Bioresource Technology 102 (2011) 9659–9667

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Bioresource Technology

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Kinetic modeling of sporulation and product formation in stationary phase byBacillus coagulans RK–02 vis-à-vis other Bacilli

Subhasish Das, Ramkrishna Sen ⇑Department of Biotechnology, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India

a r t i c l e i n f o

Article history:Received 28 April 2011Received in revised form 17 July 2011Accepted 19 July 2011Available online 24 July 2011

Keywords:Kinetic modelingBacilliSporulationStationary phase product formationModel validation

0960-8524/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.biortech.2011.07.067

⇑ Corresponding author. Tel.: +91 3222 283752; faxE-mail addresses: itssubhasish@gmail.com (S. Das)

a b s t r a c t

A logistic kinetic model was derived and validated to characterize the dynamics of a sporogenous bacte-rium in stationary phase with respect to sporulation and product formation. The kinetic constants asdetermined using this model are particularly important for describing intrinsic properties of a sporoge-nous bacterial culture in stationary phase. Non-linear curve fitting of the experimental data into themathematical model showed very good correlation with the predicted values for sporulation and lipaseproduction by Bacillus coagulans RK–02 culture in minimal media. Model fitting of literature data of spor-ulation and product (protease and amylase) formation in the stationary phase by some other Bacilli andcomparison of the results of model fitting with those of Bacillus coagulans helped validate the significanceand robustness of the developed kinetic model.

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1. Introduction

Characterizing and modeling the kinetics of sporulation andassociated product formation of commercially important microbesat stationary phase are of paramount importance, owing to the factthat no universal model is available till date. In stationary phase,the sporogenous bacteria not only produce spores, but also synthe-size some important enzymes like protease, amylase, lipase, chiti-nase, dextranase, etc. and other products like antibiotic, organicdyes, etc. (Prestidge et al., 1971; Fleming et al., 1995; Hobbset al., 1990). Therefore both phenomena follow similar kinetictrend. Although there is established way to know whether a prod-uct is growth associated or non-growth associated by Luedeking–Piret equation, there is no universal mathematical model for pre-dicting time course or describing substrate–product relationshipof stationary phase products. Hence large scale producers of thesebiologicals find difficulty in batch to batch reproducibility of prod-uct quality and fault diagnosis during batches. Therefore formula-tion and elucidation of logistic kinetic model would not onlyaddress these problems, but also will increase yield, reduce timeand cost of production of these biologicals. Some scientists (Riveraet al., 1999; Huang et al., 2003; Atehortua and Alvarez, 2007) pre-sented critical accounts on sporulation kinetics in batch and fedbatch fermentation studies and nicely described the equilibriumbetween vegetative cells, sporangia and spores in stationary phase.Huang et al. (2003) described graphically the effect of initial sub-

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strate concentration on the rate of sporulation of Bacillus subtilis,which is linearly proportional up to certain substrate concentra-tions, beyond which it is inversely proportional to initial substrateconcentration. Rho and Lee (1994) have described the effect of ini-tial substrate concentration on biomass and spore yield of Strepto-myces albidoflavus. However these works do not present anyuniversal mathematical model which can be used to describe sta-tionary phase sporulation or product formation phenomena. Liuand Tzeng (2000) fitted into the Hill model the data of rates ofsporulation of Bacillus thuringiensis at different residual substrateconcentrations obtained from only one batch using single initialsubstrate concentration. This is contrary to the more rational ap-proach and general practice of collecting residual substrate con-centration data at different initial substrate concentrations. Itwas also not clear as to how the data on different values of sub-strate concentrations (S) were obtained when the sporulation pro-cess was reported to start only after residual substrateconcentration reached the lowest constant value of about0.5 g L�1. Moreover, there was no logical explanation of selectingthe model too. Therefore, there is an urgent need to develop a suit-able and self-explanatory model to describe sporulation kineticsand related product formation kinetics in the stationary phase.

Moreover, no report was found on kinetic modeling of sporu-lation and lipase production in minimal media. Vegetable oil wasprimarily used as an anti-foaming agent, which was in turn uti-lized by the bacterium to produce more spores and lipase inthe absence of glucose (Kumar et al., 2005; Cho et al., 2009). Rel-atively excessive foaming may be attributed to the production ofa biosurfactant by Bacillus coagulans (Huszcza and Burczyk, 2003).

9660 S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667

As far as mathematical modeling of kinetics of a biochemicalprocess is concerned a better understanding of the dynamics ofgrowth and sporulation and esterase formation is not only helpfulin extracting knowledge about the process behavior, but also isessential for experimental and equipment design, processcontrol, optimization and scale-up (Atehortua and Alvarez,2007; Vlyssides et al., 2009; Gill et al., 2011; Santos et al.,2010). Though there are few reports on the response surfacebased process modeling for parameter optimization for theenhanced biomass and spore yields by the probiotic bacterium,B. coagulans RK-02 (Sen and Babu, 2005), a robust kinetic modelwould be handy to develop and design a more efficient bioproc-ess. In this work, an attempt was thus made to develop a logisticmodel for kinetics of sporulation and associated product forma-tion in stationary phase. Considering its commercial potential asan effective spore-forming probiotic, the present study thus aimsat simultaneously characterizing the sporulation dynamics andassociated lipase production kinetics of B. coagulans RK–02 bydeveloping suitable and relevant mathematical model, whichwas validated with the experimental data of other Bacillus strainsfrom literature and the values of the kinetic parameters werethen compared with those of the RK–02 strain.

