joined together. The application of this simplification has been generally performed using good sense: molecule reactivity, molecule size, etc. [16]. The equations joining the measured magnitudes (production rates or concentrations) with the magnitudes to be determined as operational conditions function (reaction rates) are the following: Rj =
NR
νij × ri
(j = 1, . . . , N)
(1)
i=1
The integration of the Eq. (1) yields the following expression, to be used with integral data (Cj versus t): Cj − Cj0 =
t NR νij ri dt i=1
(j = 1, . . . , N)
(2)
0
The proposal of kinetic equations can be performed by different methods. Differential methods are based on the application of Eq. (1) to the N key compounds of the system and integral methods are based on the application of Eq. (2) to the N key compounds. Usually the form of the functions ri have to be assumed as different functions of the species concentrations of the system, which are different for the several reactions taking place in the reaction scheme assumed. The set of equations obtained for the N key compounds by means of Eqs. (1) and (2) can be expressed by means of a matrix form, as follows: R=×r
(3)
C = × rs
(4)
Garc´ıa-Ochoa and Romero [16] proposed the use of an equation set in terms of reaction rates of the key compounds instead of the equation set usually employed in terms of productions rates (Eqs. (3) and (4)). This set of equations
585
of reaction rates can be expressed in matrix form both in differential (Eq. (5)) and integral form (Eq. (6)): r = −1 × R
(5)
rs = −1 × C
(6)
This method can be employed only when the stoichiometric matrix, , is square, that is to say, when the number of key compounds is equal to the number of reactions in the scheme assumed. This method gives several advantages both from the calculation point of view (the mathematical methods needed are simpler than in the production rates method, and the calculation of each parameter value is isolated) and from the conceptual point of view (the complex reaction network can be treated as a sum of single reactions). The application of Eq. (6) to the N key compounds of the system yields a set of independent equations. Once the kinetic equation forms of each reaction in the model are assumed, a multiple-response fit of the experimental data to the proposed model has to be employed to obtain the values of the parameters as a set. Afterwards parameter values are calculated, the model reproducibility has to be proved; when the fit is not acceptable another model has to be checked. Combinations between different kinetic equations for the same reaction scheme yield the different models. Usually more than one model can describe the system; therefore model discrimination criteria have to be established. The criteria for model discrimination can be grouped together into two main types [16]: statistical and physical criteria. Statistical criteria are based on the comparison of statistical parameters, such as the Student’s t or the Fischer’s F that are tabulated for different confidence levels and they are functions of the number of experimental data employed in the fitting and the number of parameters to be calculated. Another parameter to compare the goodness of the model fitting is the residual sum of squares; which must be as low as possible, but the values obtained under experimental error have to be considered as the same value. Physical criteria are imposed criteria such as the following: the values of all the parameters obtained have to be greater than zero, and all the parameters of the model have to show a logical tendency versus the variables studied, such as medium composition, in this case.
2. Materials and methods 2.1. Microorganism The microorganism employed was the bacterium Xanthomonas campestris NRRL B-1459 supplied by the Northern Regional Research Laboratory of the U.S. Department of Agriculture (Peoria, IL).
