# Quantitative Modeling of Optimal Physicochemical Parameters of Insect β-glucosidases from Graphical Data

Sobamfou Marius Kambire, Kouakou Eric Konan, N'guessan Louis Berenger

Kouassi, Bonito Aristide Karamoko, David Boa

► To cite this version:

Sobamfou Marius Kambire, Kouakou Eric Konan, N'guessan Louis Berenger Kouassi, Bonito Aristide Karamoko, David Boa. Quantitative Modeling of Optimal Physicochemical Parameters of Insect β-glucosidases from Graphical Data. Chemical Science International Journal, 2025, 34 (6), pp.125-137. 10.9734/CSJI/2025/v34i61010. hal-05431822

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# Quantitative Modeling of Optimal Physicochemical Parameters of Insect β-glucosidases from Graphical Data

Sobamfou Marius KAMBIRE a*, Kouakou Eric KONAN a,
N'guessan Louis Bérenger KOUASSI a,
Bonito Aristide KARAMOKO b and David BOA c

**Authors' contributions**

This work was carried out in collaboration among all authors. All authors read and approved the final manuscript.

**Article Information**

DOI: https://doi.org/10.9734/CSJI/2025/v34i61010

**Open Peer Review History:**

This journal follows the Advanced Open Peer Review policy. Identity of the Reviewers, Editor(s) and additional Reviewers, peer review comments, different versions of the manuscript, comments of the editors, etc are available here: https://pr.sdiarticle5.com/review-history/149525

**Original Research Article**

Received: 16/10/2025

Published: 24/12/2025

**ABSTRACT**

β-glucosidases are essential insect hydrolases that facilitate the digestion of polysaccharide, secondary metabolite detoxification, and chemical defense adaptation. Understanding their physicochemical behaviour is crucial for elucidating catalytic mechanisms and exploring their potential in industrial biotechnology. Traditional methods for determining optimal enzymatic parameters, such as pHopt and Top, often rely on labor-intensive experiments. This study employed a mathematical modeling approach grounded in graphical data analysis to determine the ideal parameters of insect β-glucosidases. Activity–pH and activity–temperature profiles were recreated

<footnote>a Department of Mathematics-Physics-Chemistry, University of Péléforo Gon Coulibaly, BP 1328, Korhogo, Côte d'Ivoire.</footnote>

<footnote>b European Membrane Institute, UMR 5635, University of Montpellier, 34090, Montpellier, France.</footnote>

<footnote>c Laboratory of Thermodynamics and Physico-chemistry of the Environment, University of Nangui Abrogoua, 02 BP 801 Abidjan 02, Côte d'Ivoire.</footnote>

<footnote>*Corresponding author: E-mail: mskambire@upgc.edu, mskambire@upgc.edu ci;</footnote>
from existing data using WebPlotDigitizer and then matched to both empirical and mechanistic models through nonlinear regression. The best-fitting models were chosen based on various statistical metrics, such as the coefficient of determination (R² ≥ 0.85), the AICc, PRESS, and Fisher's test. The estimated optimal pH (pHopt) ranged from 4.74 to 6.15, whereas the optimal temperature (Topt) varied from 302.9 to 319.6 K, confirming the slightly acidic and mesophilic nature of insect β-glucosidases. Activation energies (Ea) between 30.67 and 41.19 kJ mol⁻¹ and denaturation energies (Ed) between 66 and 299 kJ mol⁻¹ were calculated, indicating moderate thermal stability and functional adaptability. This novel approach provides a comprehensive framework for characterizing and optimizing enzymes by completing existing data and offering accurate predictions, ultimately highlighting the potential for insect β-glucosidases to serve as effective biocatalysts in sustainable biomass conversion.

Keywords: Insect β-glucosidases; nonlinear regression; optimal parameters; mathematical modeling.

# 1. INTRODUCTION

Enzymes play a fundamental role in biological processes by catalyzing biochemical reactions with remarkable efficiency and specificity. Among these, β-glucosidases have been extensively studied, partly because of their industrial usefulness (Vassão et al., 2018). These enzymes are critical for the full range of insect biological functions.

The dual role of β-glucosidase has been highlighted, especially when considering adaptation of insects to plant chemical defenses, thus emphasizing the complex relationship between digestive enzymes, secondary metabolites, and feeding behaviour in herbivorous insects (Huber et al., 2021). It has been emphasized that the stability of β-glucosidase activity plays a pivotal role in plant-insect interactions, particularly given that plant chemical defenses can remain effective within the digestive system of insect pests (Vassão et al., 2018). This observation provides valuable insights into the biochemical mechanisms underlying plant defense, which has major implications for the development of biological control strategies against agricultural pests.

