Characterization and immobilization of engineered sialidases from <em>Trypanosoma rangeli</em> for transsialylation

Birgitte Zeuner; Isabel González-Delgado; Jesper Holck; Gabriel Morales; María-José López-Muñoz; Yolanda Segura; Anne S. Meyer; Jørn Dalgaard Mikkelsen; Birgitte Zeuner; Isabel González-Delgado; Jesper Holck; Gabriel Morales; María-José López-Muñoz; Yolanda Segura; Anne S. Meyer; Jørn Dalgaard Mikkelsen

doi:10.3934/molsci.2017.2.140

AIMS Molecular Science

2017, Volume 4, Issue 2: 140-163. doi: 10.3934/molsci.2017.2.140

Previous Article Next Article

Research article Topical Sections

Characterization and immobilization of engineered sialidases from Trypanosoma rangeli for transsialylation

1.
Center for BioProcess Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
2.
Chemical and Environmental Engineering Group, ESCET, Universidad Rey Juan Carlos, c/Tulipán s/n, 28933 Móstoles, Madrid, Spain

Received: 13 January 2017 Accepted: 06 April 2017 Published: 13 April 2017

A sialidase (EC 3.2.1.18; GH 33) from non-pathogenic Trypanosoma rangeli has been engineered with the aim of improving its transsialylation activity. Recently, two engineered variants containing 15 and 16 amino acid substitutions, respectively, were found to exhibit significantly improved transsialylation activity: both had a 14 times higher ratio between transsialylation and hydrolysis products compared to the first reported mutant TrSA_5mut. In the current work, these two variants, Tr15 and Tr16, were characterized in terms of pH optimum, thermal stability, effect of acceptor-to-donor ratio, and acceptor specificity for transsialylation using casein glycomacropeptide (CGMP) as sialyl donor and lactose or other human milk oligosaccharide core structures as acceptors. Both sialidase variants exhibited pH optima around pH 4.8. Thermal stability of each enzyme was comparable to that of previously developed T. rangeli sialidase variants and higher than that of the native transsialidase from T. cruzi (TcTS). As for other engineered T. rangeli sialidase variants and TcTS, the acceptor specificity was broad: lactose, galactooligosaccharides (GOS), xylooligosaccharides (XOS), and human milk oligosaccharide structures lacto-N-tetraose (LNT), lacto-N-fucopentaose (LNFP V), and lacto-N-neofucopentaose V (LNnFP V) were all sialylated by Tr15 and Tr16. An increase in acceptor-to-donor ratio from 2 to 10 had a positive effect on transsialylation. Both enzymes showed high preference for formation α(2,3)-linkages at the non-reducing end of lactose in the transsialylation. Tr15 was the most efficient enzyme in terms of transsialylation reaction rates and yield of 3’-sialyllactose. Finally, Tr15 was immobilized covalently on glyoxyl-functionalized silica, leading to a 1.5-fold increase in biocatalytic productivity (mg 3’-sialyllactose per mg enzyme) compared to free enzyme after 6 cycles of reuse. The use of glyoxyl-functionalized silica proved to be markedly better for immobilization than silica functionalized with (3-aminopropyl)triethoxysilane (APTES) and glutaraldehyde, which resulted in a biocatalytic productivity which was less than half of that obtained with free enzyme.

Keywords:

Citation: Birgitte Zeuner, Isabel González-Delgado, Jesper Holck, Gabriel Morales, María-José López-Muñoz, Yolanda Segura, Anne S. Meyer, Jørn Dalgaard Mikkelsen. Characterization and immobilization of engineered sialidases from Trypanosoma rangeli for transsialylation[J]. AIMS Molecular Science, 2017, 4(2): 140-163. doi: 10.3934/molsci.2017.2.140

Related Papers:

[1]	Aziz Belmiloudi . Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models. AIMS Mathematics, 2019, 4(3): 928-983. doi: 10.3934/math.2019.3.928
[2]	Zuliang Lu, Xiankui Wu, Fei Huang, Fei Cai, Chunjuan Hou, Yin Yang . Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem. AIMS Mathematics, 2021, 6(9): 9510-9535. doi: 10.3934/math.2021553
[3]	Zahra Pirouzeh, Mohammad Hadi Noori Skandari, Kamele Nassiri Pirbazari, Stanford Shateyi . A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations. AIMS Mathematics, 2024, 9(9): 23692-23710. doi: 10.3934/math.20241151
[4]	Xin Yi, Rong Liu . An age-dependent hybrid system for optimal contraception control of vermin. AIMS Mathematics, 2025, 10(2): 2619-2633. doi: 10.3934/math.2025122
[5]	Yuanyuan Cheng, Yuan Li . A novel event-triggered constrained control for nonlinear discrete-time systems. AIMS Mathematics, 2023, 8(9): 20530-20545. doi: 10.3934/math.20231046
[6]	Xiang Wu, Yuzhou Hou, Kanjian Zhang . Optimal feedback control for a class of fed-batch fermentation processes using switched dynamical system approach. AIMS Mathematics, 2022, 7(5): 9206-9231. doi: 10.3934/math.2022510
[7]	Woocheol Choi, Young-Pil Choi . A sharp error analysis for the DG method of optimal control problems. AIMS Mathematics, 2022, 7(5): 9117-9155. doi: 10.3934/math.2022506
[8]	Qian Li, Zhenghong Jin, Linyan Qiao, Aichun Du, Gang Liu . Distributed optimization of nonlinear singularly perturbed multi-agent systems via a small-gain approach and sliding mode control. AIMS Mathematics, 2024, 9(8): 20865-20886. doi: 10.3934/math.20241015
[9]	Tainian Zhang, Zhixue Luo . Optimal harvesting for a periodic competing system with size structure in a polluted environment. AIMS Mathematics, 2022, 7(8): 14696-14717. doi: 10.3934/math.2022808
[10]	Asaf Khan, Gul Zaman, Roman Ullah, Nawazish Naveed . Optimal control strategies for a heroin epidemic model with age-dependent susceptibility and recovery-age. AIMS Mathematics, 2021, 6(2): 1377-1394. doi: 10.3934/math.2021086

Abstract

1. Introduction

In structural mechanics, particularly in the field of elasticity, there are equations that describe the behavior of thin plates under loads. These equations are often partial differential equations that govern the displacement of a thin plate. The plate equation depends on factors like material properties, geometry, and boundary conditions. In the context of geophysics, "plate equation" could refer to the equations that describe the movement and interaction of tectonic plates on the Earth's surface. Plate tectonics is a theory that explains the movement of the Earth's lithosphere (the rigid outer layer of the Earth) on the more fluid asthenosphere beneath it. In mathematics, specifically in the field of differential equations, the term "plate equation" might be used to refer to certain types of equations. For instance, in polar coordinates, Laplace's equation takes a specific form that is sometimes informally referred to as the "plate equation". Plate equation models are hyperbolic systems that arise in several areas in real-life problems, (see, for instance, Kizilova et al.^[1], Lasiecka et al. ^[2] and Huang et al.^[3]). The theory of plates is the mathematical formulation of the mechanics of flat plates. It is defined as flat structural components with a low thickness compared to plane dimensions. The advantage of the theory of plates comes from the disparity of the length scale to reduce the problem of the mechanics of three-dimensional solids to a two-dimensional problem. The purpose of this theory is to compute the stresses and deformation in a loaded plate. The equation of plates results from the composition of different subsets of plates: The equilibrium equations, constitutive, kinematic, and force resultant, ^[4,5,6].

