Cognitive optimization of microbial PHB production in an optimally dispersed bioreactor by single and mixed cultures

Abstract

Cognitive (or intelligent) models are often superior to mechanistic models for nonideal bioreactors. Two kinds of cognitive models—cybernetic and neural—were applied recently to fed-batch fermentation by Ralstonia eutropha in a bioreactor with optimum finite dispersion. In the present work, these models have been applied in simulation studies of co-cultures of R. eutropha and Lactobacillus delbrueckii. The results for both cognitive and mechanistic models have been compared with single cultures. Neural models were the most effective for both types of cultures and mechanistic models the least effective. Simulations with co-culture fermentations predicted more PHB than single cultures with all three types of models. Significantly, the predicted enhancements in PHB concentration by cognitive methods for mixed cultures were four to five times larger than the corresponding increases in biomass concentration. Further improvements are possible through a hybrid combination of all three types of models.

Appendix

The model of Lee et al. [32] for single cultures of R. eutropha

Lee et al. [32] monitored and modeled the evolutions of the concentrations of four state variables—residual biomass, PHB, glucose and ammonium chloride—and proposed the kinetic equations presented below.

$$ \frac{{{\text{d}}X_{1} }}{{{\text{d}}t}} = \mu X_{1} $$

(1)

$$ \frac{{{\text{d}}X_{2} }}{{{\text{d}}t}} = - \sigma_{1} X_{1} $$

(2)

$$ \frac{{{\text{d}}X_{3} }}{{{\text{d}}t}} = - \sigma_{2} X_{1} $$

(3)

$$ \frac{{{\text{d}}X_{4} }}{\text{dt}} = \pi {{X}}_{1} $$

(4)

$$ \frac{{{\text{d}}X_{5} }}{{{\text{d}}t}} = F_{1} + F_{2} $$

(5)

The values of the parameters used by them are listed in Table 5.

Lee et al. [32] took account of the observation of Wang and Yu [28], confirmed later by Khanna and Srivastava [25], that extremely fast synthesis of PHB, leading to high concentrations in the cells, exerts a high metabolic stress and inhibits growth. Together with their results that viable cells can produce PHB even without ammonium, they proposed the following equations for the specific rates of cell growth μ, and polymer synthesis π.

$$ \mu = \mu_{\rm m} \left[ {\frac{{X_{2} }}{{K_{\text{G}} + X_{2} + X_{2}^{2}/K_{\text{GI}} }}} \right]\left[ {\frac{{X_{3} }}{{K_{{N}} + X_{3} + X_{3}^{2}/K_{\text{NI}} }}} \right] $$

(6)

$$ \pi = \pi_{\text{m}} \left[ {1 - \frac{{X_{4}/X}}{{(X_{4}/X)_{\rm m}}}} \right]\left[ {\frac{{X_{2} }}{{K_{\text{PG}} + X_{2} + X_{2}^{2}/K_{\text{PGI}} }}} \right] \times \left[ {\frac{{X_{3} + K_{{P}} }}{{K_{PN} + X_{3} + X_{3}^{2}/K_{\text{PNI}} }}} \right] $$

(7)

The specific rate of glucose utilization is the sum of its rates for biomass synthesis, PHB formation and cell maintenance functions.

$$ \sigma_{1} = \frac{\mu }{{Y_{\text{R/G}} }} + \frac{\mu }{{Y_{{P/G}} }} + m_{\text{e}} $$

(8)

Since depletion of ammonium chloride (or sulfate) provides the stress trigger for the formation of PHB, this is utilized mainly for biomass synthesis at the specific rate:

$$ \sigma_{2} = \frac{\mu }{{Y_{\text{R/N}} }} $$

(9)

The model of Tohyama et al. [23] for co-cultures of R. eutropha and L. delbrueckii

Tohyama et al. [23] proposed the equations presented below for the kinetics without flow terms.

The rate of growth of L. delbrueckii is:

$$ r_{1} = \frac{{{\text{d}}X_{1} }}{{{\text{d}}t}} = \mu_{1} (S,P,O)X_{1} $$

(10)

and that of R. eutropha has a similar form:

$$ r_{2} = \frac{{{\text{d}}X_{2} }}{{{\text{d}}t}} = \mu_{2} (N,P,O)X_{2} $$

(11)

Glucose is utilized by L. delbrueckii at the rate

$$ r_{{S}} = \frac{{{\text{d}}S}}{{{\text{d}}t}} = - \nu_{1} (S,P,O)X_{1} $$

(12)

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

$$ r_{{P}} = \frac{{{\text{d}}P}}{{{\text{d}}t}} = \sigma_{1} (S,P,{\text{DO}})X_{1} - \nu_{2} (N,P,{\text{DO}})X_{2} $$

(13)

The specific rates in Eqs. 10–13 have the forms given below.

$$ \mu_{1} (S,P,O) = \frac{{\mu_{{{\text{m}}1}} (O)S}}{{K_{\text{S}} + S}}\left( {1 - \frac{P}{{P_{\text{m}} }}} \right)^{\text{n}} $$