2. Theory

2.1. Growth kinetics: Determination of Sopt

For modeling growth kinetics without and with substrate inhi-bition generalized model for growth kinetics as proposed by Tanet al. (1996) would be used:

l ¼ A:S

1þPn

i¼1AiðSÞi; where; i ¼ 1;2;3 and A ¼ lmax

KS: ð1Þ

For n = 1, 2 and 3, the equation becomes equivalent to that ofMonod, Haldane-Andrews and Yano models respectively. Here,A1 ¼ 1

KS(for all three models), A2 ¼ 1

KSKI1(for both Haldane–Andrews

and Yano models), A3 ¼ 1KSKIKI2

(for Yano model).Growth kinetics

data can be utilized to determine the optimum substrate concentra-tion, Sopt that corresponds to l = lmax (Bailey and Ollis, 1986). Thevalue of Sopt can be obtained by setting the derivative of Eq. (1)

equal to zero, i.e. dldS ¼ 0. Replacing Sopt for S, the value of true lmax

can be determined provided all A and Ai are known. Assuming thatthe specific rate of uptake of limiting substrate by a single cell dur-ing its exponential growth is q; the number of vegetative cells in theinoculum is N0 and the concentration of substrate that is alreadyconsumed is Sc, then the time of exhaustion of glucose in the med-ium can be calculated from Eq. (2).

s ¼ln 1þ Scl

N0q

� �l

: ð2Þ

For biomass at stationary phase vs. initial substrate concentration ageneralized equation can be derived from Monod equation. At high-er substrate concentrations there can be substrate inhibition ongrowth rate and growth phase will be elongated, while final bio-mass yield remaining the same. Therefore,

XXm¼ X0

Xm

� �KSþS0 1� ttmð Þ

KSþS0; ð3Þ

where tm = time taken by the biomass to reach maximum concen-tration, Xm, in batch cultivation and t = s. For higher S0, YX/S remainssame and s increases to a higher value than predicted (Bailey andOllis, 1986). Hence in Eq. (3), for lower substrate concentrationst � tm, and

XXm¼ X0

Xm

� � KSKSþS0

: ð4Þ

2.2. Sporulation kinetics

For modeling the sporulation kinetics, a logistic equation wasdeduced by adopting the following steps. The rate of sporulation,v(t) can be defined as number of spores formed per ml per hour(Atehortua and Alvarez, 2007). Now v(t) depends on biomass ornumber of cells present at time, t (Rivera et al., 1999; Warrinerand Waites, 1999; Piggot and Hilbert, 2004). As a part of modelingexercise, it can be assumed that (a) all cells prior to onset of spor-ulation have same mass, (b) in the broth not enough carbon sourceis available to support further growth (for lower S0) and (c) formingspores (which are not heat resistant) would not be considered asspores. Therefore,

dNs

dt/ :Nc; ð5Þ

and v ¼ dNs

dt¼ � dNc

dt; ð6Þ

where, Ns = number of spores ml�1, Nc = number of vegetative cellsml�1, and t = time (h), Negative sign indicates that the number ofvegetative cells is decreasing with time.

Now, unlike growth kinetics where lag period is small and oftennegligible for calculating specific growth rate, lag period for spor-ulation or onset of sporulation (which can be denoted by T) couldvary and be important in calculating specific rate of sporulation indifferent culture media (which we will define later). Hence insteadof Nc we would use Nc(t � T) in Eq. (5). From Taylor’s expansion,

Ncðt � TÞ ¼ NcðtÞ � TdNc

dt: ð7Þ

Combining Eqs. (5)–(7),

dNs

dt¼ �u Nc þ T

dNs

dt

� �

(where u ¼ proportionality constant)

or;dNs

dt¼ �u

Nc

1þu:T¼ �NT � Ns

kþ T; ð8Þ

where k ¼ 1u

� �

or; Ns ¼ NT 1� e�ðt�TkþTÞ

h i; ð9Þ

u ¼ 1=k can be termed as specific sporulation rate, NT = total num-ber of CFU at time T = (Nc + Ns) = maximum possible number ofspores that can be formed = Nmax. Describing sporulation rate interms of u is more meaningful than v(t) since onset of sporulation(T) is also included in the above definition of u.

Eq. (8) fits well when there is no substrate inhibition. At highersubstrate concentrations, rate of sporulation decreases (Huanget al., 2003; Rho and Lee, 1994; Liu and Tzeng, 2000). In analogyto thermodynamic explanation of substrate inhibition of growthby Tan et al. (1996), it can be assumed that each cell can have alarge number of identical functional units for binding of substrates,enzymes, enzyme-substrate complexes, etc. Now if each functionalunit on a vegetative cell has n substrate binding sites, then themore the glucose molecules will bind to these sites the more willbe the inhibition. If exactly i number of glucose molecules (i = 1,2, 3, . . . , n) bind to a functional unit then the contribution of theS term to inhibition of rate of sporulation will be of ith order. Atcertain time a functional unit can be occupied by any number ofglucose molecules ranging from 1 to n. Hence Eq. (8) can be

