Under conditions adverse to cell growth, many bacteria synthesize polyhydroxyalkanoates (PHAs) as energy storage devices. Poly-β-hydroxybutyrate (PHB) is possibly the most prominent member of the PHA family. There is growing commercial interest in PHB since many of its physical, chemical and rheological properties are comparable to those of widely used polymers such as polyethylene and polypropylene 1. While the latter polymers are synthesized chemically from petroleum sources under harsh conditions, PHB can be synthesized microbially under mild conditions. In addition, PHB can be readily biodegraded, whereas petroleum-based polymers are difficult to degrade and therefore create disposal problems 2. These advantages makes PHB a potential replacement for petroleum-based polymers in a variety of applications such as food packaging films, biodegradable carriers for medicines and insecticides, disposable cosmetic products and absorbable surgical sutures 34.

Bacteria such as Ralstonia eutropha (formerly Alcaligenes eutrophus), Alcaligenes latus and Azotobacter vivelandii may be induced to synthesize PHB by imposing a chemical stress. This is normally done by depriving the organism of a nutrient such as nitrogen or phosphorus or sulfur, which are required for cell growth 234. Of these, nitrogen is the preferred stress-creator 5678, but recent work 9 points to the possibility of limiting the supply of phosphorus to generate PHB. Even though a shortage of nitrogen induces PHB synthesis, an excessive lack of this nutrient retards cell growth 5 and promotes depolymerization of PHB 10. In addition, there should be sufficient amount of a carbon source at all times. However, similar to nitrogen, an abundance of carbon is detrimental to the growth of R. eutropha 5. Therefore, the proper supply of these two substrates is critical to the overall production of PHB.

The complexity of the metabolic network 23 and the involvement of carbon and nitrogen suggest that the feed rates of these substrates may have to be varied nonlinearly with time. This requirement is best provided by fed-batch fermentation. While the two feed rates for fermentations based on R. eutropha alone have been varied either through on-line control based on glucose or PHB or the CO₂ evolution rate 1112 or through discrete changes decided in advance 678, the rates where two cultures are employed have been controlled through the lactate concentration 13, a key intermediate in a two-culture system. Since such a mixed culture fermentation has been studied here, it will be described later in detail.

While the effects of manipulating carbon and nitrogen supply have been analyzed adequately 678911, the role of dissolved oxygen (DO) has received less attention. Nevertheless, the importance of maintaining a proper level of the DO concentration has been recognized by many investigators 567814 without quantitatively modeling its effects. Most of these studies have implicitly accommodated Kim's 15 observation that a low DO concentration favors PHB formation but inhibits cell growth, and maintained DO at around 30% of saturation. The role of DO in a metabolic context has been discussed in section 4, and it becomes more significant when two complementary cultures are used. In large (production-scale) bioreactors, disturbances in the DO level are more likely and more difficult to control than those in the biomass and the liquid substrates 1617. Since, as explained below, DO plays as important a role as carbon and nitrogen, sensitivity of the fermentation to a perturbation in the DO concentration has considerable practical importance and is therefore the subject of the present work.

R. eutropha is the most widely used organism for PHB production because it is easy to cultivate, its metabolism is well understood and it can accumulate large amounts of PHB (up to 80% of dry cell mass 34) inside the cells. As mentioned in the Introduction, the synthesis of PHB may be triggered by stress created by a shortage of nitrogen or phosphorus and adequate supply of carbon. However, an exceedingly low concentration of nitrogen and a preponderance of carbon inhibit growth 5 and promote degradation of the polymer 10. Therefore, the supply of these two substrates have to be controlled as the fermentation progresses, and this is best achieved by fed-batch operation 6789.

Fructose and glucose are the common carbon sources, and either ammonium chloride or ammonium sulfate provides nitrogen. The fermentation is aerobic and the oxygen content of the broth influences PHB formation. A low DO concentration leads to an excess of reduced co-enzymes (NADH and NADPH), thus enabling a higher carbon flux directed toward PHB synthesis for reoxidation of these co-enzymes 18. However, a severe limitation of oxygen causes formation of intermediates of the Krebs cycle that inhibit the formation of PHB 2.

