The In Silico Optimization of a Batch Reactor for D-Fructose Production Using the Cetus Process with In Situ Cofactor Quick Regeneration

Abstract

Currently, D-fructose (DF) is produced through enzymatic isomerization of beta-D-glucose (DG) under disadvantageous conditions (equilibrium conversion of 50%, costly separation, etc.). Alternatively, the two-step Cetus enzymatic process became a promising approach for producing high-purity DF. First, DG is oxidized to keto-glucose (kDG) using commercial pyranose 2-oxidase (P2Ox). To avoid the fast P2Ox inactivation by the in situ produced hydrogen peroxide, catalase is added to decompose this byproduct. The DG oxidation occurs with high conversion and selectivity, leading to kDG free of allergenic aldose compounds. Then, kDG is reduced to DF by using the NADPH cofactor and aldose reductase (ALR). This study aims to evaluate the continuous in situ regeneration of NADPH at the expense of formate decomposition in the presence of formate dehydrogenase (FDH). By adopting a kinetic model from literature, this in silico analysis determines the optimal operation of a batch reactor (BR) used in the Cetus second step to maximize the DF production and minimize the consumption of costly NADPH. Compared to its simple operation, the optimized BR with cofactor regeneration reported a 25% lower NADPH consumption, though the amount of the processed substrate is ca. 3× higher. Also, the costly enzymes (ALR, FDH) consumption is 2× smaller.

1. Introduction

Recent advances in obtaining genetically modified enzymes have allowed scientists to develop many biosynthesis processes of industrial interest, which have generally replaced the classical fine chemical synthesis processes due to the advantages of enzymatic processes: (a) they produce fewer byproducts; (b) they consume less energy; (c) they generate less environmental pollution; (d) they use smaller catalyst concentrations and much more moderate reaction conditions [1,2].

However, to optimally solve the associated engineering problems (process design, operation, control, and optimization), it is essential to produce an adequate mathematical (kinetic) model of the process. This model should preferably be based on the process mechanism to form interpretable predictions of the process behavior under variable running conditions and to compare the results to the literature data [3,4,5].

Despite their larger volumes, batch enzymatic reactors (BRs) with mechanical mixing are the most used bioreactors because they ensure a high diffusion rate of compounds in the liquid phase and easy control of temperature/pH [6,7,8].

Although the efficiency of the process has been extensively experimentally studied using various enzymes, the engineering aspect regarding the bioreactor design, including optimal operation/control, has seldom been investigated [9,10,11,12].

Concerning the reactor, an essential engineering problem pertains to the development of optimal operating policies seeking to maximize production and minimize raw material consumption while obtaining a product of high quality (fewer byproducts). This problem depends on at least two aspects: (1) the adopted technology (chemical, biochemical, or biological catalysis) and (2) the engineering analysis used to optimize the reactor operation (this study) [13,14].

In the case of a BR, its optimal operation problem can be effectively solved in silico by using an off-line (“run-to-run”) approach, with the optimal operating policy being determined by using an adequate deterministic kinetic model previously identified based on experimental data (this study, and [8,9,10,15,16,17,18,19,20,21,22]).

However, this approach is not an easy task, as it involves multiple contrary objectives and a significant degree of uncertainty in the model/constraints [23,24]. The reactor’s optimal operating policy is usually determined by using an effective optimization rule [21,25,26]. In the deterministic alternative (this study), single-/multi-objective criteria, including product selectivity/yield maximization and (raw) material consumption minimization, are usually used to obtain feasible optimal operating strategies for the analyzed reactor [24] by using specific numerical algorithms [27,28].

The a priori in silico analysis also allows for comparing the performance of various bioreactor constructive/operating alternatives, as described below [13,14].

