Mathematical Modelling Of Glucose-Responsive Behavior Of Concanavalin A-Sugar Affnity Based Hydrogels Volume 5 - Issue 1

Ruixue Yin^1,2*, Kejia Zhao¹, Hongbo Zhang¹ and Wenjun Zhang^2*

• ¹Complex and Intelligent Systems Research Center, East China University of Science and Technology, PR China
• ²Division of Biomedical Engineering, University of Saskatchewan, Canada

Received:March 7, 2022;   Published: March 17, 2022

*Corresponding author:Ruixue Yin, Complex and Intelligent Systems Research Center, East China University of Science and Technology, China

Wenjun Zhang, Division of Biomedical Engineering, University of Saskatchewan, Canada

DOI: 10.32474/MAMS.2022.05.000204

Abstract PDF

Abstract

Mathematical modeling plays an important role in studying the glucose-responsive behavior of chemically-controlled closedloop system for design and optimization. In this study, a glucose-responsive swelling model controlled by seven main equations was established and solved to investigate the glucose-responsive behavior of Con A-DexG hydrogel-based chemically-controlled closedloop system. The swelling ratio Q and diffusion coefficient D are two variables in the model, and they can help design the molecular structure of the hydrogel (i.e., a mixture of Con A and DexG) with a potential to design a personalized insulin delivery system.

Keywords: Glucose-responsive; Chemically-controlled closed-loop system; Concanavalin A; Mathematical modeling; Insulin delivery

Introduction

An artificial closed-loop system, known as the artificial pancreas, can fast respond to changes of blood sugar levels and release insulin instantly. It deems to be better therapeutic method than insulin multi-injection method and has attracted widespread attention in the fields of materials, mechanics, and biology [1-3]. A chemically controlled closed-loop system relies on the responsive materials that can quickly respond to the slight changes in the environment, resulting in physical or chemical properties change of the system [4,5]. Small changes in the environment can be temperature, pH, electrolyte concentration, biomolecules (such as glucose, enzymes, antigens, etc.) and fields (electric and magnetic fields). Glucose responsive materials that can sense blood glucose concentration and release insulin in response to changes of glucose level have a great potential to develop the chemically controlled closed-loop insulin delivery system for an improved diabetes care [6-9].

At present, three glucose-responsive materials have been investigated to develop the chemically controlled closed-loop insulin delivery system [10-12], which are glucose oxidase, concanavalin A (Con A), and phenylboronic acid. Among them, the Con A-sugar affinity-based system is the most promising one for clinical use due to its strongest specificity to glucose [13]. The first in vivo test in a live diabetic domestic pig was performed in Taylor's group by using an implantable artificial pancreas made from crosslinked dextran-Con A hydrogel [9]. Mathematical modeling plays an important role in studying the glucose-responsive behavior of chemically-controlled closed-loop system. On the one hand, the mechanism of swelling and drug releasing in hydrogel system is further improved theoretically, which promotes the development of soft matter science [14]; on the other hand, it can simulate the behavior of the system without doing a lot of experiments, thus saving the time and cost of the trial-and-error process [15]. The objective of this study is to construct a mathematical model for the glucose-responsive process of Con A-sugar affinity based chemically controlled system based on swelling and variable diffusion coefficient theory. The calculated swelling ratio Q was compared with the obtained Q by experiments to verify the model. The influence of glucose concentration as well as other physical parameters on the swelling model were discussed.

The swelling model of the Con A-DexG hydrogel

The hydrogel in this study has two major parts: Con A and DexG. DexG has carbon-carbon double bonds, and the double bonds can crosslink to form the chemical crosslinking of the hydrogel. DexG also has terminal glucose groups, and the terminal glucose groups can bind with Con A to form physical affinity crosslinking of the hydrogel. The crosslinked network absorbs water to an equilibrium swelling state, thus allowing the transport of macromolecular drugs, nutrients, and cellular wastes [16]. The Con A and DexG mixture acts as a concentrated solution at full hydration with osmotic forces causing the hydrogel swell. The combination of covalent and affinitive cross-linking bonds makes the hydrogel act like a spring against this force. When competitive glucoses are introduced into the hydrogel network, the amount of affinitive crosslink bonds decreases, leading to less resistance to osmotic forces and enhanced swelling. Flory-Rehner [17] proposed a theory to explain the relationship between crosslink density and the swelling of rubber compounds. Peppas and Merrill [18] modified the original equation to establish a polymer network swelling equation:

where  is the number average molecular mass of the polymer chain between cross-links,  is the number average molecular mass of the polymer chain,  is the partial specific volume of the polymer,  is the molar volume of water,  is the Flory-Huggins polymer-solvent interaction parameter,  is the polymer fraction of the hydrogel at equilibrium swelling,  is the polymer fraction of the hydrogel after gel formation.

