Mini-Review on the Design Principles of Biochemical Oscillators for the Continuous Ethanol Fermentation Processes

Abstract

Computational modeling and the theory of nonlinear dynamical systems allow us not only to simply describe the events of biochemical oscillators in the ethanol fermentation process but also to understand why these events occur. This article reviews results of experimental and theoretical studies about the behavior of fermentation systems for bio-ethanol production so as to understand the self-oscillatory phenomena that could affect productivity in industry. In general, Hopf bifurcation and limit cycles are the theoretical basis for the oscillations observed in continuous ethanol fermentation processes, but the underline mechanisms and causes might be different because the studied system is a collection of multi-scale oscillators. To characterize the oscillatory dynamics quantitatively, negative feedback laws are implemented. However, the stimulated oscillation through linear feedback is not adequate in describing such complex dynamics. Hence, elements of nonlinearity, auto-catalysis, and time delay are sorted out and added into the feedback loops to formulate biochemical oscillators. Then, we discuss specific examples of the various models and classify them according to the three kinds of mechanisms: nonlinear feedback, positive feedback, and delay feedback. These mechanisms and modeling work might be used as a guide for process design/operation to eliminate possible oscillations and to develop out advanced configurations that could produce bio-ethanol in a continuous, cost-effective manner.

1. Introduction

Bio-ethanol production is the first and foundation of bio-manufacturing, which is highlighted for being renewable, low-carbon, and environmentally friendly [1,2]. As a result, breakthroughs in lignocellulose pretreatment and enzymatic hydrolysis technologies have been made [3,4,5,6]. While well-recognized barriers to viable commercialization of cellulosic ethanol have been overcome, low productivity and intensive labor encountered with the fermentation process itself are now identified as a bottleneck problem [7,8].

It is critical to develop advanced process configurations to pursue improvements in productivity for fermentation itself. Processing configurations based on coupling of hydrolysis and fermentation are therefore developed, such as separate hydrolysis and fermentation (SHF), simultaneous saccharification and fermentation (SSF), hybrid hydrolysis and fermentation (HHF), and consolidated bio-processing (CBP) [9]. Yet, all above processes assume fermentation works in batch/fed-batch mode, and other methodologies, like continuous approaches, are less prevalent in industry. Hence, it is critical to explore the characteristics of fermentation working continuously.

As is shown in Figure 1a, the most common type of fermentation in industry is batch-based, in which microorganisms, that is, Saccharomyces cerevisiae, are inoculated into a tank with culturing medium, that is, glucose, KP₂PO₄, (NH₃)₂SO₄, CaCl₂, MgSO₄, etc., and after about 3–5 days of processing in a closed environment, the liquor is poured out for subsequent ethanol separation and purification. Probable causes for choosing batch/fed-batch fermentation instead of continuous mode might be due to the following: (1) mild reaction conditions and low driving force lead to extremely low growth and production rate, making it difficult to match product removal rate and form steady-state production; (2) cell membrane separates microorganism from the fermentation broth, forming an independent entity which could conduct material and energy exchange across the broth, and effective schemes are in short in industry to regulate microbial growth and metabolism; (3) quantifying the growth and metabolism kinetic parameters are challenging, and living organisms always seek to avoid hazards environment or become dormant when faced with abrupt changes, making it difficult to predict or maintain within a specific working state.

While, for batch mode production, the lag and stationary phase within a batch always demand considerable time; moreover, downtime between two batches is requested to harvest, clean, and sterilize, which decreases the overall productivity and increases labor cost. On the other hand, continuous reactors eliminate batch-to-batch intervals after reaching a steady state, but ethanol content might be lower than the corresponding batch mode because of the dilution effect, that is, the substrate conversion ratio is low due to the limited residence time, making downstream processing more difficult and costly. In order to intensify ethanol production as well as alleviate substrate run-away, common industrial practice adopts multiple fermentation tanks in series for continuous ethanol fermentation (Figure 1b), which makes the process sophisticated and the related equipment cost enormous.

In fact, process system engineers tend to design continuous systems to acquire static environments, where product quality and efficiency are easily promised [10]. However, maintaining the fermentation process in a steady state always encounters a number of problems related to the loss of its stability, especially on an industrial scale. It should be noted that traditional process systems belong to the Planck paradigm [11,12], which studies systems that are linearizable, and the working states are thermodynamically near equilibrium, indicating that the system is attracted to a stable steady state or comes to an equilibrium that counteracts the forward and reverse forces. As such, two modeling perspectives are frequently applied: (1) time-invariant system, such as a steady-state simulation of the chemical process, in which each unit module needs to be (stable-) equilibrium or dynamically balanced; (2) linearizable time-varying system, in which difference between a the state and the corresponding equilibrium is the driving force that causes the formation of “flow”, and the relationship between this “force” and “flow” is linearizable, satisfying the Onzag reciprocity relationship and the minimum entropy production theory [13].

When it comes to the ethanol fermentation processes, the above modeling methods often seem inadequate. Attention might be focused on characteristics like self-organization, self-adaptation, coordination, or morphogenesis, which are different from the linear, near-equilibrium systems; new order or structure might evolve out, and Prigogine [14] refers to the dissipative structure systems. One typical example is the biochemical oscillators. Experimental and theoretical study reveals the incidence of self-oscillatory behavior in which biomass, product, and substrate oscillate under certain conditions. During oscillations, ethanol productivity decreases due to variation of biomass content, which leads to a higher level of residue substrate being washed out compared with the steady-state counterpart. Therefore, understanding the specific mechanism causing the oscillatory behavior is critical for regulating microorganism metabolism and developing innovative configurations that could attenuate oscillations and intensify production.

