# Prediction of Viscoelastic Properties of Enzymatically Crosslinkable Tyramine–Modified Hyaluronic Acid Solutions Using a Dynamic Monte Carlo Kinetic Approach

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}O

_{2}) initiation system. More specifically, a dynamic Monte Carlo (MC) kinetic model is developed to quantify the effects of crosslinking conditions (i.e., polymer concentration, degree of phenol substitution and HRP and H

_{2}O

_{2}concentrations) on the gelation onset time; evolution of molecular weight distribution and number and weight average molecular weights of the crosslinkable polymer chains and gel fraction. It is shown that the MC kinetic model can faithfully describe the crosslinking kinetics of a finite sample of crosslinkable polymer chains with time, providing detailed molecular information for the crosslinkable system before and after the gelation point. The MC model is validated using experimental measurements on the crosslinking of a tyramine modified Hyaluronic Acid (HA-Tyr) polymer solution reported in the literature. Based on the rubber elasticity theory and the MC results, the dynamic evolution of hydrogel viscoelastic and molecular properties (i.e., number average molecular weight between crosslinks, M

_{c}, and hydrogel mesh size, ξ) are calculated.

## 1. Introduction

_{2}O

_{2}catalytic system are reported.

_{2}O

_{2}initiation system. An example of modeling of phenolic polymerization catalyzed by HRP can be found in the work of Ryu and coworkers [21]. However, the postulated kinetic mechanism did not include an enzyme deactivation reaction. In the publication of Yamagishi et al. [22], the Monte Carlo simulation of phenol–formaldehyde networks was described. In a recent publication by Kiparissides et al. [23], a comprehensive kinetic model and the method moments were employed to predict the onset gelation time of a tyramine modified Hyaluronic Acid (HA-Tyr) aqueous solution in terms of the HRP and H

_{2}O

_{2}concentrations.

_{2}O

_{2}concentrations) on the gelation onset time; evolution of gel fraction and molecular weight distribution of the crosslinked polymer chains and viscoelastic hydrogel properties. A multidimensional dynamic Monte Carlo model, based on the Gillespie’s original algorithm [26,27], is developed to calculate the dynamic evolution of molecular properties (i.e., number and weight average molecular weights, gel fraction, number of crosslinks, etc.) in a finite sample of polymer chains undergoing enzymatic crosslinking. In the MC model, each polymer chain is characterized by four internal variables—namely, the polymer chain length, i.e., degree of polymerization, the residual phenol content in the reactive polymer chains, the number of the activated phenol groups and the number of crosslinks. Note that the MC model can also provide information on the evolution of molecular weight distribution of polymer chains before and after the gelation point. The derived model is first validated using experimental kinetic measurements on a polymer–phenol conjugate system [8] (i.e., tyramine modified Hyaluronic Acid (HA-Tyr)). It is shown that the model can accurately predict the gelation onset time of the crosslinkable system in terms of the HRP and H

_{2}O

_{2}concentrations. Based on the MC-calculated hydrogel molecular properties (i.e., number of crosslinks in the hydrogel network, gel fraction, etc.), the viscoelastic properties (i.e., the storage modulus, G′), time required for the storage modulus to reach a plateau value and the polymer volume fraction in the swollen state, u

_{2,s}, are calculated for a tyramine modified Hyaluronic Acid (HA-Tyr) crosslinkable polymer solution [8]. It is shown that model predictions are in excellent agreement with the experimental measurements of Lee et al. (2008) [8] for the HA-Tyr system.

_{2}O

_{2}initiation system is postulated. In Section 3, a detailed description of the stochastic MC kinetic crosslinking model is presented. In Section 4, the derived dynamic MC model is validated using the experimental results of Lee [8] for the tyramine modified Hyaluronic Acid (HA-Tyr) system. The effects of the HPR and H

_{2}O

_{2}concentrations on the gelation onset time and the average molecular and viscoelastic properties of the synthesized hydrogels are assessed, and the model predictions are successfully compared with reported experimental measurements [8]. Moreover, predictions of the hydrogel molecular properties (i.e., average molecular weight between crosslinks, M

_{c}, and hydrogel mesh size, ξ) are presented. Finally, in Section 5, the main conclusions of the present work are summarized.

## 2. Enzymatic Crosslinking of Polymer–Phenol Conjugates

#### 2.1. The Postulated Kinetic Mechanism

_{2}O

_{2}[28].