Psychrophilic β-glucosidases from the termite symbiotic bacterium *Elizabethkingia miricola* strain BM10 are imperative for effective cellulose digestion in the termite gut at low temperatures, thus enabling the species to thrive and colonize cold habitats (Lee & Kim, 2023). This study provides a more profound understanding of symbiotic enzymatic interactions in termites and opens up prospects for the biotechnological application of these psychrophilic enzymes in industrial contexts. Digestive enzymes, such as glycosidases, are imperative for the breakdown of macronutrients in herbivorous insects, thereby conditioning nutrient assimilation and the

biological success of the species. Levels of activity measured in the midgut are directly related to the quality of the consumed food and represent a physiological response to nutritional challenges associated with a plant-based diet (Stygar et al., 2010).

It is therefore evident that glycosidases play a pivotal role in the adaptation of insects to plants, intervening both in the digestion of plant fibers and in the management of chemical defenses. It has been demonstrated that these compounds directly contribute to chemical defense against predators and parasites (He et al., 2022). This illustration of enzymatic evolution is indicative of the complexity of insect-plant interactions.

It is imperative to understand and predict the function of enzymes. This understanding is facilitated by the knowledge of the parameters that are relevant to their activity. The parameters in question include the optimum pH, optimum temperature and activation energy. The optimum pH is defined as the acid-base state in which the enzyme is most active, the optimum temperature is specified as the thermal range that maximizes activity, and the activation energy offers insight into how easily an enzyme catalyzes a chemical reaction. The interplay between these parameters is pivotal in delineating the efficiency of enzymes and in determining their suitability for biotechnological applications.

The determination of the optimum pH and temperature by visual inspection of experimental curves is subject to several limitations. Firstly, the process is subjective and imprecise. Secondly, it depends on the density of the measured data points. The employment of experimental variations or complex curves can result in approximate estimates, without the necessity of quantifying the uncertainty. The
objective of this study was to develop a robust
quantitative modeling framework for the accurate
prediction of the main physicochemical
parameters of insect β-glucosidases (optimal pH,
optimal temperature, activation energy, and
deactivation energy) from digitized published
data. The framework integrates empirical and
mechanistic models through nonlinear regression
analysis. The ultimate objective of this research
is to evaluate the suitability of these enzymes
and to optimize them for industrial biocatalytic
processes.

**2. MATERIALS AND METHODS**

**2.1 Selection of Data Sources**

A systematic screening of the published studies reporting the activity of insect β-glucosidases as a function of pH or temperature was conducted. The inclusion of articles was contingent upon the presentation of enzymatic activity profiles as graphs, with the enzymes in question being clearly of insect origin. Following a thorough examination of the extant literature, it was determined that five studies satisfied the aforementioned criteria (Ghadamyari et al., 2010; Kara et al., 2014; Kazzazi et al., 2014; Sellami & Jamoussi, 2016; Rafiei et al., 2020). The graphical datasets required for the reconstruction of activity–pH and activity–temperature curves were provided by these sources.

**2.2 Extraction of Experimental Data**

As raw numerical values were not available in
the selected studies, the data were extracted
from the published figures using WebPlotDigitizer
(version 4.6;
https://automeris.io/WebPlotDigitizer). Prior to
the advent of digitization, the axes of each graph
were calibrated using two reference points to
ensure metric consistency. The subsequent
capture of data points was conducted manually
or through the utilization of an automated curve-
detection algorithm embedded within the
software. The values obtained were exported as
numerical pairs (X, Y), corresponding to pH or
temperature (X) and enzymatic activity (Y).
Digitization introduces inherent uncertainties,
which are attributable to factors such as figure
resolution and curve thickness. In order to
minimize error, each curve was digitized twice,
and the two resulting datasets were compared to
confirm consistency. It is imperative to note that
only datasets exhibiting a maximum deviation of
5% between replicates were deemed suitable for
the modeling process.