Lee et al. [15] are one of the first to report the oscillations in chemostat culturing Zymomonas mobilis at high gravity medium condition: at a low gravity medium condition (≤10% glucose), steady states are presented for output glucose, ethanol, and biomass; when the input glucose content reaches 15–20%, sustained oscillations are observed instead [16]. Borzani [17] observed the oscillations of residual sugar, ethanol, and biomass in the continuous ethanol fermentation using S. cerevisiae and molasses. Bai et al. [18,19] focused on the sustainable oscillations characterized by long oscillation periods and large oscillation amplitudes using S. cerevisiae at a very high gravity medium (28% glucose) or under flocculating conditions. Although oscillations are observed in the continuous cultures of both S. cerevisiae [20] and Z. mobilis [16,17], the oscillation patterns are significantly different from the oscillations reported under the aerobic culturing condition (e.g., using S. cerevisiae), where periodicity of cell population distribution because of synchronizing of the rhythms on the mother and daughter cells is believed to be the primary mechanistic reason.

From molecular and reaction network perspectives, numerous studies in the oscillations of the glycolytic pathway in S. cerevisiae have been reported with respect to the Hopf bifurcations and limit cycles [21,22]. For a mechanistic analysis [23], a system biology-level study on the eukaryotic cell cycle is performed [24,25,26], which mainly focuses on the inter-relation between cyclin-dependent protein kinases (Cdks) and the APC/C [27,28]. Over the past decades, the field has pursued a detailed understanding of how the proteins regulate with each other and function collectively as a system [29,30].

To distinguish, we are able to verify that the continuous ethanol fermentation process is a collection of multi-scale oscillations. On a molecular scale, the glycolysis pathway could finish a cycle within seconds; on a super-molecular scale, the APC/Cdks would last for minutes. However, the oscillations in a yeast extract or in intact yeast cells vary significantly. Moreover, the respiratory oscillation under aerobic conditions could last for dozens of minutes to a few hours due to the fermentation condition. For industrial-scale processes, age-population distribution for cell colonies causes the transportation delay to become apparent, and the oscillation could last for days. Therefore, the principles would perform a specific function in characterizing different scales of biochemical oscillations in the ethanol fermentation process.

Based on the dynamical and control system theories, models of biochemical oscillators for the continuous ethanol fermentation process are developed. Hobley and Pamment [31] proposed one mechanism/conjecture: ethanol accumulation (i.e., the ethanol concentration history) might trigger the dormancy of the cells, and when ethanol content reaches a critical level, cell metabolism suspends; after the dilution effect improves the circumstance, cell metabolism restarts. However, Li et al. [32] argued that the effect of ethanol concentration history is negligible, but the upward ethanol concentration change rate had an intense inhibitory effect, which could be explained as follows: when the ethanol concentration attains a high level, the energy gained from the fermentation process cannot be utilized immediately by the microorganisms for duplication purposes; this is because the microorganisms need time to adjust their enzyme system to accommodate a high ethanol stress [33]. This hypothesis is in accordance with the structured model proposed by Jobses et al. [34], where it was assumed that ethanol inhibits the formation of intracellular metabolites (i.e., RNA or protein), which in turn influence the specific growth rate and cause a time lag between taking up the substrate and producing biomass through intracellular metabolites. More directly, Xiu et al. [35] suggested that the transport processes of substrate and product across the membrane should be included. This idea was investigated by including a time-delay term into the product and substrate inhibition kinetic model [36]. A similar system was also studied by [37] by including a discrete delay time between the biomass formation and the operating conditions.

To address this, the abovementioned biochemical oscillators are on the edge of driving the process to destabilize, which draws our attention because continuous ethanol fermentation is a long-term process along which set points, control objectives, physicochemical variables, and dynamic behavior can be changed by the operator, and any unexpected disturbance might cause the process to terminate. Hence, it is critical to identify elements of causes that lead the system to be internally unstable.

The focus of this paper is to help understand how the inhibitory (negative feedback) mechanisms might cause biochemical oscillators in the fermentation process. For this purpose, modeling on the continuous ethanol fermentation processes is reviewed for the routes to Hopf bifurcations and limit cycles. To characterize the oscillatory dynamics quantitatively, negative feedback laws are implemented. However, the stimulated oscillation through linear feedback is not adequate in describing such complex dynamics. Hence, elements of nonlinearity, auto-catalysis, and time delay are sorted out and added into the feedback loops to formulate biochemical oscillators. Then, we discuss specific examples of the various models and classify them according to the three kinds of mechanisms: nonlinear feedback, positive feedback, and delay feedback.

The review is organized following the schematic provided in Figure 2. Dynamical system theories on Hopf bifurcation and limit cycles are sketched in Section 2. Then, in Section 3.1, (linear) negative feedback is introduced as the basic structure and illustrated by a stimulated oscillation case, followed by analyzing elements of nonlinear, auto-catalytic, and delay in the control loop, respectively, in Section 3.2, Section 3.3 and Section 3.4, and the specific examples are provided accordingly. In the last section, we provide conclusion remarks.

2. Preliminaries of the Hopf Bifurcation and Limit Cycles

Biochemical oscillators present systems-level characteristics (e.g., periodicity, robustness [38], and entrainment [39]) that transcend the properties of individual molecules or reaction partners; hence, the topology of the reaction network should be taken into account. In contrast, the element-level characteristics of a system usually count as being linearizable, as is provided in. Specific to the dynamical theories, concepts from dynamical systems, such as Hopf bifurcation and limit cycle, are concerned. To address this, living systems are a collection of limit cycles, and it is a topic of much interest because it can adapt to the external disturbances (mainly led by uncertainty of various casual initial conditions) and survive on the edge of chaos [40]. For the mechanistic study, auto-regulation on the basis of feedback loops is detailed; moreover, elements of nonlinear, auto-catalytic, and time delay feedback are viewed as the main causes of biochemical oscillators.