BRs are commonly used for slow processes, because they are highly flexible and easy to operate for various scenarios, such as the following [14,24]: (i) a single- or a multi-objective optimization performed off-line to determine the best batch time and initial substrate/biocatalyst load [23,26,29,30,31,32,33]; (ii) a batch-to-batch (BR-to-BR) optimization, which includes a model updating step based on information acquired from the past batches to determine the optimal load of the next BR [9,16,17,18,25,28,34]; (iii) a sequence of BRs of equal volumes linked in a series (SeqBR) [11], where every BR’s content is transferred to the next one, and the reactants and biocatalyst concentrations of the latter are adjusted to levels determined off-line to ensure its optimal operation [18,28]; (iv) a fed-batch reactor (FBR) with an optimally varied feeding policy of biocatalyst/substrate(s), which is not discussed here [14,24,34,35]. Despite FBRs’ superior performance, they are difficult to operate, because they need previously prepared stocks of biocatalyst and substrate(s) at different concentrations (determined a priori in silico) to be fed for every “time-arc” of the batch (that is, an equal batch time division in which the feeding composition is constant) [36]. The time-step-wise variable optimal feeding policy of the FBR is determined off-line [14] or on-line by using a simplified, often empirical mathematical model to obtain a state parameter estimator based on on-line recorded data (such as the classical Kalman filter) [19,22,23,25,27,29,37,38,39].

D-fructose is a sweetener of high value in the food industry and medicine. As other polyols are largely used as sweeteners (e.g., sorbitol, mannitol, xylitol, erythritol), it is produced on a large scale by using chemical or biochemical catalysis [40,41].

The chemical catalysis required to produce DF (that is, hydrogenation of glucose on Ni, Fe, or Fe-Ni alloy catalysts) has the critical disadvantage of significant energy consumption, as it occurs at high pressures (30 bar) and temperatures (140 °C). One alternative is beta-D-glucose isomerization to D-fructose on an Fe/carbon black catalyst [41].

As displayed in

Table 1. Comparison between three enzymatic methods to produce D-fructose at a large scale.

Characteristics	Glucose Isomerization [a,d]	Cetus Two-Step Process [b]	Inulin Hydrolysis [c]
Number of steps	1	2	1
Conversion (%)	50 (limited by the equilibrium) [d]	99	99.5
Raw material availability	Glucose from the starch of crops, molasses, cellulose, and food processing byproducts [42,43]		Genetically modified chicory crop; cultures of Aspergillus sp.
Impurities in the product	Yes	Traces	Negligible amounts
Reaction type	Enzymatic isomerization	Enzymatic oxidation (step 1), followed by enzymatic reduction (step 2)	Enzymatic hydrolysis
Enzyme mobility	Immobilized [d]	Free (suspended)	Immobilized
Enzyme stability and other additives	Intracellular glucose-isomerase (e.g., Streptomyces murinus) of low stability; metal (Al) salts	Pyranose 2-oxidase (P2Ox) and catalase (step 1); aldose reductase and NAD(P)H (step 2); enzymes are costly	Inulinase
Temperature	50–60 °C	25–30 °C/ 25–30 °C	55 °C (40–60 °C)
Reaction time (h)	7	3–20 (step 1); 25 (step 2)	13
pH	7–8.5	6.5–7(–8.5); 7–8.5	5.5
Reaction steps	1 isomerization	2 oxidation (step 1), reduction (step 2)	1 hydrolysis
Coenzyme necessary?	No	Yes, catalase for (step 1) to prevent P2Ox quick inactivation; NAD(P)H for step 2. NAD(P)H is continuously regenerated in situ.	No
Product purification	Difficult [d]	Simple (due to high selectivity)	Simple (due to high selectivity)
Product purity	2–5% impurities [d]	High (99.9%)	High (99.9%)

[a] Process described by [44,45,46]. The raw material HFCS is obtained from yeast hydrolysis (resulting in a mixture of 42% fructose, 50% glucose, and 8% other sugars) [44]. [b] Process described by [13,47,48,49,50]. Cofactor NAD(P)H regeneration was given by [12,51,52,53,54,55,56,57]. [c] Process described by [58,59,60,61]. [d] This process suffers from a large number of disadvantages: (i) The reaction is thermodynamically limited to around 50% glucose conversion, making the subsequent fructose separation in large chromatographic columns very costly. (ii) Glucose isomerase is an intracellular enzyme with relatively poor stability, making its purification and immobilization very difficult. (iii) The amylase used to carry out the starch saccharification (to obtain the HFCS raw material) requires calcium ions for full activity, but calcium inhibits glucose isomerization, requiring its removal by ion-exchange treatment prior to glucose isomerization. (iv) The fructose product is still contaminated by several other saccharides (such as aldose, which is an allergenic compound) [7,46,58,59,60,61,62,63,64,65].