Thus, control of DO concentration is as critical as that of carbon and nitrogen. This is even more important when a pair of complementary organisms are used in place of one. The rationale for using R. eutropha in conjunction with another organism arises from the observation that R. eutropha is sluggish in metabolizing fructose and glucose but can utilize organic acids such as acetate, butyrate and lactate more easily 19. So, some investigators have used another organism such as Lactococcus lactis 20 or Lactobacillus delbrueckii 13 to convert the sugar to an organic acid, which is then utilized by R. eutropha. Such a two-culture system can generate higher concentrations of PHB than R. eutropha alone. The present analysis is based on Tohyama and Shimizu 13 because this system works well with one bioreactor whereas the L. lactis-R. eutropha combination required two stages of cultivation 20.

Now, L. delbrueckii is anaerobic whereas R. eutropha is aerobic. In a fed-batch system, L. delbrueckii is introduced at the start so as to utilize glucose and produce lactate. This requires a low concentration of DO. After sufficient amount of lactate has been generated, the reactor is inoculated with R. eutropha. At this stage there are conflicting requirements. R. eutropha requires a high DO concentration to metabolize lactate, where production of lactate by L. delbrueckii requires a low DO concentration. Thus, control of lactate concentration by manipulating DO becomes a critical factor.

The DO concentration is usually controlled by varying the stirring speed or the flow rate of the gas 561112. The implication here is that faster agitation or flow promotes better gas-to-liquid mass transfer of oxygen and thereby increases the DO level. A change in the DO level can occur due to many reasons such as a disturbance in the gaseous feed stream, fluctuations in the stirring speed and a change in the rheology of the broth. These effects become reflected in the DO concentration only after transfer of oxygen into the liquid phase. The involvement of more than one variable in determining the DO concentration and the intervention of inter-phase transport resistance may complicate, delay and attenuate the sensing of a change in the DO concentration after the occurrence of the source of disturbance. Since the DO is usually monitored and used as a measure of oxygenation, sensitivity of the fermentation to a perturbation sensed in the DO concentration is important for its performance. The sensitivity method and the fermentation model on which it is based are described next.

The analysis presented here is based on the fed-batch fermentation with L. delbrueckii and R. eutropha, studied by Tohyama et al. 2122 and modeled in their latter work. They resolved the problem of conflicting oxygen requirements by maintaining a low DO concentration (0.5 ppm) initially to favor L. delbrueckii and then increasing this (3.0 ppm) after inoculation by R. eutropha. Thereafter, since both bacteria have to function in tandem, the DO was alternated between the two levels every hour. The fermentation was run for 30 h.

In the absence of flow terms and for a completely homogeneous broth, Tohyama et al. 2122 proposed the equations presented below.

The rate of growth of L. delbrueckii is

r 1 = dX 1 dt = μ 1 ( S , P , O ) X 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOCai3aaSbaaSqaaiabigdaXaqabaGccqGH9aqpjuaGdaWcaaqaaiabbsgaKjabbIfaynaaBaaabaGaeGymaedabeaaaeaacqqGKbazcqqG0baDaaGccqGH9aqpcqaH8oqBdaWgaaWcbaGaeGymaedabeaakiabcIcaOiabbofatjabcYcaSiabbcfaqjabcYcaSiabb+eapjabcMcaPiabbIfaynaaBaaaleaacqaIXaqmaeqaaaaa@43D7@

and that of R. eutropha has a similar form:

r 2 = dX 2 dt = μ 2 ( N , P , O ) X 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOCai3aaSbaaSqaaiabikdaYaqabaGccqGH9aqpjuaGdaWcaaqaaiabbsgaKjabbIfaynaaBaaabaGaeGOmaidabeaaaeaacqqGKbazcqqG0baDaaGccqGH9aqpcqaH8oqBdaWgaaWcbaGaeGOmaidabeaakiabcIcaOiabb6eaojabcYcaSiabbcfaqjabcYcaSiabb+eapjabcMcaPiabbIfaynaaBaaaleaacqaIYaGmaeqaaaaa@43D5@