The topologically equivalent dynamical system that might formulate an autonomous oscillator is provided with the following standard form:

$$\left(\begin{array}{c}{x}_1{}^{\prime }\\ x_2{}^{\prime }\end{array}\right)=\left(\begin{array}{cc}\alpha & -1\\ 1& \alpha \end{array}\right)\left(\begin{array}{l}{x}_1\\ x_2\end{array}\right)\pm l(x_1^2+x_2^2)\left(\begin{array}{l}{x}_1\\ x_2\end{array}\right),$$

(1)

where x₁ and x₂ are the two inter-correlated elements, indicating the dimension is requested to be n ≥ 2; the Jacobian matrix has eigenvalues λ_1,2 = α ± i, and α across the imaginary axis causes the formation of Hopf bifurcation (i.e., α = 0) and a change of the stability status occurs; the “nonlinear terms” are indispensable, which leads x₁ and x₂ to interfere with each other; l represents the first Lyapunov coefficients, and when l > 0, the system undergoes sub-critical Hopf bifurcation, whereas l < 0, the system undergoes supercritical Hopf bifurcation.

A general n-dimensional system (n > 2) under Hopf bifurcation could retreat to (1) with the following system reduction and transform procedure exhibited in Figure 3, where the n-dimensional system is reduced by center manifold theory; then, coordinate change leads the 2-dimensional system to the standard form similar to (1); afterwards, the first Lyapunov coefficient is calculated to promise super-critical Hopf bifurcations.

As is depicted in Figure 4A, system (1) with only the linear part is topologically different from the original one, indicating limit cycles are un-linearizable. Moreover, when l < 0 (Figure 4B), the equilibrium at the origin is stable for α < 0, and a stable limit cycle is reproduced with radius sqrt(α) for α > 0; when l > 0 (Figure 4B), the equilibrium at the origin is stable when α < 0, and an unstable limit cycle exists with radius sqrt(α) for α > 0.

In order to make the theories of Hopf bifurcation and limit cycle directly applicable, we consider specific examples of biochemical oscillators and discuss possible routes for the formation of limit cycles. Mechanisms of relevance are nonlinear feedback, positive feedback, and delay feedback.

3. Mechanisms of the Biochemical Oscillators

3.1. Negative Feedback

Negative feedback is a concept of control engineering, and it can help stabilize regulatory systems. The intersection of the biomass concentration checkpoints is one typical example of the negative feedback loop [41,42]. For instance, in a continuous fermentation system where the limiting substrate S is constantly supplied and cells X eat substrate and reproduce (i.e., Monod-type specific growth rate μ = μ_mS/(S + k_s)), a portion of the cells are removed accompanied by the discharge effluent (Figure 5A). When the inhabit condition is accommodated, the cell duplication checkpoint is activated for the forward channel. The checkpoint arrests progression and gives the cells extra time to get the environment properly occupied. Meanwhile, the discharge effluent with different dilution rates activates a checkpoint that feeds back to decrease biomass concentration and forms a negative feedback channel.

However, when the feedback signal is strong enough (i.e., under high dilution rate), the system’s steady state can be destabilized (Figure 5B), and if the steady state is destabilized sufficiently, the system will oscillate [43,44]. To address this, the oscillation in Figure 5B is not a limit cycle: with an increase in the feedback intensity, the system begins to oscillate till reaching a critical value where sustained oscillation is stimulated; afterwards, the system will be washed out (i.e., biomass concentration approaches zero as t → ∞). It is noted that the stimulated oscillation is not a limit cycle, because a stable limit cycle is such a structure that it can adaptively attract the oscillatory trajectories regardless of the initial conditions and external inferences, whereas for the negative feedback exhibited in Figure 5B, the stimulated oscillations are on the edge of becoming destabilized, if not convergent to a steady state, when t → ∞.

According to the above analysis, negative feedback loops are not guaranteed to generate limit cycles (to address this, the stimulated oscillation exhibited in Figure 5B is not a limit cycle), but the analysis on checkpoints of the forward- and feedback-channels provides the possibility to investigate this complex phenomenon. Moreover, negative feedback loops are universal in the continuous ethanol fermentation processes. Indeed, the continuous ethanol fermentation system contains multiple negative feedback loops (

Table 1. Negative feedback caused by ethanol inhibition.

Sources	Organisms/Cites	Mechanism	References
HK and ADH	Glycolytic pathway	Direct ethanol inhibition	Larue et al., 1984 [47]
Plasma membrane ATPase	Glycolytic pathway	Affect nutrient uptake, membrane potential	Casey and Ingledew, 1986 [48]
Acetaldehyde and temperature	Fermentation by-products and stresses	Affect cell membranes and organelles	Jones, 1994 [49]
Palmitoleic acid and oleic acid	Fatty acids	Ethanol decreases membrane fluidity	You et al., 2003 [50]
Palmitric acid and stearic acid	Oxygen- and NADH-dependent desaturase	Oxygen improves the ethanol tolerance	Ryu et al., 1984 [51]
Proton gradient	Trans-membrane proton	Increases the plasma membrane permeability	Pascual et al., 1988 [52] Salguerio et al., 1988 [53] Cartwright et al., 1987 [54]
Plasma membrane ATPase	pH	H+ produced increases the proton motive force driven by ATPase	Rosa and Sa-Correia, 1992 [55]

). For instance, ethanol as the end product presents an inhibitory effect that impairs the further formation of the product and correspondingly formulates the negative feedback loops. Hence, there is a longstanding conjecture that biochemical oscillators are produced by the basic mechanism of the negative feedback circuits [45,46], and in this review, we have summarized that elements of nonlinearity, auto-catalysis, and time delay in negative feedback loops would contribute to the formation of Hopf bifurcations and generation of limit cycles.

