Introduction

The activity of the enzyme changes with temperature. In the first stage, the activity of the enzyme increases with increasing temperature. At a certain temperature, referred to as \({T_{\mathrm{opt}}}\), the activity of the enzyme is maximal. When the \({T_{\mathrm{opt}}}\) is exceeded, the activity of the enzyme decreases. The knowledge of the optimum temperature \({T_{\mathrm{opt}}}\) and the activation energy E and deactivation energy \({E_{\mathrm{D}}}\) of the enzyme enables the bioprocess optimization with the reduction of its costs at the same time. The \({T_{\mathrm{opt}}}\) value is mostly determined experimentally. The values of the activation energies E and \({E_{\mathrm{D}}}\) can be determined from the curves Arrhenius of the dependence of the logarithm of the reaction rate (lnv) on the reciprocal of temperature (\(T^{-1}\)) [1]. It has been shown that the determined values of E and \({E_{\mathrm{D}}}\) by application of the Arrhenius relationship is burdened with an error [2, 3].

Inulin consists of linear chains of 2,1-\(\beta\)-D-fructofuranose molecules terminated with a glucose residue at the reducing end. Inulin is a reserve carbohydrate \({\left( 0.5-22\%\right) }\) in many plant tubers, roots or leaves [4,5,6]. Inulin can be used for the production of many bio-based products, such as high-fructose syrups, bioethanol, fructose and glucose inulo-oligosaccharides [5, 7, 8].

Inulinases (2,1-\(\beta\)-D-fructan fructanohydrolase) are hydrolysed, catalyzing the inulin hydrolysis of about \({90-95\%}\) converse of inulin [9]. Inulinases can be classified into endo-inulinases (EC 3.2.1.153) and exo-inulinases (EC 3.2.1.80). Exo-inulinases remove terminal fructose residues from the non-reducing end of inulin, and endo-inulinases act on the internal bonds of the inulin molecule to produce inulotrioses, inulotetraoses and fructo-oligosaccharides [7, 10, 11]. Inulinases occur inter alia in plants, fungi Penicillium sp., Aspergillus niger and bacteria Pseudomonas mucidolens, Arthrobacter sp. S37 can to produce endo-inulinase [12,13,14,15,16,17,18,19,20,21].

Endo-inulinases obtained from them are characterized by high activity and high thermostability [12]. The optimum temperatures for endo-inulinases are around \(40\,^{\circ }\)C [2, 3]. They are higher for commercial endo-inulinases [14, 21] even above \(60\,^{\circ }\)C [15, 17] and \(65\,^{\circ }\)C [16, 20] for recombinant endo-inulinases.

Formation of the recombinant endo-inulinases associated with their industrial use. Commercial of preparations endo-inulinase should have economic and technical advantages including low price, high activity and stability under industrial reaction conditions [16].

There have been many publications in the literature on A. niger recombinant endo-inulinase [13,14,15,16,17] used for high-performance engineered endo-inulinanse for fructooligosaccharides production from inulin. There are also publications on recombinant endo-inulinase of various microbial origins, including Pseudomonas mucidolens into Saccharomyces cerevisiae EBY 100 [19], endo Arthrobacter sp. S37 in Yarrowia lipolytica Po1h [18] and Penicillium restrictum A191 in Escherichia coli [20]. However, processes involving recombinant endo-inulinase of various origins purposes can not be designed and optimized without knowing the values of the activation energy E and deactivation energy \({E_{\mathrm{D}}}\) and optimum temperature \({T_{\mathrm{opt}}}\). Additionally, the parameters E, \({E_{\mathrm{D}}}\) and \({T_{\mathrm{opt}}}\) for recombinant endo-inulinases A. niger [13,14,15,16,17] were determined.