_{2}O. Subsequently, the polymer–phenol conjugate is activated by the catalytic action of compound I, resulting in the formation of compound II (EII), which includes the neutralized porphyrin, Fe(IV) = O, and a phenol radical in the polymer chain (see Figure 1). The HRP-catalytic cycle returns to its initial state, Fe(III), via the reduction of compound II by the polymer–phenol conjugate, with the simultaneous formation of a phenol radical in a polymer chain and a water molecule [29,30].

#### 2.2. The Stochastic Monte Carlo Approach

_{j}denotes the stochastic reaction rate in s

^{−1}of the chemical reaction “j”, calculated in terms of the corresponding kinetic rate constant, k

_{j,MC}, in (s)

^{−}

^{1}and the total number of potential combinations of the molecules involved in the randomly selected reaction step, ${X}_{c}$ [39,40].

_{RE}is the number of distinct chemical reaction steps in the system.

_{s}different species S

_{i}(i = 1, 2, 3, …, N

_{s}). Assume that X

_{i}denotes the number of molecules of species “i” in the system. Following the original MC developments of Gillespie [26], the reaction rate for a unimolecular chemical reaction,${S}_{m}\stackrel{{k}_{m,MC}}{\to}{S}_{o}$, will be given by

_{m,MC}is the stochastic kinetic rate constant of the reaction, and ${k}_{m}$ is the respective experimental/deterministic value of the kinetic rate constant [41]. Note that, for unimolecular reactions, the numerical values of the stochastic and experimentally measured kinetic rate constants will be identical.

_{m}and S

_{n}, ${S}_{m}+{S}_{n}\stackrel{{k}_{mn,MC}}{\to}{S}_{o}$, the stochastic reaction rate will be given by

_{mn,MC}denotes the MC kinetic rate constant of the bimolecular reaction, ${k}_{mn}$ is the respective experimentally observed value of the kinetic rate constant [41] and N

_{A}is the Avogadro’s number.

_{m}, ${S}_{m}+{S}_{m}\stackrel{{k}_{mm,MC}}{\to}{S}_{o}$, the stochastic reaction rate will be given by

_{mm,MC}and ${k}_{mm}$ denote the stochastic and experimental kinetic rate constants of the reaction, respectively [41].

_{i}is the probability of occurrence of reaction “i” and is calculated by

#### 2.3. Development of a 4D MC Kinetic Crosslinking Model

_{2}O

_{2}oxidation system. In Table 2, the derived stochastic reaction rates based on the postulated kinetic mechanism (Equations (1)–(6)) are reported for the 4D MC kinetic model. To simplify the MC developments and reduce the computational time from several days to approximately one day per single MC run, it was assumed that the reaction mixture was spatially homogeneous, and the diffusional limitations were negligible.

_{R}polymer chains will be [42]

_{n}and M

_{w}) in the crosslinkable system.

_{i}(=x

_{i}MW

_{m}) represents the molecular weight of the “ith” polymer chain, x

_{i}denotes the degree of polymerization of the “ith” polymer chain, N

_{R}is the total number of polymer chains in the reactive system and MW

_{m}is the molecular weight of the repeating structural unit.

## 3. Prediction of Viscoelastic Properties of a Crosslinkable Polymer Solution

_{c}and ξ) is outlined. It is well-known that the hydrogel synthesis conditions (i.e., polymer, HRP and H

_{2}O

_{2}concentrations; degree of HA functionalization, etc.) do affect the final physical and viscoelastic properties of the hydrogel. On the other hand, the fundamental theory of rubber elasticity can provide a sound base for the calculation of hydrogel viscoelastic properties in terms of the molecular structure of the synthesized 3D hydrogel network.

_{c}. Note that the relaxation modulus, G(t), of a crosslinked network, at long times, approaches a nearly constant value, G

_{e}, representing the equilibrium shear modulus, as described by the rubber elasticity theory [44]. Moreover, the storage modulus of a crosslinkable system, G′(ω), at low frequencies, approaches the same characteristic value of G

_{e}[44]. According to the fundamental work of Treolar [45] on rubber elasticity, the equilibrium shear modulus of an ideal rubber elastic network can be calculated by the following equation:

^{3}). For the case of normal crosslinking (in which four polymer chains meet at each junction point; see Figure 3), v will be equal to twice the number of crosslinks per unit volume, v

_{o.}, (1/m

^{3}). k is the Boltzmann’s constant (J/K), T is the absolute temperature (K) and v

_{c}is the crosslinks concentration (mol/m

^{3}). R is the universal gas constant (J∙mol

^{−1}∙K

^{−1}), ρ is the density of the polymer and M

_{c}is the number average molecular weight between two crosslinks [45].