**2.3 Mathematical Modeling**

The influence of pH and temperature on enzymatic activity was analyzed through a combined empirical and mechanistic modeling approach. The selection of models was based on two key criteria: the ability to reproduce the bell-shaped activity profiles that are characteristic of β-glucosidases, and the capacity to provide physico-chemistry parameters that are interpretable, such as pHopt, Top, activation energy (Ea), and denaturation energy (Ea). In the context of pH, empirical approaches encompass the square root model (SRM) proposed by (Zwietering et al., 1992), cardinal pH model with inflection (CPMI) developed by (Rosso et al.,1995), cardinal pH model (CPM) by (Rosso et al., 1995) and a model inspired by Blanchard (Blanchard pH model, BPM). These models permit the representation of cardinal values (pHmin, pHOpt, and pHmax) but exhibit certain limitations with regard to mechanistic representation. In order to address this issue, three mechanistic models (MeM1, MeM2 and MeM3) (Jurado et al., 2004; Lundemo et al., 2017), were also considered. These models are based on the ionization of catalytic and essential groups.

In a similar manner, the effect of
temperature was the subject of investigation
through the utilization of the cardinal temperature
model with inflection (CTMI), the
cardinal temperature model (CTM), and the
Blanchard temperature model (BTM). These
models provide cardinal temperatures (Tmin, Topt,
 Tmax). Two mechanistic models, (Wojcik & Milek,
 2016) and Alexandrov-Yamagata (AYM)
 (Alexandrov & Yamagata, 2007), were selected
to describe kinetic properties in greater detail,
particularly the activation and denaturation
energies.

**2.3.1 Models describing the effect of pH on β-glucosidases activity**

The influence of pH on the activity of the studied
biocatalysts exhibits a characteristic bell-shaped
profile (Kambiré et al., 2021). This behaviour,
typified by an incremental rise in enzyme activity
commencing from strongly acidic conditions and
culminating at a maximum, is followed by a
decline within the alkaline zone (Bisswanger,
2014). The subsequent section will present the
various mathematical models that have been
utilized to describe the pH-dependent enzyme
activity.
- Square-root model (SRM)

$$A(pH) = A_{opt}^{pH} \left[ \frac{(pH-pH_{min})(1-\exp\{c(pH-pH_{max})\})}{(pH_{opt}-pH_{min})(1-\exp\{c(pH_{opt}-pH_{max})\})} \right]^2 \quad (1)$$

Where c is a dimensionless constant.

- Cardinal pH model with inflection

$$A(pH) = A_{opt}^{pH} \frac{(pH-pH_{min})(pH-pH_{max})^2}{(pH_{opt}-pH_{max})(pH_{opt}-pH_{max})(pH-pH_{opt})-(pH_{opt}-pH_{min})(pH_{opt}+pH_{max}-2pH)} \quad (2)$$

- Cardinal pH model (CPM)

$$A(pH) = A_{opt}^{pH} \frac{(pH-pH_{min})(pH-pH_{max})}{(pH-pH_{min})(pH-pH_{max})-(pH-pH_{opt})^2} \quad (3)$$

- Blanchard pH model (BPM)

$$A(pH) = A_{opt}^{pH} \left( \frac{pH_{max}-pH}{pH_{max}-pH_{opt}} \right)^{\alpha} \exp \left( -\alpha \frac{pH_{opt}-pH}{pH_{max}-pH_{opt}} \right) \quad (4)$$

$\alpha$ is the Blanchard parameter.

- Mechanistic model 1 (MeM1)

$$A(pH) = \frac{V_{max}^{opt}}{1+10^{pK_{ES1}-pH}+10^{pH-pK_{ES2}}} \quad (5)$$

- Mechanistic model 2 (MeM2)

$$A(pH) = \frac{V_{max}^{opt}}{1+10^{pK_{ES1}-2pH}+10^{pH-pK_{ES2}}} \quad (6)$$

- Mechanistic model 3 (MeM3)

$$A(pH) = \frac{V_{max}^{opt}(\theta+10^{pK_{ES2}-pH})}{1+10^{pK_{ES1}-2pH}+10^{pK_{ES2}-pH}+10^{pH-pK_{ES3}}} \quad (7)$$

Where:

$V_{opt}^{pH}$ represents the maximum velocity (activity) at optimum pH; the dissociation constants (pK) of the ionizable groups involved in catalysis are denoted as (pK<sub>ES1</sub>, pK<sub>ES2</sub>, pK<sub>ES3</sub>); pK<sub>ES1</sub> corresponds to the group that is active only when protonated and deprotonates on the acid side of the curve; pK<sub>ES2</sub> corresponds to a group that is generally active when deprotonated and acts on the basic side; pK<sub>ES3</sub> represents a third ionizable group capable of influencing activity loss at high pH; an empirical adjustment factor, denoted as $\theta$, reflects the relative fraction of activity or the residual contribution of a specific enzymatic form to the overall catalytic activity.

