The sensitivity and significance analysis of parameters in the model of pH regulation on lactic acid production by Lactobacillus bulgaricus
© Liu et al.; licensee BioMed Central Ltd. 2014
Published: 13 November 2014
The excessive production of lactic acid by L. bulgaricus during yogurt storage is a phenomenon we are always tried to prevent. The methods used in industry either control the post-acidification inefficiently or kill the probiotics in yogurt. Genetic methods of changing the activity of one enzyme related to lactic acid metabolism make the bacteria short of energy to growth, although they are efficient ways in controlling lactic acid production.
A model of pH-induced promoter regulation on the production of lactic acid by L. bulgaricus was built. The modelled lactic acid metabolism without pH-induced promoter regulation fitted well with wild type L. bulgaricus (R 2 LAC = 0.943, R 2 LA = 0.942). Both the local sensitivity analysis and Sobol sensitivity analysis indicated parameters T max , GR, K LR , S, V 0 , V 1 and d LR were sensitive. In order to guide the future biology experiments, three adjustable parameters, K LR , V 0 and V 1 , were chosen for further simulations. V 0 had little effect on lactic acid production if the pH-induced promoter could be well induced when pH decreased to its threshold. K LR and V 1 both exhibited great influence on the producing of lactic acid.
The proposed method of introducing a pH-induced promoter to regulate a repressor gene could restrain the synthesis of lactic acid if an appropriate strength of promoter and/or an appropriate strength of ribosome binding sequence (RBS) in lacR gene has been designed.
Lactobacillus delbrueckii subsp. bulgaricus has been widely applied in diary industry, especially as a starter for yogurt production. This microorganism is facultative anaerobic and can produce lactic acid. During the later period of fermentation and storage of yogurt, the production of lactic acid is dominated by L. bulgaricus . The excessive production of lactic acid during storage makes the yogurt taste too sour and this phenomenon is called post-acidification.
At industrial level, several attempts have been made to slow down post-acidification, such as using devoid of L. bulgaricus, selecting the weak post-acidification L. bulgaricus, pasteurization after fermentation and so on [2–4]. However, these methods either control the post-acidification inefficiently or kill the probiotics in yogurt. Using genetic technology to modify the bacteria has become another option to decline post-acidification. Due to lacking molecular tools, mainly caused by the absence of a reliable transformation procedure, our understanding of the physiology and genetics of L. bulgaricus is still limited .
In L. bulgaricus, the absorption of lactose is processed by lactose/galactose antiport transport system . Firstly, lactose is transported into cells by lactose permease encoded by lacS gene. Then the lactose is decomposed into glucose and galactose by β-galactosidase encoded by lacZ gene. The free galactose is pumped out of cells or stored in cells in the form of macromolecule i.e. carbohydrate gum. Glucose turns into pyruvate through glycolysis, and the glycolysis is transformed into lactic acid by the catalysis of lactate dehydrogenase [7–9]. The lacS gene and the lacZ gene constitute a lacSZ operon. In the downstream of lacZ gene, there is a lacR gene encoding a repressor which makes the lacSZ operon induced by lactose. However, in L. bulgaricus, the lacR gene has lost regulatory function due to the insertion of some gene fragments, resulting in constitutive expression of lacSZ operon .
Due to our limited knowledge, most attempts at the genetic level focused on changing the activity of one significant enzyme at a time in the metabolism of lactic acid. Druesne altered the Histidine codon to Alanine codon at the 552th locus of lactose permease, resulting in the reduction of the enzyme's activity . Early in 1990, Mollet obtained the mutant L. bulgaricus without β-galactosidase activity by spontaneous deletion . The new aforementioned organisms did produce less lactic acid but they had trouble in growing in milk independently due to the lack of energy. Adams selected two cold-sensitive mutants of the β-galactosidase from L. bulgaricus by using the expression system of E. coli . However, their later research turned back to directly mutate and screen L. bulgaricus since it was difficult to transform with L. bulgaricus.
So far, Lactobacillus delbrueckii subsp. bulgaricus ATCC 11842 has been reported successfully transformed by electroporation, although with very low reproducibility and efficiency [5, 14]. Several pH-induced promoters from Lactococcus lactis have been demonstrated, such as rcfB , P1 and P170 . Combining the two points mentioned above, we came up with a new method to build a pH-induced promoter with a repressor gene controlling the production of lactic acid. Thus, we could turn on the switch at appropriate time. Here we analyse the parameters in the kinetic model to investigate the regulation effect of pH-induced promoter on lactic acid.