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2.2. Experimental set-up and procedure Experiments were carried out in a commercial 1.5-l fermentor (B. Braun Model Biostat M). Inoculum was built up as described in a previous work [4]. The production medium employed was optimized elsewhere [3], using 40 g/l of sucrose as carbon source. The production medium without the carbon source was sterilized into the vessel. The carbon source was separately sterilized and introduced afterwards into the vessel. The pH electrode was calibrated before fermentor sterilization. For the calibration of the dissolved oxygen electrode, after polarizing at least 6 h, using oxygen desorption with nitrogen and oxygen absorption with air. Afterwards, inoculum was introduced into the vessel through a membrane septum. Temperature was controlled by means of a PID controller, using another controller for stirrer speed. At several times during the process, 10 ml of culture broth were withdrawn aseptically from the fermentor for analysis. 2.3. Analytical methods Biomass was determined as OD at 540 nm with a Spectrophotometer Shimadzu (Model UV 1603). The nitrogen source was analyzed employing a selective ammonia electrode (Orion, model 95–12). Flow cytometry has been the analytical technique employed for intracellular compounds quantification. The samples from the bioreactor were centrifuged at 4000 rpm for 30 min and the pellet was resuspended in a buffer Tris 0.1 M, NaCl 0.1 M and pH 7.4. The cells were washed three times and resuspended in the same buffer. Afterwards, cells were centrifuged at 4000 rpm for 30 min and the pellet was resuspended in 50 ml of the fixative solution (glutaraldehyde 3% in Tris buffer). Cells were incubated at 4 ◦ C for 4 h. After washing with Tris buffer, the samples were diluted to obtain a concentration between 200 and 2000 cells/l. Afterwards cells were stained for protein analysis with a fluorescein isothiocyanate 0.1 mg/ml solution and excess dye was removed by washing and rinsing in Tris buffer 0.1 M, NaCl 0.1 M, pH 7.4. For nucleic acids analysis samples were incubated at 37 ◦ C for 40 min with deoxyribonuclease solution (1 mg/ml) for RNA and with ribonuclease solution (1 mg/ml) for DNA. After incubation the enzymes were removed by centrifugation and pellet was resuspended in Iodide propidium 0.1 mg/ml solution as stain for nucleic acids and incubated for 10 min at 4 ◦ C. The fluorescence intensity data mean of FCM can be extrapolated using the corresponding calibration curve obtained in a previous work [15] and the results can be expressed finally as compound concentration in broth (g/l). 2.4. Experiments Three experiments were performed at 28 ◦ C employing different ammonium initial concentrations: 130, 257 and 475 ppm. Biomass, ammonium, DNA, RNA and intracellular protein evolution were measured. Extracellular proteins
concentration can be calculated by means of a nitrogen mass balance in the system. 2.5. Parameter calculation methods Parameter values have always been obtained by means of non-linear regression techniques [17]. This kind of regression technique was applied both in single and multiresponse way. The integral method has been employed for obtaining the parameter values for the kinetic model considered in this work. For this purpose, it was necessary to perform the integration of the equations of the models by means of numerical techniques employing a fourth-order Runge-Kutta algorithm coupled with the Marquardt algorithm [17] for multiresponse fit and Simpson algorithm for simple response regression when the ‘reaction rates method’ was applied. For statistical meaning of the fits the Fisher’s F and Student’ t-test were used. Moreover, the confidence interval of the parameters and the sum of squares of residuals referred to the data number have been calculated. 3. Results and discussion The proposal of a structured kinetic model for Xanthomonas campestris growth has been carried out following the method described above, according to the following steps. 3.1. Simplified metabolic pathway proposal First of all, a simplified scheme of reactions must be assumed, taking into account that hundreds of reactions inside the cell can be lumped together into pseudochemical reactions. The structured kinetic model proposed breaks the cell into three different lumped groups of macromolecules: DNA, RNA and proteins which represent natural grouping of biomass components. It is assumed that the composition of the cell membranes is time invariant and has no influence on metabolism. Other compounds necessary for growth such as carbon source or phosphate are assumed to be available in excess and consequently cellular growth rate is not controlled by them. In this work, it is proposed a reaction scheme of the nitrogen source metabolism into the cell (because nitrogen is assumed to be the limiting substrate in this process). The simplification of the metabolism has been carried out as in the metabolic model previously commented on [11]: the reactions forming a pathway have been assumed to obtain only one global reaction. The compounds involved in the pathway have been lumped into general compounds. Therefore, it is considered that all the nitrogen is obtained from NH4 + and all the hydrocarbons present in the pathway are assumed to be glucose. The ammonium metabolism is branched in two directions: (a) non-forming bases aminoacids synthesis; (b) forming bases aminoacids synthesis, as can be seen in Fig. 1. The total protein (intracellular and extracellular) protein has been assumed to be
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587
INTRACELLULAR PROTEIN (IPR)
r1
Non-forming bases amminoacids (aaNFB)
EXTRACELLULAR PROTEIN (EPR)
NH4+
r3
RNA
r4
RNAI
r2 Forming bases amminoacids (aaFB)
r5
DNA
Fig. 1. Simplified metabolic pathway assumed.