S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667 9661

reconstructed for higher substrate concentrations, where substrateinhibition of rate of sporulation is observed, as

dNs

dt¼ �

Nc:1T

1þ 1Tu �

Pn�10 bi½S�

i¼ Vmax

1þ K1 þPn

i¼1Bi½S�i; ð10Þ

where bi is a constant term andVmax ¼ Nc

T ; K1 ¼ b0Tu ; Bi ¼ bi

Tu ; i ¼ 1;2;3; . . . ;n.Bi can be termed ascoefficient of substrate inhibition. Although there is marked simi-larity between definitions of KS (Lobry et al., 1992) and K1, in prac-tice K1 carries entirely different meaning. From Eq. (10) (andScheme A1 and Eq. (A2) in Appendix), K1 can be defined as the low-est possible residual substrate concentration at which no glucosemolecule will bind to such active sites irreversibly and will facilitatesporulation. Hence, it can be named as constant of unsaturation orKu. Above definition says Ku is an intrinsic property of the microor-ganism, but Vmax is not (like Michaelis–Menten enzyme kinetics),since it varies with biomass (i.e., number of cells) in stationaryphase. However, definition of specific sporulation rate, u accordingto Eqs. (8) and (9), says that for S0 6 Sopt, S is negligible at t = T and uhas a constant value for different values of NT, and u is equal toumax. Hence it is an intrinsic property of a microorganism. It is tobe noted that by definition u and l are actually different. Unlikegrowth, a vegetative cell on sporulation loses divisibility, henceEq. (9). This definition (Eq. (9)) will be used in the rest of the article.For S0 > Sopt, at the onset of sporulation S ¼ ðS0 � SoptÞ. So Eq. (10)can be modified for two boundary conditions:

when S0 6 Sopt; u ¼ umax;

when S0 > Sopt; u ¼ umax

1þ Ku þPn

i¼1B S0 � Sopt� �i

: ð11Þ

2.3. Non-growth associated product formation kinetics

Product formation in stationary phase would follow same kinet-ics as that of sporulation kinetics. Hence an equation equivalent toEq. (9) is formulated:

Pt ¼ Pmax 1� e�t�TkþTð Þ

h i; ð12Þ

where Pt = concentration of product at time t, Pmax = maximumpossible product concentration, k ¼ 1

up;up ¼ specific product

formation rate. Here, Pmax ¼ YP=X � Xm. Eq. (12) can be rearrangedinto the following linear equation to account for stationary phaseproduct formation:

ln 1� Pt

Pmax

� �¼ � t

kþ Tþ T

kþ T; ð13Þ

From ln 1� PtPmax

� �vs. t least square linear fit k ¼ 1

upcan be deter-

mined. up (u in Eq. (9)) varies with S0 according to Eq. (12).

3. Materials and methods

3.1. Culture and media

A locally isolated strain, designated as B. coagulans RK–02, (Senand Babu, 2005; Sen et al., 2010) was used in this study. Cultures ofB. coagulans RK–02 were maintained on plates of Glucose Yeast Ex-tract Acetate agar (GYEA, HiMedia Laboratories Pvt. Ltd., Mumbai,India). Inoculum was prepared in GYEA medium. Final pH of themedium was adjusted to 6.9 ± 0.2. Flasks containing inoculum cul-tures were incubated in rotary shaker incubator at 180 rpm speedand 37 �C temperature.

3.2. Shake flask studies

About 108 cells were inoculated in each of 100 ml media in250 ml conical flasks in modified GMSM (glucose mineral saltmedia) (Sen et al., 2010; Das et al., 2009; Das et al., 2010) with dif-ferent glucose concentrations (g L�1): 0, 0.5, 1, 2, 4, 8, 12, 16, 20, 25,30, 35, 40, 50, 60, 70, 75, 80, 90, 100, 110 and 120. All the flaskswere incubated in the same rotary shaker incubator under sameconditions as mentioned above for all the experiments. Sampleswere collected at regular intervals and analyzed for dry weight ofbiomass, lipase, spore, nitrate, phosphate and residual glucose.

3.3. Spore count

Sample culture broth was heated at 75 �C for 30 min, followedby immediate cooling to 45 �C. This heat treatment facilitates thespores to germinate, while killing the vegetative cells. Number ofspores per ml in this heat treated mixture was determined by serialdilution and plate counting in GYEA-agar media (Sen and Babu,2005).

3.4. Microscopy of vegetative cells and spores

For quality control, Gram staining was performed for each sam-ple. This confirmed whether any contamination occurred. Olympusfluorescent microscope (Model – IX51) was used to study the mor-phologies at various stages of growth and sporulation. For sporestaining, a 0.1% aqueous solution of auramine O (4,40-Imidocarbon-yl-bis-N,N-dimethylaniline, Sigma–Aldrich, USA) was used (Bar-tholomew et al., 1965).

3.5. Assay of residual sugar in culture media

Anthrone sulfuric acid method (Yemm and Willis, 1954) wasused for the analysis of residual sugar.

3.6. Assay of lipase activity

Lipase activity was assayed by the colorimetric method of Win-kler and Stuckmann (1979)), using pNPP (p-nitrophenyl palmitate).

3.7. Assay of residual nitrate and phosphate

For assaying residual nitrate and phosphate in culture broth atdifferent time intervals K054 soil testing kit (Himedia) was used.The calibration plots were constructed by measuring absorbancesat 740 nm and 700 nm using various dilutions made from knownamount of nitrate and phosphate respectively.