(14)

$$ \nu_{1} (S,P,O) = \frac{{\mu_{1} (S,P,O)}}{{Y_{{X_{1}/S}} (O)}} + \frac{{\sigma_{1} (S,P,O)}}{{Y_{{P/S}} (O)}} $$

(15)

$$ \sigma_{1} (S,P,O) = \alpha \mu_{1} (S,P,O) + \beta (S,O) $$

(16)

$$ \mu_{2} (N,P,O) = \left( {\frac{{\mu_{{{\text{m}}2}} (O)P}}{{K_{{P}} + P + P^{2}/K_{\text{i}} }}} \right)\left( {\frac{N}{{K_{{N}} + N}}} \right) $$

(17)

$$ \nu_{2} (N,P,O) = \frac{{\mu_{2} (N,P,O)}}{{Y_{{X_{2}/P}} (O)}} $$

(18)

Note that the specific rate of lactate formation, Eq. 16, 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

$$ \beta (S,O) = \frac{{\beta_{\text{m}} (O)S}}{{K_{\text{S}} + S}} $$

(19)

Similar to Eqs. 10–12, the rate of consumption of the nitrogen source is

$$ r_{{N}} = \frac{{{\text{d}}N}}{{{\text{d}}t}} = - \nu_{3} (N,P,O)X_{2} $$

(20)

and that of PHB formation is

$$ r_{{Q}} = \frac{{{\text{d}}Q}}{{{\text{d}}t}} = \sigma_{2} (N)X_{2} $$

(21)

The specific rates in Eqs. 20 and 21 follow:

$$ \sigma_{2} (N) = \frac{{q_{\text{m}} k_{{N}} }}{{k_{{N}} + N}} $$

(22)

and

$$ \nu_{3} (N,P,O) = \frac{{\mu_{2} (N,P,O)}}{{Y_{{X_{2}/N}} (O)}} $$

(23)

Equation 20 might imply that nitrogen is consumed by R. eutropha and not by L. delbrueckii. However, this equation is a practical approximation based on the experimental observations [22, 23] that ammonium concentration changed little during the cultivation of L. delbrueckii (X ₁) compared to the changes generated by R. eutropha (X ₂). This observation is also reflected in the absence of a nitrogen term for μ in Eq. 14. Equations 17 and 22 similarly express the observations that cell growth increased with ammonium concentration while PHB was favored by reducing the ammonium concentration.

Dissolved oxygen has a critical role in a two-culture fermentation. It affects not only the net formation of lactate but also some of the kinetic parameters. Based on their experimental results, Tohyama et al. [23] proposed the equations given here for these parameters in their model.

$$ \mu_{\text{m1}} = a_{1} { \exp }( - a_{2} O) + a_{3} $$

(24)

$$ Y_{{P/S}} = b_{1} { \exp }( - b_{2} {{O}}) + b_{3} $$

(25)

$$ \beta_{\text{m}} = c_{1} { \exp }( - c_{2} ) + c_{3} $$

(26)

$$ \mu_{\text{m2}} = d_{1} { \exp }( - d_{2} {{O}}) + d_{3} $$

(27)

$$ Y_{{X_{2}/P}} = f_{1} { \exp }( - f_{2} O) + f_{3} $$

(28)

$$ Y_{{Q/P}} = g_{1} { \exp }( - g_{2} O) + g_{3} $$

(29)

The values of the empirical constants are given in Table 6. The kinetic equations of either the pure culture or the co-culture of R. eutropha and L. delbrueckii are plugged into the bioreactor model with finite dispersion. The general forms of these mass balances are as expressed below.

Table 5 Values of the parameters in the model of Lee et al. [32]

Table 6 Values of the parameters in the model of Tohyama et al. [23]

$$ \tau \frac{{\partial \bar{y}}}{\partial t} = \frac{1}{Pe}\frac{{\partial^{2} \bar{y}}}{{\partial z^{2} }} - \frac{{\partial \bar{y}}}{\partial z} \pm \tau \bar{r} $$

(30)

Here τ is the residence time in the bioreactor, i.e. the reciprocal of the dilution rate, and z the dimensionless radial distance from the central axis. $\bar{y}$ is the vector of concentrations and ${\bar{\text{r}}}$ the corresponding vector of reaction rates give above [23, 32]. The sign of the last term is positive for the concentrations of the bacteria and the product PHB, and negative for the substrates. Equation 30 is solved under the following initial and boundary conditions.

$$ t = 0:\bar{y} = \bar{y}_{0} \forall z \; ({\text{homogeneity}}) $$

(31)

$$ z = 0:\frac{{\partial \bar{y}}}{\partial z} = 0\;({\text{symmetry}}) $$

(32)

$$ z = 1:\frac{{\partial \bar{y}}}{\partial z} = 0\;({\text{no outflow}}) $$

(33)