formed by non-forming bases aminoacids and the nucleic acids (DNA and RNA) by forming bases aminoacids synthesis. There are some main assumptions in this procedure: • It is considered that there is a transformation of some molecules of RNA bases to DNA bases. This consideration has been made due to two main assumptions: (a) the deoxyribonucleotides are obtained from ribonucleotides by means of an oxido-reduction reaction; (b) the uracile molecule (RNA specific component) is transformed into a timine one (DNA specific component). These RNA bases have been named as RNAI . • In the reduction of RNAI into DNA (transformation of ribonucleotides into deoxyribonucleotides), there is a lost of an oxygen atom for each RNAI base; therefore, an intermediary enzymatic pathway is involved and grouped in an hypothetical compound called ‘INT’ that yields and oxidised compound, named as ‘INT-O’. The molecular formulas assumed for the intracellular compounds considered (RNA, DNA and proteins) have
been obtained as average of the molecular formulas of the different components—aminoacids or bases—assumed as forming those molecules. Because X. campestris is able to synthesize all the 20 aminoacids, proteins (both, intracellular and extracellular) have been considered formed by non-forming bases aminoacids, with an average molecular formula taking into account 18 non-forming bases aminoacids, yielding: C5.6 H10.1 O2.3 N1.4 S0.1 . On the other hand, forming-bases aminoacids (glycine and aspartic acid), which average molecular formula considered has been C3 H5.5 O3 N, are the reactant considered to take part in the synthesis of RNA. Therefore, RNA has been considered as bases (adenine, guanine, cytosine and uracile) obtained from the forming-bases aminoacids, its average molecular formula being: C9.5 H11.75 O12.75 N3.75 P3 . In a similar way, DNA has been considered as formed by adenine, guanine, timine and cytosine, yielding the following average molecular formula C9.75 H12.5 O11.75 N3.75 P3 . All these assumptions yield a reaction scheme formed by five lumped reactions (Eqs. (7)–(10))
• Non-forming bases aminoacids (r1 ): 0.933C6 H12 O6 + 1.4NH4 + + 0.216NAD+ + 0.1CoASH + ATP ↓ C5.6 H10.1 O2.3 N1.4 S0.1 + 0.216(NADH, H+ ) + 0.1CoA + 3.29H2 O + (ADP + Pi )
(7)
• Forming bases aminoacids (r2 ): 0.5C6 H12 O6 + NH4 + + 2.25NAD+ → C3 H5.5 O3 N + 2.25(NADH, H+ )
(8)
• RNA bases synthesis (r3 and r4 ): 1.08C6 H12 O6 + C3 H5.5 O3 N + 2.74NH4 + + 3.5ATP + 4.83NAD+ + 0.25GTP + 3C10 H12 O13 N5 P3 ↓ C9.5 H11.75 O12.75 N3.75 P3 +5.73H2 O+3.5(ADP+Pi )+4.83(NADH+H+ )+0.25(GDP+Pi )+3C10 H12 O10 N5 P2
(9)
• DNA synthesis (r5 ): 0.041C6 H12 O6 + 0.258(NADPH, H+ ) + C9.5 H11.75 O12.75 N3.75 P3 + 1.24INT ↓ C9.75 H12.5 O11.75 N3.75 P3 + 0.258NADP+ + 1.24INT − O.
(10)
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
Compound concentration (g/L)
588
Key
Compound
Curve
A RNA DNA IPR X
1 2 3 4 5
1.00
0.75
5
0.50
4 0.25
3
1
2
0.00 0
10
20
30
40
50
60
t (h) Fig. 2. Experimental results (points) and reproduction obtained using the Kinetic Model 5 for the experiment carried out using an initial ammonium concentration of 130 ppm.
Afterwards, a stoichiometric study of the X. campestris growth involving nine species (NH4 + , cofactors, ATP, non forming bases aminoacids, forming bases aminoacids, glucose, RNA, RNAI and DNA) has been performed showing that there are five key compounds: NH4 + , non forming bases aminoacids (intracellular and extracellular proteins), RNA, RNAI and DNA were chosen to be analysed. Therefore, the five stoichiometric relationships considered (Eqs. (7)–(10)) among these nine species are independent. As indicated above, flow cytometry has been the analysis technique employed for intracellular compounds quantification (DNA, RNA and intracellular protein). Afterwards, the extracellular protein concentration has been obtained by means of a nitrogen balance.