_{p}is the number of original or primary molecules before crosslinking, Flory argued that N

_{p}-1 intermolecular linkages are required to link the primary molecules together into a single ramified structure in which there are no closed loops. After this point, each additional crosslink will produce one closed loop or two network chains. Moreover, Flory argued that only these additional crosslinks are effective in the network formation [45,46]. Thus, for the calculation of ${G}_{e},$the following equation can be used:

^{−3}), ${N}_{p}$ is the number of primary molecules before crosslinking (m

^{−3}) and${M}_{c}$ is the number average molecular weight between two crosslinks for an ideal network [45].

_{A}, and the number average molecular weight of the primary molecules, M

_{n}.

_{e}for a nonideal network:

_{e}and Equations (16), (17), (20) and (21), the average molecular weight between two crosslinks, M

_{c}, can be calculated.

_{e}was calculated from Equation (21) in terms of the concentration of crosslinks, v

_{c}(mol/m

^{3}), obtained from the solution of the 4D Monte Carlo kinetic model describing the enzymatic crosslinking of the HA-Tyr polymer solution in the presence of the HRP/H

_{2}O

_{2}oxidation system. Moreover, the average molecular weight between two crosslinks, M

_{c}, will be given by the following equation:

_{c}. Peppas and Merrill [51] derived the following equation for the calculation of ${u}_{2,s}$:

_{1}is the Flory’s interaction parameter of the polymer–solvent, V

_{1}is the molar volume of the solvent and ${u}_{2,r}$ is the polymer volume fraction in the relaxed state. By multiplying the right-hand side term of Equation (23) with the correction parameter g, one can account for the effect of crosslinks undergoing some fluctuations, the effect of chain entanglements and intramolecular crosslinks that do not contribute to the network’s stiffness, as described by Tripathi and Tobita [20,52].

_{m}is the molecular weight of a repeating structural unit in a chain, λ is a backbone bond factor, C

_{n}is the Flory characteristic ratio and $l$ is the length of the bond. In the case of hyaluronic acid, $l$ is the bond length from glycosidic oxygen to glycosidic oxygen in a monosaccharide [56].

_{c}) for a tyramine modified hyaluronic acid crosslinkable solution [8]. Moreover, the calculated values for the polymer volume fraction in the swollen state (${u}_{2,s}$) and mesh size (ξ) are reported for the same system. Finally, model predictions on the dynamic evolution of the storage modulus (G′) are compared with experimental measurements for the HA-Tyr system [8].

## 4. Comparison of Model Predictions with Experimental Results

_{2}O

_{2}initiation system. The experimental results of Lee et al. (2008) [8] were used to estimate the kinetic model parameters and validate the MC model predictions. To reduce the computational time of the MC simulations, it was assumed that the initial sample of the HA-Tyr polymer chains was monodisperse. However, MC simulations carried out with an initial polydisperse sample of polymer yielded similar results.

_{1}varied in the range of (4.14∙10

^{5}–2∙10

^{7}) [28,57,58,59,60,61], and k

_{2}and k

_{3}varied in the range of (3.71∙10

^{3}–6.79∙10

^{6}) [58,59,60] and (1.45∙10

^{4}–4.56∙10

^{6}) [58,59,60,61], respectively, depending on the substrate type [59,60]. However, the numerical value of k

_{2}is generally an order of magnitude larger than k

_{3}[59,60]. In the present study, the numerical values of the kinetic model parameters (i.e., k

_{1}, k

_{2}, k

_{3}, k

_{4}, k

_{5}) were estimated by fitting the predictions of a deterministic kinetic model (using the method of moments [62]) to the experimental measurements of Lee et al. [8] on the gelation onset time of a crosslinkable hyaluronic acid–tyramine aqueous solution under different crosslinking conditions (i.e., HRP and H

_{2}O

_{2}concentrations). Subsequently, the estimated values of the kinetic rate constants (k

_{1}, k

_{2}, k

_{3}, k

_{4}, k

_{5}) were used in the present 4D stochastic Monte Carlo kinetic simulation model. The unknown value of the kinetic rate constant k

_{6}(see Chemical Reaction 6) was estimated by fitting the 4D MC predictions to the experimental measurements of Lee et al. [8] on the storage modulus G′ and time required for G′ to reach its plateau value.

