Indian Journal of Chemical Technology Vol. 12, May 2005, pp. 322-326
Batch kinetics and modeling of poly-β-hydroxy butyrate synthesis from Azotobacter vinelandii using different carbon sources R Dhanasekar* & T Viruthagiri Biochemical Engineering Laboratory, Department of Chemical Engineering, Annamalai University Annamalai Nagar 608 002, India Received 5 April 2004; revised received 14 January 2005; accepted 11 February 2005 A mathematical model is presented for the batch production of poly-β-hydroxy butyrate, a well known biopolymer, from Azotobacter vinelandii utilizing different carbon sources, sucrose and cheese whey. The effect of substrate concentrations and inoculum concentrations on cell mass and P(3HB) production are studied in a batch experiment. Substrate utilization kinetics is also studied. Set of differential equations for cell mass rate, product formation rate, and substrate utilization rate as functions of initial cell concentration, cell concentration at any time and stationary cell concentrations, are used to predict the system behaviour. The kinetic pattern obtained from the batch experiments are used to simulate the models. Keywords: P(3HB), Azotobacter vinelandii, sucrose, cheese whey, kinetic modeling IPC Code: C08G61/00
Poly-β-hydroxy butyrate P(3HB) is an intracellular carbon and energy storage material synthesized by a great variety of bacteria. P(3HB) has various physical/chemical characteristics such as biodegradability, piezoelectric effect and optical activity1. It has been drawing considerable industrial interest for a wide range of applications. It is found in procaryotic and eucaryotic organisms, which seems to function as a component of an ion channel through cell membranes2. Recently, it has been detected in relatively large amounts in human blood plasma. Therefore, it is highly plausible that implanting P(3HB) in mammalian tissues would not be toxic. Azotobacter vinelandii, a growth associated P(3HB) producer, accumulates P(3HB) up to 75% of dry cell weight without limitation of any nutrients3. Moreover, A.vinelandii can utilize sucrose directly as its carbon source. As sucrose is usually less expensive than glucose4, it provides an opportunity to reduce the costs of P(3HB) production greatly. Furthermore, it can also utilize cheap substrates such as cheese whey and cane molasses, which may reduce the cost of production considerably. The application of mathematical models for the design, operation, and optimization of chemical ___________ *For correspondence (E-mail:
[email protected])
reactors and integral plants is widespread. In modeling biological systems, however, the problems to be overcome are concerned with the complexity. Unstructured models have been used which describe only the quantity of the biological phase. In situations where the cell population composition changes significantly and where these composition changes influence kinetics, structured models are preferred. Only few authors have described metabolic modeling for poly-β-hydroxy butyrate production5,6. Recently, unstructured models were used for P(3HB) production from sugars by a mixed culture of Lactobacillus delbrueckii and A. eutrophus7. In the present study, the Logistic model and Leudeking Piret model8 have been used to describe the batch growth kinetics of P(3HB). The simulated results from the models are compared with the experimental data and are discussed. Experimental Procedure Microorganism and growth medium
A. vinelandii (MTCC 124*) was grown in a sucrose medium composed of 0.81 mM MgSO4; 0.58 mM CaSO4; 50 μM ferric citrate and 1 μM Na2MoO4 in 0.5 mM potassium phosphate buffer (pH, 7.2), containing 2% (w/v) sucrose, 15 mM ammonium acetate, and 0.1% (w/v) peptone, cheese whey media
DHANASEKAR & VIRUTHAGIRI: BATCH KINETICS & MODELLING OF POLY-β-HYDROXY BUTYRATE
contained 30% (v/v) whey in Burk's12 buffer (pH, 7.2). The culture medium was inoculated with a 4% (v/v) inoculum, pregrown for 24 h in sucrose medium or cheese whey medium13 as appropriate, and incubated at 28 to 30oC with vigorous aeration (50 mL of culture per 500 mL flask, shaken at 225 rpm in a shaker) for 48 h. Analytical methods