Glucose is utilized by L. delbrueckii at the rate

r S = dS dt = − ν 1 ( S , P , O ) X 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOCai3aaSbaaSqaaiabbofatbqabaGccqGH9aqpjuaGdaWcaaqaaiabbsgaKjabbofatbqaaiabbsgaKjabbsha0baakiabg2da9iabgkHiTiabe27aUnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaee4uamLaeiilaWIaeeiuaaLaeiilaWIaee4ta8KaeiykaKIaeeiwaG1aaSbaaSqaaiabigdaXaqabaaaaa@43E8@

Lactate is the product of glucose consumption and it is the carbon substrate for R. eutropha, so its net rate of formation is

r P = dP dt = σ 1 ( S , P , O ) X 1 − ν 2 ( N , P , O ) X 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOCai3aaSbaaSqaaiabbcfaqbqabaGccqGH9aqpjuaGdaWcaaqaaiabbsgaKjabbcfaqbqaaiabbsgaKjabbsha0baakiabg2da9iabeo8aZnaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaee4uamLaeiilaWIaeeiuaaLaeiilaWIaee4ta8KaeiykaKIaeeiwaG1aaSbaaSqaaiabigdaXaqabaGccqGHsislcqaH9oGBdaWgaaWcbaGaeGOmaidabeaakiabcIcaOiabb6eaojabcYcaSiabbcfaqjabcYcaSiabb+eapjabcMcaPiabbIfaynaaBaaaleaacqaIYaGmaeqaaaaa@5007@

The specific rates in Eqs. (1)–(4) have the forms given below.

μ 1 ( S , P , O ) = μ m 1 ( O ) S K S + S ( 1 − P P m ) n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiVd02aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqqGtbWucqGGSaalcqqGqbaucqGGSaalcqqGpbWtcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabeY7aTnaaBaaabaGaeeyBa0MaeGymaedabeaacqGGOaakcqqGpbWtcqGGPaqkcqqGtbWuaeaacqqGlbWsdaWgaaqaaiabbofatbqabaGaey4kaSIaee4uamfaaOWaaeWaaeaacqaIXaqmcqGHsisljuaGdaWcaaqaaiabbcfaqbqaaiabbcfaqnaaBaaabaGaeeyBa0gabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabb6gaUbaaaaa@4DAC@

ν 1 ( S , P , O ) = μ 1 ( S , P , O ) Y X1/S ( O ) + σ 1 ( S , P , O ) Y P / S ( O ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5BC1@

σ₁ (S, P, O) = αμ₁ (S, P, O) + β(S, O)

μ 2 ( N , P , O ) = ( μ m 2 ( O ) P K P + P + P 2 / K i ) ( N K N + N ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiVd02aaSbaaSqaaiabikdaYaqabaGccqGGOaakcqqGobGtcqGGSaalcqqGqbaucqGGSaalcqqGpbWtcqGGPaqkcqGH9aqpjuaGdaqadaqaamaalaaabaGaeqiVd02aaSbaaeaacqqGTbqBcqaIYaGmaeqaaiabcIcaOiabb+eapjabcMcaPiabbcfaqbqaaiabbUealnaaBaaabaGaeeiuaafabeaacqGHRaWkcqqGqbaucqGHRaWkcqqGqbaudaahaaqabeaacqaIYaGmaaGaei4la8Iaee4saS0aaSbaaeaacqqGPbqAaeqaaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGaeeOta4eabaGaee4saS0aaSbaaeaacqqGobGtaeqaaiabgUcaRiabb6eaobaaaiaawIcacaGLPaaaaaa@5361@

ν 2 ( N , P , O ) = μ 2 ( N , P , O ) Y X 2 / P ( O ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyVd42aaSbaaSqaaiabikdaYaqabaGccqGGOaakcqqGobGtcqGGSaalcqqGqbaucqGGSaalcqqGpbWtcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabeY7aTnaaBaaabaGaeGOmaidabeaacqGGOaakcqqGobGtcqGGSaalcqqGqbaucqGGSaalcqqGpbWtcqGGPaqkaeaacqqGzbqwdaWgaaqaaiabbIfayjabikdaYiabc+caViabbcfaqbqabaGaeiikaGIaee4ta8KaeiykaKcaaaaa@4999@