_{1}= 10

^{7}(L∙mol

^{−1}s

^{−1}), k

_{2}= 8∙10

^{5}(L∙mol

^{−1}s

^{−1}), k

_{3}= 2.2∙10

^{4}(L∙mol

^{−1}s

^{−1}), k

_{4}= 7.3∙10

^{2}(L

^{2}mol

^{−2}s

^{−1}), k

_{5}= 10

^{12}(L∙mol

^{−1}s

^{−1}) and k

_{6}= 50 (L∙mol

^{−1}s

^{−1}).

_{w}) and in the solution phase (M

_{w,sol}) are depicted with respect to the crosslinking time. In particular, three different samples of approximately 0.1, 0.25 and 1 million polymer chains (e.g., 90 kDa) were selected to simulate the crosslinking kinetics of a HA-Tyr solution (i.e., 1.75% w/v) in the presence of 0.124 units/mL HRP and 728-μM H

_{2}O

_{2}[8]. As can be clearly seen in all the simulated cases, the overall M

_{w}exhibits a very steep increase after the gelation onset time, followed by a plateau value (continuous green, red and blue lines in Figure 4). On the other hand, the M

_{w,sol}exhibits an initial increase up to the gelation onset time, followed by a subsequent decrease to a plateau value (see the broken green, red and blue lines in Figure 4). Note that the observed increase in M

_{w}is characteristic of the onset formation of a three-dimensional hydrogel network. It is also worth mentioning that the sample size does not affect the calculation of the gelation onset time (i.e., for different HRP and H

_{2}O

_{2}concentrations), as well as the time evolution of M

_{w,sol}. However, the sample size does affect the final plateau values of M

_{w}. In fact, as the sample size increases, the MC calculated plateau value of M

_{w}increases. The last result is attributed to the fact that, when the sample size increases, the sum of the molecular weight of the final cluster of chains increases, and thus, the final molecular weight of the system increases.

_{n}) in the system and sol phase (M

_{n,sol}) (Figure 5a), as well on the overall polydispersity index in the system (PDI) (Figure 5b), are depicted for three different samples of polymer chains. As can be seen, the M

_{n}value increases with the crosslinking time. Note that the effect of sample size on the MC calculated values is less pronounced than that seen for M

_{w}in Figure 4. On the other hand, the time evolution of PDI after the gelation onset time does depend on the initial sample size due primarily to its effect on M

_{w}, as seen before. Finally, the effect of the sample size on the calculated M

_{n,sol}values appears to be insignificant (Figure 5a).

_{w}shows a very steep increase (i.e., the value of M

_{w}increases by several orders of magnitude) characterizes the gelation onset time and the formation of a 3D polymer network. Thus, the gelation onset time is defined as the time at which the M

_{w,sol}attains a maximum. From that time on, the gel fraction starts increasing while the polymer fraction in the solution decreases as polymer chains from the solution are connected to the growing polymer network. Regarding the calculation of gel and sol fractions in the Monte Carlo-based methods, different approximations have been proposed in the literature [20,36,63]. In the present MC algorithm, the largest polymer chain in the simulated sample is treated as gel, while all the remaining polymer chains in the sample are considered to be in the solution phase [36,63]. Thus, the molecular weight and the mass fraction of the largest chain and the molecular weight and mass fraction of the remaining chains is bookkept during the whole simulation in order to find the maximum M

_{w,sol}.

_{2}O

_{2}are depicted for the three different initial sample sizes. As can be seen, the initial number of polymer chains in the sample does not affect the MC calculated values of the gel and sol polymer fractions. Note that, until the gelation onset time, the values of the gel and sol polymer fractions remain constant and equal to 0% and 100%, respectively. Accordingly, after the gelation onset time, the gel fraction progressively increases to its final value (i.e., 100%), while the respective value for the sol fraction decreases to 0%, which marks the complete incorporation of all polymer chains into the network.

_{2}O

_{2}on the gelation onset time.

_{2}O

_{2}on the gelation onset time for the HA-Tyr crosslinkable system are depicted. As can be seen, there is an excellent agreement between the MC model predictions and reported experimental results on the gelation onset time, especially in the case of the HRP variation, as shown in Figure 8a. In the case of the H

_{2}O

_{2}variation (Figure 8b), the MC model predictions exhibit some small deviation from the corresponding experimental measurements, especially at very low and very high H

_{2}O

_{2}concentrations. The observed differences can be attributed to (i) the accuracy of the experimental measurements at very small and high gelation times, (ii) the number of MC simulations performed for each selected H

_{2}O

_{2}concentration, (iii) the sample size and (iv) the general validity of the postulated kinetic mechanism over an extended range of variation of the H

_{2}O

_{2}concentration. Note that, as the initial sample size increases, the MC simulation results do exhibit a smaller run-to-run variability. Moreover, as the number of MC simulations per case increases, the calculated mean value of the gelation onset time exhibits a smaller deviation from the experimentally observed value. In the present study, the plotted results represent the average value of two MC simulations per case (Figure 8, Figure 9, Figure 10 and Figure 11). In all MC simulations, the initial sample size contained circa 10

^{6}polymer chains.