Determination the parameters E, \({E_{\mathrm{{\bf D}}}}\), \({T_{\mathrm{{\bf opt}}}}\) for recombinant endo-inulinase activity

The activity of recombinant endo-inulinase in the inulin hydrolysis reaction changes with temperature. At temperatures below the optimum temperature \({T_{\mathrm{opt}}}\), the endo-inulinase activity increases with increasing temperature. After exceeding the optimum temperature \({T_{\mathrm{opt}}}\), the enzyme activity decreases with increasing temperature. Both recombinant endo-inulinase activity increase or decrease are described by the first-order equations due to the enzyme concentration, like in recombinant exo-inulinase [23, 24].

In particular studying recombinant endo-inulinase activity during the inulin hydrolysis, it is assumed that the change in substrate concentration \(C_{\mathrm{S}}\) during reaction time t describes by the first-order equations

$$\begin{aligned} \frac{\mathrm{d}C_{\mathrm{S}}}{\mathrm{d}t}=-k C_{\mathrm{E}} \end{aligned}$$
(1)

where k is the kinetic constant of the enzymatic reaction (min\(^{-1}\)) and \({C_{\mathrm{E}}}\) is the concentration of the active enzyme (M).

Next, the change in recombinant endo-inulinase dimensionless activity A it is also assumed that by the first-order kinetics [23,24,25] with the following equation

$$\begin{aligned} \frac{\mathrm{d}A}{\mathrm{d}t}=-k_{\mathrm{D}}A \end{aligned}$$
(2)

where \(k_{\mathrm{D}}\) is the deactivation process kinetic constants (min-1). Dimensionless activity of enzyme A is expressed by the equation

$$\begin{aligned} A=\frac{C_{\mathrm{E}}}{C_{\mathrm{E}0}} \end{aligned}$$
(3)

where \(C_{\mathrm{E}0}\) is initial concentration of the active enzyme (M).

The kinetic constants k and also, the deactivation constants \(k_{\mathrm{D}}\) depend on temperature T according the equation Arrhenius. The initial assumptions described by equations Eqs. 1, 2 and 3 allow to present the final form of the solution Eq. (4), which was presented in greater detail in the earlier work [26]. The Eq.(4) describes the change in the absolute activity of the enzyme A vs. temperature T

$$\begin{aligned} A=\frac{\mathrm{exp}\left( \frac{\left( T_{\mathrm{opt}}-T\right) E_{\mathrm{D}}B}{R T T_{\mathrm{opt}}\left( \mathrm{exp}B-1\right) }\right) \left( 1-\mathrm{exp}\left( -B \mathrm{exp}\left( \frac{\left( T-T_{\mathrm{opt}}\right) E_{\mathrm{D}}}{RTT_{\mathrm{opt}}}\right) \right) \right) }{1-\exp \left( -B\right) } \end{aligned}$$
(4)

where \({T_{\mathrm{opt}}}\) is the optimum temperature at which recombinant endo-inulinase activity shows maximum activity (K), T is temperature (K), R is gas constant (8.314 J mol\(^{-1}\) K\(^{-1}\)), \({E_{\mathrm{D}}}\) is the activation energy of the deactivation process (J mol\(^{-1}\)) and dimensionless parameter B is determined from the relationship

$$\begin{aligned} B=k_{\mathrm{D}0} t \exp \left( -\frac{E_{\mathrm{D}}}{R T_{\mathrm{opt}}}\right) \end{aligned}$$
(5)

where t is an inulin hydrolysis time of recombinant endo-inulinases (min) and \({k_{\mathrm{D}0}}\) is a pre-exponential factor the kinetic constant of the deactivation process of recombinant endo-inulinase (min\(^{-1}\)).

When the value of the deactivation energy \({E_{\mathrm{D}}}\) and the parameter B were known, the activation energy E is calculated from the equation

$$\begin{aligned} E=E_{\mathrm{D}}-\frac{E_{\mathrm{D}} B}{\exp B-1} \end{aligned}$$
(6)

