Dynamics of product inhibition on lactic acid fermentation
Introduction
Lactic acid has been widely used in food, industrial, pharmaceutical and cosmetic applications. As a general purpose food additive, l-lactic acid is generally recognized as safe (GRAS) [1]. Interest in biodegradable lactide polymers has accelerated research on the production of pure L-lactic acid as a bulk raw material [1]. Lactic acid-derived polymers are becoming increasingly important because of their applications within drug delivery systems and their biodegradable and thermoplastic nature meaning that they can be produced as high volume biodegradable plastics for packaging and other applications. Polylactic acid shows great potential in a wide range of material uses such as food packaging, garbage bags, and agricultural plastic sheeting, which is due to its mechanical properties [2].
In 1987, the world production of lactic acid averaged approximately equal proportions being produced by chemical synthesis and fermentation processes [3]. Now, however, all lactic acid is produced using mainly biotechnological means, enzymic or fermentation, and in the latter case, with bacteria species, such as Bacillus or Lactobacillus [4], [5], [6].
More economic is the fed-batch fermentation where the substrate, that is, feed rate profile is varied during the process and the final product is removed at the end of the process. Fed-batch operations may be operated in a variety of ways, by regulating the feed rate in a predetermined manner, or by using feedback control. The most commonly used are constantly fed, exponentially fed, and extended fed-batch. In industry, lactic acid is usually produced by using batch fermentation. Some work has been done with lactic acid fermentation in fed-batch culture [7], [8]. But the final concentration of lactic acid is relatively low.
It is obvious that a model-based efficient approach is necessary to ensure uniform operation and maximum productivity with the lowest possible cost in fed-batch processes, without requiring a human operator. Optimizations of bioprocesses are performed, based on precise and robust mathematical models. In general, models of bioprocesses are described by a set of differential equations derived from mass balances. During the practical industry product, manufacturers always consider how to keep a sustainable and steady output of lactic acid. Therefore, we need to investigate the effect of substrate input on the output of the lactic acid.
Lactic acid is a primary metabolite, and it is well known that the production of lactic acid is strictly dependent on cell growth and the final biomass. Membrane cell-recycle bioreactor (MCRB) could solve these problems satisfactorily. The efficiency of the MCRB was demonstrated in a number of previous studies on the enhancement of lactic acid productivity [9], [10].
According to Fig. 1 [11], we consider the following model of the lactic acid fermentation in membrane bioreactor.where x is the biomass concentration (g/l); S is the substrate concentration (g/l); S0 is the initial substrate concentration (g/l); P is the lactic acid concentration (g/l); Q1 is the volumetric bleed flow-rate (l/h); V is the reactor volume; δ1 and δ2 are the yield coefficients (g/g); m is the maintenance coefficient (h−1); μ is the maximal specific growth rate of biomass (h−1); KD is the death coefficient (h−1); Ks is the kinetic coefficient.
This paper is arranged like this: in Section 2, by using the qualitative theory of ordinary differential equations, we investigate the behavior of the ordinary system which models the process of continuous input of substrate. In Section 3, by using Floquet theorem for the impulsive equations, small-amplitude perturbation skills, we get the local stability of the biomass-free periodic solution. In Section 4, we prove the system (3.1) is permanent if some conditions are satisfied. Finally, we give a brief discussion.
Section snippets
Qualitative analysis for system (1.1)
Let U(t) = δ2S + P, the upper derivative of U(t) along a solution of (1.1) is described aswe haveTherefore, we only consider the limit system of (1.1) as follows:Next, we give the following Lemma 2.1. Lemma 2.1 The system (2.1) is uniformly bounded. Proof This proof will have two cases. When KD > δ1m, we have
The model of impulsive perturbation
With an impulsive perturbation, the Eq. (2.1) becomeswhere T is the impulsive period, n = {1, 2, …}. Other parameters have the same meanings as system (1.1).
By the basic theories of impulsive differential equations in Bainov and Simeonov [12]; Lakshmikantham [13], the solution of system (3.1) is unique and piecewise continuous in (nT, (n + 1)T], n ∈ N for any initial
Permanence
Firstly we show that all solutions of (3.1) are uniformly ultimately bounded. Theorem 4.1 There exists a constant M > 0 such that S(t) ⩽ M, x(t) ⩽ M, P(t) ⩽ M for each positive solution (S(t), x(t), P(t)) of (3.1) with t large enough. Proof Define a function and the upper right derivative of W(t) along a solution of (3.1) is described aswhere . We obtainBy the definition of W(t), we obtain
Discussion
The model for the lactic acid fermentation process is developed and used by Boudrant et al. [11]. To simulate the oscillatory behavior of an experimental fermentor,we incorporate continuous input substrate and impulsive input substrate, respectively. Firstly, by using the qualitative theory of ordinary differential equations, we prove the biomass-free equilibrium point is globally asymptotically stable if R1 < 1, which is simulated in Fig. 3. The uniquely positive equilibrium point is globally
Acknowledgement
This work is supported by the National Natural Science Foundation of China (No.10971001) and Henan Science and Technology Department (Nos. 082102140025 and 092300410228).
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