Following this, there are a wide number of works devoted to the analysis and control of the academic model of hyperbolic systems, the so-called plate equations, For example, the exact and the approximate controllability of thermoelastic plates given by Eller et al. ^[7] and Lagnese and Lions in ^[8] treated the control of thin plates and Lasiecka in ^[9] considered the controllability of the Kirchoff plate. Zuazua ^[10] treated the exact controllability for semi-linear wave equations. Recently many problems involving a plate equations were considered by researchers. Let us cite as examples the stabilization of the damped plate equation under general boundary conditions by Rousseau an Zongo ^[11]; the null controllability for a structurally damped stochastic plate equation studied by Zhao ^[12], Huang et al. ^[13] considerrd a thermal equation of state for zoisite: Implications for the transportation of water into the upper mantle and the high-velocity anomaly in the Farallon plate. Kaplunov et al. ^[14] discussed the asymptotic derivation of 2D dynamic equations of motion for transversely inhomogeneous elastic plates. Hyperbolic systems have recently continued to be of interest to researchers and many results have been obtained. We mention here the work of Fu et al. ^[15] which discusses a class of mixed hyperbolic systems using iterative learning control. Otherwise, for a class of one-dimension linear wave equations, Hamidaoui et al. stated in ^[16] an iterative learning control. Without forgetting that for a class of second-order nonlinear systems Tao et al. proposed an adaptive control based on an disturbance observer in ^[17] to improve the tracking performance and compensation. In addition to these works, the optimal control of the Kirchoff plate using bilinear control was considered by Bradly and Lenhart in ^[18], and Bradly et al. in ^[19]. In fact, in this work we will talk about a bilinear plate equation and we must cite the paper of Zine ^[20] which considers a bilinear hyperbolic system using the Riccati equation. Zine and Ould Sidi ^[19,22] that introduced the notion of partial optimal control of bilinear hyperbolic systems. Li et al. ^[23] give an iterative method for a class of bilinear systems. Liu, et al. ^[24] extended a gradient-based iterative algorithm for bilinear state-space systems with moving average noises by using the filtering technique. Furthermore, flow analysis of hyperbolic systems refers to the problems dealing with the analysis of the flow state on the system domain. We can refer to the work of Benhadid et al. on the flow observability of linear and semilinear systems ^[25], Bourray et al. on treating the controllability flow of linear hyperbolic systems ^[26] and the flow optimal control of bilinear parabolic systems are considered by Ould Sidi and Ould Beinane on ^[27,28].

For the motivation the results proposed in this paper open a wide range of applications. We cite the problem of iterative identification methods for plate bilinear systems ^[23], as well as the problem of the extended flow-based iterative algorithm for a plate systems ^[24].

This paper studies the optimal control problem governed by an infinite dimensional bilinear plate equation. The objective is to command the flow state of the bilinear plate equation to the desired flow using different types of bounded feedback. We show how one can transfer the flow of a plate equation close to the desired profile using optimization techniques and adjoint problems. As an application, we solve the partial flow control problem governed by a plate equation. The results open a wide way of applications in fractional systems. We began in section two by the well-posedness of our problem. In section three, we prove the existence of an optimal control solution of (2.3). In section four, we state the characterization of the optimal control. In section five we debate the case of time bilinear optimal control. Section six, proposes a method for solving the flow partial optimal control problem governed by a plate equation.

2. Well-posedness of the problem

Consider $\Theta$ an open bounded domain of $I\!\!R^2$ with $C^2$ boundary, for a time $m$ , and $\Gamma = \partial\Theta \times (0, m)$ . The control space time set is such that

$\begin{equation} Q \in U_{p} = \{ Q\in L^{\infty}([0,m];L^{\infty}(\Theta)){\mbox{ such that }} -p\leq Q(t) \leq p\}, \end{equation}$

(2.1)

with $p$ as a positive constant. Let the plate bilinear equation be described by the following system

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial }^2 {u} }{{\partial t}^2} } + \Delta^{2} u = Q(t)u_{t}, & & (0,m) \times \Theta, \\ u(x,0) = u_0(x) ,\; {\frac{{\partial } u }{{\partial t}} } (x,0) = u_1(x), & & \Theta,\\ u = \frac{\partial u}{\partial \nu} = 0, & & \Gamma, \end{array} \right. \end{equation}$

(2.2)

where $u_{t} = \frac{\partial u}{\partial t}$ is the velocity. The state space is $H_{0}^{2}(\Theta)\times L^{2}(\Theta)$ , (see Lions and Magenes ^[29] and Brezis ^[30]). We deduce the existence and uniqueness of the solution for (2.2) using the classical results of Pazy ^[31]. For $\lambda > 0$ , we define $\nabla u$ as the flow control problem governed by the bilinear plate equation (2.2) as the following:

$\begin{equation} \min\limits_{Q\ \in U_{p} }C_{\lambda}(Q), \end{equation}$

(2.3)

where $C_{\lambda}$ , is the flow penalizing cost defined by

$\begin{align} C_{\lambda}(Q) = \frac{1}{2}\Big\|\nabla u-u^{d}\Big\|^{2}_{(L^2(0,m;L^2(\Theta)))^{n}} + \frac{\lambda}{2}\int_{0}^{m} \int_{\Theta} Q^2(x,t)dxdt\\ = \frac{1}{2}\sum\limits_{i = 1}^{n} \Big\|\frac{\partial u}{\partial x_{i}} - u_{i}^{d}\Big\|^{2}_{L^2(0,m;L^2(\Theta))} + \frac{\lambda}{2}\int_{0}^{m} \int_{\Theta} Q^2(x,t)dxdt, \end{align}$

(2.4)

where $u^d = (u_{1}^{d}, ....u_{n}^{d})$ is the flow target in $L^2(0, m;L^2(\Theta))$ . One of the important motivations when considering the problem (2.3) is the isolation problems, where the control is maintained to reduce the flow temperature on the surface of a thin plate (see El Jai et al. ^[32]).