Based on equations Eq. (4) and Eq. (6), the values of deactivation energy \({E_{\mathrm{D}}}\), optimum temperatures \({T_{\mathrm{opt}}}\) and parameter B were determined in SigmaPlot 14.5 by using a method of nonlinear estimation [24, 26,27,28,29,30,31] in which the residual sum of squares RSS (Eq. (7)) was minimized

$$\begin{aligned} RSS\left( E_{\mathrm{D}},B,T_{\mathrm{opt}}\right) =\sum _{i=0}^n \frac{1}{A^2} \left( A-A_{\mathrm{cal}}\left( E_{\mathrm{D}},B, T_{\mathrm{opt}}\right) \right) ^2 \end{aligned}$$
(7)

where A - recombinant endo-inulinase of various origins activity determined experimentally, \(A_{\mathrm{cal}}\left( E_{\mathrm{D}},B,T_{\mathrm{opt}}\right)\) - recombinant endo-inulinase of various origins activity calculated from Eq. (4).

Equations from Eq. (4) to Eq. (6) are used to determine the parameters E, \({E_{\mathrm{D}}}\) and \({T_{\mathrm{opt}}}\) of inulin hydrolysis by endo-inulinase Aspergillus niger [24], by exo-inulinases A. niger [27] and by recombinanat exo-inulinases A. niger [28], olive oil hydrolysis by lipase [3] and hydrolysis of starch by α-amylase Bacillus spp. [26] and porcine pancreas [29].

Results and discussion

The previously study presented the values optimum temperatures \({T_{\mathrm{opt}}}\), the activation energies E and deactivation energies \({E_{\mathrm{D}}}\) for inulin hydrolysis by endo-inulinases A. niger [24]. The main aim of this study was to determine the parameters \({T_{\mathrm{opt}}},\ {E}\), \({E_{\mathrm{D}}}\) for inulin hydrolysis by recombinant endo-inulinases A. niger [13,14,15,16,17]. Additionally, these parameters were analyzed for recombinant endo-inulinases of various origins [18,19,20]. Table 1 shows the conditions for measuring recombinant endo-inulinases activity, such as concentration of inulin, pH buffer and type, and time measurement. Measurements of endo-inulinase activity were determined using the Nelson-Somogyi method at 610 nm [13, 16] and dinitrosalicylic acid at 540 nm [15, 17,18,19,20].

Table 1 Conditions for measuring of recombinant endo-inulinases activity
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Volkov et al. [13] analyzed the recombinant endo-inulinase obtained from gene A. niger strain into Penicillium canescens A3. Used strains have been obtained from the Laboratory of Enzyme Biotechnology in Bach Institute of Biochemistry in Russian Academy of Sciences.

Endo-inulinase gene from A. niger sp. F4 was loned and expressed in Yarrowia lipolytica Po1h by Liu et al. [15]. Used strains has been obtained from the Ocean University of China. Recombinant endo-inulinases A. niger degraded inulin with a high degree of polymerization.

High-yield endo-inulinase for fructooligosaccharides production from inulin was constructed by Mao et al. [16]. Used strains have been obtained also from the Ocean University of China. An inulin binding module (IBM) from a cycloinulinooligosaccharide fructanotransferase of Bacillus macerans CFC1 was fused into either N-(IBM-Endo) or C-(Endo-IBM) terminal of an endo-inulinase A. niger was fused.

Gen endo-inulinase from A. niger CICIM F0620 has been cloned and expressed in Pichia pastoris by He et al. [17]. Used strains have been obtained from the State Key Laboratory of Food Science and Technology at Jiangnan University in China. The recombinant endo-inulinase activity was 4.18 times that observed using the native gene.

Endo-inulinase gene from Arthrobacter sp. S37 was cloned and expressed in Yarrowia lipolytica Po1h by Li et al. [18]. Endo Arthrobacter sp. S37 strain (collection number 2E001892) and Y. lipolytica Po1h strain (collection number 2E0189) were used from the Marine Microorganisms Culture Collection of China (MCCC). The endo-inulinase activity produced by the recombinant Y. lipolytica was very high. The optimum pH of the recombinant endo-inulinase Y. lipolytica was 4 while, the native endo-inulinase Arthrobacter sp. was optimum in pH at 7.