Note that the specific rate of lactate formation, Eq. (7), has a constitutive component, β, and a growth-related component, αμ₁. This arises because glucose is utilized by L. delbrueckii for growth as well as lactate synthesis. The constitutive rate has the form

β ( S,O ) = β m ( O ) S K S + S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqOSdiMaeiikaGIaee4uamLaeeilaWIaee4ta8KaeiykaKIaeyypa0tcfa4aaSaaaeaacqaHYoGydaWgaaqaaiabb2gaTbqabaGaeiikaGIaee4ta8KaeiykaKIaee4uamfabaGaee4saS0aaSbaaeaacqqGtbWuaeqaaiabgUcaRiabbofatbaaaaa@3FEC@

Similar to Eqs. (1)–(3), the rate of consumption of the nitrogen source is

r N = dN dt = − ν 3 ( N , P , O ) X 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOCai3aaSbaaSqaaiabb6eaobqabaGccqGH9aqpjuaGdaWcaaqaaiabbsgaKjabb6eaobqaaiabbsgaKjabbsha0baakiabg2da9iabgkHiTiabe27aUnaaBaaaleaacqaIZaWmaeqaaOGaeiikaGIaeeOta4KaeiilaWIaeeiuaaLaeiilaWIaee4ta8KaeiykaKIaeeiwaG1aaSbaaSqaaiabikdaYaqabaaaaa@43D0@

and that of PHB formation is

r Q = dQ dt = σ 2 ( N ) X 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOCai3aaSbaaSqaaiabbgfarbqabaGccqGH9aqpjuaGdaWcaaqaaiabbsgaKjabbgfarbqaaiabbsgaKjabbsha0baakiabg2da9iabeo8aZnaaBaaaleaacqaIYaGmaeqaaOGaeiikaGIaeeOta4KaeiykaKIaeeiwaG1aaSbaaSqaaiabikdaYaqabaaaaa@3EEC@

The specific rates in Eqs. (11) and (12) follow

σ 2 ( N ) = q m k N k N + N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aaSbaaSqaaiabbkdaYaqabaGccqGGOaakcqqGobGtcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabbghaXnaaBaaabaGaeeyBa0gabeaacqqGRbWAdaWgaaqaaiabb6eaobqabaaabaGaee4AaS2aaSbaaeaacqqGobGtaeqaaiabgUcaRiabb6eaobaaaaa@3DB2@

and

ν 3 ( N , P , O ) = μ 2 ( N , P , O ) Y X 2 / N ( O ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyVd42aaSbaaSqaaiabiodaZaqabaGccqGGOaakcqqGobGtcqGGSaalcqqGqbaucqGGSaalcqqGpbWtcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabeY7aTnaaBaaabaGaeGOmaidabeaacqGGOaakcqqGobGtcqGGSaalcqqGqbaucqGGSaalcqqGpbWtcqGGPaqkaeaacqqGzbqwdaWgaaqaaiabbIfayjabikdaYiabc+caViabb6eaobqabaGaeiikaGIaee4ta8KaeiykaKcaaaaa@4997@

While Tohyama et al. 22 have discussed the model in detail, certain salient features may be noted here. Equation (11) might imply that nitrogen is consumed by R. eutropha and not by L. delbrueckii. This may appear implausible. In fact, Eq. (11) is a practical approximation based on the experimental observations 2122 that ammonia concentration changed little during the cultivation of L. delbrueckii (X₁) compared to the changes generated by R. eutropha (X₂). This observation is also reflected in the absence of a nitrogen term for μ in Eq. (5). Equations (8) and (13) similarly express the observations that cell growth increased with ammonium concentration while PHB was favored by reducing the ammonium concentration.

Lactate is the critical intermediate linking the growth of L. delbrueckii and R. eutropha. As the model shows, the carbon substrate (glucose) and DO concentrations control the production of lactate whereas ammonium sulfate and DO control its consumption. This establishes the central role of DO and consequently the importance of any perturbations in this concentration.

The degree of dispersion in a stirred reactor may be characterized by the Peclet number

Pe = uL/D_e

where the 'characteristic length' L is usually the diameter of the vessel. For finite dispersion, the kinetics of Eqs. (1)–(4), (11) and (12) may be incorporated into the mass balances for a fed-batch bioreactor 2324 to obtain the model presented below.