3. Existence of solution

Lemma 3.1. If $(u_{0}, u_{1}) \in H_{0}^{2}(\Theta)\times L^{2}(\Theta)$ and $Q \in U_{p}$ , then the solution $(u, u_{t})$ of (2.2) satisfies the following estimate:

$\Big\| (u, u_{t}) \Big\|_{{\cal C}(0,m; H_{0}^{2}(\Theta)\times L^{2}(\Theta))} \leq T (1+ \eta m)e^{\eta K m},$

where $T = \Big\| (u_{0}, u_{1}) \Big\|_{ H_{0}^{2}(\Theta)\times L^{2}(\Theta)}$ and $K$ is a positive constant ^[18,19].

Using the above Lemma 3.1, we prove the existence of an optimal control solution of (2.3).

Theorem 3.1. $(u^{*}, Q^*)\in {\cal C}([0, m];\; \; H_{0}^{2}(\Theta)\times U_{p})$ , is the solution of (2.3), where $u^{*}$ is the output of (2.2) and $Q^*$ is the optimal control function.

Proof. Consider the minimizing sequence $(Q_{n})_{n}$ in $U_{p}$ verifying

$C^{*} = \lim\limits_{n\rightarrow +\infty}C_{\lambda}(Q_n) = \inf\limits_{Q\in L^{\infty}(0,m; L^{\infty}(\Theta))} C_{\lambda}(Q).$

We choose $\bar{u}^{n} = (u^{n}, \frac{\partial u^{n}}{\partial t})$ to be the corresponding state of Eq (2.2). Using Lemma 3.1, we deduce

$\begin{equation} \Big\| u^{n}(x,t) \Big\|_{H_{0}^{2}(\Theta)}^{2} + \Big\| u_{t}^{n} (x,t) \Big\|_{ L^{2}(\Theta)}^{2} \leq T_{1} e^{\eta K m} {\mbox{ for }} 0\leq t\leq m {\mbox{ and }} T_{1}\in I\!\!R^{+}. \end{equation}$

(3.1)

Furthermore, system (2.2) gives

$\Big\| u^{n}_{tt}(x,t) \Big\|_{H^{-2}(\Theta)}^{2} \leq T_{2} \Big\| u_{t}^{n} (x,t) \Big\|_{ L^{2}(\Theta)}^{2} {\mbox{ with }} T_{2}\in I\!\!R^{+}.$

Then easily from (3.1), we have

$\begin{equation} \Big\| u^{n}_{tt}(x,t) \Big\|_{H^{-2}(\Theta)}^{2} \leq T_{3} e^{\eta K m} {\mbox{ for }} 0\leq t\leq m {\mbox{ and }} T_{3}\in I\!\!R^{+}. \end{equation}$

(3.2)

Using (3.1) and (3.2), we have the following weak convergence:

$\begin{equation} \begin{array}{lll} Q_{n} \rightharpoonup Q^{*}, & & L^{2}(0,m ; L^{2}(\Theta)),\\ u^{n} \rightharpoonup u^{*}, & & L^{\infty}(0,m ; H_{0}^{2}(\Theta)),\\ u_{t}^{n} \rightharpoonup u^{*}_{t}, & & L^{\infty}(0,m ; L^{2}(\Theta)),\\ u_{tt}^{n} \rightharpoonup u^{*}_{tt}, & & L^{\infty}(0,m ; H^{-2}(\Theta)),\\ \end{array} \end{equation}$

(3.3)

From the first convergence property of (3.3) with a control sequence $Q_{n} \in U_{p}$ , easily one can deduce that $Q^{*} \in U_{p}$ ^[20].

In addition, the mild solution of (2.2) verifies

$\begin{align} \int_{0}^{m} u^{n}_{tt}f(t)dt+ \int_{0}^{m} \int_{\Theta}\Delta u^{n} \Delta f(t) dx dt = \int_{0}^{m} Q^{*}u_{t}^{n} f(t)dt, \forall f \in H_{0}^{2}(\Theta). \end{align}$

(3.4)

Using (3.3) and (3.4), we deduce that

$\begin{align} \int_{0}^{m} u^{*}_{tt}f(t)dt+ \int_{0}^{m} \int_{\Theta}\Delta u^{*} \Delta f(t) dx dt = \int_{0}^{m} Q^{*}u_{t}^{*} f(t)dt, \forall f \in H_{0}^{2}(\Theta), \end{align}$

(3.5)

which implies that $u^{*} = u(Q_{*})$ is the output of (2.2) with command function $Q^{*}$ .

Fatou's lemma and the lower semi-continuous property of the cost $C_{\lambda}$ show that

$\begin{align} C_{\lambda}(Q^{*}) \leq \frac{1}{2}\lim\limits_{k\rightarrow +\infty}\Big\|\nabla u^{k}-u^{d}\Big\|^{2}_{(L^2(0,m;L^2(\Theta)))^{n}} + \frac{\lambda}{2}\lim\limits_{k\rightarrow +\infty} \int_{0}^{m} \int_{\Theta} Q_{k}^2(x,t)dxdt\\ \leq \liminf\limits_{k\rightarrow +\infty} C_{\lambda}(Q_{n})\\ = \inf\limits_{Q \in U_{p}} C_{\lambda}(Q), \end{align}$

(3.6)

which allows us to conclude that $Q^{*}$ is the solution of problem (2.3). □

4. Characterization of solution

We devote this section to establish a characterization of solutions to the flow optimal control problem (2.3).

Let the system

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } v }{{\partial t^{2}}} } = -\Delta^{2}v(x,t)+Q(x,t) v_{t} + d(x,t) v_{t}, & & (0, m)\times \Theta,\\ v(x,0) = v_{t}(x,0) = v_0(x) = 0, & & \Theta, \\ v = \frac{\partial v}{\partial \nu } = 0, & & \Gamma, \end{array} \right. \end{equation}$

(4.1)

with $d \in L^{\infty}(0, m; L^{\infty}(\Theta))$ verify $Q+\delta d \in U_{p}, \quad \forall \delta > 0$ is a small constant. The functional defined by $Q \in U_{p}\mapsto \bar{u}(Q) = (u, u_{t}) \in {\cal C}(0, m; H_{0}^{2}(\Theta)\times L^{2}(\Theta))$ is differentiable and its differential $\overline{v} = (v, v_{t})$ is the solution of (4.1) ^[21].

The next lemma characterizes the differential of our flow cost functional $C_{\lambda}(Q)$ with respect to the control function $Q$ .