Endo-inulinase gen from Pseudomonas mucidolens was cloned and expressed into S. cerevisiae EBY 100 by Hyun-Chul et al. [19]. Used strains have been obtained from own the Laboratory of Department of Biotechnology and Bioengineering, Dong-Eui University, Korea. The recombinant endo-inulinase S. cerevisiae was achieved the highest yield 71.2 of oligosaccharides from inulin after 30 h of reaction.

Endo-inulinase gene from Penicillium restrictum A191 was cloned and expressed in E. coli by Puratos [20]. Used strains by Puratos [20] has been obtained from the BCCM/MUCL Agro-food and Environmental Fungal Collection, Université Catholique de Louvain, Belgium, the number MUCL 42612. The optimum temperature of the recombinant endo-inulinase E. coli was \(65\,^{\circ }\)C.

The effect of temperature on the recombinant endo-inulinases activity

Based on experimental data on the change in the activity of recombinant endo-inulinase from A. niger [13,14,15,16,17] and of recombinant endo-inulinase of various origins [18,19,20] values of deactivation energy \({E_{\mathrm{D}}}\), B parameter and optimum temperature \({T_{\mathrm{opt}}}\) are determined from Eq. (4) and listed in Table 2. Figures 1–9 show the experimental data on the activity of recombinant endo-inulinases.

Fig. 1
figure 1

The activity of recombinant endo-inulinase A. niger: (\(\bullet\)) by measurements Volkov et al. [13], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 2
figure 2

The activity of recombinant endo-inulinase A. niger: (\(\bullet\)) by measurements Megazyme [14], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 3
figure 3

The activity of recombinant endo-inulinase A. niger F4: (\(\bullet\)) by measurements Liu et al. [15], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 4
figure 4

The activity of recombinant endo-inulinase A. niger F4 (IBM-Endo): (\(\bullet\)) by measurements Mao et al. [16], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 5
figure 5

The activity of recombinant endo-inulinase A. niger F4 (Endo-IBM): (\(\bullet\)) by measurements Mao et al. [16], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 6
figure 6

The activity of recombinant endo-inulinase A. niger CICIM F0620: (\(\bullet\)) by measurements He et al. [17], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 7
figure 7

The activity of recombinant endo-inulinase Yarrowia lipolytica Po1h: (\(\bullet\)) by measurements Li et al. [18], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 8
figure 8

The activity of recombinant endo-inulinase Saccharomyces cerevisiae EBY 100: (\(\bullet\)) by measurements Hyun-Chul et al. [19], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Fig. 9
figure 9

The activity of recombinant endo-inulinase Escherichia coli: (\(\bullet\)) by measurements Puratos [20], (−) from Eq. (4), (\(--\)) 95% confidence bond

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Knowing the deactivation energy \({E_{\mathrm{D}}}\) and the parameter B values, the activation energy E is calculated from Eq. (6). The obtained results \({T_{\mathrm{opt}}}\), \({E_{\mathrm{D}}}\), B and E for recombinant endo-inulinases A. niger and different origins activity were shown and presented according to the increasing values \({T_{\mathrm{opt}}}\) in Table 2.

Table 2 The values of parameters estimated recombinant endo-inulinases from various origin
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Table 3 shows the statistical data calculated for obtained the parameters \({E_{\mathrm{D}}}\), \({T_{\mathrm{opt}}}\) and B of recombinant endo-inulinases.

Table 3 The statistical data for the parameters \({E_{\mathrm{D}}}\), \({T_{\mathrm{opt}}}\) and B of recombinant endo-inulinases
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The residual sum of squares RSS were obtained below 0.153, the high regression coefficients \(R^2\) were above 0.95 for recombinant endo-inulinase A. niger. F-Fisher test values were in the range from 53.92 to 932.77 with a low probability value P were below 0.0013.

For recombinant endo-inulinase of various origins the residual sum of squares RSS were obtained below 0.088, the high regression coefficients \(R^2\) were above 0.97. The statistical data in Table 3 confirmed that it was appropriate to apply Eq.(4) when determining the parameters \({E_{\mathrm{D}}}\), \({T_{\mathrm{opt}}}\) and B. F-Fisher test values were in the range from 72.07 to 349.48 with a low probability value P below 0.0007.