3.2. Nonlinear Feedback

3.2.1. The Illustrative Example of Nonlinear Feedback

To assess oscillators and clocks induced by nonlinear feedback, the best studied of these feedback loops centers on the cyclin-dependent protein kinase (CDK1) and the anaphase-promoting complex (APC) [44]. During interphase, a succession of biochemical events contributes to the activation of CDK1, labeled as CDK1_on, which drives the cell into mitosis and brings about the activation of APC in a switch-like manner [56,57]. In turn, APC polyubiquitylates and tags cyclin B for degradation by the proteasome, which causes the deactivation of CDK1, labeled as CDK1_off. As a consequence, a negative feedback loop is formulated and is exhibited in Figure 6A. For the two components CDK1 and APC, the four switch-like state changes constitute the oscillating loop and form the Boolean model (Figure 6C). For instance, when the system starts in an interphase-like state with APC_off/CDK1_off, the complete cycle goes as APC_off/CDK1_off → APC_off/CDK1_on → APC_on/CDK1_on → APC_on/CDK1_off.

Haase et al. [58,59] have found that in S. cerevisiae strains, the CDK1-APC oscillator (Figure 6A,C) is linked to and cooperates with a transcriptional circuit that can independently generate oscillations. To describe the complexity, one more component is added to the system, for example, a protein like Polo-like kinase 1 (Plk1) [44], which is activated by CDK1 and, in turn, contributes to the activation of APC, as is shown in Figure 6B. The negative feedback loop for the CDK1-Plk1-APC system is more complicated than the CDK1-APC oscillator (Figure 6C) because there are eight (2 × 2 × 2) possible states. It is noted from Figure 6D that not all of the possible states are in the flip cycle: despite the six states in the loop, for example, APC_off/CDK1_off/Plk1_off → APC_off/CDK1_on/Plk1_off → APC_off/CDK1_on/Plk1_on → APC_on/CDK1_on/Plk1_on → APC_on/CDK1_off/Plk1_on → APC_on/CDK1_off/Plk1_off, the two initial states APC_off/CDK1_off/Plk1_on and APC_on/CDK1_on/Plk1_off would finally flip into the closed-loop.

The abovementioned Boolean analysis is simple and appealing, but it is not quantitative. From practical viewpoints, we are interested in design principles that would generate biochemical oscillators; however, the Boolean models (Figure 6A,B) yield oscillations even though we know that real negative feedback loops do not always oscillate. Specific to the current case, even if a single CDK1 molecule could activate APC, which in turn deactivates CDK1, a cell contains numerous CDK1, and APC molecules would not promise to flip all particles simultaneously.

To tackle the problem, chemical kinetic theory is adopted for describing the dynamics of such a lumped system, where a set of ordinary differential equations could be constructed based on the conservation law. Here, Plk1 is inserted as an intermediary between CDK1 and APC (Figure 7A). The reference model is provided as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{dCDK{1}^{*}}{dt}}={\alpha }_1-{\beta }_{1}CDK{1}^{*}{\displaystyle \frac{AP{C}^{*n1}}{K_1^{n1}+AP{C}^{*n1}}}\\ {\displaystyle \frac{dPlk{1}^{*}}{dt}}={\alpha }_2(1-Plk{1}^{*}){\displaystyle \frac{CDK{1}^{*n2}}{K_2^{n2}+CDK{1}^{*n2}}}-{\beta }_{2}Plk{1}^{*}\\ {\displaystyle \frac{dAP{C}^{*}}{dt}}={\alpha }_3(1-AP{C}^{*}){\displaystyle \frac{Plk{1}^{*n3}}{K_3^{n3}+Plk{1}^{*n3}}}-{\beta }_{3}AP{C}^{*}\end{array}\right.,$$

(2)

where the parameters for system (2) are cited as α₁ = 0.1, α₂ = 3, α₃ = 3, β₁ = 3, β₂ = 1, β₃ = 1, K₁ = 0.5, K₂ = 0.5, K₃ = 0.5, n₁ = 8, n₂ = 8, and n₃ = 8 [44]. For any arbitrary initial conditions with respect to CDK1, APC, and Plk1, the system approaches the same pattern of oscillations eventually: the increasing and decreasing trend of all three states is unanimous; however, there is a slight difference in peaking time-instant, for example, CDK1 activity peaks first; this is in accordance with the Boolean model exhibited in Figure 6 since the activation of CDK1 leads the cell to mitosis, which causes Plk1 and at last APC activity (Figure 7C). In the phase plane view shown in Figure 7B, a stable limit cycle is generated that generates a closed circle of states with all trajectories spiraling in or out toward, similar to the one exhibited in Figure 4B.

From the theory provided in Section 2, the stability status of system (2) could be evaluated by computing the eigenvalues of the system Jacobian. The eigenvalues of the matrix are −5.29 and 0.88 ± 3.47i with the provided parameters. The different peaking time-instant could be explained through the distribution of the eigenvalues: the interrelation of Plk1 and APC in system (2) leads to the generation of a limit cycle, and they share similar structure as system (1) after proper mapping, while CDK1 falls into a stable manifold, indicating that the oscillatory trajectory of CDK1 is because of the entrainment effect. Hence, nonlinearity of the 2nd- and 3rd-equations in (2) dominates the generation of the limit cycle and should be properly addressed.

3.2.2. Nonlinearity Induced Limit Cycles in the Feedback

The concept of “nonlinearity” is ambiguous and difficult to assess even with the mathematical model being precisely offered. To narrow down the domain of discussion, polynomial-type nonlinear feedback, in contrast with the linear one, is focused in the feedback loop. It is noticed that the well-known Oregonator model (Figure 8) characterizing the Belousov–Zhabotinsky reaction system shares identical structure as the CDK1-Plk1-APC loop (Figure 6B), while the formulas contain only the polynomial type expressions. Therefore, we are able to assess the nonlinearity-induced limit cycles in the feedback loop by taking Oregonator as the representative example.