In Figs. 2–4 the tendencies of the experimental data points obtained in this work can be observed during time fermentation. As can be seen in all of them, the intracellular protein concentration is always higher than nucleic acids concentration, and DNA concentration is always higher than RNA concentration. Theoretical biomass (considered as a sum of DNA, RNA and intracellular proteins) is close to experimental biomass values. Moreover, no maximum in intracellular compounds evolution is observed, therefore, pseudosteady state assumptions can be considered for both forming bases aminoacids and RNAI species. These assumptions yield two simplified reaction schemes, which are going to be considered in this work (see Table 1):
1.75 Compoound concentration (g/L)
Key 1.50 1.25
Compound
Curve
A RNA DNA IPR X
1 2 3 4 5
5
1.00
4 0.75 0.50
3
1
0.25
2
0.00 0
10
20
30
40
50
60
t (h) Fig. 3. Experimental results (points) and reproduction obtained using the Kinetic Model 5 for the experiment carried out using an initial ammonium concentration of 257 ppm.
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
589
Compound concentration (g/L)
1.75 Key
Compound
Curve
A RNA DNA IPR X
1 2 3 4 5
1.50 1.25 1.00
Key
Curve without Inhibition term with Inhibition term
5
0.75
4 0.50
1
3
0.25
2
0.00 0
10
20
30
40
50
t (h) Fig. 4. Experimental results (points) and reproduction obtained using the Kinetic Model 5 for the experiment carried out using an initial ammonium concentration of 457 ppm: (a) without inhibition term and (b) with inhibition term.
• Reaction Scheme 1 (RS-1): pseudosteady state is assumed for both intermediary compounds. The scheme is formed by three reactions (r1 , r3 and r5 ). • Reaction Scheme 2 (RS-2): pseudosteady state is assumed only for RNAI . The reaction scheme involves four reactions (r1 , r2 , r3 and r5 ). 3.2. Kinetic equations assumption The following step in the kinetic modeling is the proposal of kinetic equations for the different reactions involved in the two reaction schemes considered. Because there is no information in literature about the kinetic expressions for the reactions considered in Xanthomonas campestris growth,
different kinetic equations have been checked using the ‘reaction rates method’ [16], that allows to test different expressions to find the best one in each case. The application of this method yields both the best kinetic equations for each reaction and the initial values of the kinetic parameters to be employed in a multiple response non-linear fitting. The application of this method needs the knowledge of the kinetic model expressed as production rates of the key compounds. The application of the pseudosteady state assumption to the scheme RS-1 yields the following relationships between the reaction rates of the scheme considered: dCRNAI = RRNAI = 0 ⇒ RRNAI = r4 − r5 ⇒ r4 = r5 dt (11)
Table 1 Kinetic Models proposed Reaction scheme RS-1
aaNFB
r1 r3
NH4 +
RNA
r5 DNA RS-2
r1
aaNFB
NH4 +
RNA r3 r2 aaFB
r5 DNA
Kinetic equation
Reference
r 1 , r3 , r5 = k i C X C A r1 , r3 , r5 = ki CX CA /(Ki + CA ) r1 = k i C X C A r3 , r5 = ki CX CA /(Ki + CA ) r1 = ki CX CA /(Ki + CA ) r 3 , r5 = k i C X C A
Model 1 Model 2 Model 3
r 1 , r3 , r5 = k i C X C A r2 = k i C A r1 , r3 , r5 = ki CX CA /(Ki + CA ) r2 = k i CX CA r1 = ki CX CA /(Ki + CA ) r 3 , r5 = k i C X C A r2 = k i C A r1 = k i C X C A r3 , r5 = ki CX CA /(Ki + CA ) r2 = ki CA
Model 4 Model 5 Model 6 Model 7
Model 8
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F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
dCaaNFB = RaaNFB = 0 ⇒ RaaNFB dt = r2 − r3 − r4 ⇒ r2 = r3 + r4 = r3 + r5
(12)