3.8. Nonlinear curve fitting

For the estimation of the constants in Eqs. (1), (4), (9), (11) and(12) with experimental data OriginLab OriginPro 8 software wasused. Prior to model fitting growth kinetic constraints (Eq. (1))were set according to literature reported elsewhere (Gokulakrish-nan et al., 2006). YX/S was determined by plotting (Xm � X0) vs.(S0 � S) data and determining tangent of the plot. For estimationof specific sporulation rate (or specific product formation rate) con-stants from Eqs. (9) and (12) all parameters were set positive. Nmax

was initiated with maximum Nc value (experimentally deter-mined) of corresponding S0 data, but was not set fixed. T and kwere initiated with 1. From the final fitting data only mean valuewith one or two decimal places were noted. For higher S0 (>Sopt)Nmax = NT was kept fixed with experimentally determined NT valuein the stationary phase. YP/X was determined from tangent ofexperimentally obtained Pmax vs. Xm plot for S0 6 Sopt. For

9662 S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667

estimation of rate constants in Eq. (12), for S0 < Sopt, Pt vs. S0 datawere directly fitted into the equation and parameters were deter-mined in the same way as u was determined for sporulation kinet-ics except setting Pmax constant equal to experimental maxima. Forhigher S0(>Sopt) Pmax was calculated from the relationPmax ¼ YP=X :Xm, and was set fixed. For estimation of parameters inEq. (11) all parameters were set to be positive. Lower and higherbound of Sopt were set by taking the lowest value of Sopt (say F)determined from Eq. (1) and new Sopt was set with constraints asF 6 Sopt < 3F. umax was initiated with maximum u in the datasheetand other terms were initiated with 1. Different sets of initialparameter values were assigned to avoid biasness in estimationof kinetic parameters.

4. Results and discussion

4.1. Time courses of the production of biomass, lipase and spore withrespect to different substrate concentrations

The initial studies on the growth of B. coagulans RK-02 indicatedthat biomass production was facilitated in media containing initialglucose concentrations (S0) of 4 g L�1 and above. But biomassgrowth was found to be inhibited beyond an initial glucose concen-tration of 20 g L�1. Thus the time courses of biomass and residualglucose concentration were studied at initial glucose concentra-tions of 4 g L�1 and 20 g L�1 and the corresponding results werepresented in Fig. 1(a) and (b) respectively.

Both Fig. 1(a) and (b) showed that the biomass increased expo-nentially until glucose concentration in the media was depleted toapproximately 0.4 g L�1 at about 8 h for S0 = 4 g L�1 and at about14 h for S0 = 20 g L�1. Since oil is also a carbon source, therefore,as soon as glucose was consumed, lipase was found to be producedin the broth (Fig. 1a). However, there was no change in growth pro-file because oil was used as anti-foaming agent at very low concen-trations, good enough to induce lipase production. (Media withlower S0 namely 0, 0.5, 1 g L�1 were exempted for lipase studywhere presence of even slightest of oil might affect biomass pro-file). For higher S0 values (S0 > 20 g L�1) about 4–6 h lag phase wereobserved. For S0 < 12 g L�1 practically there was no lag phase. Bio-mass concentration at the beginning of the stationary phase in-creased proportionally with the increase in initial glucoseconcentration up to 20 g L�1 and remained almost constant at amaximum value of 2 g L�1 thereafter (Fig. A2). This may be due

Fig. 1. Time courses of biomass, lipase activity, spore concentration, and residual gluco4 g L�1 and (b) 20 g L�1.

to exhaustion of nitrogen source and/or phosphates (Fig. 1) andother salts in culture media or substrate inhibition or productionof some toxic metabolites. Thus excess residual substrate thatwas not being converted to biomass would be accumulating inmedium and delay sporulation. The exhaustion of nitrogen sourcein media for such higher S0 indicated that the culture entered intothe stationary phase while glucose was still present in the media.The average growth yield was estimated to be YX/S = 0.116 g/g(S.E ± 0.002 g/g) of glucose for up to about S0 = 16 g L�1, beyondwhich yield was gradually decreasing up to S0 = 20 g L�1 (Fig. A2).Single-factor-at-a-time approach such as this study can be adoptedto identify the key components in the culture medium. Optimiza-tion of process conditions including medium composition andenvironmental conditions by response surface modeling or artifi-cial neural network modeling coupled with genetic algorithm orMonte-Carlo optimization algorithm is more likely to enhance bio-mass yield (YX/S) and product yield significantly (Sen, 1997; Senand Babu, 2005; Sivapathasekaran et al., 2010).

4.2. Estimation of Sopt from growth kinetics studies

The specific growth rates (l) for each S0 values were determinedfrom linear tangent of ln(X/X0) vs. time (h) plot. It was observed thatspecific growth rate (l) increased with increase in S0 values up toS = 14 g L�1 (Fig. 2) corresponding to S0 = 20 g L�1, above whichlde-creased gradually with further increase in S0 (or S). For determiningthe parameters of growth kinetics without substrate inhibition forS 6 14 g L�1, i.e., S0 6 20 g L�1 in l vs. S (residual substrate concen-tration in mid-exponential phase) plot Eq. (1), n = 1 and for entire re-gion with substrate inhibition Eq. (1), n = 2 and n = 3 were used(Fig. A2, Table 1). Fitting of data into all the models showed highlmax

and low KS values, which are typical of Bacillus species. However, in acontinuous culture of B. coagulans using optimized complex med-ium at 52 �C, Payot et al. (2006) reported a lmax value of 0.28 h�1,which is quite lower than that of the RK-02 culture in GMS medium.A relatively higher value of lmax coupled with a lower value of KS

adequately justifies the basis for the selection of the glucose mineralsalts medium for the growth of the culture. A lower value of KS wasalso determined by fitting X vs. S0 data into Eq. (4) (Fig. A3). Theadvantage of this equation is that it allows the researcher to use S0

instead of S thereby making the study less tedious. However consid-eration of lag phase using Eq. (3) would result in a better curve fittingand more accurate parameter estimation. It was observed that

se, nitrate and phosphate concentrations for an initial glucose concentration of (a)

Fig. 2. (a) Non linear curve fitting of number spores per ml vs. time (h) data in Eq. (9) for determination of specific sporulation rate (u, h�1), onset of sporulation (T, h) andnumber of cfu ml�1 in stationary phase (Nmax). (b) Simulated sporulation kinetics by Eq. (11) with different values of n. Specific rate of spore formation (u, h�1) vs. initialsubstrate concentration (g L�1) data were fitted in this substrate inhibition model for spore formation.