By Field–Koros–Noyes (FKN) kinetics, the Oregonator is modeled as follows:

$$\left\{\begin{array}{l}\epsilon {\displaystyle \frac{dx}{dt}}=qy-xy+x(1-x)\\ \delta {\displaystyle \frac{dy}{dt}}=-qy-xy+f z\\ {\displaystyle \frac{dz}{dt}}=x-z\end{array}\right.,$$

(3)

where the parameters are ε = 0.04, δ = 0.0004, and q = 0.0008 [60], and the continuation parameter is f. Similar to the canonical CDK1-APC case (Figure 6A,B), the Oregonator (3) can be reduced to 2-dimensional by eliminating the stable manifold while maintaining the topological structure of the limit cycle, and one can refer to Figure 1 or [61] for the detailed reduction procedure.

To formulate a feedback loop for (2), the general form is adopted as follows:

$${\displaystyle \frac{dx}{dt}}=Ax+Bg(Cx,\mu ),$$

(4)

where A is an n × n matrix, B is an n × l matrix, C is an m×n matrix, and g: R^m → R^l. μ is an adjusting parameter representing the feedback intensity. Equation (4) is equivalent to a multi-loop feedback system composed of a linear operator G and a nonlinear part g. The details of the conduct are provided in Box 1 for interested readers.

Box 1. The closed-loop equivalence of the dynamic system.

First, nonlinear part of the general multi-dimensional system is incorporate into the input

$${\displaystyle \frac{dx}{dt}}=Ax+BDy+B[g(Cx,\mu )-Dy]$$

Next, set y = C x, and Laplace transform leads to

$$\begin{array}{l}G(s,\mu )=C{[sI-(A+BDC)]}^{-1}B\\ u=g(Cx,\mu )-Dy\\ y=-e=Cx\end{array}$$

We can choose D = 0 if A ≠ 0, and D = In if A = 0, and the feedback system is provided as follows

        

Now, the nonlinear feedback loop is constructed, but it differs from Figure 5 in that a limit cycle is constructed that can adaptively adjust to the destined trajectory, whereas the stimulated oscillation in Figure 5B (linear feedback) might be vulnerable to external disturbances. The theories for bifurcating out Hopf points and progressing to limit cycles are provided in Box 2, where nonlinearity-induced limit cycles in the feedback are addressed.

The appearing property of the nonlinear-feedback structure represented in Box 1 is that one can design the limit cycle by decomposing the system to a classic forward (G(s)) and a nonlinear feedback (g) channel, where G(s) contains only the primary (rational) information of each species, that is, CDK1 actives APC directly, while g contains mutual information between species, that is, CDK1 actives Plk1, and Plk1 contributes to the activation of APC, which in turn inactivates CDK1. Quantifying g is a challenging task, and when the experimental information of frequency ω and amplitude θ is provided, g = −I/N(θ) could be designed where the intersection points for G(s) = −I/N(θ) indicate the generation of the limit cycle.

Box 2. Criterion for the onset of Hopf bifurcation.

By linearizing the feedback path in the closed-loop, the generalized Nyquist criterion is adopted to assess the characteristic polynomials

By aid of the graphic explaination, the characteristic gain loci G(iω) should intersect the amplitude loci I/N(θ),

                

The describing function schematic provides

$$\begin{array}{l}y=G(i\omega )u\approx -G(i\omega )N(\theta )y\\ \Rightarrow G(i\omega )=-I/N(\theta )\end{array}$$

3.3. Auto-Catalytic/Positive-Plus Feedback

3.3.1. The Illustrative Example of Positive-Plus Feedback

Because the biochemical oscillating systems are internally unstable, many design strategies adopt inserting a positive element into the feedback loop. In reaction kinetics, the positive element manifests itself by auto-catalysis. To address this, the signal of the auto-catalytic element could be amplified after long-standing feedback, that is, a small change in input into a large change in output. As a result, auto-catalytic feedback loops might function as bistable or hysteretic switches. Therefore, a system with an auto-catalytic element is potentially unstable because of the positive-plus-feedback mechanism, and the design of biochemical oscillators might be realized by this mechanism.

Elements of auto-catalytic reaction are universal in the metabolic pathway and specific to yeast S. cerevisiae; during glycolysis, two adenosine triphosphate (ATP) investments cause the formation of four ATPs (Figure 9A). The Lotka model [62] is one of the earliest to describe the possible dynamics for the auto-catalytic system. To illustrate the mechanism in the feedback loop, we translate ATP change in glycolysis into an ecological system, where the topological structure is identical. Assume the grass G is constantly supplied and animal A eats grass and reproduces; animal B eats animal A and reproduces/dies. The Boolean analysis provides that a cycle is formulated. When the species B is abundant and A is limited, A would cause B to decrease, and subsequently, a low quantity of B allows A to increase, which is followed by an increase in B, and this continues with a self-adjusting pattern. Depending on the resources and species type, the system might switch between two alternatives, namely, A dominate and B dominate (bistability), or once A reproduces with a fast speed, the number of A maintains at a high level till a small increase in B triggers the massive decrease (hysteresis).

It is noticed that the rhythm of yeast is a collection of multi-scale limit cycles [63]. The research on glycolytic oscillations was conducted in both the yeast extracts and intact cells. For the former, the metabolites in cell-free extracts of yeast when glucose was continuously provided and a period of about 1 min were found for other metabolites of the glycolytic pathway as well [64]. For the latter, the carbon dioxide evolution (CER) and the percentage of budding cells oscillated with a period of about 60 min in aerobic glucose-limited continuous cultures [65]. It is noticed that the budding oscillations are partly related to the cell cycle in the fermentation process, where age-population of the cells needs to be taken into account [66]. Further experiments on yeast colonies were shown to have some oscillatory behavior, and oscillations have periods of several days and were supposed to be involved in guiding the spatial growth of yeast colonies [67].

Though the glycolytic oscillations are complicated, attempts have been made for a kinetic simulation/analysis. The earliest ones were performed by Higgins [68]. The simulations included inhibition of phosphofructokinase by fructose 1,6-bisphosphate and activation by AMP. Richter et al. [69] presented a model where a myokinase reaction was included to catalyze the reaction from AMP and ATP to two ADP as well as an ATPase to remove the accumulating ATP. Simplified models were described by Selkov [70] based only on the auto-catalytic properties of glycolysis. More recently, the inter-cellular synchronization between glycolyzing cells [22,71] and the synchronization of short-period oscillations [72] were analyzed with kinetic models.

