A STOCHASTIC MODEL FOR MICROBIAL FERMENTATION PROCESS UNDER GAUSSIAN WHITE NOISE ENVIRONMENT

. In this paper, we propose a stochastic model for the microbial fermentation process under the framework of white noise analysis, where Gaussian white noises are used to model the environmental noises and the speciﬁc growth rate is driven by Gaussian white noises. In order to keep the regularity of the terminal time, the adjustment factors are added in the volatility coeﬃcients of the stochastic model. Then we prove some fundamental properties of the stochastic model: the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on the initial values.


1.
Introduction. In recent years there has been an increasing attention in microorganisms and their activities. This is due to their lower cost, higher production and no pollution in industrial applications [13,14,2]. In order to understand the exact mechanism of microbial fermentation, many works have been done to develop mathematical models simulating the actual microbial fermentation processes. We take the production of L-glutamic acid (LGA) by Corynebacterium glutamicum as an example, where LGA is produced commercially under aerobic fermentation conditions, the C. glutamicum undergo stresses, and the specific growth rate of C. glutamicum is continuously decreasing with the increase of product concentration (i.e. product inhibition occurs) and a low level of dissolved oxygen concentration ( [5,18,4,7,8]). On the base of formal kinetic approach, Bona and Moser [3] proposed a mathematical model for L-glutamic acid production process, which included formal growth inhibition with product and production repression with substrate. Suresh et al. [16] developed the model in [3] by considering the growth-associated product formation conditions. Tian et al [17] presented a mathematical model of bioprocess with product inhibition and impulse effect. In these models, the microbial fermentation process are described by deterministic differential equations.
The deterministic microbial fermentation model requires that the parameters involved are completely known. However, in the real-world system, some parameters of the microbial fermentation model may fluctuate due to the environmental noises, which are usually random. Therefore stochastic influences are considered by many authors in the model to conform with the actual fermentation. For example, Soboleva et al [15] presented a stochastic model for calculating the time-dependent probability distribution of the microbial population size under arbitrary changes of temperature through time. Albert et al [1] gave a stochastic Listeria monocytogenes growth model in milk at the farm bulk tank stage. Kutalik et al [9] presented deterministic models of the lag and subsequent growth of a bacterial population and analyzed their connection with stochastic models for the lag and subsequent generation times of individual cells. Li and Mao considered a non-autonomous stochastic Lotka-Volterra competitive system in [10]. Wang et al [19,20] formulated a stochastic model for the bioconversion of glycerol to 1,3-propanediol by Klebsiella pneumoniae in batch culture, where the microbial fermentation process is described by the stochastic differential equations driven by five-dimensional Brownian motion. Considering the actual microbial fermentation process, the stochastic models in [19,20] may lose the regularity of the terminal time, for the initial value of the substrate concentration may be greater than the maximum concentrations of substrate and no product comes into being at the beginning of the microbial fermentation process. The regularity of the terminal time is an essential factor for the rationality of the stochastic system [11]. If the terminal time is irregular, it is possible that the terminal time → 0 occurs, which is irrational for a stochastic microbial fermentation model. Therefore, the stochastic models in [19,20] require to be modified to ensure the regularity of the terminal time, which motivates our work.
In this paper we propose a stochastic microbial fermentation model under the framework of white noise analysis. Gaussian white noises are used to model the environmental noises and the specific growth rate is driven by Gaussian white noises. In order to ensure the regularity of the terminal time, we add two adjustment factors in the volatility coefficients of the stochastic model. Then, in the proposed stochastic model, we prove the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on the initial value, which are fundamental properties for the further analysis of the stochastic microbial fermentation model.
The paper is organized as follows: in Section 2 we formulate a stochastic model for the microbial fermentation process, where Gaussian white noise is used to model the environmental noises and the specific growth rate is driven by Gaussian white noise. Some properties of the stochastic microbial fermentation model are considered in Section 3, such as the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on the initial value. Finally we conclude this paper in Section 4.