### 2.3.2 Models describing the effect of temperature on β-glucosidases activity

As with pH, a bell-shaped curve is obtained when enzyme activity is plotted as a function of temperature, showing an increase with rising temperature, reaching a maximum, and then declining (Kambiré et al., 2021). Four mathematical models (Eqs. 8, 9, 10, 11, and 13) were employed in this study to analyze the experimental data, with these models explicitly providing T<sub>opt</sub> and other parameters such as activation energy (E<sub>a</sub>) and denaturation energy (E<sub>d</sub>).
- Cardinal temperature model with inflection (CTMI)

$$A(T) = A_{opt}^{T} \frac{(T-T_{max})(T-T_{min})^{2}}{(T_{opt}-T_{min})[(T_{opt}-T_{min})(T-T_{opt})-(T_{opt}-T_{max})(T_{opt}+T_{min}-2T)]} \quad (8)$$

- Cardinal temperature model (CTM)

$$A(T) = A_{opt}^{T} \frac{(T-T_{min})(T-T_{max})}{(T-T_{min})(T-T_{max})-(T-T_{opt})^{2}} \quad (9)$$

- Blanchard model (BM)

$$A(T) = A_{opt}^{T} \left( \frac{T_{max}-T}{T_{max}-T_{opt}} \right)^{\alpha} \exp \left( -\alpha \frac{T_{opt}-T}{T_{max}-T_{opt}} \right) \quad (10)$$

- Wojcik-Milek model (WMM)

$$A(T) = A_{opt}^{T} \frac{\exp\left(\frac{(T_{opt}-T)E_{d}\beta}{RTT_{opt}(\exp\beta-1)}\right)\left\{1-\exp\left[-\beta\exp\left(\frac{E_{d}(T-T_{opt})}{RTT_{opt}}\right)\right]\right\}}{1-\exp(-\beta)} \quad (11)$$

The activation energy $E_a$ is given by Eq. (9):

$$E_a = E_d - \frac{E_d \beta}{\exp \beta - 1} \quad (12)$$

- Alexandrov-Yamagata model (AYM)

$$A(T) = A_{opt}^{T} \frac{E_{d} \exp[E_{a}(T-T_{opt})/TRT_{opt}]}{E_{d}-E_{a}(1-\exp[E_{d}(T-T_{opt})/TRT_{opt}])} \quad (13)$$

## 2.4 Statistical Analysis and Nonlinear Regression

The selection of the model was determined by a number of statistical criteria. The model is deemed satisfactory when it exhibits a coefficient of determination (R²) or an adjusted coefficient of determination (R²adj) greater than 0.85. Models with a lower value of corrected Akaike information criterion (AICc) or sum of squares of predicted residual errors (PRESS) are considered more effective. Following a comprehensive evaluation of the available models, it was determined that the model with the highest Fisher's test (F-test) value was to be selected as the optimal model. The nonlinear regressions were performed using SigmaPlot version 15.0 (Systat Software Inc., San Jose, CA, USA) with 200 iterations, a step size of 1, and a tolerance of 10⁻¹².

## 2.5 Methodological Considerations and Limitations

The use of digitized graphical data presents limitations, including potential loss of resolution and reliance on image quality. However,

repeated digitization and cross-validation minimized these effects. Furthermore, empirical models reproduce trends but do not explicitly represent biochemical mechanisms, whereas mechanistic models depend on assumptions about ionizable residues or thermal inactivation pathways. The combination of both approaches provides a balanced and robust methodology.

# 3. RESULTS AND DISCUSSION

## 3.1 Influence of pH on β-glucosidases Activity

The pH-activity profiles of insect β-glucosidases from five species (*Dociostaurus maroccanus*, *Tuta absoluta*, *Glyphodes pyloalis*, *Galleria mellonella*, and *Leptinotarsa decemlineata*) are presented in Fig. 1. The curves obtained from published data were fitted using mechanistic and empirical models. The selection of models exhibiting optimal statistical performance (defined as an R² threshold of ≥ 0.85, lowest AICc and PRESS values, and highest F-test) was conducted for each enzyme (see Table 1). For *D. maroccanus* and *T. absoluta*, mechanistic models (MeM2 and MeM1, respectively)
provided the most suitable fit. Conversely, the CPMI (empirical model) exhibited a superior capacity to delineate the enzymatic behaviour of