Obviously, the glycolic oscillations are of foundational importance for the rhythm of yeast. Here, the Brusselator (Figure 9B) is adopted to mimic the oscillatory dynamics of the glycolic oscillations (Figure 9C). This is a conceptional model (i.e., this model is simple in math) developed by Prigogine [73]. One can refer to the more complicated yet psychologically reasonable models in the literature. Given in the space-time domain, the Brusselator could be provided as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }X}{\mathsf{\partial }t}}=A-(B+1)X+X^{2}Y+D_1{\nabla }^{2}X\\ {\displaystyle \frac{\mathsf{\partial }Y}{\mathsf{\partial }t}}=BX-X^{2}Y+D_2{\nabla }^{2}Y\end{array}\right.,$$

(5)

where D_i (i = 1, 2) is the Fick’s diffusion constant, and ▽² is the Laplace operator. When the in-homogeneity in space is unconsidered, that is, ▽² = k², the oscillatory trajectories are solved by balancing harmonics on both sides of Equation (5), and the intrinsic frequency could be obtained as follows:

$$i\omega ={\displaystyle \frac{1}{2}}\left(B-1-k^2{D}_1-A^2-k^2{D}_2\pm \sqrt{B-1-k^2{D}_1+A^2+k^2{D}_2-4A^{2}B}\right),$$

(6)

Equation (6) could be used as the criterion for the existence of limit cycles. The result is shown in Figure 9D. One difference in comparison with the nonlinear feedback produced by the CDK1-Plk1-APC circuit exhibited in Figure 7C is that the system approaches the opposed pattern of oscillations, that is, with X approaching the peak and Y maintaining at the minimum. In the phase plane view, the oscillations for the auto-catalytic-plus-negative feedback model are sharp, and there are distinct slow and fast changes that might relate to the abovementioned properties like bi-stability and hysteresis.

3.3.2. Auto-Catalytic Elements in the Feedback

The typical response of a linear (or more precisely, linearizable, as is shown in Figure 5) feedback loop without an auto-catalytic element is a gradual, progressively slowing approach to a steady state. However, till now, the role of auto-catalytic elements in generating oscillations has not been discussed exclusively. Here, the case of culturing S. cerevisiae is concerned, where the growth of yeast is performed by feeding it with glucose, minerals, and vitamins, and oscillations have been observed for both cells and sugar substrate [74]. The dynamic model is provided as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{dS}{dt}}=D(S_{in}-S)-{\displaystyle \frac{\mu }{Y}}X\\ {\displaystyle \frac{dX}{dt}}=-DX+\mu X.\end{array}\right.,$$

(7)

where S is the substrate, X is the cell content, D is the dilution rate, S_in is the substrate input concentration, μ is the specific growth rate of cells, and Y is the yield factor representing the cells reproduced per gram of sugar substrate.

For ease of analysis, system (7) is shifted to the dimensionless one as follows:

$$\begin{array}{l}\left\{\begin{array}{l}{\displaystyle \frac{du}{d\tau }}=u\left(g(v)-1\right)\hspace{1em}\hspace{1em}u(0)=u_0\\ {\displaystyle \frac{dv}{d\tau }}=1-v-{\displaystyle \frac{g(v)}{Y_{xs}}}u\hspace{1em}v(0)=v_0\end{array}\right.\\ with:\hspace{1em}\\ \tau =Dt;\hspace{1em}u=X/S_{in};\hspace{1em}v=S/S_{in};\hspace{1em}g(v)=\mu /D\end{array},$$

(8)

where (u^*, v^*) is the initial condition. When Y_xs is taken as a constant, we are able to shift system (8) to 1-dimentional, and the form of the Abel equation indicates that the solution cannot be periodic. The result is in accordance with the simulations presented in Figure 5.

However, variation of the substrate-to-cell yield Y is reported to be related to the generation of oscillations [17]; hence, theoretical analysis is performed by extending Y as a function of substrate, which in fact expresses the effect of substrate inhibition [75]. When the concept of maintenance in the ethanol fermentation system is introduced [76], one is able to formulate the auto-catalytic element using the linear relation Yxs = Y₀ + Y₁S; with a fixed amount of cell produced, low substrate content could increase the consumption of substrate further.

Therefore, extending Y_xs as a linear function of substrate, biochemical oscillators are possible. The theories for bifurcating out Hopf points and progressing to limit cycles are provided in Box 3, where auto-catalysis-induced limit cycles in the feedback are addressed. Mention α > 1 always holds to make g(v^*) = 1 valid. In Box 3, assuming that the pair λ_1,2 = ±iω exists, one has α > β + 1, and the washout point (0, 1) is a saddle. Moreover, stability of the Hopf bifurcation (Γ = 0) is determined. Y₀/Y₁ < R evolves out of sustained self-oscillations, and the bifurcation is supercritical.

Box 3. Criterion of auto-catalytic element in the feedback.