, the biocatalytic routes to producing D-fructose are by far more convenient due to a large number of advantages: they consume less energy, since they occur under ambient conditions; they produce less waste due to the high yields and selectivity; and the product is free of allergenic compounds.

Glucose isomerization. “Starting from the high-fructose syrup (HFCS) obtained from starch [44], after rough/fine filtration, ion exchange, and evaporation, a DG isomerization step leads to a high fructose syrup (HFS, of 42–55% D-fructose)” [44,66,67,68]. However, this industrial process has many disadvantages, as mentioned in [d] of Table 1.

Inulin hydrolysis. This promising alternative produces DF of high purity. However, the used inulinase is still expensive, while the whole development process is limited by the production capacity of GMO chicory crops and of the cultures of Aspergillus sp. (Table 1).

The two-step Cetus process: For the production of high-purity DF with high yields, the following steps are necessary [48,49]: Step 1: DG is converted to kDG in the presence of commercial pyranose 2-oxidase (P2Ox, from Coriolus sp. expressed in E. coli) at 25–30 °C and pH = 6–7, with a high conversion and selectivity [13,69,70]. Catalase is added to decompose the resulting H₂O₂, thus avoiding the quick P2Ox inactivation. Step 2: The obtained kDG is then reduced to DF by using commercial (recombinant human or animal, expressed in E. coli) aldose reductase (ALR) (EC 1.1.1.21) and NAD(P)H as the cofactor (proton donor) at 25 °C and pH = 7 (Figure 1) [47]. The resulting NAD(P)⁺ is regenerated (in situ or externally) and re-used [51,52,70] (Figure 1). Being costly, NADPH should be recovered. The cofactor (NADPH or NADH) regeneration can be achieved in several ways [51,53,54,70,71,72]. For instance, Gijiu et al. [12] realized this step by using an in situ alternative at the expense of the enzymatic degradation of ammonium formate. The same route will also be followed here, being simple and less expensive.

This study aims to analyze and optimize Step 2 of the Cetus process with continuous in situ regeneration of the cofactor NADPH (Figure 1). Thus, by adopting an adequate kinetic model from the literature, the in silico analysis will evaluate the performance of this alternative by comparing it with the baseline process experimentally studied in a BR, without NADPH regeneration. The obtained results will finally allow the optimization of the initial BR load by using a nonlinear programming (NLP) procedure, seeking DF production maximization in the presence of various technological constraints, by limiting the consumption of raw materials.

This study presents a significant number of novel aspects: (i) The engineering evaluation of the Cetus process. (ii) The in silico engineering analysis of the Cetus process with continuous in situ NADPH regeneration is a first in the literature. (iii) The method with which this BR optimization problem was successfully solved, limiting the consumption of costly enzymes and the cofactor, is a model that can be followed to solve similar multi-enzymatic processes. (iv) Prior to this study, there were very few enzymatic processes analyzed in the literature from an engineering point of view that also accounted for the cofactor during the optimization procedure. (v) The scientific value of this study is not virtual, as the numerical analysis is based on the kinetic model of Maria and Ene [47], constructed and validated using the extensive experimental data sets shown in Figure 2, and based on the kinetic model of Maria [55] for NADPH regeneration, validated using the extensive experiments of Slatner et al. [51]. (vi) The in silico analysis suggests that a BR optimally operated using a policy determined by applying an NLP procedure can lead to high performance, that is, total conversion, with the reaction occurring quantitatively with half the enzyme consumption (FDH, ALR) and 25% less NADPH. (vii) The biocatalyst and cofactor (ALR and NADPH concentrations) play major and combined roles as control variables in BR optimization (an option seldom discussed in the literature). (viii) The in silico (model-based) optimal operation of enzymatic reactors is a very important engineering tool because it can lead to consistent economic benefits, as proven by the results presented in this study.

2. The Experimental Enzymatic Reactor

The analyzed BR is that used by Maria and Ene [47] to derive the kinetic model of the Cetus process’s first step and second step (analyzed here). The BR characteristics are presented in

Table 2. Nominal (non-optimal) operating conditions of the experimental BR with suspended ALR and NADPH used by Maria et al. [47] to investigate kDG conversion to D-fructose. DSn = data set number “n”. Notations: S = substrate (kDG); P = product (D-fructose, DF); A = NADPH; A(+) = NADP⁺; E = ENZ = ALR.