G. pyloalis, G. mellonella, and L. decemlineata, indicating a more symmetrical response around pH<sub>opt</sub>.

<table><caption>Table 1. Parameters estimated from the model selected to fit β-glucosidases activity versus pH</caption><thead><tr><th rowspan="3">Parameters</th><th colspan="5">β-glucosidase from</th></tr><tr><th>Dociostaurus maroccanus</th><th>Tuta absoluta</th><th>Glyphodes pyloalis</th><th>Galleria mellonella</th><th>Leptinotarsa decemlineata</th></tr><tr><th>MeM2</th><th>MeM1</th><th>CPMI</th><th>CPMI</th><th>CPMI</th></tr></thead><tbody><tr><td>Model selected</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>A<sup>pH</sup><sub>opt</sub> (U mg<sup>-1</sup>)</td><td>---</td><td>---</td><td>0.19 ± 0.01</td><td>20.98 ± 0.77</td><td>5.71 ± 0.08</td></tr><tr><td>pH<sub>min</sub></td><td>---</td><td>---</td><td>3.45 ± 0.12</td><td>2.01 ± 0.18</td><td>1.87 ± 0.037</td></tr><tr><td>pH<sub>opt</sub><sup>(a)</sup></td><td>6.15 ± 0.21</td><td>5.03 ± 0.28</td><td>6.14 ± 0.14</td><td>5.10 ± 0.13</td><td>4.74 ± 0.03</td></tr><tr><td>pH<sub>max</sub></td><td>---</td><td>---</td><td>10.78 ± 0.20</td><td>9.29 ± 0.13</td><td>7.89 ± 0.02</td></tr><tr><td>V<sup>pH</sup><sub>opt</sub> (U mg<sup>-1</sup>)</td><td>8.94 ± 0.79</td><td>189.26 ± 26.43</td><td>---</td><td>---</td><td>---</td></tr><tr><td>pK<sub>ES1</sub></td><td>6.07 ± 0.21</td><td>3.55 ± 0.28</td><td>---</td><td>---</td><td>---</td></tr><tr><td>pK<sub>ES2</sub></td><td>6.23 ± 0.21</td><td>6.51 ± 0.28</td><td>---</td><td>---</td><td>---</td></tr><tr><td>Statistic</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>R<sup>2</sup></td><td>0.9278</td><td>0.9181</td><td>0.9534</td><td>0.9669</td><td>0.9945</td></tr><tr><td>R<sup>2</sup><sub>adj</sub><sup>(b)</sup></td><td>0.9072</td><td>0.8636</td><td>0.9334</td><td>0.9421</td><td>0.9927</td></tr><tr><td>AICc</td><td>12.29</td><td>82.06</td><td>-72.19</td><td>39.40</td><td>-31.9859</td></tr><tr><td>PRESS</td><td>2116.13</td><td>7714.07</td><td>0.01</td><td>5336.05</td><td>0.4407</td></tr><tr><td>F-test</td><td>67.21</td><td>43.10</td><td>113.24</td><td>286.10</td><td>1701.42</td></tr></tbody></table>

MeM1: Mechanistic model 1; MeM2: Mechanistic model 2; CPMI: cardinal pH model with inflection; (a) pH<sub>opt</sub> = (pK<sub>ES1</sub> + pK<sub>ES2</sub>)/2; (b): Adjusted R-squared; AICc: Corrected Akaike information criterion; PRESS: Predicted residual sum of squares statistic; F-test: Fisher coefficient.

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Activity of β-glucosidases versus pH (models fitted to experimental data from the literature). β-glucosidase from: (a) *D. maroccanus*, (b) *T. absoluta*, (c) *G. pyloalis*, (d) *G. mellonella*, (e) *L. decemlineata*</figcaption></figure>

The calculated $pH_{opt}$ ranges from 4.74 to 6.15. The observed range of values corresponds to the typical range of values for insect β-glucosidases, the peak activity of which is usually seen in a mildly acidic environment, corresponding to the digestive tract's natural conditions (Delkash-Roudsari et al., 2015; Kara et al., 2014; Kazzazi et al., 2014; Sellami & Jamoussi, 2016; Singh et al., 2016). Specifically, *L. decemlineata* exhibited the lowest pH value for optimal activity (4.74), whereas *D. maroccanus* and *G. pyloalis* demonstrated the highest values (6.15 and 6.14, respectively).