Table 1Estimated parameters of various growth kinetic models without and with substrate inhibition explaining uninhibited region and entire region respectively.

Model Equation Parameters R2 Expression foroptimum S0

Sopt

(g L�1)True lmax

(h�1)

Tan et al. (1996):n = 1

l ¼ AS1þA1 S

A = 0.28 ± 0.04, lmax = 0.64 h�1, A1 = 0.43 ± 0.08, KS = 2.3 g L�1 0.98

Tan et al. (1996):n = 2

l ¼ AS1þA1 SþA2 S2

A = 0.28 ± 0.04, apparent lmax = 0.93 h�1, A1 = 0.30 ± 0.07,KS = 3.34 g L�1, A2 = 0.01

0.96 ðA2Þ�1=2 8.48 0.52

Tan et al. (1996):n = 3

l ¼ AS1þA1 SþA2 S2þA3 S3

A = 0.28 ± 0.04, apparent lmax = 0.83 h�1, A1 = 0.33 ± 0.09, KS = 3.0 g L�1,A2 = 0.01, A3 = 0.00005

0.96 0:5ðA3Þð1=3Þ 13.57 0.5

S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667 9663

substrate inhibition was a major problem with higher substrate con-centrations. In a study on Lactobacillus cultures, Rao et al. (2004)demonstrated that the maximum specific growth rate could be in-creased by optimizing the concentration of NaCl and the culturepH, without any significant interaction between these two parame-ters. Such an optimization strategy may be useful in reducing thedoubling time of a culture. An optimal substrate concentration forachieving maximum specific growth rate was determined to be8.48 and 13.57 g L�1 using Eq. (2) for n = 2 and n = 3, respectively(Table 1).

4.3. Studies on sporulation kinetics

For lower S0 values, as soon as glucose was consumed in themedium, sporulation started and the number of spores increasedrapidly within next few hours, before sporulation rate gradually

Table 2Time of exhaustion of limiting substrate (s, h) from Eq. (2) and prediction of onset of spor

Initial substrateconcentration, S0 (g L�1)

Predicted time (s, h) of limitingsubstrate exhaustion (Eq. (2))

Averagemeasured lnet

0.5 – –1 7.9 0.08 ± 0.012 10 0.16 ± 0.034 10.8 0.242 ± 0.038 10.5 0.32 ± 0.0312 10 0.38 ± 0.0116 9.12 0.46 ± 0.0320 9.67 0.46 ± 0.07

decreased. After glucose was exhausted in the media, it took about6–8 h to clearly detect the presence of B. coagulans spores (Fig. 1aand b, Table 2). Appearance of spores (NS) in broth showed anexponential relationship with time. This nature of graph is ex-plained by Eq. (9). For S0 = 4 g L�1, the ascending part of graphwas very steep and after 30 h, the spores reached a saturationconcentration. Microscopic analysis of the culture samples olderthan 30 h showed only spores were present in culture and no traceof vegetative cells were found. Whereas for S0 = 20 g L�1, slowerrate of spore formation was observed and saturation was notobserved even after 36 h of fermentation (Fig. 1). As soon as sub-strate, either carbon or nitrogen source or phosphate, is consumed,a cell starts the process of sporulation (Piggot and Hilbert, 2004).Naturally, question arises that even when almost no detectableglucose was present in media with S0 = 20 g L�1 after 15 h, how alarge population of cells could maintain vegetative form at 36 h

ulation (T, h, using Eq. (9)).

Residual concentration oflimiting substrate (g L�1)(measured)

Predicted onset of sporulation (T(h),from Eq. (9))

After 10 h After 12 h

0.39 ± 0.02 0.42 ± 0.03 5.88 ± 0.60.36 ± 0.02 0.35 ± 0.05 11.95 ± 1.110.37 ± 0.03 0.35 ± 0.03 11.90 ± 0.890.35 ± 0.04 0.34 ± 0.02 17.3 ± 1.20.39 ± 0.02 0.39 ± 0.03 19.85 ± 1.050.52 ± 0.02 0.48 ± 0.02 21.19 ± 0.961.27 ± 0.70 0.72 ± 0.12 23.78 ± 2.56

3.896 ± 0.91 1.67 ± 0.56 24.012 ± 3.86

9664 S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667

(which was vindicated from microscopic study). Probably glucosemolecules bind to the active sites (Tan et al., 1996) and are trans-ported across cell membrane slowly. In case demand of glucose in-side cell is lesser than available glucose molecules a relativelyslower rate of glucose transport would occur. This would resultin longer residence of glucose molecules on active sites. Thusalthough glucose wont be detectable in medium, available glucosein cells would delay sporulation.

Eq. (2) was used to predict the time of consumption (s) of glu-cose in broth (Table 2). For low S0, s remained almost the same.Non-linear curve fitting of time vs. number of spores per ml datain Eq. (9) resulted in the prediction of specific sporulation rate(u) and time (T) of onset of sporulation in a culture (Table 2).Unlike equation for determination of specific growth ratel ¼ 1

t lnð XX0Þ

h iwhich cannot predict onset of growth, nor maximum

biomass formation in the corresponding medium of particular S0,Eq. (9) can precisely predict onset of sporulation (or of a productformation) and maximum spore/product concentration. This is aunique feature of Eq. (9).