Therefore, the production rates of the key compounds according to this scheme are the following: dCA = −1.4r1 − r2 − 2.74r3 − 2.74r4 dt = −1.4r1 − 3.74r3 − 3.74r5
(13)
dCRNA = r3 dt dCDNA = r5 dt
(14) (15)
The application of Eq. (6) to the N key compounds of both simplified schemes yields a set of independent equations. The reaction rates method has been applied in integral and differential form for the two reaction schemes considered. The expressions obtained for the reaction rates of reactions 1, 3 and 5 are the same in both cases, for scheme RS-1 and for scheme RS-2. These kinetic equations in integral form are the following: t rS1 = (18) r1 dt = (CIPR − CIPR0 ) + CEPR
0
t
rS3 =
0 t
r3 dt = CRNA − CRNA0
(19)
r5 dt = CDNA − CDNA0
(20)
dCIPR dCAPR = r1 − (16) dt dt Assuming pseudosteady state only for RNAI lumping species in the scheme RS-2, the same relationship between the reaction rates expressed by Eq. (11) is obtained. The kinetic model expressed as production rates of the key compounds is now formed by Eqs. (14)–(16) together with the following instead of Eq. (13):
rS5 =
dCA = −1.4r1 − r2 − 2.74r3 − 2.74r4 dt = −1.4r1 − r2 − 2.74r3 − 2.74r5
Three kinds of kinetic equations have been checked for each reaction: (17)
0.016
0
Nevertheless, for scheme RS-2 it is necessary to include also the reaction 2: t rS2 = r2 dt = −(2.74(CRNA − CRNA0 ) 0
+ 2.74(CDNA − CDNA0 ) + 1.4(CIPR − CIPR0 ))
First order power-law equation :
ri = kCA
(21)
(22)
0.040 1
0.012 3
0.030 2
0.008 2
0.020
0.004
0.010
1 0.000 0
10
20
30 t (h)
(a)
40
50
60
0.000
0
10
20
0.016 1
0.020
3
40
Curve Kinetic equation [22] 1 2 [23] 3 [24]
0.012 0.015
2
3
0.008
30
(b)
2
0.010
0.004 0.005
1 0.000 0
(c)
10
20
30 t (h)
40
50
60
0.000
(d)
0
10
20
30
40
50
60
t (h)
Fig. 5. Results obtained when the ‘reaction rates’ method is applied in differential form to the experiment carried out using 257 ppm of initial ammonium concentration employing different kinetic expressions for the four reactions involved in the reaction schemes assumed (RS-1 and RS-2): (a) r1 ; (b) r2 ; (c) r3 and (d) r5 .
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594 0.4
3
591
1
0.20 2 0.3
1
0.15
2
0.2
0.10 Curve Kinetic equation 1 [22] 2 [23] 3 [24]
0.1
0.05 0.00
0.0 0
10
20
30
40
50
60
t (h)
(a)
0
10
20
(b)
30
40
0.35
0.30
1
0.20
2
0.30
2
1
0.25
0.15
0.20
0.10
0.15
0.05
0.10
0.0
60
3
3
0.25
50
t (h)
0.0 0
10
20
(c)
30 t (h)
40
50
0
60
(d)
10
20
30 t (h)
40
50
60
Fig. 6. Results obtained when the ‘reaction rates’ method is applied in integral form to the experiment carried out using 257 ppm of initial ammonium concentration employing different kinetic expressions for the four reactions involved in the reaction schemes assumed (RS-1 and RS-2): (a) r1 ; (b) r2 ; (c) r3 and (d) r5 .
Second order power-law equation : Hyperbolic equation :
ri = kCX
ri = kCX CA
CA CA + K i
(23) (24)
As commented above, the reaction rates method can be applied both in integral and differential form. As an example, Figs. 5 and 6 show the results obtained when the ‘reaction rates’ method is applied to the experimental data using an initial ammonium concentration of 257 ppm. Fig. 5 shows the results obtained when this calculation method is applied in differential form: solid squares are the sum of derivatives of experimental values in each case and the tendencies presented by the kinetic equations proved (Eqs. (22)–(24)), are presented as solid lines. In Fig. 5 can be observed that three of these reaction rates (r1 , r3 and r5 ) show an autocatalytic behavior that can be explained only by means of kinetic equations involving biomass concentration (Eqs. (23) and (24)). On the other hand, the reaction number 2 does not show such tendency; therefore, only Eq. (22) must be more adequate for kinetic modeling of this reaction.