Mechanistic parameters ($pK_{ES1}$ and $pK_{ES2}$) obtained for *D. maroccanus* (6.07 and 6.23) and *T. absoluta* (3.55 and 6.51) confirm the involvement of at least two ionizable groups in catalysis, where one acts as a general acid and the other as a general base. These observations are consistent with the catalytic mechanism commonly proposed for glycosidases, which involves alternating proton transfer between two active-site residues (Lundemo et al., 2017).

The robustness of the derived models is confirmed by statistical criteria. $R^2$ values were found to be highly significant (ranging from 0.92 to 0.99), and the PRESS values were found to be particularly low for *G. pyloalis* and *L. decemlineata* (0.01 and 0.44, respectively), thus demonstrating the accuracy of the fitting procedure. The slightly negative AICc values obtained for some species provide further support for the quality of fit. A negative AICc value is indicative of a close match between the experimental data and the predicted values, suggesting a robust model devoid of evidence of overfitting.

It is evident that the accurate reproduction of experimental trends and the provision of mechanistic insights into acid-base interactions that govern the catalysis of insect β-glucosidases can be achieved through the accurate fitting of pH-activity data.

## 3.2 Influence of Temperature on β-glucosidases Activity

Fig. 2 illustrates the impact of temperature on enzyme activity, while Table 2 provides a concise overview of the parameters derived from the optimal fits. The experimental data were then compared with a range of kinetic models (see Fig. 2).

For *D. maroccanus* and *T. absoluta*, Alexandrov-Yamagata model (AYM) proved to be the most satisfactory, yielding high adjusted $R^2$ values of 0.99 and 0.98, respectively. Wojcik-Milek model (WMM) demonstrated the optimal fit for *G. pyloalis* and *G. mellonella*, while the Blanchard temperature model (BTM) exhibited the most suitable fit for *L. decemlineata*. The observed optimal temperatures ($T_{opt}$) range from 302.9 K (29.8 °C) in *T. absoluta* to 319.6 K (46.6 °C) in *G. mellonella*, confirming the mesophilic profile typical of insect β-glucosidases. (Dehghanikhah et al., 2014; Kazzazi et al., 2014; Riseh et al., 2012; Uchima et al., 2012; Koffi et al., 2012).

In addition to the optimal temperature, the mechanistic models (WMM and AYM) yield the activation energies of the enzymatic reaction. The values range from 30.67 kJ·mol⁻¹ for *D. maroccanus* to 41.19 kJ·mol⁻¹ for *G. pyloalis*.
<table><caption>Table 2. Parameters estimated from the model selected to fit β-glucosidases activity versus temperature</caption><thead><tr><th rowspan="2">Parameters</th><th colspan="5">β-glucosidase from</th></tr><tr><th>Dociostaurus maroccanus</th><th>Tuta absoluta</th><th>Glyphodes pyloalis</th><th>Galleria mellonella</th><th>Leptinotarsa decemlineata</th></tr></thead><tbody><tr><td colspan="6"><strong>Models selected</strong></td></tr><tr><td></td><td><strong>AYM</strong></td><td><strong>AYM</strong></td><td><strong>WMM</strong></td><td><strong>WMM</strong></td><td><strong>BTM</strong></td></tr><tr><td>A<sup>pH</sup><sub>opt</sub> (U mg<sup>-1</sup>)</td><td>10.47 ± 0.82</td><td>138.94 ± 3.41</td><td>0.20 ± 0.01</td><td>26.11 ± 0.99</td><td>5.24 ± 0.13</td></tr><tr><td>T<sub>opt</sub> (K)</td><td>317.36 ± 1.88</td><td>302.94 ± 1.05</td><td>316.34 ± 0.88</td><td>319.65 ± 0.73</td><td>318.94 ± 0.85</td></tr><tr><td>T<sub>max</sub> (K)</td><td>---</td><td>---</td><td>---</td><td>---</td><td>348.80 ± 2.95</td></tr><tr><td>α(BTM)</td><td>---</td><td>---</td><td>---</td><td>---</td><td>2.38 ± 0.61</td></tr><tr><td>β(WWM)</td><td>---</td><td>---</td><td>0.36 ± 0.08</td><td>0.27 ± 0.08</td><td>---</td></tr><tr><td>E<sub>a</sub> (kJ mol<sup>-1</sup>)</td><td>30.67</td><td>35.28 ± 7.08</td><td>41.19 ± 4.61</td><td>38.07 ± 5.37</td><td>---</td></tr><tr><td>E<sub>d</sub> (kJ mol<sup>-1</sup>)</td><td>276.45</td><td>66.49 ± 9.73</td><td>240.37 ± 27.12</td><td>298.58 ± 50.43</td><td>---</td></tr><tr><td colspan="6"><strong>Statistic<sup>c</sup></strong></td></tr><tr><td>R<sup>2</sup></td><td>0.9715</td><td>0.9895</td><td>0.9962</td><td>0.9804</td><td>0.9805</td></tr><tr><td>R<sup>2</sup><sub>adj</sub><sup>a</sup></td><td>0.9954</td><td>0.9816</td><td>0.9905</td><td>0.9608</td><td>0.9740</td></tr><tr><td>AICc</td><td>2.31</td><td>59.02</td><td>-56.02</td><td>67.85</td><td>-23.03</td></tr><tr><td>PRESS</td><td>74.34</td><td>649.42</td><td>0.01</td><td>423.70</td><td>1.10</td></tr><tr><td>F-test</td><td>70.18</td><td>842.61</td><td>358.11</td><td>340.98</td><td>708.03</td></tr></tbody></table>