Lemma 4.1. Let $Q \in U_{p}$ and the differential of $C_{\lambda}(Q)$ can be written as the following:

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \sum\limits_{i = 1}^{n} \int _{\Theta}\int_{0}^{m} \frac{\partial v(x,t)}{\partial x_{i}}(\frac{\partial u}{\partial x_{i}} - u_{i}^{d} )dtdx+ \varepsilon\int_{\Theta}\int_{0}^{m} d Q dt dx.\\ \end{align}$

(4.2)

Proof. Consider the cost $C_{\lambda}(Q)$ defined by (2.4), which is

$\begin{align} C_{\lambda}(Q) = \frac{1}{2} \sum\limits_{i = 1}^{n}\int_{\Theta}\int_{0}^{m}(\frac{\partial u}{\partial x_{i}} - u_{i}^{d} )^2 dt dx + \frac{\lambda}{2} \int_{\Theta}\int^{m}_{0} Q^2(t) dt dx. \end{align}$

(4.3)

Put $u_{k} = z(Q+k d)$ , $u = u(Q)$ , and using (4.3), we have

$\begin{align} \lim\limits_{k \longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \lim\limits_{\beta\longrightarrow 0}\sum\limits_{i = 1}^{n} \frac{1}{2}\int_{\Theta}\int_{0}^{m}\frac{( \frac{\partial u_{k}}{\partial x_{i}} - u_i^d)^{2}-( \frac{\partial u}{\partial x_{i}}-u_i^d)^{2}}{k}dt dx\\ + \lim\limits_{k \longrightarrow 0}\frac{\lambda}{2}\int_{\Theta} \int_{0}^{m}\frac{(Q+k d)^2-Q^2}{k}(t)dt dx. \end{align}$

(4.4)

Consequently

$\begin{align} \lim\limits_{k \longrightarrow 0} \frac{C_{\lambda}(Q +kd)-C_{\lambda}(Q )}{k} \\ = \lim\limits_{k \longrightarrow 0}\sum\limits_{i = 1}^{n}\frac{1}{2}\int_{\Theta}\int^{m}_{0}\frac{( \frac{\partial u_{k}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}})}{k}( \frac{\partial u_{k} }{\partial x_{i}}+ \frac{\partial u}{\partial x_{i}}-2u_i^d) dt dx \\ + \lim\limits_{k \longrightarrow 0} \int_{\Theta}\int_{0}^{m}(\lambda d Q+ k \lambda d^2)dt dx\\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial v(x,t)}{\partial x_{i}}( \frac{\partial u(x,t)}{\partial x_{i}}-u_i^d)dt dx+\int_\Theta \int_{0}^{m}\lambda d Q dt dx. \end{align}$

(4.5)

□ We define the following family of adjoint equations for system (4.1)

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } {w_{i}}}{{\partial t^{2}}} } +\Delta^{2} w_{i} = Q^*(x,t) (w_{i})_{t}+( \frac{\partial u}{\partial x_{i}} - u_{i}^{d}), & & (0, m)\times\Theta, \\ w_{i}(x,m) = (w_{i})_{t}(x,m) = 0, & & \Theta, \\ w_{i} = \frac{\partial w_{i}}{\partial \nu } = 0, & & \Gamma. \end{array} \right. \end{equation}$

(4.6)

Such systems allow us to characterize the optimal control solution of (2.3).

Theorem 4.1. Consider $Q \in U_{p}$ , and $u = u (Q)$ its corresponding state space solution of (2.2), then the control solution of (2.3) is

$\begin{equation} Q(x,t) = max(-p, min(\frac{-1}{\lambda}(u_{t})(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}),p)), \end{equation}$

(4.7)

where $w = (w_{1}....w_{n})$ with $w_{i} \in C ([ 0, T ]; H^{2}_{0}(\Theta))$ is the unique solution of (4.6).

Proof. Choose $d\in U_{p}$ such that $Q + k d \in U_{p}$ with $k > 0$ . The minimum of $C_{\lambda}$ is realized when the control $Q$ , verifies the following condition:

$\begin{equation} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q )}{k}. \end{equation}$

(4.8)

Consequently, Lemma 4.1 gives

$\begin{align} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} \\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial v(x,t)}{\partial x_{i}}( \frac{\partial u(x,t)}{\partial x_{i}}-u_i^d)dt dx+\int_\Theta \int_{0}^{m}\lambda d Q dt dx.\\\\ \end{align}$

(4.9)

Substitute by equation (4.6) and we find

$\begin{align} 0 \leq \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial v(x,t)}{\partial x_{i}}( {\frac{{\partial^{2} } {w_{i}}(x,t) }{{\partial t^{2}}} } +\Delta^{2} w_{i}(x,t)-Q(x,t)( w_{i})_{t}(x,t))dt dx+\int_\Theta \int_{0}^{m}\lambda d Q dt dx\\ \\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial }{\partial x_{i}}( {\frac{{\partial^{2} } v}{{\partial t^{2}}} } +\Delta^{2} v-Q(x,t) v_{t})w_{i}(x,t)dt dx+\int_\Theta \int_{0}^{m}\lambda d Qdt dx\\ \\ = \sum\limits_{i = 1}^{n}\int_\Theta \int_{0}^{m}\frac{\partial }{\partial x_{i}}(d(x,t) u_{t})w_{i}dt dx+\int_\Theta \int_{0}^{m}\lambda dQ dt dx\\ \\ = \int_\Theta \int_{0}^{m}d(x,t)[u_{t}(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}(x,t)}{\partial x_{i}})+\lambda Q dt dx].\\ \\ \end{align}$

(4.10)

It is known that if $d = d(t)$ in a chosen function with $Q+k d \in U_{p}$ , using Bang-Bang control properties, one can conclude that

$\begin{equation} Q(x,t) = max(-p, min(\frac{-u_{t}}{\lambda}(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}),p)) = max(-p, min( \frac{-u_{t}}{\lambda}Div(w), p)), \end{equation}$

(4.11)

with $Div(w) = \sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}$ . □

5. Flow time bilinear control problem

Now, we are able to discuss the case of bilinear time control of the type $Q = Q(t)$ . We want to reach a flow spatial state target prescribed on the whole domain $\Theta$ at a fixed time $m$ .

In such case, the set of controls (2.1) becomes

$\begin{equation} Q \in U_{p} = \{ Q\in L^{\infty}([0,m]){\mbox{ such that }} -p\leq Q(t) \leq p {\mbox{ for }} t \in (0, m)\}, \end{equation}$

(5.1)

with $p$ as a positive constant.

The cost to minimize is

$\begin{align} C_{\lambda}(Q) = \frac{1}{2}\Big\|\nabla u(x,m)-u^{d}\Big\|^{2}_{(L^2(\Theta))^{n}} + \frac{\lambda}{2}\int_{0}^{m} Q^2(t) dt\\ = \frac{1}{2}\sum\limits_{i = 1}^{n} \Big\|\frac{\partial u}{\partial x_{i}}(x,m) - u_{i}^{d}\Big\|^{2}_{L^2(\Theta)} + \frac{\lambda}{2}\int_{0}^{m} Q^2(t)dt, \end{align}$

(5.2)

where $u^d = (u_{1}^{d}, ....u_{n}^{d})$ is the flow spatial target in $L^2(\Theta)$ . The flow control problem is

$\begin{equation} \min\limits_{Q\ \in U_{p} }C_{\lambda}(Q), \end{equation}$

(5.3)

where $C_{\lambda}$ is the flow penalizing cost defined by (5.2), and $U_{p}$ is defined by (5.1).