_{e}could be approximated by the experimentally measured plateau value of G′(ω) at the applied low frequency ω (i.e., 1 Hz).

_{2}O

_{2}concentration on the G′ equilibrium (plateau) value (blue circles with a blue dashed line) and the time required for G′ to reach its plateau value (blue squares with a blue dashed line) are depicted. The red discrete points show the respective experimental measurements [8]. As can be seen, as the H

_{2}O

_{2}concentration increases, the equilibrium value of G′ increases up to a maximum value. This is followed by a subsequent decrease in G′ at higher H

_{2}O

_{2}concentrations. Similarly, the time required for G′ to reach its plateau value exhibits an initial increase with the H

_{2}O

_{2}concentration, followed by a subsequent decrease at higher H

_{2}O

_{2}concentrations. Note that the observed decrease in the values of the G′ and plateau time at high H

_{2}O

_{2}concentrations (i.e., above circa 1000 μM) is due to enzyme deactivation (Equation (4)) and the subsequent decrease of the respective rates of termination by combination (Equation (5)) and cyclization (Equation (6)) reactions.

_{2}O

_{2}concentrations, the MC calculated concentration of crosslinks, v

_{c}, is less than 1 (i.e., below its critical cut-off value). This means that the application of Equation (21) is not valid for the network correction term (1 − ρ/(v

_{c}∙M

_{n})), accounting for loose chain ends, is negative and, thus, the calculated value of G

_{e}. It is apparent that the MC calculated results for G′ at equilibrium and corresponding plateau time do deviate from the respective experimental values at low and high H

_{2}O

_{2}concentrations, for the hydrogel network under those conditions is far from ideal [45,46]. Moreover, the postulated kinetic mechanism may not be applicable over such an extended range of variation of H

_{2}O

_{2}concentration.

_{c}) can be estimated for g = 1. Moreover, from the application of Equation (22) and the MC calculated values of the crosslinks concentration, v

_{c}, the M

_{C}values can be calculated. In Figure 11, the experimentally determined values of M

_{c}(red discrete points) are compared with the respective values of M

_{c}calculated by the Monte Carlo method (blue squares with a blue dashed line) for different H

_{2}O

_{2}concentrations. Note that, as the H

_{2}O

_{2}concentration increases, the M

_{c}value initially decreases up to a minimum value, followed by a subsequent increase in M

_{c}at higher H

_{2}O

_{2}concentrations. It is worth mentioning that the results of Figure 11 on the M

_{c}variation are in full agreement with the calculated values of G′ at equilibrium and the calculated variation of v

_{c}with respect to the variation of the H

_{2}O

_{2}concentration. Moreover, from the calculated values of M

_{c}and Equations (23) and (24), the polymer volume fraction in the swollen state, u

_{2,s}, and the mesh size, ξ, can be calculated. In Figure 11, the variation of the mesh size, ξ, with respect to the H

_{2}O

_{2}concentration is depicted (black squares with a black dashed line).

_{g}, and a linear approximation of the correction factor g′ with respect to the concentration of crosslinks, v

_{c}(i.e., g′ = 0.8426∙v

_{c}− 0.0693), the dynamic evolution of the storage modulus of the crosslinkable HA-Tyr system, G′(t), was calculated using the following equation:

_{2}O

_{2}and 0.025 units/mL HRP) is depicted [8]. The MC dynamic predictions of G′ (blue continuous line) shown in Figure 12 are in good agreement with the respective experimental measurements (blue discrete points) of Lee et al. [8]. Note that the MC calculated values of G′ are plotted after the gelation onset time at which the first gel fraction appears (i.e., the value of w

_{g}is different from zero). In Figure 12, the experimental measurements of loss modulus (G″) (blue discrete points) are also depicted [8]. Note that the numerical value of g′, calculated by the postulated linear approximation of g′ in terms of crosslinks concentration at a time approximately equal to 5000 s (i.e., corresponding to the plateau value of G′ [8]), was equal to 0.53. This value is equal to the product of network correction factor g (see inset Figure 9 for an HRP concentration of 0.025 units/mL) times the term (1 − ρ/(v

_{c}∙M

_{n}) (see Equation (21)); that is, g′ = g(1 − ρ/(v

_{c}∙M

_{n}) = 0.73∙0.7274 = 0.53.