The reduced metabolic network
The mathematical model
Variables used in the model
Concentration of lactose out of the cells
Concentration of lactose in the cells
Concentration of glucose in the cells
Concentration of pyruvate in the cells
Concentration of lactic acid out of the cells
LR, LS, LZ
Concentration of LacR protein, lactose permease, β-galactosidase in the cells
Concentration of the mRNA of lacR gene
Concentration of the mRNA of lacS and lacZ gene
Concentration of cell mass
Each entity, described in Table 1 is given as total amount in the population.
The model does not take energy metabolism into consideration.
All reactions are modelled by mass action principles, except for enzyme reactions which obey Michaelis-Menten equation, and transcription which obeys saturation kinetics.
The level of translation product is assumed to be proportional to mRNA levels according to Dockery and Keener .
There is no delay in synthesis of either component or delay because of protein transportations.
All substances are subject to degradation.
The metabolism of lactose
As mentioned before, the metabolism of lactose is described by Michaelis-Menten equation. Since lactose transport is reversible, a term was included to count for lactose efflux dependent on the internal lactose concentration . The V max for lactose permease and β-galactosidase is associated with the concentration of enzymes, so we use K cat and the concentration of enzymes to estimate the V max . Since the lac operon is usually induced by lactose, we assume that the repressor LacR protein could react with lactose. Thus we need to take the reduction into consideration when evaluating the concentration of intracellular lactose and LacR protein. As for the utilization of sugar, L. bulgaricus belongs to the type of homofermentation. Theoretically, this microorganism could totally convert glucose to lactic acid, but the conversion of glucose is about 80%-90% in practice . Therefore, the estimated proportion of pyruvate transformed into lactic acid is 80%.
The gene regulation
The transcription and decomposition of lactose are tightly regulated at the genetic level: production of the enzymes can be decreased at the transcriptional level by regulatory protein LacR binding at appropriate DNA sites. We set M to stand for the transcript concentration of lacR gene. Since lacS and lacZ are located in the same operon, it is assumed that both genes are transcribed at same rate. We used N to describe the concentration of the two genes' mRNA. When describing the equations, the mRNAs are assumed to be transcribed at two distinct rates: basal one when there is no regulation and a higher/lower rate when being regulated .
The transcription of lacR gene
Transcription of lacS and lacZ gene
Combining the metabolism of lactose with gene regulation
List of parameters used in the model
Michaelis constant for lactose permease
2.6 × 10-4 M
Patrick et al. (1997) 
rate constant for lactose transportation
1.29 × 105h-1
Patrick et al. (1997)
Michaelis constant for β-galactosidase
9.8 × 10-4M
Moez et al. (2009) 
hydrolysis rate for lactose
1.65 × 105h-1
Moez et al. (2009)
Michaelis constant for glycolysis
1 × 10-4M
L. Lactis Marcel et al. (2002) 
maximum reaction speed when enzymes was saturated by substrate
L. Lactis Marcel et al. (2002)
Michaelis constant for lactate dehydrogenase
L. Lactis Marcel et al. (2002)
maximum reaction speed when lactate dehydrogenase was saturated by pyruvate
L. Lactis Marcel et al. (2002)
LacS synthesis rate constant
Kennell et al. (1977) 
LacS decay rate constant
Kennell et al. (1977)
LacZ synthesis rate constant
1.13 × 103h-1
Kennell et al. (1977)
LacZ decay rate constant
Kennell et al. (1977)
LacR synthesis rate constant
LacR decay rate constant
d M , d N
mRNA decay rate constant
Varmus et al. (1970)  and Patrick et. al. (1997)
lacR, lacSZ gene concentration
2.5 × 10-9 M
Cheng et al. (2001) 
steepness of the smooth switch function
Akyol et al. (2008) 
LacR/lactose binding rate constant
7.2 × 104 M-1·h-1
basal rate of lacR gene transcription for mRNA
increasing constant of lacR gene transcription for mRNA
basal rate of lacSZ gene transcription for mRNA
Obtained by 'fitting'
maximum rate of lacSZ gene transcription
6 × 104 h-1
sensitivity of lacSZ gene transcription to LacS and LacZ
saturation constant for growth of L. bulgaricus on extracellular lactose
9.82 × 10-3 M
Concepcion et al. (2000) 
maximum lactic acid concentration to inhibit the growth of L. bulgaricus
Concepcion et al. (2000)
maximum growth rate
Concepcion et al. (2000)
We set stoichiometric constant of R 4 with two since two molecules of pyruvate are formed from one molecule of glucose. For the stoichiometric constant of R 5 , it stands for the proportion of pyruvate which is converted to lactic acid.