We focus on the set of singularity points {(u*, v*): u[g(v) − 1] = 0; 1 – v = [g(v)/Y_xs]u}, and perturbation around (u*, v*) provides the linearized system,

$$\left\{\begin{array}{l}{\displaystyle \frac{du}{d\tau }}\approx \left(g(v^{*})-1\right)(u-u^{*})+u^{*}\left(g\prime (v^{*})\right)(v-v^{*})=u^{*}g\prime (v^{*})(v-v^{*})\\ {\displaystyle \frac{dv}{d\tau }}\approx {\displaystyle \frac{g(v^{*})}{Y_{xs}(v^{*})}}(u-u^{*})+\left(-1-\left[{\displaystyle \frac{d}{dv}}\left({\displaystyle \frac{g}{Y_{xs}}}\right)(v^{*})\right]u^{*}\right)(v-v^{*})\end{array}\right.$$

For g(v) = αv/(β + v) and Y = Y₀ + Y₁S, eigenvalues of the Jacobian provides,

$$\begin{array}{l}\lambda ={\displaystyle \frac{1}{2}}\left(-\mathrm{\Gamma }\pm \sqrt{{\mathrm{\Gamma }}^2-4{\displaystyle \frac{{(\alpha -1)}^2-(\alpha -\beta )\beta }{\alpha \beta }}}\right)\\ with:\\ \hspace{1em}\mathrm{\Gamma }=1+{\displaystyle \frac{(\alpha -1)(\alpha -1-\beta )}{\alpha \beta }}-{\displaystyle \frac{B{S}_{in}(\alpha -1-\beta )}{Y_0(\alpha -1)+Y_1{S}_{in}\beta }};\\ \hspace{1em}\alpha ={\mu }_m/D;\hspace{0.33em}\beta =K_s/S_{in}\end{array}$$

Then, the criterion is obtained,

$$\mathrm{\Gamma }=0\hspace{1em}\Rightarrow \hspace{1em}{\displaystyle \frac{Y_0}{Y_1}}={\displaystyle \frac{\alpha \beta (1-\beta )-\beta (1+\beta )}{(\alpha -1)\left({(\alpha -1)}^2+\beta \right)}}{S}_{in}=R$$

Γ > 0: stable; Γ < 0 unstable; Γ = 0: the center where Hopf bifurcation might take place.

Based on the rhythm of yeast, the example in this subsection could be categorized as the “respiratory oscillations” since they are routinely monitored by following the dissolved oxygen. Mathematically, when the auto-catalytic element is introduced with substrate inhibition, that is, Yxs = Y₀ + Y₁S, the above “physical” explanation is simulated even though the role of oxygen is only implicit. The simulation results are provided in Figure 10, where two types of the specific growth rates are analyzed. With the high dilution rate D in (7), both Monod- (Figure 10A,B) and Haldane-type kinetics (Figure 10C,D) are detected with biochemical oscillators [77].

3.4. Delay Feedback

3.4.1. The Illustrative Example of Delay Feedback

As discussed in previous sections, Plk1 in the CDK1-APC loop or the auto-catalytic element in continuous ethanol fermentation might add a type of time delay to the system. The formation of oscillations is reported [33,43] to be related to delayed negative-feedback loops [37], and one of the best examples is yeast grown under high dilution rate. Nutrient-limited conditions exhibit robust, highly periodic oscillations in the form of respiratory bursts, which could be detected by the optical density (OD 600) [65,79].

To illustrate the essential requirements for delay feedback-induced oscillations, first, negative feedback is necessary to carry the dynamical system back to the ‘starting point’ of its oscillation; second, the negative feedback signal must be sufficiently delayed in time so that the system does not settle on a stable steady state [80]. However, delay is always related to the distributive characteristics, such as colony and age distribution of the cell population, and needs to be evaluated explicitly. To account for this, age distribution actually causes the population behavior to delay, compared with the single-cell scenarios. The complicated math is provided in Figure 11 for interested readers.

When concerned about the distributive factors like age population, modeling efforts are concentrated on the relationships between single cells and population behavior while keeping the complexity of the model low. Jobses et al. [34] analyzed the sustained oscillations in biomass, ethanol, and glucose concentrations in the continuous ethanol fermentation using Z. mobilis and pointed out that the theoretical possibility of such a system is a delayed response of Z. mobilis on ethanol inhibition, which is modeled by introducing a reproduction-related K-component and turns the structural model dual compartmental, and

Table 2. Comparison of the structured and unstructured models.

Expression of Ethanol Inhibition
Direct time-delay effect	Indirect time-delay effect by inserting intermediates
$\left\{\begin{array}{l}S(t)+X(t)\to P(t)+aX(t)\\ P(t+\tau )\hspace{0.33em}\underrightarrow{inhibits}\hspace{0.33em}S(t)+X(t)\end{array}\right.$	$\left\{\begin{array}{l}S(t)+E(t)\to bE(t)\\ S(t)+E(t)\to P(t)+aX(t)\\ P(t)\hspace{0.33em}\underrightarrow{inhibits}\hspace{0.33em}{S}_{to}E(t)\end{array}\right.$

provides the difference against the explicit time delay. Cazzador [83] considered the asymmetric reproduction of the budding yeast; Daugulis et al. [19,84] integrated the ethanol inhibition history, etc. The above modeling techniques express time delay by means of a series of connected conductions, and relevance to the pure time delay is studied in [85].

3.4.2. Time Delay in the Feedback

In this section, pure time delay in the feedback loop is assessed. The model of analysis is provided as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{dS}{dt}}=D(S_{in}-S)-\sigma X\\ {\displaystyle \frac{dX}{dt}}=-DX+\mu X\\ {\displaystyle \frac{dP}{dt}}=-DP+\rho X\\ S(0)=S_0,X(0)=X_0,P(0)=P_0.\end{array}\right.,$$

(9)

The complexity of system (9) lies in the fact that the kinetics describing changes correlated with microbial metabolism could be quite sophisticated, and similar to the Michaelis–Menten equation, the first well-known model characterizing growth rate of the biomass constrained by one limiting substrate is developed by Monod. While, for the fermentation with S. cerevisiae, high ethanol concentration inhibits metabolism of the biomass, and the specific growth rate μ(S, Pτ) could be described as follows [36]:

$$\begin{array}{l}\mu (S,P_{\tau })={\displaystyle \frac{{\mu }_{m}S}{S+K_s}}\cdot {\displaystyle \frac{K_i}{P_{\tau }+K_i}}\\ P_{\tau }:=P(t-\tau )\end{array},$$

(10)

where K_s is the half saturation growth rate of biomass; K_i represents the inhibitory constant; and P_τ is the delayed state of ethanol concentration, which indicates that ethanol inhibition manifests itself after τ-time interval. Time delay seen as a bifurcation parameter is reported [86,87] for producing Hopf bifurcations and limit cycles.