## 5. Conclusions

_{2}O

_{2}oxidation system. The model was applied to the investigation of gelation kinetics of HA-Tyr crosslinkable solutions. It was shown that the 4D MC model can predict the gelation onset time of the HA-Tyr solution under different crosslinking conditions (i.e., HRP and H

_{2}O

_{2}concentrations). The MC kinetic model provided detailed on the molecular and structural properties of the HA-Tyr polymer chains (i.e., Number Chain Length Distribution, NCLD, bivariate Chain Length–Number of Crosslinks Distribution, M

_{n}and M

_{w}in the solution and gel phases, crosslinks concentration, gel mass fraction, etc.) before and after the gelation point. Based on the fundamental rubber elasticity theory and the MC calculated crosslinks concentration, the equilibrium storage modulus (G′) and the time required to reach the plateau value of G′ were calculated for the HA-Tyr crosslinkable system. Model predictions were successfully compared with respective experimental measurements reported by Lee et al. [8]. Moreover, based on the MC results, the molecular weight between crosslinks (M

_{c}), as well as the hydrogel mesh size, ξ, were predicted for the same system. Note that the calculated values of M

_{c}and ξ can be further utilized to estimate the hydrogel diffusion coefficient of various solutes. Finally, it was shown that, using the present MC model, one can calculate the dynamic evolution of storage modulus for the HA-Tyr system. It is believed that the present 4D MC kinetic model can be used for the optimal design of a hydrogel having desired viscoelastic and molecular properties for as specific biomedical application.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

C | concentration of polymer in the solution |

C_{n} | Flory’s characteristic ratio |

E | Horseradish Peroxidase (HRP) |

g | network correction factor |

G | relaxation modulus (Pa) |

G′ | storage modulus (Pa) |

G_{e} | equilibrium shear modulus treated by rubberlike elasticity theory (Pa) |

G_{x,m,a,c} | largest polymer chain with x monomeric units, m residual phenol groups, a activated phenol groups and c crosslinks |

k | Boltzmann’s constant (J/K) |

k_{j} | kinetic rate constant of “j” reaction |

k_{j,MC} | stochastic kinetic rate constant of “j” reaction |

M_{c} | number average molecular weight between two crosslinks |

M_{i} | molecular weight of the “ith” polymer chain |

M_{n,sol} | number average molecular weight in the solution (g/mol) |

M_{w} | weight average molecular weight in the system (g/mol) |

MW_{m} | molecular weight of a repeating structural unit |

M_{w,sol} | weight average molecular weight in the solution (g/mol) |

N_{A} | Avogadro’s Number |

NCLD | number chain length distribution |

N_{E} | the number of enzyme species E |

N_{H2O2} | total number of hydrogen peroxide species |

N_{R} | total number of polymer chains |

N_{p} | number of primary molecules before crosslinking |

N_{RE} | total number of reactions |

rand1 | randomly generated number uniformly distributed in the range of [0, 1] |

rand2 | randomly generated number uniformly distributed in the range of [0, 1] |

${r}_{E}^{2}$ | mean square end-to-end distance of a strand |

$\overline{{r}_{0}^{2}}$ | mean square end-to-end distance of a non-constrained strand |

P_{i} | probability of reaction “i” |

R | the universal gas constant (J∙mol^{−1}K^{−1}) |

R_{j} | the rate of the “j” chemical reaction |

T | absolute temperature (K) |

u_{2,r} | polymer volume fraction at the relaxed state |

u_{2,s} | polymer volume fraction at the equilibrium swollen state |

V | volume of the mixture |

V_{1} | molar volume of the solvent |

w_{g} | gel mass fraction |

x_{i} | degree of polymerization of the “ith” polymer chain |

X_{c} | total number of possible combinations of reactive species in a reaction |

Greek Symbols | |

Δt | the time step between two reactions |

ν | number of network chains or segments per unit volume (m^{−3}) |

v_{c} | the number of moles of crosslinks per unit volume (mol∙m^{−3}) |

v_{e} | number effective network chains or segments per unit volume (m^{−3}) |

v_{o} | number of crosslinks per unit volume (m^{−3}) |

λ | backbone bond factor |

ξ | mesh size of the network |

ρ | polymer density (kg∙m^{−3}) |

χ_{1} | Flory interaction parameter of the polymer-solvent |

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**Figure 1.**(

**a**) Activation of polymer–phenol conjugates in the presence of the HRP enzyme and hydrogen peroxide. (

**b**) Isomerization of phenolic radicals and crosslinking reactions between different types of “live” polymer chains and enolization of crosslinked polymer chains (adapted from reference [23]).