AYM: Alexandrov and Yamagata model; WMM: Wojcik and Milek model; BTM: Blanchard temperature model;

AICc: Corrected Akaike information criterion; PRESS: Predicted Residual Sum of Squares; F-test: Fisher's coefficient.

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Activity of β-glucosidases versus temperature (models fitted to experimental data from the literature). β-glucosidase from: (a) *Dociostaurus maroccanus*, (b) *Tuta absoluta*, (c) *Glyphodes pyloalis*, (d) *Galleria mellonella*, (e) *Leptinotarsa decemlineata*</figcaption></figure>

The values obtained in this study are lower than those reported in the existing literature for insect β-glucosidases. The reported range in the literature is typically between 37.90 and 115.81 kJ mol⁻¹ (Fagbohoun et al., 2020; Kambiré et al., 2021; Otsuka et al., 2020). It is possible that these discrepancies may be attributable to the models that have been used. These models are designed to fit experimental activities to idealized kinetic equations. However, it has been demonstrated that these equations do not always account for thermal inactivation phenomena. In contrast, the equilibrium model is able to account for such phenomena. (Daniel & Danson, 2013). It is also possible that these differences may be attributable to variations in experimental conditions or structural properties that are specific to enzymes from different insect species. Notwithstanding, the models utilized yield compelling results (Vasic-Racki et al., 2003; Kambiré et al., 2021).

In this context, low activation energy (Ea < 35 kJ mol⁻¹) has been demonstrated to facilitate the rapid hydrolysis of plant glycosides (protoxins). This process results in the release of toxic aglycones, which is typically a response to predators or a rapid detoxification process (Huber et al., 2021; Pentzold et al., 2014; Poreddy et al., 2015). In contrast, a high activation energy (Ea > 45 kJ mol⁻¹) has been shown to confer increased stability to the enzyme, thereby allowing sustained activity during periods of prolonged digestion (e.g. in the alkaline digestive tract or under proteolytic stress). This increased stability has been demonstrated to promote the breakdown of complex polysaccharides for the efficient absorption of nutrients (Vassão et al., 2018).

The denaturation energies, derived from mechanistic models, range from 66 to 299 kJ mol⁻¹. These values, which are substantially higher than the activation energies, suggest that thermal inactivation requires a significantly greater amount of energy than activation. This indicates the moderate stability of these enzymes at typical temperatures.

The robustness and accuracy of the model were validated through the utilization of rigorous statistical metrics, which are outlined below: The R-squared value was found to range from 0.96 to 0.99, with negative Akaike's Information Criterion (AICc) values observed for *G. pyloalis* and *L. decemlineata*, and with PRESS values falling 1.1.

## 3.3 Comparison with Previous Studies

The study's parameters were compared to literature values (see Table 3), with consistent results being generated, particularly for *T. absoluta*'s β-glucosidase, with an optimal pH of 5.03 and an optimal temperature of 302.9 K (29.9 °C). This closely matches previously published data of 5.0 and 303 K (30 °C). This consistency serves to validate the estimation method employed, which is founded upon digital extraction and nonlinear analysis.