To address time delay-induced Hopf bifurcations, system (9) is revisited first. The equilibrium (S^*, X^*, and P^*) is stable if all roots of the corresponding characteristic equation of the linearized equation have negative real parts [88]. By system reduction, the eigenvalues are λ_1,2 = −D/Y_xs, and λ₃ = μ − D. For very large dilution rate D, the system would washout, but for whatever case with a constant Y_xs, system (9) would not produce Hopf bifurcation when τ = 0.

Next, the system with τ ≠ 0 is studied. Similarly, we can assess the stability status by analyzing the eigenvalues of the system Jacobian, only that the characteristic equation contains exponential polynomials as a result of delay, that is,

$$\begin{array}{l}{(\lambda +D)}^2(\lambda +a+b{e}^{-\lambda \tau })=0\\ with:a={\displaystyle \frac{X^{*}}{Y_{xs}}}\cdot {\displaystyle \frac{\mathsf{\partial }{\mu }^{*}}{\mathsf{\partial }S}};\hspace{0.33em}b=-Y_p{X}^{*}{\displaystyle \frac{\mathsf{\partial }{\mu }^{*}}{\mathsf{\partial }{P}_{\tau }}}\end{array},$$

(11)

Other than the double roots λ = −D, the transcendental equation λ + a + be^−λτ = 0 determines the occurrence of Hopf bifurcations. The math for bifurcating out Hopf points with Equation (11) is provided in Box 4, where time delay-induced limit cycles in the feedback is addressed.

Box 4. Analysis on the transcendental equation.

Since a and b are positive, λ + a + be − λτ is always positive for λ ≥ 0 when τ ≠ 0, indicating the transcendental equation has no real and non-negative root. With varying τ, a pair of eigenvaluse crossing the imaginary axis when,

$$\left.\begin{array}{l}a+b\cos (\omega \tau )=0\\ \omega -b\sin (\omega \tau )=0\end{array}\right\}\Rightarrow {\omega }^2=b^2-a^2$$

If b < a, the zero would not reach the imaginary axis, and switch of stability will not happen for any τ > 0. Otherwise, the critical frequency is obtained, as well as τ_c

$$\begin{array}{l}{\omega }_c=\sqrt{b^2-a^2}\\ {\tau }_c={\displaystyle \frac{1}{{\omega }_c}}\arccos \left({\displaystyle \frac{a}{-b}}\right)+{\displaystyle \frac{2n\pi }{{\omega }_c}}\end{array}$$

that when τ > τ_c > 0, oscillations are obtained. Moreover,

$$sign\left\{\mathrm{Re}\left({\left.{\displaystyle \frac{d\lambda }{d\tau }}\right|}_{\lambda =i{\omega }_c}\right)\right\}>0$$

needs for bifurcating out unstable eigenvalues with the increase of τ, i.e., for any a τ > τ_c, a pair of conjugate roots λ_1,2 = α(τ) ± iω(τ) exists for the transcendental equation, where α(τ) > 0.

Heuristically, delay tends to destabilize. From the above analysis, unstable roots Re(λ) > 0 might emerge because of the transcendental Equation (11), and varying τ might cause zeros of the characteristic equation to across the imaginary axis, and the systems (9) and (10) might change stability status. By taking τ as a continuation parameter, one can examine the location of the roots and direction of motion as they cross the imaginary axis.

Based on the delay feedback mechanism, it is possible to investigate the oscillatory behavior for systems (9) and (10). First, the batch [89] (Figure 12A) and chemostat [90] (Figure 12B) experimental data are adopted to estimate parameters of the process. With the provided regression parameters, τ_c = 20.97 h is calculated, which is less than the internal delay τ = 21.72 h, indicating that oscillations are produced for the system. Then, the analysis is conducted with varying inputs, and domains of washout, stable/oscillatory, and unstable regions are provided in Figure 12C. In between the Dc and Db, when τ > τ_c, oscillations are produced; otherwise, the system is stable (Figure 12D).

4. Conclusions

The study of oscillations in cell biology provides valuable information about the organization of cellular processes. Biochemical oscillators are one of the simplest cases of complex systems, and they are always on the edge of driving the process to destabilize. Therefore, a study on the mechanisms that cause internal instability might provide insights into eliminating these oscillations, facilitating process design/operation for constructing a continuous ethanol fermentation process in a cost-effective manner.

In this mini-review, negative feedback laws are focused as the tool to describe possible oscillations. However, the stimulated oscillations through linear feedback (or gain control) are not internally stable ones; hence, they are not proper for characterizing biochemical oscillators. We have sorted out elements of nonlinearity, auto−catalysis, and time delay as the three scenarios that might cause the feedback loops to oscillate, and specific examples are provided for each case.

To address this, these mechanisms are directly applicable for modeling the biochemical oscillators universal in ethanol fermentation processes. Concerning that the abovementioned biochemical oscillators are on the edge of driving the process to destabilize, which draws our attention because continuous ethanol fermentation are long-term processes along which set-points, control objectives, physicochemical variables, and dynamic behavior can be changed by the operator, and any unexpected disturbance might cause the process to terminate. Hence, it is critical to identify elements of causes that lead the system to be internally unstable, which might be used as a guide for process design/operation to eliminate possible oscillations and to develop out advanced configurations that could produce bio-ethanol in a continuous, cost-effective manner.

Given the limitations inherent in our literature review process, our analysis primarily reflects our current knowledge and understanding of this subject matter, for example, there might be other elements causing the formation of oscillations in the feedback loop. Consequently, it is possible that not all relevant works in this field have been included, and we acknowledge the potential for misinterpretation of specific findings. Nevertheless, we aspire to have introduced a novel perspective on this topic and offer a resourceful reference point for our fellow researchers to build upon and further explore.