Parameter	Nominal Initial Value		Remarks
Data set # 1 (DS1)	[S]o = [kDG]o	35 mM	Other species initial conc. [P]o = 0; [A(+)]o = [NADP(+)]o = 0; [EA]o = 0
	[A]o = [NADPH]o	35 mM
	[E]o = [ALR]o	0.0048 U/mL
Data set # 2 (DS2)	[S]o = [kDG]o	35 mM
	[A]o = [NADPH]o	35 mM
	[E]o = [ALR]o	0.00257 U/mL
Data set # 3 (DS3)	[S]o = [kDG]o	35 mM
	[A]o = [NADPH]o	6 mM
	[E]o = [ALR]o	0.0055 U/mL
Data set # 4 (DS4)	[S]o = [kDG]o	15 mM
	[A]o = [NADPH]o	35 mM
	[E]o = [ALR]o	0.006 U/mL
Temperature, pH	25 °C, 7 [47,51] (optimal)		pH buffer
Optimization limits of initial loads	[S]o ∈ [5–100], mM [47,73] [NADPH]o ∈ [5–80], mM [74,75]		[E]o ∈ [0.003–0.1] U/mL [47,76] [FDH] ∈ [100–2000] (U/L) [12,51]
NADPH regeneration	[HCOO]o = [kDG]o		Similar to Maria [55]; Slatner et al. [51]
	[CO2]o = 0; [FDH]o = 1000 U/L (adopted as an average)		[FDH]o should be determined using optimization
Reactor volume (L)	1		Up to 3 L capacity
Batch time (tf) (h)	24		For DS1–DS4
Solubility in water	DG (kDG)	5–7 M	(25–30 °C) [77]
	DF	ca. 22.2 M	25 °C, pH = 7 [https://en.wikipedia.org/wiki/Fructose, accessed on 14 August 2025]
CO2 solubility [CO2] *	31.3 (mM) at (25 °C)		[78,79]
DG (kDG) water solution viscosity	1–3 cps (for <0.3 M) 1000 cps (4.5 M, 30 °C) vs. 1094 cps (molasses, 38 °C)		[57,80]

(*) denotes the saturation value.

[13,47]. The reactor operation is completely automated, with tight control of the pH, temperature, and mixing intensity.

3. Kinetic Model of Biocatalytic Process

The reaction scheme of the two coupled enzymatic reactions is presented in Figure 1. In the main reaction (R1), kDG is reduced to D-fructose (DF) by using suspended ALR (aldose reductase) and the cofactor NADPH (nicotinamide adenine dinucleotide phosphate—reduced form) as the proton donor. In parallel, the NADPH cofactor is continuously regenerated in situ through the reaction (R2), at the expense of ammonium formate (HCOO) degradation in the presence of suspended FDH (formate dehydrogenase), according to a similar reaction used by Slatner et al. [51] for the same purpose.

For the main reaction R1, the process kinetic model proposed by Maria et al. [47] is based on their proposed reaction pathway, shown in Figure 1. The reaction rate expressions are given in

Table 3. The overall reactions considered in the kinetic model proposed by Maria and Ene [47] (that is, the main reaction R1 scheme in Figure 1—right) for the kDG enzymatic reduction to DF using the NADPH cofactor and suspended aldose reductase (commercial recombinant ALR obtained by expressing human 1-316aa plasmids in E. coli; enzyme source: ATGEN, Cat. no. ALR-0901).

The Overall Reaction R1 Shown in Figure 1 and Its Associated Side Reactions

${\mathrm{C}}_6{\mathrm{H}}_{10}{\mathrm{O}}_6 \left(S\right) + \mathrm{NADPH} \left(A\right)+ {\mathrm{H}}^{+}\underset{(\mathrm{phosphate} \mathrm{buffer})}{\stackrel{\mathrm{ALR} (E), r_P}{\to }}{\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6 \left(P\right) + {\mathrm{NADP}}^{+}(A^{+})$ ${y\mathrm{NADPH} + \mathrm{ALR}}_{\underset{k_{-d}}{\leftarrow }}^{\stackrel{k_d}{\to }}(\mathrm{ALR}•{\mathrm{NADPH}}_{\mathrm{y}})$ $\mathrm{ALR}\left(E\right)\stackrel{r_i}{\to }\mathrm{inactiveALR}\left(\mathit{Ein}\right)$