**Figure 3.**Schematic representation of a hydrogel network: (

**A**) intramolecular loop, (

**B**) free polymer chain ends and (

**C**) polymer chain entanglement.

**Figure 4.**Effect of the initial number of polymer chains on the time evolution of the weight average molecular weight (M

_{w}) in the system and sol phase (M

_{w,sol}) for a HA-Tyr solution (i.e., 1.75% w/v, MW = 90 kDa, 0.124 units/mL HRP and 728-μM H

_{2}O

_{2}) [8].

**Figure 5.**Effect of the initial number of polymer chains on the time evolution of the number of average molecular weights (M

_{n}) in the system and sol phase (M

_{n,sol}) (

**a**) and Polydispersity Index in the system (

**b**) for a HA-Tyr solution (i.e., 1.75% w/v, MW = 90 kDa, 0.124 units/mL HRP and 728-μM H

_{2}O

_{2}) [8].

**Figure 6.**Effect of the initial sample size on the time evolution of the sol and gel mass fractions for a HA-Tyr solution (i.e., 1.75% w/v, MW = 90 kDa, 0.124 units/mL HRP and 728-μM H

_{2}O

_{2}) [8].

**Figure 7.**Effect of the initial sample size on the time evolution of the crosslinks concentration for a HA-Tyr solution (i.e., 1.75% w/v, MW = 90 kDa, 0.124 units/mL HRP and 728-μM H

_{2}O

_{2}) [8].

**Figure 8.**Comparison of the MC model results with reported experimental measurements on the gelation onset time for the HA-Tyr system (i.e., 1.75% w/v, MW = 90 kDa) [8]: (

**a**) effect of the HRP concentration on the gelation onset time (H

_{2}O

_{2}concentration: 728 μM) and (

**b**) effect of the H

_{2}O

_{2}concentration on the gelation onset time (HRP concentration: 0.062 units/mL).

**Figure 9.**Effect of the HRP concentration on the G′ equilibrium value and time required for G′ to reach its plateau value for the HA-Tyr crosslinkable solution (i.e., 1.75% w/v, MW = 90 kDa and H

_{2}O

_{2}concentration = 728 μM). Blue squares with a blue dashed line and blue circles with a blue dashed line represent the MC calculated values, and red circles and squares denote the respective experimental measurements of Lee et al. [8]. All MC simulations were conducted with an initial polymer chain population of ~10

^{6}.

**Figure 10.**Effect of the H

_{2}O

_{2}concentration on the G′ equilibrium value and time required for G′ to reach its plateau value for the HA-Tyr crosslinkable solution (i.e., 1.75% w/v, MW = 90 kDa and HRP concentration = 0.062 units/mL). Blue squares with a blue dashed line and blue circles with a blue dashed line represent the MC calculated values, and red circles and squares denote the respective experimental measurements of Lee et al. [8]. All MC simulations were conducted with an initial polymer chain population of ~10

^{6}.

**Figure 11.**Effect of the H

_{2}O

_{2}concentration on the molecular weight between crosslinks, M

_{c}, for the HA-Tyr crosslinkable solution (i.e., 1.75% w/v, MW = 90 kDa and HRP concentration = 0.062 units/mL). All MC simulations were conducted with an initial polymer chains population of ~ 10

^{6}.

**Figure 12.**Comparison of MC results (blue line) with the reported experimental measurements (blue dots) on storage modulus during gelation for the HA-Tyr crosslinkable system [8]. The MC simulation results were obtained with an initial polymer chains population of ~10

^{6}. The vertical dash line marks the gelation onset time as predicted by the 4D MC algorithm. The horizontal dash line denotes the equilibrium value of G′ as reported by Lee et al. [8]. The experimental measurements (blue squares) of the loss modulus (G″) are also depicted.

**Table 1.**Selected papers on HRP/H

_{2}O

_{2}crosslinked polymer–phenol systems for biomedical applications.