Corollary 5.1. The solution of the flow time control problem (5.3) is

$\begin{align} Q(t) = max(-p, min(\int_{\Theta}\frac{-u_{t}}{\lambda} (\sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}})dx,p)) \end{align}$

(5.4)

with $u$ as the solution of (2.2) perturbed by $Q(t)$ and $w_{i}$ as the solution of

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } {w_{i}}}{{\partial t^{2}}} } +\Delta^{2} w_{i} = Q(t) (w_{i})_{t}, & & (0, m)\times\Theta, \\ w_{i}(x,m) = ( \frac{\partial u}{\partial x_{i}}(x,m) - u_{i}^{d}), & & \Theta, \\ (w_{i})_{t}(x,m) = 0, & & \Theta, \\ w_{i} = \frac{\partial w_{i}}{\partial \nu } = 0, & & \Gamma. \end{array} \right. \end{equation}$

(5.5)

Proof. Similar to the approach used in the proof of Theorem 4.1, we deduce that

$\begin{align} 0\leq \int_{0}^{m}d(t)[\int_\Theta u_{t}(\sum\limits_{i = 1}^{n}\frac{\partial w_{i}(x,t)}{\partial x_{i}})dx+\lambda Q ]dt, \end{align}$

(5.6)

where $d(t)\in L^{\infty}(0, m)$ , a control function such that $Q+k d \in U_{p}$ with a small positive constant $k$ . □

Remark 5.1. (1) In the case of spatiotemporal target, we remark that the error $(\frac{\partial u}{\partial x_{i}}(x, t) - u_{i}^{d})$ between the state and the desired one becomes a the change of velocity induced by the known forces acting on system (4.6).

(2) In the case of a prescribed time $m$ targets, we remark that the error $(\frac{\partial u}{\partial x_{i}}(x, m) - u_{i}^{d})$ between the state and the desired one becomes a Dirichlet boundary condition in the adjoint equation (5.5).

6. Partial flow optimal control problem

This section establishes the flow partial optimal control problem governed by the plate equation (2.2). Afterward we characterize the solution. Let $\theta\subset \Theta$ be an open subregion of $\Theta$ and we define

$\begin{array}{lll} \widetilde{P_{\theta}}: (L^2(\Theta)) & \longrightarrow & (L^2(\theta)) \\ \qquad \qquad u & \longrightarrow & \widetilde{P}_{\theta} u = u|_{\theta}, \end{array}$

and

$\begin{array}{lll} P_{\theta}: (L^2(\Theta))^{n} & \longrightarrow &(L^2(\theta))^{n} \\ \qquad \qquad u & \longrightarrow & P_{\theta} u = u|_{\theta}. \end{array}$

We define the adjoint of $P_{\theta}$ by

$P_{\theta}^{*}u = \left\{\begin{array}{lll} u \, \, in \, \, \Theta, \\ 0 \in \Theta \setminus \theta. \end{array} \right.$

Definition 6.1. The plate equation (2.2) is said to flow weakly partially controllable on $\theta \subset \Theta$ , if for $\forall \, \, \beta > 0$ , one can find an optimal control $Q\in L^{2}(0, m)$ such that

$|P_\theta \nabla u_Q(m)-u^d||_{(L^2(\theta))^{n}}\leq \beta,$

where $u^d = (z_{1}^{d}, ...., u_{n}^{d})$ is the desired flow in $(L^2(\theta))^{n}$ .

For $U_{p}$ defined by (5.1), we take the partial flow optimal control problem

$\begin{equation} \min\limits_{Q \in U_{p}} C_{\lambda}(Q), \end{equation}$

(6.1)

and the partial flow cost $C_{\lambda}$ is

$\begin{align} C_{\lambda}(Q) = \frac{1}{2}\Big\|P_\theta\nabla u(m)-u^{d}\Big\|^{2}_{(L^2(\theta))^{n}} + \frac{\lambda}{2} \int_{0}^{m} Q^2(t) dt\\ = \frac{1}{2}\sum\limits_{i = 1}^{n} \Big\|\widetilde{P}_{\theta}\frac{\partial u(T)}{\partial x_{i}} - u_{i}^{d}\Big\|^{2}_{(L^2(\theta))} + \frac{\lambda}{2} \int_{0}^{m} Q^2(t) dt. \end{align}$

(6.2)

Next, we consider the family of optimality systems

$\begin{equation} \left \{ \begin{array}{lll} {\frac{{\partial^{2} } {w_{i}} }{{\partial t^{2}}} } = \Delta^{2} w_{i}+Q(t) (w_{i})_{t},& & (0, m)\times\Theta,\\ w_{i}(x,m) = ( \frac{\partial u(m)}{\partial x_{i}} - \widetilde{P}_{\theta}^{*}u_{i}^{d}), & & \Theta, \\ (w_{i})_{t}(x,m) = 0, & & \Theta, \\ w_{i}(x,t) = \frac{\partial w_{i}(x,t)}{\partial \nu } = 0, & & \Gamma. \end{array} \right. \end{equation}$

(6.3)

Lemma 6.1. Let the cost $C_{\lambda}(Q)$ defined by (6.2) and the control $Q(t) \in U_{p}$ be the solution of (6.1). We can write

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q )}{k} = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[ \int_{0}^{m}\frac{\partial^{2} w_{i}}{\partial t^{2}} \frac{\partial v(x,t)}{\partial x_{i}} dt+ \int_{0}^{m}w_{i} \frac{\partial }{\partial x_{i}}( \frac{\partial^{2} v}{\partial t^{2}}) dt\right]dx\\ + \int_{0}^{m}\lambda d Q dt, \end{align}$

(6.4)

where the solution of (4.1) is $v$ , and the solution of (6.3) is $w_{i}$ .