Rate expressions of the reactions displayed in Figure 1—right, corresponding to the mechanism of the overall reaction R1

$r_P={\displaystyle \frac{k_p[E_t][S] \left(\frac{[A]}{K_R{K}_A}-\frac{1}{K_{eq}{K}_{AP}}\frac{[A^{+}][P]}{[S]}\right)}{1+\frac{[A]}{K_A}+\frac{[A][S]}{K_R{K}_A}+\frac{[A^{+}]}{K_{AP}}}}$, (successive Bi-Bi mechanism) $r_d=k_d[A][E]$; $r_{-d}=k_{-d}[E^{*}A]$; $r_i=k_i[E]$ $K_A={\displaystyle \frac{[E][A]}{[EA]}}={\displaystyle \frac{k_{-a}}{k_a}}$; $K_R={\displaystyle \frac{[EA][S]}{[EAS]}}={\displaystyle \frac{k_{-r}}{k_r}}$; $K_{eq}={\displaystyle \frac{[E{A}^{+}][P]}{[EAS]}}={\displaystyle \frac{k_p}{k_{-p}}}$; $K_{AP}={\displaystyle \frac{[E][A^{+}]}{[E{A}^{+}]}}={\displaystyle \frac{k_{ap}}{k_{-ap}}}$

E = aldose reductase (ALR); A = NADPH; A⁺ = NADP⁺; S = kDG (substrate); P = D-fructose (product).

. The nine associated rate constants in

Table 4. The rate constants of the kinetic model in Table 3 estimated by Maria et al. [47] using the four data sets presented in Figure 2.

Rate Constant	Value	Rate Constant	Value
$k_p$, mM/min/(U/mL)	3.9 106	$k_d$, 1/(mM min)	2.07·106
$K_A$, mM	65.41	$k_{-d}$, 1/min	858.23
$K_R$, mM	1.24	$y$, mM/(U/mL)	1.48·104
$K_{eq}$, mM	1427	$k_i$, 1/min	7.01·10−2
$K_{AP}$, mM	0.886

are those identified by Maria et al. [47] using four kinetic data sets (DS1–DS4 in Table 2) recorded in batch experiments plotted in Figure 2. An extensive and reasoned/documented discussion about this kinetic model is given by Maria and Ene [47]. To maximize the recorded kinetic information, these runs were carried out by using long batch times of 24 h and by varying the initial enzyme/reactant/cofactor ratios in the range of kDG ∈ [15–35] mM, NADPH ∈ [6–35] mM, and ALR ∈ [0.0026–0.006] U/mL.

The overall reduction reaction r_P of R1 is given in Table 3. It follows a successive Bi-Bi mechanism, being accompanied by the reversible binding of ALR to NADPH to form an inactive complex (E*A_y) and by the deactivation of the enzyme ALR [47].

The model rate constants have been estimated using these four sets of experimental kinetic curves (Figure 2). A weighted least-squares criterion has been used as the statistical estimator, because the standard measurement errors of the species are very different [4]. The obtained kinetic model of Maria et al. [47] was proven to be adequate in a statistical sense (see the model predictions vs. experimental points in Figure 2).

The continuous in situ regeneration of the reduction reaction cofactor (NADPH here) is a very common technique to ensure high conversion of the main enzymatic reaction throughout the entire batch. There are several alternatives used to realize such an objective, well discussed in the literature [70,72,81,82,83,84]. The NADPH regeneration reaction R2 displayed in Figure 1 was adopted by analogy with the NADH regeneration in the D-fructose reduction to mannitol, extensively studied experimentally by Slatner et al. [51]. Based on these experimental data, Maria [55] proposed a kinetic model for cofactor regeneration and estimated its rate constants. In this study, by preserving this similarity (that is, using the FDH enzyme and the same HCOO substrate under the same reaction conditions), the same kinetic model was adopted for NADPH regeneration while keeping the same relative rate constants, with adapted units for kc2, as presented in

Table 5. The kinetic model proposed by Maria [55] for the reaction (R2) in Figure 1, that is, the continuous in situ regeneration of the cofactor NADPH at the expense of ammonium formate (HCOO) degradation in the presence of FDH. Rate constants have been estimated to match the experimental kinetic data of Slatner et al. [51] and extrapolated when using NADPH instead of NADH under the same reaction conditions and the same FDH.