2.
A stochastic model for the microbial fermentation process under uncertainty. A microbial fermentation model is composed of three equations: a growth equation, a substrate equation, and a product equation. Taking the production of L-glutamic acid (LGA) by Corynebacterium glutamicum as an example, they describe the kinetic behaviors of the concentration of bacteria X 1 (t), the sugar (glucose) X 2 (t), and LGA X 3 (t), respectively. In this paper, we use the microbial fermentation model developed by Bona and Moser (1997) [3] and Suresh et al.(2009) [16] as follows: where µ(X 2 , X 3 ) is the specific growth rate and it is given by µ max ∈ R is the maximal specific growth rate. K 2 ∈ R is the substrate saturation constant and K 3 ∈ R denotes growth inhibition constant by product. q 1,2 , q 3,2 , q 3,1 ∈ R yield coefficient biomass from substrate, coefficient product from substrate and the growth-associated constant, respectively. The initial conditions are given by The modeling of the system (1)-(3) requires that the parameters involved are completely known. However, in the real-world system, the specific growth rate µ(X 2 , X 3 ) may fluctuate due to some environmental noises, which are random.
This paper is designated to propose a realistic stochastic model for the microbial fermentation process. We use the Gaussian white noises to model the environmental noises. Let W (t) be a 1−dimensional (1-parameter) Gaussian singular white noise process defined on a Hida distribution space (S) * . Assume that the specific growth rate µ(X 2 , X 3 ) is subject to the Gaussian white noises W (t) as follows: Then the deterministic microbial fermentation model (1)-(3) is improved as the stochastic model driven by Gaussian white noises: where (We refer to [6] for more information about the framework of Gaussian white noise.) The stochastic model (4)-(6) can be transformed into the stochastic differential equations driven by Brownian motion under the framework of Gaussian white noise, for the full information of X 1 (t) is available at time t. Let (Ω, F, P ) be a probability space and let {B t } t≥0 be a standard Brownian Motion defined on it. F t = σ(B s ; 0 ≤ s ≤ t) denotes the natural filtration generated by {B t } t≥0 . Then we have Proof of Theorem 2.1. It suffices to prove Eq. (7), for the proofs of Eq.(8) and Eq. (9) are similar. Let X 1 (t)δB(t) be Skorohod integral of X 1 (t) (see Definition 2.5.1. in [6]). Then, by the relation of the Wick product to Skorohod integral (see Theorem 2.5.9 in [6]), Eq.(4) can be written as Since the full information of In view of the relation between Skorohod integral and Itô integral (see Proposition 2.5.4 in [6] ), we can write Eq.(10) as Eq. (7) on [0, T ].
For the sake of simplicity, we define the systemic state X(t) of a microbial fermentation process as Then, by the stochastic model (7)- (9), the systemic state X(t) is of the following form (11) Now we consider the terminal time of the stochastic system (11). The microbial fermentation process X(t) is restricted on an interval [0, T ] and on a Borel set G. In reality, G is of the following form where the constants ρ 1 , ρ 2 and ρ 3 denote the maximum concentrations of biomass, substrate and product in the bioreactor medium (gl −1 ), respectively. Let τ G be the first exiting time of G for X(t), i.e., Then the terminal time is T ∧ τ G for the stochastic system (11). The regularity of the terminal time T ∧τ G is an essential factor for the rationality of the stochastic system (11). If the terminal time T ∧ τ G is regular, it is impossible that the event T ∧ τ G → 0 occurs. In the following, we define the regularity of the terminal time.
Definition 2.2. [11] Let T ∈ (0, ∞). The terminal time T ∧ τ G is called regular for the stochastic system (11) if there exists a constant δ > 0 such that τ G > δ almost surely for all initial values x ∈ G.
Considering the actual microbial fermentation process, we find that the terminal time is possibly irregular in the stochastic system (11), which is due to two facts: • It is possible that the initial value of the substrate concentration X 2 (t) is greater than ρ 2 . Then, depending on (8), the probability of X 2 (∆t) > ρ 2 is greater than 0 as ∆t → 0 . • It is usually assumed that X 3 (0) = x 3 = 0, for no product comes into being at the beginning of the microbial fermentation process. According to (9), the probability X 3 (∆t) < 0 is greater than 0 as ∆t → 0.
In both of the above cases, the terminal time T ∧ τ G loses the regularity in the stochastic system (11), for the probability of τ G → 0 is greater than 0. Consequently, It is indispensable that the stochastic system (11) is improved to ensure the regularity of the terminal time.
We make improvement on the stochastic system (11) by adding respectively adjustment factors in the volatility coefficient of (8) and Eq.(9), where χ E is the indicator function of an event E. As such, the stochastic system (11) is modified as Thus we propose the stochastic microbial fermentation model (12) described by the stochastic differential equation (12). It is obvious that the state X(t) of the stochastic system (12) is an Itô diffusion.
3. The properties of the stochastic microbial fermentation model. In this section, we discuss some properties of the stochastic model (12), such as the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on initial value. These properties are fundamental for the further analysis of the stochastic microbial fermentation model. Firstly we consider the regularity of the terminal time for the stochastic system (12).
Next we discuss the existence and the uniqueness of a solution for the stochastic system (12) on [0, T ∧ τ G ] in G. In the theory of stochastic differential equations, Lipschitz continuity of the coefficients is typically required to ensure the existence and the uniqueness of a solution. Therefore, we begin by stating a lemma which shows the coefficients is Lipschtiz continuous in stochastic system (12). Note that we may extend, for technical reasons, the discussions on [0, T ] due to the fact that for some constant D.
We are now able to obtain the existence and the uniqueness of a solution for the stochastic system (12). Theorem 3.3. Let T > 0. Then the stochastic system (12) has a unique t−continuous solution X(t, ω) on [0, T ] in G, with the properties that and Proof of Theorem 3.3. We have obtained the drift coefficient b(·) and the volality coefficient σ(.) of (12) are Lipschitz continuous on [0, T ] in Lemma 3.2. It is obvious that the stochastic system (12) is an Itô diffusion. In the case of Itô diffusion, Lipschitz continuity of b(·) and σ(.) implies that b(·) and σ(·) are at most linear growth on [0, T ], i.e.
At the end of this section, we give the continuous dependence of the solution on the initial value in the stochastic system (12)    holds for any fixed 2 ≤ p < ∞, where C is a constant that depends only on T and p.
Then we have (23), which completes the proof.
Remark 2. The indicator function in the modified stochastic microbial fermentation model (12) services several purposes: 1) keep the regularity of the terminal time; 2) guarantee the existence and uniqueness of a solution and the continuous dependence of the solutions on initial value; 3) make the expectation value of X(t) still satisfy the deterministic model (1)-(3) which is an essential and necessary requirement when adding stochastic terms. There should be other ways of modifying the stochastic microbial fermentation model which service these purposes in similar manner, and will be part of our future work.

Conclusion.
In this paper, taking the production of LGA by Corynebacterium glutamicum as an example, we propose a continuous-time stochastic model for the microbial fermentation process under the framework of Gaussian white noise analysis. Then we prove the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on initial value. These properties are fundamental for the further analysis of the stochastic microbial fermentation model. Based on this stochastic model, the control problems of a microorganism culture process, such as feedback controls, are stochastic optimal control problems. Therefore it is a demanding task to deal with the stochastic optimal control problems in the stochastic microbial fermentation model. We leave this for future research.