For initial glucose level ranging within 4–12 g L�1 it wasobserved that there was about 7–10 h gap between total glucoseconsumption in medium (s) and onset of sporulation (T) (Table 2).This time gap (T � s) can be termed as preparatory phase for sporu-lation. For a strain of B. subtilis, (T � s) was reported to be about 7 h(Piggot and Hilbert, 2004). For S0 < 4 g L�1, (T � s) was predictedmuch lower than 7 h. It was also noted that glucose was never totallyexhausted in these media, probably to maintain proper osmoticpressure (Table 2). These observations probably indicate sporulationstarted in the media when glucose level was as low as about 0.5 g L�1

(Liu and Tzeng, 2000). In other words the culture did not wait forcomplete consumption of glucose in media to start sporulating.For higher S0 (above Sopt), greater time gaps between onset ofsporulation and time of glucose exhaustion (T � s) were observed.Above Sopt, measured s went on increasing although it was predictedthat s would remain almost the same for substrate concentrations.This discrepancy was because Eq. (2) was formulated from Monodmodel and not substrate inhibition models like Haldane or Yanomodels.Whereas for lower S0 values rate of sporulation remain high,for higher S0 values (S0 > 20 g L�1) opposite trend is observed. About103–104 spores per ml were present in all the broth samples withhigher S0 at 24 h and the rates of spore formation was very low insubsequent hours. This trend for B. coagulans RK-02 was similar to

Table 3Estimation of sporulation kinetics constants in Eq. (11) using specific sporulation rate (u)

Model Equation

Eq. (11), n = 2 When S0 6 Sopt u ¼ umax, when S0 > Sopt, u ¼ umax

1þKuþB1ðS0�Sopt ÞþB

Eq. (11), n = 3 When S0 6 Sopt u ¼ umax, when S0 > Sopt, u ¼ u1þKuþB1ðS0�Sopt ÞþB2

Eq. (11), n = 4 When S0 6 Sopt u ¼ umax, when S0 > Sopt, u ¼ 1þKuþB1ðS0�Sopt ÞþB2

that for other sporeformers as described in literature (Huang et al.,2003; Rho and Lee, 1994; Liu and Tzeng, 2000). Hence for describingsuch S0 vs. u relationship application of Eq. (11) becomes necessary.Characterization of sporulation kinetics with the newly derivedkinetic model would unleash underlying facts of the physiologicalchanges and further understanding about the phenomena. AlthoughEq. (11) uses S0 as variable but in practice u is varied against residualsubstrate concentration S. In this equation a new meaning of Ku

presents itself. For S0 6 Sopt, a negative value of S results inKu ¼ �

Pni¼1BiS and consequently, u ¼ umax. Negative S means that

residual substrate concentration is lower than cellular substrateconcentration. When extracellular and intracellular glucoseconcentrations become almost equal, bacteria cannot absorbglucose by endosmosis process anymore. Glucose is taken up byactive transport mechanism across cell membrane (Kajikawa et al.,1997), resulting in higher intracellular glucose concentration thanthat of extracellular and hence, S becomes negative. Therefore, thisnegative S is substrate gradient constant due to substrate crisis inthe medium and can be denoted as Scri.

By non-linear curve fitting of number of spores per ml vs. time(h) data in Eq. (9), the specific sporulation rate (u, h�1), the time ofonset of sporulation (T, h) and maximum possible number ofspores per ml (Nmax), for S0 = 4 g L�1 were determined (Fig. 2a).Similarly for all S0, the specific sporulation rate (u) and also T weredetermined using Eq. (9). Predicted Nmax using Eq. (9) was almostsimilar to experimentally determined number of CFU per ml in sta-tionary phase (NT). Beyond 50 h spore concentration in culture didnot decrease or increase and those data were used as well resultingin very good non-linear curve fitting with high regression coeffi-cient (in Fig. 2a time span was shown only up to 100 h). Non-linearcurve fitting of u and S0 data (Fig. 2b, Table 3) into Eq. (11) forn = 2, 3, 4 helped in evaluating the maximum specific sporulationrate, umax to be 1.5 � 107 h�1 and optimum initial substrate con-centration to be about 12 g L�1. In other words when C/N ratio islower than 4.156 g/g, sporulation starts rapidly ðu ¼ umaxÞ as glu-cose is exhausted in medium. For higher C/N ratio, exhaustion ofnitrogen source triggers sporulation; but presence of glucosemakes the process slow (hence inhibition constants—Ku and Bi

terms). High umax indicates that after exhaustion of carbon sourcein medium sporulation occurs immediately after glucose isexhausted in medium, provided S0 < 12 g L�1. For different valuesof n, the equations were fitted very well with the data, except for

vs. initial substrate concentration (S0) data.

Parameters Adjusted R2

2ðS0�Sopt Þ2Umax = 1.5 � 107 h�1

Sopt = 11.99 g L�1

Ku = 0.00015 g L�1

B1 = 0.0033 g L�1

B2 = 44.52 g L�1

Scri = �3.5439 g L�1

0.96

max

ðS0�Sopt Þ2þB3ðS0�Sopt Þ3Umax = 1.5 � 107 h�1

Sopt = 11.73 g L�1

Ku = 0.012 g L�1

B1 = 0.082 g L�1

B2 = 0.4 g L�1

B3 = 10.73 g L�1

Scri = no real solution

0.96

umax

ðS0�Sopt Þ2þB3ðS0�Sopt Þ3þB4ðS0�Sopt Þ4Umax = 1.5 � 107 h�1

Sopt = 11.99 g L�1

Ku = 0.28 g L�1

B1 = 6 � 10�17 g L�1

B2 = 0.61 g L�1

B3 = 4.33 g L�1

B4 = 155 g L�1

Scri = �2.774

0.96

Table 4Estimation of lipase production kinetics constants in Eq. (11) using specific lipaseproduction rate (up) vs. initial substrate concentration (S0) data.