Firstly, we did not take lacR gene into consideration. This is just the case of wild type L. bulgaricus which lacSZ operon is constitutive expression. Since most of the parameters in this reduced model have been reported, V 2 is the only one needs to be modified. We used SBToolbox in Matlab to set up the reduced model and simulated the metabolism process . We compared the concentration changes of extracellular lactose and lactic acid within 8 hours with the data from Fatama, where the conditions used were 43 °C, 8% of skim milk concentration (without pH value control) and 4% of inoculum ratio . After that, we added lacR gene. This time we did not have sufficient data for full parameterization of the model. However, we did have enough information to estimate their relative sizes and give the qualitative nature of our investigations. The parameter values are listed in Table 2.
Sensitivity and significance analysis of parameters
We still used SBToolbox to perform local sensitivity analysis and Sobol's method for global sensitivity analysis. The local sensitivity analysis investigates how small changes in a single parameter value could affect the model output. The method is based on the partial differentiation of the output with respect to the input parameters . Herein, the partial differentiation is evaluated numerically by introducing a 1% increment from the specific parameter value. We chose Sobol sensitivity analysis method to calculate global sensitivity since the sensitivity index quantified the overall effects of a parameter, in combination with any other parameters, on the model output . The number of simulation to carry out is 10000 times. We first set the number as 1000 according to Schmidt . Since the results are varied among different simulations, we increased the number to 10000 and got stable outcomes. During the above sensitivity analysis, we chose lactic acid concentration as the model output.
After figuring out the sensitivity parameters to the model output, we carried out different simulations with each sensitivity parameter at two levels, a relative low value and a relative high value. The metabolism of lactic acid and changing of pH values was carefully considered. Those results indicate how pH values affect lactic acid production under different situations and help us to understand how to design LacR regulation in biology experiments in future.
Results and discussion
Reduced model without lacR gene
Sensitivity analysis of parameters
Significant analysis of parameters
In sensitivity analysis, sensitivity does not mean importance, since sensitive parameters are always fixed inherently and unadjustable. We need to combine the sensitivity results with operability, especially for the design of biology experiments. Although T max , K LZ , V 2 , d LS , d LR , GR, d M , d N and u max perform high sensitivity in both methods, these parameters are usually inherently. K m·LP value is adjustable, but to decrease its value would generate fatal disadvantage on the growth of the bacteria. So we do not include this parameter into consideration. Due to the same reason, we exclude K m·GLYC , which is the Michaelis constant for glycolysis. As for the parameter S, which means sensitivity of lacSZ gene transcription to LacS and LacZ proteins, it is difficult to measure and adjust in experiments. Therefore, the parameters which exhibit great influence on the production of lactic acid in both methods are K LR , V 0 and V 1 , with K LR representing RBS strength of lacR gene, and V 0 together with V 1 reflecting characters of the pH-induced promoter.
Other two important features for the promoter are the threshold pH level around which it would start and whether the promoter would keep turning on when the pH value decreased to 3.5 or lower. Akyol  and Madsen  only described the status of pH-induced promoters along with the pH declined to 5.5. Further experiments are needed to test the characteristics of the promoters. This model neglect the effect of lower pH on the promoter, assuming that it would still turn on with the decrease of pH value.
In conclusion, we propose a new method to control the production of lactic acid by L. bulgaricus which is to build a pH-induced switch (promoter) to turn on repressor gene. The proposed method overcomes the disadvantage in bacterial growth by directly changing one of the enzyme related to lactic acid metabolism. Then, we build a reduced model of pH-induced promoter regulation on the production of lactic acid and adjust the model with data from wild type L. bulgaricus. After that we carry out sensitivity analysis of the parameters and figure out three significant ones in the model. To make our switch work well, we need to find an appropriate strength of promoter and/or an appropriate strength of RBS in lacR gene. The values of those two parameters should between the high level and low level we have set in the analysis, so that when the pH value declines to the promoter's starting point, the lacR gene could express moderately to inhibit the production of lactic acid.
This work and its publication is partially supported by the China Natural Science Foundation (Grant Number: 30771371, 31271781), the National High-tech R&D Program of China (863 Program) (Grant Number: 2001AA231091, 2004AA231071), Heilongjiang Province Science Foundation (Grant Number: 2004C0314), Heilongjiang Province key scientific and technological project (Grant Number: WB07C02), HIT Science Foundation (Grant Number: HIT. 2003. 38), National MOST special fund (Grant Number: KCSTE- 2000-JKZX- 021, NCSTE- 2007- JKZX- 022, 2012EG111228).
This article has been published as part of BMC Bioinformatics Volume 15 Supplement 13, 2014: Selected articles from the 9th International Symposium on Bioinformatics Research and Applications (ISBRA'13): Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/15/S13.
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