Proof. The functional $C_{\lambda}(Q)$ given by (6.2), is of the form:

$\begin{align} C_{\lambda}(Q) = \frac{1}{2} \sum\limits_{i = 1}^{n}\int_{\theta}(\widetilde{P}_{\theta}\frac{\partial u}{\partial x_{i}} - u_{i}^{d} )^2 dx + \frac{\lambda}{2} \int^{m}_{0} Q^2(t) dt. \end{align}$

(6.5)

Choose $u_{k} = u(Q+k d)$ and $u = u(Q)$ . By (6.5), we deduce

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \lim\limits_{k\longrightarrow 0}\sum\limits_{i = 1}^{n} \frac{1}{2}\int_{\theta}\frac{(\widetilde{P}_\theta \frac{\partial u_{k}}{\partial x_{i}}-u_i^d)^{2}-(\widetilde{P}_\theta \frac{\partial u}{\partial x_{i}}-u_i^d)^{2}}{k}dx\\ + \lim\limits_{k\longrightarrow 0}\frac{\lambda}{2}\int_{0}^{m}\frac{(Q+k d)^2-Q^2}{k}dt. \end{align}$

(6.6)

Furthermore,

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} \\ = \lim\limits_{k\longrightarrow 0}\sum\limits_{i = 1}^{n}\frac{1}{2}\int_{\theta}\widetilde{P}_{\theta}\frac{( \frac{\partial u_{k}}{\partial x_{i}}- \frac{\partial u}{\partial x_{i}})}{k}(\widetilde{P}_{\theta} \frac{\partial u_{k} }{\partial x_{i}}+ \widetilde{P}_{\theta} \frac{\partial u}{\partial x_{i}}-2u_i^d)dx \\ + \lim\limits_{k\longrightarrow 0}\frac{1}{2}\int_{0}^{m}(2\lambda d Q+k \lambda d^2)dt\\ = \sum\limits_{i = 1}^{n}\int_\theta \widetilde{P}_{\theta} \frac{\partial v(x,m)}{\partial x_{i}} \widetilde{P}_{\theta}( \frac{\partial u(x,m)}{\partial x_{i}}-\widetilde{P}_{\theta}^{*}u_i^d)dx+\int_{0}^{m}\lambda dQ dt \\ = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}_{\theta} \frac{\partial v(x,m)}{\partial x_{i}} \widetilde{P}_{\theta}w_{i}(x,m)dx+\lambda \int_{0}^{m}dQ dt. \end{align}$

(6.7)

Using (6.3) to (6.7), we conclude

$\begin{align} \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k} = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[ \int_{0}^{m}\frac{\partial^{2} w_{i}}{\partial t^{2}} \frac{\partial v(x,t)}{\partial x_{i}} dt+ \int_{0}^{m}w_{i} \frac{\partial }{\partial x_{i}}( \frac{\partial^{2} v}{\partial t^{2}}) dt\right]dx\\\\ + \int_{0}^{m}\lambda d Qdt.\\ \end{align}$

(6.8)

□

Theorem 6.1. Consider the set $U_{p}$ , of partial admissible control defined as (5.1) and $u = u (Q)$ is its associate solution of (2.2), then the solution of (6.1) is

$\begin{equation} Q_{\varepsilon}(t) = max(-p,min( \frac{-1}{\lambda}\widetilde{P}_{\theta}(u_{t})(\widetilde{P}_{\theta} Div(w)),p)), \end{equation}$

(6.9)

where $Div(w) = \sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}}$ .

Proof. Let $d \in U_{p}$ and $Q + k d \in U_{p}$ for $k > 0$ . The cost $C_{\lambda}$ at its minimum $Q$ , verifies

$\begin{equation} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k}. \end{equation}$

(6.10)

From Lemma 6.1, substituting $\frac{\partial^{2} v}{\partial t^{2}}$ , by its value in system (4.1), we deduce that

$\begin{align} 0 \leq \lim\limits_{k\longrightarrow 0} \frac{C_{\lambda}(Q+k d)-C_{\lambda}(Q)}{k}\\ = \sum\limits_{i = 1}^{n}\int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[\int_{0}^{m} \frac{\partial v}{\partial x_{i}}\frac{\partial^{2} w_{i} }{\partial t^{2}}dt+ \int_{0}^{m}(-\Delta^{2} \frac{\partial v}{\partial x_{i}}+Q(t) \frac{\partial }{\partial x_{i}}(v_{t}) + d(t)\frac{\partial }{ \partial x_{i}}(u_{t}))w_{i} dt\right]dx\\ + \int_{0}^{m}\lambda d Q dt, \end{align}$

(6.11)

and system (6.3) gives

$\begin{align} 0 \leq \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta}\left[\int_{0}^{m} \frac{\partial v}{\partial x_{i}}(\frac{\partial^{2} w_{i} }{\partial t^{2}}-\Delta^{2} w_{i}-Q(t) (w_{i})_{t})dt+d(t) \frac{\partial }{\partial x_{i}}(u_{t})w_{i} dt\right]dx+\int_{0}^{m}\lambda d Q dt.\\ \\ = \sum\limits_{i = 1}^{n} \int_\theta \widetilde{P}^*_{\theta}\widetilde{P}_{\theta} \int_{0}^{m} (h(t)u_{t})\frac{\partial w_{i}}{\partial x_{i}} dt+ \int_{0}^{m}\lambda d Q dt.\\ \\ = \int_{0}^{m} h(t)\int_\theta\left[ u_{t}\widetilde{P}^*_{\theta}\widetilde{P}_{\theta} \sum\limits_{i = 1}^{n}\frac{\partial w_{i}}{\partial x_{i}} + \lambda Q\right] dx dt, \end{align}$

(6.12)

which gives the optimal control

$\begin{equation} Q_{\varepsilon}(t) = max(-p,min( \frac{-1}{\lambda}\widetilde{P}_{\theta}(u_{t})(\widetilde{P}_{\theta} Div(w)),p)). \end{equation}$

(6.13)

□

7. Conclusions

This paper studied the optimal control problem governed by an infinite dimensional bilinear plate equation. The objective was to command the flow state of the bilinear plate equation to the desired flow using different types of bounded feedback. The problem flow optimal control governed by a bilinear plate equation was considered and solved in two cases using the adjoint method. The first case considered a spatiotemporal control function and looked to reach a flow target on the whole domain. The second case considered a time control function and looks to reach a prescribed target at a fixed final time. As an application, the partial flow control problem was established and solved using the proposed method. More applications can be examined, for example. the case of fractional hyperbolic systems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was funded by the Deanship of Scientific Research at Jouf University under Grant Number (DSR2022-RG-0119).

Conflict of interest

The authors affirm that they have no conflicts of interest to disclose.