Also, Figs. 1-9 present standard deviation errors for experimental points, with the 95% confidence bands for the obtained curves.

The values of activation energy E

The valeues of activation energies E were obtained in the range from \({22.08 \pm 13.94}\) kJ mol\(^{-1}\) to \({62.62 \pm 17.24}\) kJ mol\(^{-1}\) and in the range from \({27.21 \pm 6.95}\) kJ mol\(^{-1}\) to \({92.69 \pm 15.31}\) kJ mol\(^{-1}\) for recombinant endo-inulinase A. niger and various origins, respectively.

Table 2 shows, that the values E estimated for recombinant endo-inulinases from A. niger F4 (Endo-IBM) - Fig. 5, was lowest about 60% than recombinant endo-inulinases from A. niger F4 (IBM-Endo) - Fig. 4 [16].

For the commercial endo-inulinases A. niger in previous work [24], it was found that E is in the range from \({23.53 \pm 3.20}\) kJ mol\(^{-1}\) to \({50.66 \pm 3.61}\) kJ mol\(^{-1}\).

The endo-inulinase Arthrobacter sp. S37 in Yarrowia lipolytica Po1h [18] is characterized by the highest the value E of among analyzed cases.

The values of deactivation energy \({E_{\mathrm{D}}}\)

The values of deactivation energies \({E_{\mathrm{D}}}\) of recombinant endo-inulinase A. niger were obtained in the range from \({143.24 \pm 9.87}\) kJ mol\(^{-1}\) to \({301.95 \pm 95.81}\) kJ mol\(^{-1}\) (Table 2). These values \({E_{\mathrm{D}}}\) are higher than the values obtained for non-recombinant endo-inulinases A. niger shown in previous work [24] which the values \({E_{\mathrm{D}}}\) were in the range from \({88.42 \pm 5.03}\) kJ mol\(^{-1}\) to \({142.87 \pm 2.75}\) kJ mol\(^{-1}\).

The values \({E_{\mathrm{D}}}\) of recombinant endo-inulinases from A. niger were higher about 40% (IBM-Endo) and 100% (Endo-IBM) [16] than the values \({E_{\mathrm{D}}}\) obtained for non-recombinant endo-inulinase A. niger equals \({150.89 \pm 40.72}\) kJ mol\(^{-1}\) calculated from Eq. (4).

The deactivation energy \({E_{\mathrm{D}}}\) of the commercial inulinase of Aspergillus niger (Fructozyme LTM, Novozymes, Denmark) for marked \({k_{\mathrm{D}}}\) Ricca et al. [22] was calculated from the Arrhenius equation as equal to 313.47 kJ mol\(^{-1}\).

The values of deactivation energies \({E_{\mathrm{D}}}\) for recombinant endo-inulinases of various origins were obtained ine the range from \({168.58 \pm 7.78}\) kJ mol\(^{-1}\) to \({295.80 \pm 18.78}\) kJ mol\(^{-1}\) (Table 2).

In most of analyzed cases, the recombinant endo-inulinases A. niger have higher values of \({E_{\mathrm{D}}}\), so they are more thermally stable than, that no-recombinant endo-inulinases A. niger [24].

When the \({E_{\mathrm{D}}}/{E}\) relationship is known for an A. niger and of various origins recombinant endo-inulinases, one can choose the enzyme with the highest thermal stability. The higher the \({E_{\mathrm{D}}}/{E}\) value of a given recombinant endo-inulinases, the more stable the enzyme is. The better activity, thermostability was obtained for the C-terminal fusion (Endo-IBM) recombinant endo-inulinase A. niger [16]. In this work, the value of the \({E_{\mathrm{D}}}/{E}\) is equal to 13.67 but also this value is almost 2.5 yields higher than form for the N-terminal fusion (IBM-Endo) recombinant endo-inulinase A. niger [16]. Among recombinant endo-inulinase of various origins, the most stable is Pseudomonas mucidolens into S. cerevisiae [19] with the value \({E_{\mathrm{D}}}/{E}\) equals 10.87, but with low value \({T_{\mathrm{opt}}}\) equals \({319.57 \pm 0.57}\) K.