Fig. 6 shows the results obtained when the ‘reaction rates’ method is applied in integral form; in this case the reactions 1, 3 and 5 show a sigmoid tendency (which can be obtained for functions such as those of Eqs. (23) and (24)); nevertheless, reaction 2 presents a tendency to saturation (only achieved from functions such as those of Eq. (22)). The same tendencies have been observed when this method has been applied to all the experiments and the same conclusions than those from Figs. 5 and 6 can be reached. 3.3. Kinetic Models discrimination The combinations between the different kinetic equations previously obtained (for reactions 1–3 and 5) and the different reaction schemes assumed (RS-1 and RS-2) yield eight different models, as can be seen in Table 1. Experimental data have been fitted to all these models using a nonlinear multiple-response algorithm joined to a subroutine for integration, as indicated above.
592
Table 2 Kinetic parameter values and statistical parameters values obtained for Kinetic Models 5–8 based on RS-2 Model 5
CN0 = 257 ppm
CN0 = 475 ppm
Model 7
Model 8
Parameter
Value
F
SSR
Parameter
Value
F
SSR
Parameter
Value
F
SSR
Parameter
Value
F
SSR
k1
0.571 ± 0.021
986.2
0.75
k1 K1
0.151 ± 0.002 0.089 ± 0.001
1925
0.52
k1 K1
0.102 ± 0.003 0.182 ± 0.001
1125
0.96
k1 k3
0.801 ± 0.001 0.271 ± 0.002
1399
0.89
k3
0.141 ± 0.015
k3 K3
0.029 ± 0.002 0.039 ± 0.002
k3
0.080 ± 0.002
K3 k5
0.042 ± 0.003 0.054 ± 0.004
k5
0.195 ± 0.008
k5
0.060 ± 0.002
k5
0.119 ± 0.001
K5
0.079 ± 0.002
k2
0.039 ± 0.004
k5 k2
0.080 ± 0.002 0.057 ± 0.020
k2
0.038 ± 0.020
k2
0.035 ± 0.001
k1
0.571 ± 0.030
k1 K1 k3
0.100 ± 0.001 0.131 ± 0.020 0.203 ± 0.002
k1 K1 k3
0.100 ± 0.005 0.042 ± 0.003 0.302 ± 0.002
k1 k3 K3
0.300 ± 0.002 0.018 ± 0.002 0.050 ± 0.002
1989
0.095
k3
0.152 ± 0.030
K3 k5
0.048 ± 0.001 0.039 ± 0.003
k5
0.340 ± 0.002
k5 K5
0.039 ± 0.003 0.110 ± 0.001
k5 k2
0.199 ± 0.020 0.039 ± 0.020
K5 k2
0.112 ± 0.008 0.054 ± 0.002
k2
0.055 ± 0.002
k2
0.076 ± 0.002
k1
0.611 ± 0.030
k1 K1
0.130 ± 0.002 0.110 ± 0.002
k1 K1
0.100 ± 0.002 0.062 ± 0.001
k1 k3
0.531 ± 0.001 0.017 ± 0.002
611.2
0.93
k3
0.182 ± 0.002
k3 K3
0.026 ± 0.001 0.037 ± 0.001
k3
0.199 ± 0.005
K3 k5
0.057 ± 0.002 0.035 ± 0.002
k5
0.202 ± 0.010
k5
0.048 ± 0.002
k5
0.200 ± 0.001
K5
0.100 ± 0.003
k2
0.065 ± 0.003
K5 k2
0.099 ± 0.008 0.071 ± 0.005
k2
0.059 ± 0.002
k2
0.060 ± 0.001
F (95% confidence) = 2.49
2725
295.1
0.06
0.75
F (95% confidence) = 2.66
3825
185.3
0.005
0.95
F (95% confidence) = 2.85
2352
211.8
0.10
1.2
F (95% confidence) = 2.49
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
CN0 = 130 ppm
Model 6
F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
The selection of the best kinetic model must be carried out employing discrimination criteria. The first criteria employed to discriminate the kinetic models have been statistical criteria: Student’s t-test, Fischer F test and minimum square sum of residuals (SSR) have been employed in this work. The kinetic models obtained from scheme RS-1 (Model 1 to Model 4) do not fulfill statistical criteria (in most experiments there is not algorithm convergence). On the other hand, all the models obtained from scheme RS-2 (Models 5–8)—see Table 2—fulfill statistical criteria: Student’s t and Fischer’s F are always higher than those ones tabulated for 95% of confidence level and the values of the SSR obtained are significantly lower in all cases. The second discrimination criteria employed to discern among all the models that fulfill the previous criteria have been physical ones. Firstly, all the parameters obtained must be positive. Secondly, parameter values must show a logical tendency with operational conditions: initial ammonium concentration, in this case; thus similar values of the parameters must be obtained when this condition is changed. In Table 2 it can be observed that the four models corresponding to the RS-2 yield positive parameter values for all their kinetic parameters. On the other hand, for Model 5 the parameter values are very similar for runs carried out with different initial ammonium concentration, while those values obtained applying Model 7 are different from one to another run, thus, Model 7 does not fulfill the criteria about tendency in parameter values. 