$HCO{O}^{-}+NAD{P}^{+}\stackrel{FDH}{\to }C{O}_2\uparrow +NADPH$

$R2={\displaystyle \frac{kc2\cdot [FDH][HCOO][NAD{P}^{+}]}{KM2+K_{HC}[HCOO]+K_{NADP}[NAD{P}^{+}]}}$, (mM/min)

kc2 = 0.1387, 1/min/(U/L); KM2 = 8.8047 × 10−2 mM; KHC = 5.0061 × 10−2; KNADP = 90.181

. The Michaelis–Menten constants adopted (FDH) (EC 1.2.1.2, or EC 1.17.1.9 from Candida boidinii) match the literature data, as follows: (i) The KM2 rate constant of 0.088 (mM) is comparable to the 0.09 mM result of Chenault and Whitesides [70], the 0.09–0.8 mM result of Jiang et al. [82], or the 0.083–0.92 mM result of Brenda [85]. (ii) kc2 = 0.1387 1/min/(U/L) = 2.31 (1/s) (for nominal FDH = 1000 U/L) is comparable to the 0.26–3.7 (1/s) result of Brenda [85] or the 1.07–8.8 (1/s) result of Jiang et al. [82] under the same NADPH regeneration conditions (25 °C, pH 7).

4. Dynamic Model of Enzymatic Batch Reactor

For in silico simulation of the key species dynamics in the BR, a classical ideal model was adopted [6] using the usual hypotheses: (i) isothermal and iso-pH; (ii) perfectly mixed liquid phase (with no concentration gradients), ensured by continuous mechanical mixing; (iii) constant liquid volume, with its increase due to pH-controlling additives being negligible.

In a general form, the enzymatic BR dynamic model is presented in Equation (1), including the mass balances of the six key species of the kDG reduction (reaction R1 in Figure 1), most of them being observable. The species mass balance takes the following form:

$${\displaystyle \frac{d{C}_i(t)}{dt}}=\pm {\mathrm{r}}_{\mathrm{i}}\left(C_o,k,t\right){ ; C}_{i,o}=C_i\left(t=0\right)$$

(1)

The index “i” refers not only to the species [kDG, P, NADPH, NADP(+), ALR, E*A] of reaction R1 but also to the species [HCOO, CO₂, FDH] of reaction R2. The detailed dynamic model of the BR is presented in

Table 6. Key species mass balances of the batch bioreactor BR model, including the bioprocess kinetic model of Maria et al. [47], completed with NADPH regeneration, similarly to the model of Maria [55]. The reaction rate expressions are given in Table 3 and Table 5, while the associated rate constants are given in Table 4 and Table 5.

Key Species Mass Balances in the BR (Corresponding to Equation (1))

The Main Experimental Conditions in Table 2

$-{\displaystyle \frac{dS}{dt}}={\displaystyle \frac{dP}{dt}}=r_P$; ${\displaystyle \frac{d{A}^{+}}{dt}}=r_P-R2$ ${\displaystyle \frac{dA}{dt}}=- r_P-y{ r}_d+y{ r}_{-d}+R2$ ${\displaystyle \frac{dE}{dt}}= -{ r}_d+{ r}_{-d}-{ r}_i$ ${\displaystyle \frac{d(E^{*}A)}{dt}}={ r}_d-{ r}_{-d}$ ${\displaystyle \frac{d[HCOO]}{dt}}=-R2$; ${\displaystyle \frac{d[C{O}_2]}{dt}}=+R2$

Liquid volume = 1 L Phosphate buffer, pH = 7; 25 °C Initial concentrations are in the following ranges: [kDG] = 15–35 mM; [NADPH] = 6–35 mM; Initial [ALR] = 2.6–6 U/L; [HCOO]o = [kDG]o [56]; [FDH] = 100–2000 U/L. If [CO2] ≥ [CO2] *, then [CO2] ≈ [CO2] *, and the excess leaves the system. FDH inactivation is neglected. Notations: S = substrate (kDG); P = product (fructose); A = NADPH; A(+) = NADP(+); E = ALR. The units are in mM, min, and U/L. (*) denotes the saturation concentration of Table 2.

.