References

[1]	Bode L (2012) Human milk oligosaccharides: Every baby needs a sugar mama. Glycobiology 22: 1147-1162. doi: 10.1093/glycob/cws074
[2]	Kunz C, Meyer C, Collado MC, et al. (2016) Influence of gestational age, secretor and Lewis blood group status on the oligosaccharide content of human milk. J Pediatr Gastroenterol Nutr in press.
[3]	ten Bruggencate SJM, Bovee-Oudenhoven IMJ, Feitsma AL, et al. (2014) Functional role and mechanisms of sialyllactose and other sialylated milk oligosaccharides. Nutr Rev 72: 377-389.
[4]	Holck J, Larsen DM, Michalak M, et al. (2014) Enzyme catalysed production of sialylated human milk oligosaccharides and galactooligosaccharides by Trypanosoma cruzi trans-sialidase. New Biotechnol 31: 156-165. doi: 10.1016/j.nbt.2013.11.006
[5]	Wilbrink MH, ten Kate GA, van Leeuwen SS, et al. (2014) Galactosyl-lactose sialylation using Trypanosoma cruzi trans-sialidase as the biocatalyst and bovine κ-casein-derived glycomacropeptide as the donor substrate. Appl Environ Microbiol 80: 5984-5991. doi: 10.1128/AEM.01465-14
[6]	Wilbrink MH, ten Kate GA, Sanders P, et al. (2015) Enzymatic decoration of prebiotic galacto-oligosaccharides (Vivinal GOS) with sialic acid using Trypanosoma cruzi trans-sialidase and two bovine sialoglycoconjugates as donor substrates. J Agric Food Chem 63: 5976-5984. doi: 10.1021/acs.jafc.5b01505
[7]	Scudder P, Doom JP, Chuenkova M, et al. (1993) Enzymatic characterization of β-D-galactoside α2,3-transsialidase from Trypanosoma cruzi. J Biol Chem 268: 9886-9891.
[8]	Pereira ME, Zhang K, Gong Y, et al. (1996) Invasive phenotype of Trypanosoma cruzi restricted to a population expressing trans-sialidase. Infect Immun 64: 3884-3892.
[9]	Paris G, Ratier L, Amaya MF, et al. (2005) A sialidase mutant displaying trans-sialidase activity. J Mol Biol 345: 923-934. doi: 10.1016/j.jmb.2004.09.031
[10]	Jers C, Michalak M, Larsen DM, et al. (2014) Rational design of a new Trypanosoma rangeli trans-sialidase for efficient sialylation of glycans. PLoS One 9: e83902. doi: 10.1371/journal.pone.0083902
[11]	Pontes-de-Carvalho LC, Tomlinson S, Nussenzweig V (1993) Trypanosoma rangeli sialidase lacks trans-sialidase activity. Mol Biochem Parasitol 62: 19-25. doi: 10.1016/0166-6851(93)90173-U
[12]	Amaya MF, Buschiazzo A, Nguyen T, et al. (2003) The high resolution structures of free and inhibitor-bound Trypanosoma rangeli sialidase and its comparison with T. cruzi trans-sialidase. J Mol Biol 325: 773-784. doi: 10.1016/S0022-2836(02)01306-2
[13]	Buschiazzo A, Tavares GA, Campetella O, et al. (2000) Structural basis of sialyltransferase activity in trypanosomal sialidases. EMBO J 19: 16-24. doi: 10.1093/emboj/19.1.16
[14]	Pierdominici-Sottile G, Palma J, Roitberg AE (2014) Free-energy computations identify the mutations required to confer trans-sialidase activity into Trypanosoma rangeli sialidase. Proteins 82: 424-435. doi: 10.1002/prot.24408
[15]	Zeuner B, Luo J, Nyffenegger C, et al. (2014) Optimizing the biocatalytic productivity of an engineered sialidase from Trypanosoma rangeli for 3'-sialyllactose production. Enzyme Microb Technol 55: 85-93. doi: 10.1016/j.enzmictec.2013.12.009
[16]	Michalak M, Larsen DM, Jers C, et al. (2014) Biocatalytic production of 3′-sialyllactose by use of a modified sialidase with superior trans-sialidase activity. Process Biochem 49: 265-270. doi: 10.1016/j.procbio.2013.10.023
[17]	Zeuner B, Holck J, Perna V, et al. (2016) Quantitative enzymatic production of sialylated galactooligosaccharides with an engineered sialidase from Trypanosoma rangeli. Enzyme Microb Technol 82: 42-50.
[18]	Nyffenegger C, Nordvang RT, Jers C, et al. (2017) Design of Trypanosoma rangeli sialidase mutants with improved trans-sialidase activity. PLoS One 12: e0171585. doi: 10.1371/journal.pone.0171585
[19]	Kasche V (1986) Mechanism and yields in enzyme catalysed equilibrium and kinetically controlled synthesis of β-lactam antibiotics, peptides and other condensation products. Enzyme Microb Technol 8: 4-16. doi: 10.1016/0141-0229(86)90003-7
[20]	van Rantwijk F, Woudenberg-van Oosterom M, Sheldon RA (1999) Glycosidase-catalysed synthesis of alkyl glycosides. J Mol Catal B-Enzym 6: 511-532.
[21]	Hansson T, Andersson M, Wehtje E, et al. (2001) Influence of water activity on the competition between β-glycosidase catalysed transglycosylation and hydrolysis in aqueous hexanol. Enzyme Microb Technol 29: 527-534. doi: 10.1016/S0141-0229(01)00421-5
[22]	Zeuner B, Jers C, Mikkelsen JD, et al. (2014) Methods for improving enzymatic trans-glycosylation for synthesis of human milk oligosaccharide biomimetics. J Agric Food Chem 62: 9615-9631. doi: 10.1021/jf502619p
[23]	Mateo C, Palomo JM, Fernandez-Lorente G, et al. (2007) Improvement of enzyme activity, stability and selectivity via immobilization techniques. Enzyme Microb Technol 40: 1451-1463.
[24]	Rodrigues RC, Ortiz C., Berenguer-Murcia A, et al. (2013) Modifying enzyme activity and selectivity by immobilization. Chem Soc Rev 42: 6290-6307.
[25]	Barbosa O, Ortiz C, Berenguer-Murcia A, et al. (2014) Glutaraldehyde in bio-catalysts design: a useful crosslinker and a versatile tool in enzyme immobilization. RSC Adv 4: 1583-1600. doi: 10.1039/C3RA45991H
[26]	Grimsley GR, Scholtz JM, Pace CN (2009) A summary of the measured pK values of the ionizable groups in folded proteins. Protein Sci 18: 247-251.
[27]	Mateo C, Abian O, Bernedo M, et al. (2005) Some special features of glyoxyl supports to immobilize proteins. Enzyme Microb Technol 37: 456-462. doi: 10.1016/j.enzmictec.2005.03.020
[28]	Mateo C, Palomo JM, Fuentes M, et al. (2006) Glyoxyl agarose: A fully inert and hydrophilic support for immobilization and high stabilization of proteins. Enzyme Microb Technol 39: 274-280. doi: 10.1016/j.enzmictec.2005.10.014
[29]	Barbosa O, Ortiz C, Berenguer-Murcia A, et al. (2015) Strategies for the one-step immobilization-purification of enzymes as industrial biocatalysts. Biotechnol Adv 33: 435-456.