3.4. Model validation The kinetic model selected in this work has finally been Model 5 and experimental data reproductions are shown in Figs. 2–4. In these figures it can be observed the good agreement between experimental data and model predictions. Nevertheless, the reproduction obtained for the experiment carried out using the highest ammonium initial concentration (450 ppm) could be improved. In Fig. 4 it can be observed that when such ammonium concentration is employed the microorganism does not consume the entire nitrogen source, but X. campestris grows less than in the experiment carried out using 257 ppm of initial ammonium concentration. This can be due to an inhibitory effect of ammonium on X. campestris growth; this effect has been taken into account in the Model 5 into the reaction production rate of intracellular proteins (Eq. (25)) assuming the inhibitory expression proposed by Wayman and Tseng [18], which consider a critical concentration of the inhibitory substrate (CAc ), in this work it was assumed to be 350 ppm. dCIPR dCEPR = r1 − − (kI (CA0 − CAc )) (25) dt dt A new non-linear multiple-response fit has been carried out to obtain the value of the inhibitory parameter considered (kI ). Fig. 4 shows the reproduction of the experimental data obtained by means of the Kinetic Model 5 including the
593
inhibitory effect of ammonium showing a better reproduction of both biomass and intracellular proteins experimental data (dotted lines).
4. Conclusions A method has been described to determine a structured kinetic model for X. campestris growth according to the steps quoted above. The use of the ‘method of reaction rates’ allows forming an idea about the kinetic equations to be considered in the simplified reaction scheme previously assumed. The method applied in this work for model discrimination, based on statistical and physical criteria, has yielded a structured model able to describe X. campestris growth with a reduced number of equations and parameters. The final structured kinetic model obtained to describe Xanthomonas campestris growth is formed by the following set of five differential equations, with the parameter values and confidence intervals indicated: (0.584 ± 0.060)CA CX dCA = 18 −1.4 dt 136.9 (0.048 ± 0.020)CA (0.158 ± 0.026)CA CX − − 2.74 103.5 475.25 (0.199 ± 0.020)CA CX (26) − 2.74 463.0 dCRNA = (0.158 ± 0.026)CA CX dt dCDNA = (0.199 ± 0.020)CA CX dt dCIPR dCEPR = (0.584 ± 0.060)CA CX − dt dt ⇒ CA0 ≤ 0.350
(27) (28)
(29a)
dCIPR dCEPR = (0.584 ± 0.060)CA CX − dt dt − (0.080 ± 0.001)(CA0 −0.350) ⇒ CA0 ≥ 0.350 (29b) dCX dCRNA dCDNA dCIPR = + + (30) dt dt dt dt This model is able to fit very well data obtained in different runs carried out with different initial nitrogen concentrations, as shown in Figs. 2–4. Moreover, it is also able to predict different growth rates when that operational condition changes.
Acknowledgments The Ministry of Science & Technology, “Programa de Procesos y Productos Quimicos del Plan Nacional de I+D”,
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F. Garcia-Ochoa et al. / Enzyme and Microbial Technology 34 (2004) 583–594
has supported this work under contract with reference PPQ2001-1361-C02-01.
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