V F ∂ X 1 ∂ t = ∂ 2 X 1 ∂ z 2 − Pe ∂ X 1 ∂ z + V F r 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGwbGvaeaacqqGgbGraaWaaSaaaeaacqGHciITcqqGybawdaWgaaqaaiabigdaXaqabaaabaGaeyOaIyRaeeiDaqhaaOGaeyypa0tcfa4aaSaaaeaacqGHciITdaahaaqabeaacqaIYaGmaaGaeeiwaG1aaSbaaeaacqaIXaqmaeqaaaqaaiabgkGi2kabbQha6naaCaaabeqaaiabikdaYaaaaaGccqGHsislcqqGqbaucqqGLbqzjuaGdaWcaaqaaiabgkGi2kabbIfaynaaBaaabaGaeGymaedabeaaaeaacqGHciITcqqG6bGEaaGccqGHRaWkjuaGdaWcaaqaaiabbAfawbqaaiabbAeagbaakiabbkhaYnaaBaaaleaacqaIXaqmaeqaaaaa@50FC@

V F ∂ X 2 ∂ t = ∂ 2 X 2 ∂ z 2 − Pe ∂ X 2 ∂ z + V F r 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGwbGvaeaacqqGgbGraaWaaSaaaeaacqGHciITcqqGybawdaWgaaqaaiabikdaYaqabaaabaGaeyOaIyRaeeiDaqhaaOGaeyypa0tcfa4aaSaaaeaacqGHciITdaahaaqabeaacqaIYaGmaaGaeeiwaG1aaSbaaeaacqaIYaGmaeqaaaqaaiabgkGi2kabbQha6naaCaaabeqaaiabikdaYaaaaaGccqGHsislcqqGqbaucqqGLbqzjuaGdaWcaaqaaiabgkGi2kabbIfaynaaBaaabaGaeGOmaidabeaaaeaacqGHciITcqqG6bGEaaGccqGHRaWkjuaGdaWcaaqaaiabbAfawbqaaiabbAeagbaakiabbkhaYnaaBaaaleaacqaIYaGmaeqaaaaa@5104@

V F ∂ P ∂ t = ∂ 2 P ∂ z 2 − Pe ∂ P ∂ z + V F r P MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGwbGvaeaacqqGgbGraaWaaSaaaeaacqGHciITcqqGqbauaeaacqGHciITcqqG0baDaaGccqGH9aqpjuaGdaWcaaqaaiabgkGi2oaaCaaabeqaaiabikdaYaaacqqGqbauaeaacqGHciITcqqG6bGEdaahaaqabeaacqaIYaGmaaaaaOGaeyOeI0IaeeiuaaLaeeyzauwcfa4aaSaaaeaacqGHciITcqqGqbauaeaacqGHciITcqqG6bGEaaGccqGHRaWkjuaGdaWcaaqaaiabbAfawbqaaiabbAeagbaakiabbkhaYnaaBaaaleaacqqGqbauaeqaaaaa@4DD0@

V F ∂ Q ∂ t = ∂ 2 Q ∂ z 2 − Pe ∂ Q ∂ z + V F r Q MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGwbGvaeaacqqGgbGraaWaaSaaaeaacqGHciITcqqGrbquaeaacqGHciITcqqG0baDaaGccqGH9aqpjuaGdaWcaaqaaiabgkGi2oaaCaaabeqaaiabikdaYaaacqqGrbquaeaacqGHciITcqqG6bGEdaahaaqabeaacqaIYaGmaaaaaOGaeyOeI0IaeeiuaaLaeeyzauwcfa4aaSaaaeaacqGHciITcqqGrbquaeaacqGHciITcqqG6bGEaaGccqGHRaWkjuaGdaWcaaqaaiabbAfawbqaaiabbAeagbaakiabbkhaYnaaBaaaleaacqqGrbquaeqaaaaa@4DD8@