The disparities observed for *D. maroccanus*, *G. mellonella*, and *L. decemlineata* (ΔpH ≤ 0.5; ΔT ≤ 5 K) are presumably attributable to the reduced precision of the graphical method that has been conventionally employed in earlier studies. Consequently, optimal parameters reported in the literature, which are often derived from visual readings, are considered less reliable.
<table><caption>Table 3. Optimal pH and temperature (pH<sub>opt</sub> and T<sub>opt</sub>) of insect β-glucosidases: present work versus previous reports</caption><thead><tr><th rowspan="3">Enzyme</th><th colspan="4">Parameter</th></tr><tr><th colspan="2">pH<sub>opt</sub></th><th colspan="2">T<sub>opt</sub> (K)</th></tr><tr><th>Present work</th><th>Previous work</th><th>Present work</th><th>Previous work</th></tr></thead><tbody><tr><td>Dociostaurus maroccanus</td><td>6.15</td><td>5</td><td>317.36</td><td>328 (Rafiei et al., 2020)</td></tr><tr><td>Tuta absoluta</td><td>5.03</td><td>5</td><td>302.94</td><td>303 (Sellami &amp; Jamoussi, 2016)</td></tr><tr><td>Glyphodes pyloalis</td><td>6.14</td><td>5.5</td><td>316.34</td><td>318 (Ghadamyari et al., 2010)</td></tr><tr><td>Galleria mellonella</td><td>5.10</td><td>5.5</td><td>319.65</td><td>315 (Kara et al., 2014)</td></tr><tr><td>Leptinotarsa decemlineata</td><td>4.74</td><td>4-4.5</td><td>318.94</td><td>323 (Kazzazi et al., 2014)</td></tr></tbody></table>

The findings of this study demonstrate unequivocally that the estimation of parameters from digitized graphical data, in conjunction with nonlinear adjustments, results in the attainment of accurate and reproducible physicochemical constants. This modeling approach effectively overcomes the limitations of conventional visual reading methods, which often lack precision and are difficult to reproduce.

Moreover, modeling contributes to a more profound comprehension of biochemical processes. The enhancement and refinement of enzyme assays can be facilitated through the utilization of models (Vasic-Racki et al., 2003).

The combination of empirical and mechanistic models offers an essential complementary perspective: the former describe the overall trend and facilitate the extraction of key parameters. Mechanistic models offer a more detailed understanding of enzyme behaviour, particularly through the use of catalytic pK values and energy parameters (Ea, Ed).

The findings demonstrate that β-glucosidases from insects are well-suited to mildly acidic and mesophilic environments, reflective of the typical digestive conditions in phytophagous insects. Consequently, mathematical modeling of enzyme activity has been demonstrated to serve as a reliable substitute for conventional graphical reading methods. The process yields precise values and facilitates a more profound comprehension of the underlying molecular mechanisms. The strong correlation coefficients and consistent findings, which are in accordance with existing data in certain cases, validate the reliability of this method and its potential

application in the study of other insect enzymes or industrial biotechnology processes.

## 4. CONCLUSION

The present study demonstrates the efficacy of mathematical modeling in determining the optimal physicochemical parameters of insect β-glucosidases, as based on digitized graphical data. Reliable estimates of optimum pH, optimum temperature, activation energy, and denaturation energy were obtained by applying both empirical and mechanistic models to enzyme activity profiles extracted from the literature. The models demonstrated an exact replication of the experimental trends, thereby validating their reliability and capacity to make predictions. The results indicate that insect β-glucosidases demonstrate optimal activity under conditions of slight acidity and mesophilic conditions, which is consistent with the physiological environment of insect digestive systems. Mechanistic models further provided valuable insight into the catalytic behaviour of these enzymes, emphasizing the contribution of ionizable residues and the energy required for activation and denaturation. This study underscores the promise of mathematical modeling as a cost-effective and reproducible alternative to conventional experimental methods for enzymatic characterization. Beyond its methodological advantages, the approach offers a deeper understanding of enzyme function and stability, which can inform the rational design and optimization of biocatalysts for industrial or environmental applications. It is recommended that future research extend the current modeling framework to other enzyme families, or alternatively integrate molecular dynamic simulations in order to establish a more robust correlation between kinetic behaviour and
structural features. This approach signifies a
promising advancement in the realm of
quantitative enzymology and the rational
utilization of insect-derived enzymes in
biotechnology.

DISCLAIMER (ARTIFICIAL INTELLIGENCE)

Author(s) hereby declare that NO generative AI
technologies such as Large Language Models
(ChatGPT, COPILOT, etc.) and text-to-image
generators have been used during writing or
editing of this manuscript.

COMPETING INTERESTS

Authors have declared that no competing
interests exist.

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