[30]	Zucca P, Fernandez-Lafuente R, Sanjust E (2016) Agarose and its derivatives as supports for enzyme immobilization. Molecules 21: 1577.
[31]	Calandri C, Marques DP, Mateo C, et al. (2013) Purification, immobilization, stabilization and characterization of commercial extract with β-galactosidase activity. J Biocatal Biotransformation 2: 1-7.
[32]	Hartmann M, Kostrov X (2013) Immobilization of enzymes on porous silicas – benefits and challenges. Chem Soc Rev 42: 6277-6289. doi: 10.1039/c3cs60021a
[33]	Bernal C, Urrutia P, Illanes A, et al. (2013) Hierarchical meso-macroporous silica grafted with glyoxyl groups: opportunities for covalent immobilization of enzymes. New Biotechnol 30: 500-506. doi: 10.1016/j.nbt.2013.01.011
[34]	Bernal C, Sierra L, Mesa M (2014) Design of β-galactosidase/silica biocatalysts: Impact of the enzyme properties and immobilization pathways on their catalytic performance. Eng Life Sci 14: 85-94. doi: 10.1002/elsc.201300001
[35]	González-Delgado I, Segura Y, Morales G, et al. (2017) Production of high galacto-oligosaccharides by Pectinex Ultra SP-L: optimization of reaction conditions and immobilization on glyoxyl-functionalized silica. J Agric Food Chem 65: 1649-1658. doi: 10.1021/acs.jafc.6b05431
[36]	Liu Y, Li Y, Li XM, et al. (2013) Kinetics of (3-aminopropyl)triethoxysilane (APTES) silanization of superparamagnetic iron oxide nanoparticles. Langmuir 29: 15275-15282. doi: 10.1021/la403269u
[37]	Gunda NSK, Singh M, Norman L, et al. (2014) Optimization and characterization of biomolecule immobilization on silicon substrates using (3-aminopropyl)triethoxysilane (APTES) and glutaraldehyde linker. Appl Surf Sci 305: 522-530.
[38]	Zhang D, Hegab HE, Lvov Y, et al. (2016) Immobilization of cellulase on a silica gel substrate modified using a 3-APTES self-assembled monolayer. SpringerPlus 5: 48.
[39]	Nordvang RT, Nyffenegger C, Holck J, et al. (2016) It all starts with a sandwich: Identification of sialidases with trans-glycosylation activity. PLoS One 11: e0158434. doi: 10.1371/journal.pone.0158434
[40]	Alva V, Nam SZ, Söding J, et al. (2016) The MPI bioinformatics Toolkit as an integrative platform for advanced protein sequence and structure analysis. Nucleic Acids Res 44: W410-W415. doi: 10.1093/nar/gkw348
[41]	Sayle R, Milner-White EJ (1995) RasMol: Biomolecular graphics for all. Trends Biochem Sci 20: 374-376. doi: 10.1016/S0968-0004(00)89080-5
[42]	Fersht AR, Serrano L (1993) Principles of protein stability derived from protein engineering experiments. Curr Opin Struct Biol 3: 75-83. doi: 10.1016/0959-440X(93)90205-Y
[43]	Torrez M, Schultehenrich M, Livesay DR (2003) Conferring thermostability to mesophilic proteins through optimized electrostatic surfaces. Biophys J 85: 2845-2853. doi: 10.1016/S0006-3495(03)74707-9
[44]	Hagiwara Y, Sieverling L, Hanif F, et al. (2016) Consequences of point mutations in melanoma-associated antigen 4 (MAGE-A4) protein: Insights from structural and biophysical studies. Sci Rep 6: 25182. doi: 10.1038/srep25182
[45]	Lu Y, Zen KC, Muthukrishnan S, et al. (2002) Site-directed mutagenesis and functional analysis of active site acidic amino acid residues D142, D144 and E146 in Manduca sexta (tobacco hornworm) chitinase. Insect Biochem Mol Biol 32: 1369-1382. doi: 10.1016/S0965-1748(02)00057-7
[46]	Cha J, Batt CA (1998) Lowering the pH optimum of D-xylose isomerase: the effect of mutations of the negatively charged residues. Mol Cells 8: 374-382.
[47]	Joshi MD, Sidhu G, Pot I, et al. (2000) Hydrogen bonding and catalysis: A novel explanation for how a single amino acid substitution can change the pH optimum of a glycosidase. J Mol Biol 299: 255-279. doi: 10.1006/jmbi.2000.3722
[48]	Hirata A, Adachi M, Sekine A, et al. (2004) Structural and enzymatic analysis of soybean β-amylase mutants with increased pH optimum. J Biol Chem 279: 7287-7295.
[49]	Amaya MF, Watts AG, Damager I, et al. (2004) Structural insights into the catalytic mechanism of Trypanosoma cruzi trans-sialidase. Structure 12: 775-784. doi: 10.1016/j.str.2004.02.036
[50]	Vandekerckhove F, Schenkman S, Pontes de Carvalho L, et al. (1992) Substrate specificity of the Trypanosoma cruzi trans-sialidase. Glycobiology 2: 541-548.
[51]	Bridiau N, Issaoui N, Maugard T (2010) The effects of organic solvents on the efficiency and regioselectivity of N-acetyl-lactosamine synthesis, using the β-galactosidase from Bacillus circulans in hydro-organic media. Biotechnol Prog 26: 1278-1289.
[52]	Thiem J, Sauerbrei B (1991) Chemoenzymatic syntheses of sialyloligosaccharides with immobilized sialidase. Angew Chem Int Ed Engl 30: 1503-1505. doi: 10.1002/anie.199115031
[53]	Ajisaka K, Fujimoto H, Isomura M (1994) Regioselective transglycosylation in the synthesis of oligosaccharides: comparison of β-galactosidases and sialidases of various origins. Carbohydr Res 259: 103-115. doi: 10.1016/0008-6215(94)84201-9
[54]	Marques ME, Mansur AAP, Mansur HS (2013) Chemical functionalization of surfaces for building three-dimensional engineered biosensors. Appl Surf Sci 275: 347-360.
[55]	Ferreira L, Ramos MA, Dordick JS, et al. (2003) Influence of different silica derivatives in the immobilization and stabilization of a Bacillus licheniformis protease (Subtilisin Carlsberg). J Mol Catal B-Enzym 21: 189-199.
[56]	Thomä-Worringer C, Sørensen J, López-Fandiño R (2006) Health effect and technological features of caseinomacropeptide. Int Dairy J 16:1324-1333. doi: 10.1016/j.idairyj.2006.06.012
[57]	Koshland D (1953) Stereochemistry and the mechanism of enzymatic reactions. Biol Rev 28: 416-436.

This article has been cited by:

Fatima Hussain, Suha Shihab, Transforming Controlled Duffing Oscillator to Optimization Schemes Using New Symmetry-Shifted G(t)-Polynomials, 2024, 16, 2073-8994, 915, 10.3390/sym16070915

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)