5. BR Simulation and Optimization

5.1. Nominal BR Simulation and Selection of Control Variables

By using the reactor model in Table 6, accounting for in situ NADPH regeneration, the species dynamics were simulated for all four batch experiments DS1–DS4 of Maria et al. [47], with the initial conditions shown in Table 2. The results, plotted in Figure 2, reveal the following: (1) Regenerated NADPH remains at a low but effective concentration during the batch, thus ensuring a high process efficiency. (2) Cofactor regeneration is efficient enough, so residual [NADPH] at the batch end is higher than when regeneration does not occur. (3) Due to continuous cofactor regeneration, the conversion is complete. (4) Efficient NADPH recovery leads to quick and practically total HCOO decomposition. (5) In turn, the quick HCOO decomposition leads to an abrupt rise in [CO₂], quickly reaching its saturation level, with the excess leaving the system. Only in the DS4 case does the lower [HCOO] produce [CO₂] below its saturation level. (6) As pointed out by Maria and Ene [47], the suspended ALR enzyme (ENZ) suffers significant inactivation during the batch, even if its level is sufficient to ensure the progress of the process during the whole batch. As reviewed by Maria et al. [47], a more efficient but costly alternative is to use an immobilized ALR.

Analyzing the main reactions in the process in Figure 1 and the reactor model in Table 6 reveals that the chosen control variables are those with the highest influence on BR efficiency, that is, [S]o = [KDG]o, [A]o = [NADPH]o, [E]o = [ALR]o, [FDH]o (right column in Table 6).

5.2. Single-Objective Function Optimization (NLP) of the BR

Optimal operation of the BR in operation mode involves in silico determination of the optimal initial load that ensures product [P] (D-fructose) maximization in the presence of multiple technological constraints.

Optimization of the BR operation involves finding its initial load with the four key species (control variables). In mathematical terms, for a single objective function, this optimization problem can be written as the maximization of [P] (D-fructose) production, that is,

Find [KDG]o, [NADPH]o, [ALR]o, [FDH]o, such that
Max Ω, where Ω = [P(t)]

(2)

The problem in Equation (2) can be solved by using a common nonlinear programming (NLP) optimization rule [4], seeking to determine the extreme of the objective function in the presence of multiple constraints given in Section 5.3. In Equation (2), the time-varying P(t) is in fact a multi-variable function P(C(t), C_o,k)(t), evaluated by using the process/reactor model in Equation (1) over the whole batch time (t) ∈ [0, t_f] (Figure 2), with the initial condition of C_j,0 = C_j(t = 0) searched during optimization using the iterative numerical rule.

Because the enzymatic process kinetic model in Equation (1), the optimization objective in Equation (2), and the problem constraints in Equation (3) are all highly nonlinear, the formulated problem in Equation (2) translates into a difficult NLP with a multimodal objective function and a non-convex searching domain. To obtain the global feasible solution with enough precision, the multimodal optimization solver MMA of Maria [4] has been used, having been proven to be very effective for solving such difficult NLP problems. The MMA is an adaptive random search that automatically adapts the random search direction and step length by considering the search history in generating the new trial point distribution. To increase the reliability in locating the problem’s global optimum, the MMA search was repeated several times, each time using a randomly chosen starting point in the defined feasible domain with Equation (3).

5.3. Optimization Problem Constraints

The abovementioned formulated NLP problem, Equation (2), must account for the following constraints:

(a)

The BR model in Equation (1);

(b)

To limit the excessive consumption of raw materials (especially the costly enzymes), feasible searching limits are imposed on the control/decision variables (Table 2), based on the previous trials of Maria et al. [47,73], and on literature information [74,75,76]. In mathematical terms, the constraints (b) translate to

$$C_{i,o,\mathit{min}}\le { C}_{i,o}\le { C}_{i,o,\mathit{max}}$$

(3)

index ‘i’ = KDG, NADPH, ALR, FDH

6. Optimization Results and Discussion

The BR optimization problem results are as follows:

-

A comparison of the key species dynamics is shown in Figure 2 for the nominal operation of the BR (Table 2), with or without the use of in situ NADPH regeneration. The same comparison is repeated quantitatively in

Table 7. The productivity and raw material consumption of the analyzed BR in Table 2, when operated in various modes.