The values of optimum temperature \({T_{\mathrm{opt}}}\)

The values of optimum temperature \({T_{\mathrm{opt}}}\) for recombinant endo-inulinase from A. niger were determined in the range from \({328.67 \pm 1.32}\) K to \({335.94 \pm 1.22}\) K [13,14,15,16,17]. This values \({T_{\mathrm{opt}}}\) are higher even \(18\,^{\circ }\)C than determined for no-recombinant endo-inulinases from A. niger [24]. The estimated values of the optimum temperatures \({T_{\mathrm{opt}}}\) for recombinant endo-inulinase of various origins [18,19,20] were found to be in the range from \({319.41 \pm 0.85}\) K to \({338.53 \pm 0.45}\) K.

Based on the analyses, it was observed that recombinant endo-inulinase A. niger at Fig. 1, Fig. 3, Fig. 4 and recombinant endo-inulinase E. coli at Fig. 9, are characterized by the determined optimal temperature values \({T_{\mathrm{opt}}}\), correspond to the optimum temperature determined experimentally. Additionally, the statistical parameters (F–Fisher test values and probability value P) for these measurements are the highest. Whereas for recombinant endo-inulinase Aspergillus niger (Endo-IBM) at Fig. 5, recombinant endo-inulinase Yarrowia lipolytica at Fig. 7 and recombinant endo-inulinase S. cerevisiae at Fig. 8, estimated optimum temperatures \({T_{\mathrm{opt}}}\) are about \(3.5\,^{\circ }\)C lower than the values determined experimentally. In the applied method of determining the parameters \({E_{\mathrm{D}}}\), \({T_{\mathrm{opt}}}\), B with Eq. (4) all measuring points that contribute to the final \({T_{\mathrm{opt}}}\) result are included.

The highest optimum temperature \({T_{\mathrm{opt}}}\) has been determined for the recombinant endo-inulinase Penicillium restrictum A191 in E. coli by measurements Puratos [20]. That value \({T_{\mathrm{opt}}}\) is most highest than those described in the available literature. Higher values \({T_{\mathrm{opt}}}\) obtained for recombinant endo-inulinases Aspergillus niger. The maximal activities of recombinant endo-inulinase A. niger F4 by expressed Bacillus macerans CFC1 [16] and A. niger CICIM F0620 in Pichia pastoris [17] are at temperatures \({335.94 \pm 1.22}\) K and \({334.81 \pm 1.08}\) K, respectively.

Conclusions

The values of parameters the activation energies E and the deactivation energies \({E_{\mathrm{D}}}\) and the optimum temperatures \({T_{\mathrm{opt}}}\) of inulin hydrolysis by recombinant endo-inulinases from Aspergillus niger and from various origins were determined. The differences in the estimated values of the activation energy E are equal to about 40 kJ mol\(^{-1}\) and about 70 kJ mol\(^{-1}\) for recombinant endo-inulinases from A. niger and from various origins, respectively. The differences in the estimated values of deactivation energy \({E_{\mathrm{D}}}\) are equal to about 160 kJ mol\(^{-1}\) and about 130 kJ mol\(^{-1}\) for recombinant endo-inulinase A. niger and from various origins. For the optimum temperatures \({T_{\mathrm{opt}}}\), the difference between the obtained values is about \(7.5\,^{\circ }\)C and about \(19\,^{\circ }\)C for recombinant endo-inulinases from A. niger and from various origins. The recombinant endo-inulinases A. niger were characterized by the higher values of optimum temperature \({T_{\mathrm{opt}}}\) as well as higher values of energy \({E_{\mathrm{D}}}\) compare to the values of recombinant endo-inulinases from various origins and also no-recombinant endo-inulinases A. niger. Knowledge of parameters E, \({E_{\mathrm{D}}}\), \({T_{\mathrm{opt}}}\) for recombinant endo-inulinase from a various origin of yeast and bacteria strains will allow design, modeling and optimization of inulin hydrolysis.