The optical density of all cultures were measured with a spectronic-20D spectrophotometer at 500 nm, with blanks of the appropriate growth media. Curves relating OD to dry weight were constructed by harvesting cultures at room temperature, washing with distilled water, and resuspending the cells in distilled water. Portions were dried at 100°C and weighed, while further samples were suspended in medium to give a series of optical densities up to 1.0. P(3HB) was assayed by the method of Law and Slepecky14. Liquid culture (1 mL) containing up to 2 mg cellular material, was placed in a centrifuge tube, and 9 mL of alkaline hypochlorite solution was added. The mixture was incubated for about 24 h, the resulting suspension was centrifuged and the supernatant liquid was decanted. The residue was washed twice with 10 mL portions of distilled water, acetone, and diethyl ether, before it was dissolved in concentrated sulphuric acid. OD was read at 235 nm against a similarly treated medium blank. Purified polymer samples were used as standards. Results and Discussion
μ S dX = m X Ks + S dt
μ=
μm S Ks + S
where, μm is the maximum growth rate achievable when S >> Ks, while the concentrations of all other essential nutrients are unchanged. Ks is that value of the limiting nutrient concentration at which the specific growth rate is half its maximum value. It is the division between the lower concentration range, where μ is linearly dependent on S, and the higher range, where μ becomes independent of S. The above equation can be written as
… (1)
integrating the equation with the limits X = Xo when t = 0; X = Xt when t = t ⎛ μm S X t = X 0 exp ⎜⎜ ⎝ Ks + S
⎞ t ⎟⎟ ⎠
… (2)
Under optimal growth conditions and when the inhibitory effects of substrate and product play no role, the rate of cell growth follows the well known exponential relationship, dX = μ0 X dt
… (3)
where μo is a constant defined as the initial specific growth rate. Eq. (3) implies that X increases with time regardless of substrate availability. In reality, the growth of cell is governed by a hyperbolic relationship and there is a limit to the maximum attainable cell mass concentration. In order to describe such growth kinetics, the Logistic equation is introduced, dX/dt =μoX(1-X/Xm)
… (4)
integrating Eq. (4), with the initial condition, X = X0 at t=0
Modelling
Monod equation relates the specific growth rate μ and growth limiting substrate concentration and is given by,
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Xt = X 0 e
μ0 t
[1–X0/Xm (1– e
μo t
)]
… (5)
where Xm is the stationary population size. The integrated form of Eq. (5) is called a Logistic equation which gives a sigmoidal variation of X that may empirically represent both an exponential and a stationary phase; the two phases over which most P(3HB) is produced. Product formation is described by Leudeking-Piret kinetics. The product formation rate has been shown to depend upon both the instantaneous biomass concentration X and growth rate dX/dt in a linear fashion, dP/dt = αdX/dt + βX
… (6)
where α and β are empirical constants which may vary with fermentation conditions.
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INDIAN J. CHEM. TECHNOL., MAY 2005
The product formation with time is estimated by solving Eq. (6) with dX/dt and Xt being defined in Eqs (4) and (5), Pt = P0+α [ X 0 e μ 0 t / (1–X0/Xm (1 - e μ 0 t)) – X0] + β (Xm/μ0 ) ln [1- X0 /Xm (1–e μ 0 t)]
… (7)
Eq. (7) was derived from a classical study of Leudeking and Piret on the lactic acid fermentation by Lactobacillus delbrueckii8. This two-parameter kinetic expression had proven extremely useful and versatile in fitting product formation data from many different fermentations, including biopolymers9,10. The substrate utilization kinetics are given by the following equation, which considers substrate conversion to cell mass and product and substrate consumption for maintenance, dS/dt = -1/Yx/s(dX/dt) – 1/Yp/s (dP/dt) + keX
… (8)
in Eq. (8) the last term refers to the substrate used to support cell maintenance, even in the absence of growth. From Eqs (6) and (8), it follows, -dS/dt = (1/Yx/s + α/Yp/s) dX/dt + (β/Yp/s + ke)X … (9) or -dS/dt = γ dX/dt + η X
