^{1}

Mathematical modeling and parameter estimation are critical steps in the optimization of biotechnological processes. In the 1,3-propanediol (1,3-PD) production by glycerol fermentation process under anaerobic conditions, 3-hydroxypropionaldehyde (3-HPA) accumulation would arouse an irreversible cessation of the fermentation process. Considering 3-HPA inhibitions to cells growth and to activities of enzymes, we propose a novel mathematical model to describe glycerol continuous cultures. Some properties of the above model are discussed. On the basis of the concentrations of extracellular substances, a parameter identification model is established to determine the kinetic parameters in the presented system. Through the penalty function technique combined with an extension of the state space method, an improved genetic algorithm is then constructed to solve the parameter identification model. An illustrative numerical example shows the appropriateness of the proposed model and the validity of optimization algorithm. Since it is difficult to measure the concentrations of intracellular substances, a quantitative robustness analysis method is given to infer whether the model is plausible for the intracellular substances. Numerical results show that the proposed model is of good robustness.

Microbial conversion of glycerol to 1,3-propanediol (1,3-PD) is particularly attractive in that the process is relatively easy and does not generate toxic byproducts. 1,3-PD has numerous applications in polymers, cosmetics, foods, lubricants, and medicines. Industrial 1,3-PD production has attracted attention as an important monomer to synthesize a new type of polyester, polytrimethylene terephthalate (PTT) [

Anaerobic metabolism pathways of glycerol by

It is critical to formulate the fermentation process using a precise mathematical model in the optimization of biotechnological processes. An excess kinetic model for substrate consumption and product formation was established in previous studies [

Robustness is one of the fundamental characteristics of biological systems. By saying that a system is robust we imply that a particular function or characteristic of the system is preserved despite changes in the operating environment [

Considering 3-HPA inhibitions to cells growth and to activities of the enzymes GDHt and PDOR in glycerol metabolism, we propose a novel mathematical model to describe 1,3-PD production by

This paper is organized as follows. In Section

During continuous fermentation of glycerol metabolism by

the transport of extracellular glycerol across cells membrane by passive diffusion and by glycerol transport facilitator;

intracellular 1,3-PD is expected to be diffused from the intracellular environment to the extracellular medium in the fermentative broth.

Let

Since 3-HPA is a toxic intermediary metabolite, its inhibition to the specific cellular growth rate is introduced besides substrate and products inhibitions. Therefore, the specific cellular growth rate

The specific consumption rate of substrate

The governing equations based on mass balance [

The reductive pathway is emphasized because 3-HPA is the key intermediate for 1,3-PD production. 3-HPA accumulation during fermentation process can cause growth cessation and low product formation [

The intracellular 1,3-PD concentration depends on the conversion of 3-HPA catalyzed by PDOR whose activity is inhibited by the substrate, the diffusion from the intercellular to the extracellular and the dilution effect on cell growth, so whose variation can be formulated by

Now, let

To begin with, we introduce some symbols which will be used below. Let

For the system (

the set

the absolute difference between extracellular and intracellular 1,3-PD and that of glycerol concentration is bounded, that is,

Under the assumptions (H3) and (H4), we can easily verify the following properties of the velocity vector field

For any

For any

For given

Then, the existence and uniqueness of the solution for the system (

For any

The proof can be obtained from Properties

Given

From Theorem

The feasible parameter set

In view of the compactness of

Now, we determine the kinetic parameter

Let

In particular, the stable state of a dynamical system (

For given

Since the orders of magnitude for concentrations involved are different, we adopt the average relative error between the computational values

To determine the parameter values of the system (

Following the above properties, we can conclude the following theorem.

For

In

Obviously, the process

As a result,

Since

We have the following steps:

According to the actual continuous fermentation process, the initial state

The optimal parameter vector in the system (

Conditions | Optimal parameter vector ^{∗} |
---|---|

(16.93, 0.015, 375.9, 0.134, 9.5, 162.8, | |

| |

Substrate-limited condition | |

| |

| |

| |

(36.37, 0.287, 456, 0.878, 0.5, 162.8, | |

0.036, 0.087, 2.235, 2.216, 11.25, 282.1, | |

Substrate-sufficient condition | 39.12, 0.629, 9.841, 3.854, 0.006678, |

13.09, 10.77, 0.590, 62.83, 7.833, 17.39, | |

5.624, 0.353, 0.6145, 0.836, 8.094, 1.254, 0.138) |

The concentrations of biomass change with respect to fermentation time under substrate-limited conditions for

The concentrations of glycerol change with respect to fermentation time under substrate-limited conditions for

The concentrations of 1,3-PD change with respect to fermentation time under substrate-limited conditions for

The concentrations of biomass change with respect to fermentation time under substrate-sufficient conditions for

The concentrations of glycerol change with respect to fermentation time under substrate-sufficient conditions for

The concentrations of 1,3-PD change with respect to fermentation time under substrate-sufficient conditions for

For robust biological systems, we expect that mathematical models that attempt to explain these systems should also be robust. Robustness of the model is analyzed in this section. In view of the glycerol dissimilation mechanism, we assume that

for each

Robustness can be defined as a system’s characteristic that maintains one or more of its functions under external and internal perturbations [

Let

Let

Based on the above analyses, a mathematical definition of biological robustness can be stated as follows.

The robustness measurement of a model

On the basis of the definition of robustness, we develop an Algorithm

We have the following steps:

According to Algorithm

The robustness of the system (

Conditions | Substrate-limited | Substrate-sufficient |

| ||

Values | 3.97268 | 2.70685 |

Robustness simulations of intracellular glycerol concentration under substrate-limited conditions for

Robustness simulations of 3-HPA concentration under substrate-limited conditions for

Robustness simulations of intracellular 1,3-PD concentration under substrate-limited conditions for

Robustness simulations of intracellular glycerol concentration under substrate-sufficient conditions for

Robustness simulations of 3-HPA concentration under substrate-sufficient conditions for

Robustness simulations of intracellular 1,3-PD concentration under substrate-sufficient conditions for

Glycerol bioconversion to 1,3-PD by

The supports of the Natural Science Foundation for the Youth of China (nos. 11001153, 11201267), the Tianyuan Special Funds of the National Natural Science Foundation of China (no. 11126077), the Shandong Province Natural Science Foundation of China (nos. ZR2010AQ016, ZR2011AL003, and BS2012DX025) and the Fundamental Research Funds for the Central Universities (no. DUT12LK27) are gratefully acknowledged.