Model Parameters Goodness of model fitting

Eq. (11), n = 1 Umax = 2.25 � 1013 h�1

Sopt = 11.92 g L�1

Ku = 3.7 � 10�32 g L�1

B1 = 209.9 g L�1

Adjusted R2 = 0.92

Eq. (11), n = 2 Umax = 2.25 � 1013 h�1

Sopt = 11.31 g L�1

Ku = 1.4 � 10�22 g L�1

B1 = 3.1 � 10�9 g L�1

B2 = 36.3 g L�1

Adjusted R2 = 0.92

Eq. (11), n = 3 Umax = 2.25 � 1013 h�1

Sopt = 11.39 g L�1

Ku = 1.2 � 10�18 g L�1

B1 = 0.13 g L�1

B2 = 8.3 g L�1

B3 = 60.6 g L�1

Adjusted R2 = 0.91

S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667 9665

n = 1 (data not shown). For n = 3, no real solution for Scri was ob-tained. Hence, n is an even positive integer for which Eq. (11) holdstrue, probably indicating glucose molecules enter into cells in pairsduring active transport. For different values of n in Eq. (11), Ku, B1,B2, B3, B4 values varied a lot, but there was something common forall n. Ku was very small. This indicates only very low residual sub-strate concentration will cause no glucose molecule to bind to theactive sites irreversibly, when sporulation will occur at maximumrate. It was observed in general, Ku < B1 < B2 < B3, which indicatethat for binding more glucose molecules to the active sites moreenergy or more residual glucose concentration was needed. Forn > 4, identifiability problem was encountered and those estimatedparameter values were not considered. Therefore the model shouldbe as complex as necessary. The higher the values of Bi terms, thelower will be the sporulation rates. High Sopt value will cause morebiomass to form before the culture reaches stationary phase.Therefore biomass yield will increase. High umax indicates highspore formation rate to reach saturation. High value of Ku indicatessporulation will start in broth easily, because the glucose trans-porters in the cell membrane of the bacterium have low affinityfor the substrate. Now the simplest form of Eq. (11), to which theu vs. S0 data fitted very well (i.e., for n = 2), predicted Ku to be verylow. Therefore GMS medium, containing glucose as limiting sub-strate, was good at supporting sporulation of the bacterium.

4.4. Lipase production kinetics

As soon as glucose was exhausted, the probiotic bacteriumstarted producing lipase to consume vegetable oil as an alternativecarbon source, which was primarily used as antifoam agent (Daset al., 2010), (Fig. 1). It was observed for S0 < Sopt after exhaustionof glucose, lipase activity immediately increased and reachedsaturation (Fig. 1a). Hence high up values were obtained by fittinglipase activity (Pt) vs. time (t) data into Eq. (12). With increase inbiomass in stationary phase (for increasing S0) Pmax increased pro-portionally with a ratio equal to YP/X (=0.0997, R2 = 0.91), up toS0 = Sopt. For S0 > Sopt had there been used some potent proteaseinhibitor the expected Pmax for could reach a value equal to XT.YP/X.For this region after exhaustion of glucose increase in lipase activ-ity was slow, because of residual glucose interfering with lipaseproduction (Fig. 1b) and low up values were obtained by fittingPt versus t data into Eq. (12) (Fig. 3a). The model fitting of experi-mental data was in very good correlation with predicted data in Eq.

Fig. 3. (a) Least square fit line fitting of lipase activity (U ml�1) vs. time (h) data for S0 = 2lipase production kinetics by proposed model (Eq. (11)) with different values of n. Specififitted in this substrate inhibition model for product formation.

(12). For all other substrate concentrations up values were deter-mined similarly. For low S0, glucose depletion in media andpredicted onset of lipase production results went hand in hand.For high S0 possibly depletion of nitrate or phosphate in media inpresence of lipase inducer (i.e., vegetable oil) lipase production istriggered. The biological meaning of the maximum productconcentration (Pmax) is that the maximum concentration of theproduct (here, lipase) that is obtained in a batch culture ofB. coagulans RK–02 with respect to a particular initial substrateconcentration within a specified time of experimentation. As thefermentation proceeds, the concentration of the product of ourinterest (lipase) increases in the broth with time and reaches amaximum after a certain period of fermentation process for allS0 < SOpt. On the contrary, for all S0 > Sopt, infinite time is requiredto achieve Pmax. Pmax is expressed in absolute terms (i.e. it is nota specific value). A biochemical engineer can be curious to knowwhen product formation reaches 100% mark. Logically the end-point cannot be determined due to exponential nature of the timecourse of product formation in a batch. However for practical pur-pose sometimes it would be more economical to obtain 90–95%product than 100% and the corresponding time can be determinedusing Eq. (12). For S0 > Sopt, fitting Pt vs t data into Eq. (13) producessame result as for Eq. (12). For S0 < Sopt where saturation is reachedrapidly model Eq. (13) cannot be used, because the data points in

0 g L�1 to determine specific product formation rate up using Eq. (12). (b) Simulatedc rate of lipase formation (up) vs. initial substrate concentration (S0, g L�1) data were

Table 5The literature data on time course of sporulation and product formation in stationary phase cultures of Bacilli (Prestidge et al., 1971; Fleming et al., 1995; Liu and Tzeng, 2000)fitted into Eq. (9) or 12 to determine specific sporulation rate u (h�1) or specific product formation rate up (h�1).