Bioreactor Operation			Raw Material Consumption (a,b,c)						DF Prod, mmol
			kDG, mmol		NADPH, mmol	Final NADPH, mmol	ALR, (U)	FDH, (U)
BR Non-optimal experiments [47]	Without NADPH regeneration, Figure 2 (d) (very poor)	DS1	35		35	0.18	4.8	-	11
		DS2	35		35	0.18	2.57	-	11.1
		DS3	35		6	0.03	5.5	-	4.9
		DS4	15		35	0.29	6	-	7.8
	With NADPH regeneration, Figure 2 (d) (good)	DS1	35		35	1.25	4.8	1000	35
		DS2	35		35	1.06	2.57	1000	35
		DS3	35		6	0.5	5.5	1000	35
		DS4	15		35	1.19	6	1000	15
BR optimal initial load, within limits in Table 2	With NADPH regeneration Figure 3 (e,f) (best)	kDG	100	100	26	1.17	3.38	440	100
		NADPH	26
		ALR	3.38
		FDH	440

(a) The reactor liquid initial volume of 1 L (Table 2). (b) The displayed digits come from the numerical simulations. (c) The initial load concentration multiplied by the liquid volume. (d) The BR experimental nominal data set—points #1 to #4 (DS1–DS4 in Table 2, Figure 2) of Maria and Ene [47]. (e) The BR optimal policy (initial load) was obtained by using search intervals in Table 2. (f) The units of the initial load are [kDG], mM; [NADPH], mM; [ALR], [FDH] (U/L).

.

-

The NLP optimal operating policy of the analyzed BR is given in Table 2. The optimal species dynamics are plotted in Figure 3, while its efficiency in quantitative terms is given in Table 7.

-

A comparison of all BR operating alternatives in terms of P production and consumption of raw materials (based on the initial load) is presented in Table 7.

By analyzing these results and the operating alternatives in Table 7, several conclusions can be derived:

(1)

The non-optimal DS1–DS4 BR experimental runs defined in Table 2 perform much better if the NADPH is regenerated in situ. Thus, the realized yields (4.9/35, 11/35, 7.8/15 in Table 7) are very low if NADPH is not regenerated, though the yields are 100% if NADPH is regenerated. This is a major reason to use the in situ cofactor regeneration for this process.

(2)

The non-optimal BR operation (DS1–DS3) using in situ NADPH regeneration resulted in the high consumption of enzymes as a result of the operating alternatives in Table 7. This sub-optimal operation can be improved by applying an NLP procedure using the optimization objective in Equation (2), subjected to the constraints in Section 5.3. Thus, one obtains the optimal BR operation in Table 7 (last row), with the species dynamics plotted in Figure 3. Compared to the experimental nominal, non-optimal BR operation (DS1–DS4), with or without cofactor regeneration, the optimized BR with cofactor regeneration resulted in a 25% lower consumption of NADPH, though the amount of the processed substrate is approximately 3× higher. Moreover, the consumption of costly enzymes (ALR, FDH) is roughly half.

(3)

By analyzing the NLP optimal operating policy of the BR, shown in Table 7 and Figure 3, the following conclusion can be derived: the P-productivity increases with the initial substrate [kDG, NADPH] concentrations if enough enzymes (ALR, FDH) are present and if ALR (and FDH) is not deactivated too fast. To better fulfill such a condition, the best alternative could be to use more stable enzymes, that is, immobilized on suitable porous supports [86,87,88] (not investigated here).

(4)

For sufficiently stable (immobilized) enzymes (ALR, FDH), DF production maximization clearly depends on the available amount of substrate (kDG) and cofactor (NADPH). As the kDG results from Step 1 of the Cetus process [73], a more realistic optimization must concomitantly consider both linked Cetus processes. Some trials have already been conducted [89].

7. Conclusions

In conclusion, the in silico, off-line optimization of a BR operation can significantly improve its efficiency due to its high flexibility, with an easily adaptable process model [90], and the effectiveness of the optimization rules applied, that is, single-objective NLP (used here) or multi-objective techniques (not considered here; see [12,89,91]).

The nominal, non-optimal BR operation without cofactor regeneration demonstrated very poor performance. By comparison, the optimized BR with cofactor regeneration showed 25% lower consumption of NADPH, though the amount of the processed substrate was approximately 3× higher. Moreover, the consumption of costly enzymes (ALR, FDH) was roughly halved.

Thus, the in silico BR optimization analysis appears to be fully justified by the obtained economic benefits.