V F ∂ S ∂ t = ∂ 2 S ∂ z 2 − Pe ∂ S ∂ z + V F r S + F C F S f MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGwbGvaeaacqqGgbGraaWaaSaaaeaacqGHciITcqqGtbWuaeaacqGHciITcqqG0baDaaGccqGH9aqpjuaGdaWcaaqaaiabgkGi2oaaCaaabeqaaiabikdaYaaacqqGtbWuaeaacqGHciITcqqG6bGEdaahaaqabeaacqaIYaGmaaaaaOGaeyOeI0IaeeiuaaLaeeyzauwcfa4aaSaaaeaacqGHciITcqqGtbWuaeaacqGHciITcqqG6bGEaaGccqGHRaWkjuaGdaWcaaqaaiabbAfawbqaaiabbAeagbaakiabbkhaYnaaBaaaleaacqqGtbWuaeqaaOGaey4kaSscfa4aaSaaaeaacqqGgbGrdaWgaaqaaiabboeadbqabaaabaGaeeOrayeaaOGaee4uam1aaSbaaSqaaiabbAgaMbqabaaaaa@557C@

V F ∂ N ∂ t = ∂ 2 N ∂ z 2 − Pe ∂ N ∂ z + V F r N + F N F N f MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGwbGvaeaacqqGgbGraaWaaSaaaeaacqGHciITcqqGobGtaeaacqGHciITcqqG0baDaaGccqGH9aqpjuaGdaWcaaqaaiabgkGi2oaaCaaabeqaaiabikdaYaaacqqGobGtaeaacqGHciITcqqG6bGEdaahaaqabeaacqaIYaGmaaaaaOGaeyOeI0IaeeiuaaLaeeyzauwcfa4aaSaaaeaacqGHciITcqqGobGtaeaacqGHciITcqqG6bGEaaGccqGHRaWkjuaGdaWcaaqaaiabbAfawbqaaiabbAeagbaakiabbkhaYnaaBaaaleaacqqGobGtaeqaaOGaey4kaSscfa4aaSaaaeaacqqGgbGrdaWgaaqaaiabb6eaobqabaaabaGaeeOrayeaaOGaeeOta40aaSbaaSqaaiabbAgaMbqabaaaaa@5560@

dV dt = F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGKbazcqqGwbGvaeaacqqGKbazcqqG0baDaaGccqGH9aqpcqqGgbGraaa@3424@

Here F = F_S + F_N is the total inflow rate. The total inflow enters as a term in all equations since it dilutes the concentrations appropriately. However, only Eqs. (20) and (21) contain the inflow (or feed) concentrations because the flow stream contains only the carbon (S) and nitrogen (N) substrates.

Equations (16)–(22) are subject to the following initial and boundary conditions:

t = 0: X₁ = X₁₀, X₂ = X₂₀, P = P₀ = 0, Q = Q₀ = 0, S = S₀, N = N₀, V = V₀

z = 0 : ∂ X 1 ∂ z = ∂ X 2 ∂ z = ∂ P ∂ z = ∂ Q ∂ z = ∂ S ∂ z = ∂ N ∂ z = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOEaONaeyypa0JaeGimaaJaeiOoaOtcfa4aaSaaaeaacqGHciITcqqGybawdaWgaaqaaiabigdaXaqabaaabaGaeyOaIyRaeeOEaOhaaOGaeyypa0tcfa4aaSaaaeaacqGHciITcqqGybawdaWgaaqaaiabikdaYaqabaaabaGaeyOaIyRaeeOEaOhaaOGaeyypa0tcfa4aaSaaaeaacqGHciITcqqGqbauaeaacqGHciITcqqG6bGEaaGccqGH9aqpjuaGdaWcaaqaaiabgkGi2kabbgfarbqaaiabgkGi2kabbQha6baakiabg2da9KqbaoaalaaabaGaeyOaIyRaee4uamfabaGaeyOaIyRaeeOEaOhaaOGaeyypa0tcfa4aaSaaaeaacqGHciITcqqGobGtaeaacqGHciITcqqG6bGEaaGccqGH9aqpcqaIWaamaaa@5E6C@

z = 1: X₁ = X₂ = P = Q = 0, S = S_f, N = N_f

Given that z = 0 corresponds to the central axis of the reaction vessel and z = 1 the periphery at the point of introduction of the feed stream, Eq. (24) expresses symmetry around the impeller shaft.