Name of the culture B. subtilis in modifiedmedium of Donnellaet al.

B. licheniformis DN 286 inminimal medium with 2%maltose

B. licheniformis DN286 in BHImedium

B. licheniformis HWL 10 inminimal medium withmaltose 2%

B. thuringiensisYim303-G2 in GYSmedium

References Prestidge et al. (1971) Fleming et al. (1995) Fleming et al.(1995)

Fleming et al. (1995) Liu and Tzeng (2000)

Sporulationkineticparameters

T (h) 12.51 ± 0.46 22.6 ± 6.7 20.4 ± 4.3 16.1 ± 3.1u (h�1) 2 � 108 6.23 � 10�5 2.95 � 10�5 1.15 � 107

R 0.87 0.91 0.96 0.90Proteases Onset of

production,T (h)

Protease A: 6.1Protease B: 6.3Protease C: 6.1

Serine protease: 17.8 ± 2.1 Serine protease: 15.9 ± 1.8

up (h�1) Protease A: 0.17Protease B: 0.12Protease C: 0.13

Serine protease: 1.08 � 107 Serine protease: 1.07 � 107

R Protease A: 0.99Protease B: 0.98Protease C: 0.98

0.97 0.98

Amylase T (h) 20.9 ± 3.9up (h�1) 3.28 � 106

R 0.96

9666 S. Das, R. Sen / Bioresource Technology 102 (2011) 9659–9667

saturation level (i.e., Pmax) cannot calculated using Eq. (13). This isthe limitation of Eq. (13). The up and S0 data were fitted into Eq.(11) (Fig. 3b, Table 4) which fitted well with high coefficient ofregression according to the trend of the product formation forn = 1, 2 and 3. For n = 4 and above the experimental data did notconverge into the corresponding equation for different sets of data.Probably increase in number of data points may help prediction ofhigher Bi terms (i.e., n = 4 and above). The Sopt value was almostsame for lipase production and for sporulation kinetics. NegligibleKu for lipase production kinetics (even lower than that for sporula-tion kinetics) values indicated that with increase in residual glu-cose there is practically negligible chance that glucose moleculewould not bind to active sites. Hence as the glucose moleculeswould bind to these sites rate of lipase production would be inhib-ited. Nearly zero Ku value indicated that at least one value ofScri � 0. This clearly indicated lipase production can start wheninternal and external glucose level becomes equal (for S0 < Sopt).In the contrary, as far as sporulation is concerned a cell can waitfurthermore until by active transport mechanism absorption ofmore glucose from medium is possible. As it happened for sporula-tion kinetics about Bi terms a very similar trend was observed forlipase: Ku < B1 < B2 < B3 and the explanation goes the same.

4.5. Sporulation and product formation kinetics of B. subtilis, B.licheniformis and B. thuringiensis

To validate robustness of the model for other Bacillus speciesdata were obtained from previous literature (Prestidge et al.,1971; Fleming et al., 1995; Liu and Tzeng, 2000). Time course ofspore concentration data of B. subtilis, B. licheniformis, and B. thur-ingiensis were fitted into Eq. (9) and corresponding values of T andu were obtained (Table 5). All the data fitted well into the equationwith handsome correlation coefficients between observed and pre-dicted values. High u values of B. subtilis and B. thuringiensis denotethat the cultures become saturated with spore at a fast rate of spor-ulation. The two media used for B. licheniformis DN 286 cultivationwere good for biomass growth (Fleming et al., 1995), but sporula-tion rates were slow.

In order to check the validity of the robustness of the model bydetermining the specific product formation rate up (h�1), the liter-ature data on time course (Prestidge et al., 1971; Fleming et al.,1995) of product formation in stationary phase of some sporoge-nous bacteria were fitted into Eq. (12) and the results of model fit-

ting are given in Table 5. These data fitted very well into Eq. (12)thus establishing the robustness of the model. An interesting find-ing that emerges by comparing sporulation kinetics with productformation kinetics is that a medium, which is good for sporulation,does not always guarantee good product formation in the station-ary phase and vice versa. This alternatively corroborates the factthat a robust kinetic model like the one developed in this studyis absolutely necessary to adequately characterize the dynamicsof sporulation and non-growth associated product formation in astationary phase culture.

5. Conclusions

A logistic and robust mathematical model to analyze and de-scribe the kinetics of stationary phase phenomena of a microbialculture was developed. The kinetic model (Eqs. (9) or (12)) canadequately explain the trends of the dynamics of sporulationand/or product formation in stationary phase of growth of a spo-rogenous probiotic bacterium. Eq. (11) provides some veryimportant definitions of the characteristic kinetic constants of aspore-forming bacterium. This theory can be extended for fittingthis model into cell death kinetics of non-sporeforming microor-ganisms as well, which should essentially follow same kinetictrend.

Ackowledgements

The authors would like to thank Dr. Vidya Prabhakar Kodali forhelping in spore concentration determination and Shailesh Khark-wal for helping in lipase assay. The authors also acknowledge thefinancial assistance received from the CSIR, Govt. of India througha sponsored project (Grant No. 37(1230)/05/EMR-II).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.biortech.2011.07.067.

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