… (10)
where
γ = 1/Yx/s + α/Yp/s η = β/Yp/s + ke
… (11) … (12)
similarly, the substrate concentration with time is estimated by solving Eq. (8) with dX/dt and Xt being defined in Eqs (4) and (5), St = S0 - γ {X0e μ 0 t /[1 – X0 / Xm (1 – e μ 0 t )– X0} –η (Xm/μ0) ln [1– X0 / Xm(1 - e μ 0 t)] … (13) where the last term on the right hand side signifies that St may continue to diminish due to product formation and for maintenance even after X reaches Xm (stationary phase). Modified form of Logistic growth equation11 is also tested for batch growth kinetics,
dX/dt = μ0 [1 – (X/Xm)r] X
… (14)
Eq. (14) is integrated by using partial fraction method with the initial condition, X = X0 (t = 0) gives, X r 0e μ 0 rt
Xt = 1−
X r0 X rm
(1 − e
μ 0 rt
… (15) )
The cell mass concentration with respect to time depends on the initial and final cell mass concentrations, which varies with microorganisms used and fermentation conditions. The Logistic model, modified Logistic model, and Monod models were used for representing the experimental data obtained in this study. In this effort the model parameters (Table 1) were first evaluated by solving Eqs (5), (7) and (13) using C language computer programming. These values were then used to simulate the profiles of biomass, product, and substrate concentration during the fermentation. Fig. 1 shows that there is an excellent agreement between the experimental data and the simulation results, and the logistic model appeared to provide adequate representation of growth and fermentation kinetics of Azotobacter vinelandii. The predicted values from the logistic model were then compared with those of modified logistic model and Monod model. The models which describe the biomass growth rate were used together with Eqs (7) and (13) to predict the dynamics of fermentation. In general, the Monod equation, which involves a substrate concentration term in the biomass growth rate equation, was inadequate in representing the experimental data. As shown in Fig. 1, the model predicted lower biomass accumulation rates during the exponential growth phase. On the other hand, the simulation results for sucrose consumption rate was lower than the actual data. It also predicted that the sucrose utilization is not significant during the exponential growth phase, hence lower biomass and P(3HB) accumulation. The simulation results using the logistic model compared favorably as shown in Fig. 1 with the experimental data. However, the logistic model was less accurate in representing the actual concentration of biomass, product and substrate. In comparison, the modified logistic model fitted the experimental data
DHANASEKAR & VIRUTHAGIRI: BATCH KINETICS & MODELLING OF POLY-β-HYDROXY BUTYRATE
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Table 1⎯Values of the kinetic parameters Constants
μ0 (h-1) μm (h-1) Ks (g/L) r X0 (g/L) Xm (g/L) S0
α β γ η σ2
11
A.eutrophus Logistic Modified model logistic model 0.30 4.17 0.15 -
0.185 5.81 3.93 0.15 -
Monod model 0.067 0.12 0.09 0.48 0.019 5.90 0.0435 85.23
A.vinelandii (sucrose) Logistic Modified model Logistic model 0.18 0.09 2.62 25 g/L 0.39 0.012 8.7 0.015 0.2846
0.18 0.88 0.09 2.62 0.415 0.0155 8.15 0.008 0.0035
Monod model 0.062 0.17 0.06 0.7 0.009 0.4914
A.vinelandii (whey) Logistic Modified model logistic model 0.144 0.06 1.38 30% (v/v) 0.11 0.025 3.24 ×10-5
0.144 0.95 0.06 0.067 0.028 2.05 × 10-4
Fig. 1⎯Comparison of the experimental data (dots) for P(3HB) production using sucrose as substrate and simulations from (⎯⎯⎯) Logistic, (- - - -) Monod, (--------) Modified Logistic models. X0 = 0.09 g/L; S0 = 24.0g/L
Fig. 2⎯Comparison of the experimental data (dots) for P(3HB) production using sucrose as substrate and simulations from (⎯⎯⎯) Logistic, (- - - -) Monod, (--------) Modified Logistic models. X0 = 0.31 g/L; S0 = 24.0 g/L