The swelling ratio can be calculated by the ratio of the polymer fraction of the hydrogel after gel formation to the polymer fraction of the hydrogel at equilibrium swelling, shown in equation 2.

The volume increase caused by swelling will be regarded as a decrease in the polymer composition, so the swelling ratio will be greater than 1.

The competitive binding between DexG (DeG) and glucose (Glu) with Con A can be expressed as the given equations according to the ligand competition theory [16].

The binding process of both equations (3) and (4) are in a dynamic equilibrium, and the equilibrium constants are given by:

K_DeG = [Con A-DeG] / [Con A] [DeG], (5)

K_Glu = [Con A-Glu] / [Con A] [Glu]. (6)

The total number of cross-links was determined by finding the respective number of affinity and covalent bonds. The number of covalent bonds was assumed to equal the number of methacrylate groups measured for DexG molecules as it was assumed that each bind to a Con A molecule. Affinity cross-links were calculated by solving the binding equations 3- 6. It was assumed that there was only one dextran binding to one Con A tetramer. Although this is not strict, the model is greatly simplified.

According to the binding equations 3-6, the equilibrium constants of DexG-Con A and Con A-glucose binding are given by

where  is the initial concentration of Con A,  is the initial concentration of DexG,  is the initial concentration of glucose that is set to 0.022mol/L based on the concentration of blood sugar after meal in diabetic patients,  is the concentration of resultant DexG-Con A complex,  is the concentration of resultant Glu-Con A complex, and k₁ and k₂ are the equilibrium constants of DexG-Con A and Con A-glucose binding reactions, respectively.  In order to calculate , Equation 8 needs to be rearranged and substituted into Equation 7. This leads to a quadratic equation, i.e.:

By calculating the concentration of covalent bonds and affinity cross-links, the average molecular weight between crosslinks can be calculated:

where c_p is the concentration of polymer DexG in the hydrogel, and c_covalent can be calculated by the following equation:

where the branching degree is characterized as 4.8% and the DS is characterized as 20.6% by NMR spectrum.

The concentrations of the components are governed by the degree of swelling; therefore, calculation of the swelling must be done iteratively. The swollen polymer fraction can be used to calculate the mesh size of the hydrogel [19]

where C_n is the Flory characteristic ratio, M_r is the molecular mass of repeat unit, l is the unit length along the polymer backbone.

Then, the mesh size can be used to calculate the variable diffusion coefficient D based on the deviation from the liquid phase diffusivity [20,21]

where D₀ is the liquid phase diffusivity of the solute, a is the hydrodynamic radius of the solute.

Results and Discussion

Parameter’s calculation and validation

By using MATLAB software, the parameters can be calculated according to equations 1, 2, 7-13. The initial values and parameters calculated are shown in Table 1. For the model validation, the calculated swelling ratio Q was compared with the obtained Q by experiments in our previous study [22]. At the glucose level of 0.022 mol/L, the measured equilibrium swelling ratio Q of glucose-responsive Con A-DexG hydrogel samples by weighting method was about 4.75, while the calculated Q was 6.625 by the above mathematical model. The values are on the same order of magnitude. It is noted that the weighting method to get the swelling data also has significant operating error; therefore, using the swelling model of Con A-DexG hydrogel to get the swelling ratio is reasonable, and it is another effective way to obtain the swelling and diffusion behavior besides experiments.

Influence of glucose concentration on swelling and diffusion behavior

The parameter values at different glucose levels were calculated by MATLAB programming to discuss the influence of glucose concentration on the calculation results of the model. The range of glucose concentration was 0-0.06 mol/L.

(1) c__DexGConA

As it can be seen from Figure 1.  The concentration of generated DexG-Con A complex decreases with increase of glucose level. According to the ligand competition equilibrium equations 3-4, with the increase of the glucose concentration, the chemical equilibrium of equation 4 shifts to the right side of the equation, generating more Con A-Glu complex. Meanwhile, the amount of Con A binding sites decreases, which also promoting the left shift of equation 3, leading to the decrease of DexG-Con A complex.

(2)