In their experiments, Tohyama et al. 22 applied discrete injections of glucose and ammonium to control lactate concentration at a set point. This, however, does not generate a truly optimum feed policy, which can be derived by Pontryagin's maximum principle. However, the maximum principle is susceptible to singularities that necessitate difficult manipulations 12. The chemotaxis algorithm provides a simpler, more practical alternative that generates solutions close to the optimum solution, and it has been effective in improving a fed-batch PHB fermentation with R. eutropha alone 24. Here the feed rate (of either substrate) is expressed as a polynomial function of time:

F i ( t ) = ∑ k = 0 M a k ( t / T ) k ; i = S or N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOray0aaSbaaSqaaiabbMgaPbqabaGccqGGOaakcqqG0baDcqGGPaqkcqGH9aqpdaaeWbqaaiabbggaHnaaBaaaleaacqqGRbWAaeqaaOGaeiikaGIaeeiDaqNaei4la8IaeeivaqLaeiykaKYaaWbaaSqabeaacqqGRbWAaaGccqGG7aWocqqGPbqAcqGH9aqpcqqGtbWucqqGGaaicqqGVbWBcqqGYbGCcqqGGaaicqqGobGtaSqaaiabbUgaRjabg2da9iabicdaWaqaaiabb2eanbqdcqGHris5aaaa@4D51@

For a fully dispersed broth, D_e → ∞ and hence Pe → 0. When there is no dispersion, called segregated or plug flow, D_e → 0 and Pe → ∞. Both these are idealized limits. Since production-scale reactors have finite non-zero values of Pe, the effect of Pe on the sensitivity is part of the current analysis. Given a value of Pe, the mass balances for a fed-batch bioreactor may be written by incorporating the kinetics presented above.

The complete model was solved numerically for Pe = 0.01, 20 and 60, using the parameter data in Table 1. The significance of these choices is explained later. Then the DO value was perturbed both positively and negatively, and the model solved again. Let y_l, y¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafeyEaKNbaebaaaa@2D66@ and y_u denote the (time-dependent) outputs of a variable y at the lower, the average and the upper values of DO.

The effect of a disturbance in the DO level on an output variable y may be quantified by the response coefficient, defined as 25

δ y = O ¯ y ¯ ∂ y ∂ ( O ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiTdq2aaSbaaSqaaiabbMha5bqabaGccqGH9aqpjuaGdaWcaaqaaiqbb+eapzaaraaabaGafeyEaKNbaebaaaWaaSaaaeaacqGHciITcqqG5bqEaeaacqGHciITcqGGOaakcqqGpbWtcqGGPaqkaaaaaa@3B15@

Using the perturbed values mentioned above, Eq. (27) may be approximated as 26

δ y = O ¯ y ¯ [ y ¯ − y 1 − 2 h 2 h 2 ] y 1 + [ y 1 − y ¯ + h h 2 ] y ¯ + [ y ¯ − y 1 2 h 2 ] y u MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6277@

where h is the distance between two consecutive values of the DO concentration. Since Eqs. (27) and (28) generate dimensionless numbers, it becomes possible to compare the responses of different variables. Obviously, the higher the coefficient the more sensitive is that variable.

Previous studies, for both PHB 24 and other fermentations 2027, have shown that a finite optimum degree of dispersion generated higher amounts of the product than complete dispersion or complete segregation. For fed-batch fermentation with R. eutropha, the productivity of PHB was maximum for dispersion corresponding to Pe ≈ 20 24. Therefore, sensitivity of the fermentation to DO perturbations was determined for this value of Pe and compared with the two asymptotic limits corresponding to Pe = 0.01 and Pe = 60. The lowest limit corresponds to a case of complete dispersion that is typical of laboratory-scale bioreactors such as that used by Tohyama et al. 21, while the latter extreme (Pe = 60) indicates the absence of any significant dispersion, as in segregated flow. (Ideally, Pe = 0 for complete dispersion but this created numerical difficulties, and hence a small finite value of Pe was used). To calculate the response coefficients according to Eqs. (16) and (17), perturbations were applied to the DO concentration in both the lean phase (DO = 0.5 ppm) and the rich phase (DO = 3 ppm). Disturbances during the lean phase did not have a significant effect on the performance, i.e. the response coefficients were close to zero. So the results presented are all for the rich phase. When the DO concentration is 0.5 ppm, the coefficients may be low because this phase is mainly to replenish lactate by reducing oxygen availability to favor glucose metabolism by L. delbrueckii. Ralsonia, the oxygen-dependent partner in the mixed culture, is less active during this phase.