better because the additional parameter r present in Eq. (15). The modified logistic model predicts the overall fermentation more accurately. As shown in Fig. 2, the simulated results obtained from the Monod, logistic and modified logistic models are also compared with the experimental data obtained by changing the initial cell concentration. The models were also applied for representing the kinetic data of A. vinelandii utilizing cheese whey as substrate. Fig. 3 presents the comparison of the simulation results derived from Monod, logistic and modified logistic models, and the data obtained for A. vinelandii utilizing cheese whey. The logistic model was as good as the modified logistic model in
describing the product concentration data. However, the modified logistic model represented the biomass concentration data better than the logistic model. In this case, also, the Monod model was inadequate in representing the experimental data (Fig. 3). Fig. 4 shows the comparison of the simulation results derived from Monod, logistic, and modified logistic models, and the experimental data obtained for A. vinelandii utilizing cheese whey with higher initial cell concentration. It is obvious from Eq. (4), that, the growth rate is considered to be independent of the concentration of the growth limiting substrate. Consequently, the logistic and modified logistic models are not expected
INDIAN J. CHEM. TECHNOL., MAY 2005
326
Fig. 3⎯Comparison of the experimental data (dots) for P(3HB) production using cheese whey as substrate and simulations from (⎯⎯⎯) Logistic, (- - - -) Monod, (---------) Modified Logistic models. X0 = 0.06 g/L ; S0 = 30% (v/v)
to be successfully applied in situations where the rate of substrate addition limits the rate of growth, the model must be incorporated with the substrate concentration like Monod equation for describing the microbial growth rate. The logistic and modified logistic models make possible a generalized batch kinetics of A. vinelandii growth during the biosynthesis of poly-β-hydroxyl butyrate. The simulation results are useful to predict the dynamics of A. vinelandii growth and P(3HB) accumulations, and initial cell concentrations. Only four response curves are shown in Figs (1-4) but the models have been checked using 19 different experiments which were carried out for both sucrose and cheese whey, by varying the substrate concentration, and the initial cell concentration. All of the experimental results were found to be in good agreement with the theoretical predictions. It shows that the logistic model for growth of biomass and Leudeking-Piret model for intracelluler polymer accumulation is close to reality.
Fig. 4⎯Comparison of the experimental data (dots) for P(3HB) production using cheese whey as substrate and simulations from (⎯⎯⎯) Logistic, (- - - -) Monod, (--------) Modified Logistic models. X0 = 0.3 g/L; S0 = 30% (v/v) X = = Xo Xm = = Yx/s Yp/s = Greek symbols = α = β = γ = η = μo = μm
Ks ke Po, Pt
= = =
r So, St
= =
Monod constant (g/L) maintenance co-efficient (gsub / gcell L) product concentration at initial and at anytime 't' (g/L) exponent in equations 15, 17 & 19 substrate concentration at initial and at anytime 't' (g/L)
constant in equation (6) constant in equation (6) constant defined in equation (11) constant defined in equation (12) initial specific growth rate (1/h) maximum specific grow rate (1/h)
References 1 2 3 4 5 6 7 8
Nomenclature
biomass concentration (g/L) initial biomass concentration (g/L) maximum biomass concentration (g/L) biomass yield based on substrate (g/g) product yield based on substrate (g/g)
9 10 11 12 13 14
Holmes P A, Phys Technol, 16 (1985) 32. Reusch R N & Sadoll H L, J Bacteriol, 156 (1988) 778. Page W J & Knosp O, Appl Environ Microbial, 55 (1989) 1334. Yamane T, Biotechnol Bioeng, 41 (1993) 165. Van Aalst-van Leeuwen M A, Pot M A, Van Loosdrecht M C M & Heijnen J J, Biotechnol Bioeng, 55 (5) (1997) 773. Leaf T A & Srienc F, Biotechnol Bioeng, 57 (5) (1998) 557. Katoh T, Yuguchi D, Yoshii H, Shi H & Shimizu K, J Biotechnol, 67 (1999) 113. Leudeking R & Piret E L, J Biochem Microbiol Technol Eng, 1 (1959) 393. Weiss R M & Ollis D F, Biotechnol Bioeng, 22 (1980) 859. Klimek J & Ollis D F, Biotechnol Bioeng, 22 (1980) 2321. Mulchandani A, Luong J H T & Leduy A, Biotechnol Bioeng, 32 (1988) 639. Page W J & Sadofit H L, J Bacteriol, 125 (1976) 1080. Dhanasekar R, Viruthagiri T & Sabarathinam P L, Indian J Chem Technol, 8 (2001) 68. Law J H & Slepecky R A, J Bacteriol, 82 (1961) 33.