Figures 1 to 6 display the temporal variations of the response coefficients to a perturbation in the DO concentration for each of the three values of Pe considered. It is instructive to analyze these in four groups. First consider the two bacterial species. Their response coefficients (Figures 1 and 2) show opposite trends at each Peclet number. This difference may be attributed to the dissimilar affinities of the organisms to oxygen. While L. delbrueckii grows in the absence of oxygen, R. eutropha is an aerobe and thus requires DO. For quantitative comparisons, the minimum and maximum coefficients for each concentration variable and each Peclet number have been compiled in Table 2. The coefficients for L. delbrueckii and R. eutropha also differ by an order of magnitude, and this difference is continued between the two principal substrates, glucose and ammonium sulfate (Figures 3 and 4).

Like R. eutropha vis-à-vis L. delbrueckii, ammonium sulfate is present in much smaller concentrations than glucose, and their response coefficients too differ by an order of magnitude (Table 2). These differences and their contrasting profiles (Figures 3 and 4) illustrate the dynamic effects of the metabolic roles of oxygen and nitrogen in the PHB synthesis network. L. delbrueckii converts glucose first to pyruvate and then to lactate by utilizing NADPH. R. eutropha metabolizes this lactate to acetyl-CoA, which serves as a precursor for PHB through a sequence of three enzymatic reactions 2313. Under heterotrophic conditions, R. eutropha generates its ATP requirement through the TCA cycle. With lactate, this occurs either through a glyoxylate shunt or from pyruvate via a phosphoenolpyruvate synthase reaction 213. At high ammonium concentration, the NADPH is preferentially utilized for the reaction from α-ketoglutarate to glutamic acid and glutamin, thus reducing the availability of NADPH for PHB synthesis. Limiting the ammonium concentration blocks the synthesis of amino acids, decreases the flow of NADPH through the glyoxylate pathway, and thereby facilitates PHB synthesis.

The effect of DO is similar to that of ammonium. A low DO concentration also leads to an excess of NADH and NADPH, thus promoting PHB formation 13. However, a severe shortage of oxygen in the medium retards PHB biosynthesis 28, just as strong starvation of nitrogen inhibits cell growth 5 and thus diminishes the PHB concentration. By contrast, glucose utilization by L. delbrueckii is relatively unaffected by oxygen availability. Moreover, while nitrogen favors cell growth up to small concentrations and is then inhibitory, glucose and fructose have a positive effect over a much wider range before inhibition begins 514.

These differences may explain partly the contrasting profiles for glucose (Figure 3) and ammonium sulfate (Figure 4). Another factor that may account for the differences between Figures 1 and 2 and between Figures 3 and 4 is the difference in the magnitudes of the pair of concentrations in each group. According to Table 1 and the concentration profiles obtained by Tohyama et al. 2122, L. delbrueckii and glucose have much larger concentrations than R. eutropha and ammonium sulfate. Therefore a disturbance in the DO concentration is likely to have a smaller effect on the former two than on the latter pair of variables. The possibility of large variables to function effectively as inertial sinks for disturbances is supported by similar observations with Klebsiella oxytoca 29 and Escherichia coli 30 cultures.

While the response coefficients of lactate (Figure 5) and PHB (Figure 6) qualitatively follow the same trends as the other variables, it is significant that the coefficients of lactate are one to two orders of magnitude larger than those of others. As explained before, lactate is produced by L. delbrueckii and consumed by R. eutropha. Tohyama et al. 2122 observed that as the initial concentration of lactate increases, so does its inhibitory effect on both L. delbrueckii and R. eutropha. So they recommended maintaining a low lactate concentration at about KPKi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqqGlbWsdaWgaaWcbaGaeeiuaafabeaakiabbUealnaaBaaaleaacqqGPbqAaeqaaaqabaaaaa@3101@. Now, recall that