Material | Authors |
---|---|

HA-Tyr | Kurisawa, Chung, Yang, Gao and Uyama 2005 [6] |

Dextran-Tyr | Jin, Hiemstra, Zhong and Feijen 2007 [7] |

HA-Tyr | Lee, Chung and Kurisawa 2008 [8] |

Dextran-Tyr | Jin, Moreira Teixeira, Dijkstra, Zhong, Blitterswijk, Karperien and Feijen 2010 [9] |

Carboxymethylcellulose-tyramine | Ogushi, Sakai and Kawakami 2007 [10] |

Carboxymethylcellulose-phenolic hydroxyl groups (CMC-Ph) | Sakai, Ogushi and Kawakami 2009 [3] |

Gelatin-hydroxyphenylpropionic acid (Gtn–HPA) | Wang, Chung, Chan and Kurisawa 2010 [11] |

Dextran-tyramine (Dex-TA)/Hyaluronic acid-tyramine (HA-TA) conjugates | Wennink, Niederer, Bochynska, Teixeira, Karperien, Feijen and Dijkstra 2011 [12] |

HA-Tyr | Ren, Gao, Kurisawa and Ying 2015 [13] |

CMCH-Tyr | Bi, Liu, Kang, Zhuo and Jiang 2019 [14] |

Equation | Stochastic Reaction Rate | MC Simulation Algorithm |
---|---|---|

1 | ${R}_{1}={k}_{1,MC}{N}_{E}{N}_{{H}_{2}{O}_{2}}$ | ${N}_{E}={N}_{E}-1$ ${N}_{{H}_{2}{O}_{2}}={N}_{{H}_{2}{O}_{2}}-1$ ${N}_{{E}_{I}}={N}_{{E}_{I}}+1$ |

2 | ${R}_{2}={k}_{2,MC}{N}_{{E}_{I}}{\displaystyle {\displaystyle \sum}_{i=1}^{{N}_{RE}}}{m}_{i}{R}_{{x}_{i},{m}_{i},{a}_{i},{c}_{i}}$ | ${N}_{{E}_{I}}={N}_{{E}_{I}}-1$ Selection of ${R}_{x,m,a,c}$ ${R}_{x,m,a,c}\to {R}_{x,m-1,a,c}$ ${N}_{{E}_{II}}={N}_{{E}_{II}}+1$ |

3 | ${R}_{3}={k}_{3,MC}{N}_{{E}_{II}}{\displaystyle {\displaystyle \sum}_{i=1}^{{N}_{RE}}}{m}_{i}{R}_{{x}_{i},{m}_{i},{a}_{i},{c}_{i}}$ | ${N}_{{E}_{II}}={N}_{{E}_{II}}-1$ Selection of ${R}_{x,m,a,c}$ ${R}_{x,m,a,c}\to {R}_{x,m-1,a,c}$ ${N}_{E}={N}_{E}+1$ |

4 | ${R}_{4}=\frac{1}{2}{k}_{4,MC}{N}_{{E}_{II}}{N}_{{H}_{2}{O}_{2}}\left({N}_{{H}_{2}{O}_{2}}-1\right)$ | ${N}_{{E}_{II}}={N}_{{E}_{II}}-1$ ${N}_{{H}_{2}{O}_{2}}={N}_{{H}_{2}{O}_{2}}-2$ ${N}_{{E}_{IV}}={N}_{{E}_{IV}}+1$ |

5 | ${R}_{5}=\frac{1}{2}{k}_{5,MC}{\displaystyle {\displaystyle \sum}_{i=1}^{{N}_{R}-1}}\left({a}_{i}{R}_{{x}_{i},{m}_{i},{a}_{i},{c}_{i}}{\displaystyle {\displaystyle \sum}_{l=i+1}^{{N}_{R}}}{a}_{l}{R}_{{y}_{l},{n}_{l},{b}_{l},{d}_{l}}\right)$ | Selection of ${R}_{x,m,a,c}$ Selection of ${R}_{y,n,b,d}$ Production of ${R}_{x+y,m+n,a+b-2,c+d+1}$ Removal of ${R}_{x,m,a,c}$ Removal of ${R}_{y,n,b,d}$ |

6 | ${R}_{6}={k}_{6,MC}\left(\frac{1}{2}{a}_{i}\left({a}_{i}-1\right){G}_{x,m,a,c}\right)$ | Selection of ${G}_{x,m,a,c}$ ${G}_{x,m,a,c}\to {G}_{x,m,a-2,c+1}$ |

**Table 3.**Effect of the initial sample size on the computational time needed for the kinetic simulation of crosslinking of the HA-Tyr chains with 0.124 units/mL HRP and 728-μM H

_{2}O

_{2}and PDI = 1 [8].

No of polymer chains | 55,214 | 103,527 | 248,465 | 255,367 | 517,635 | 1,028,368 |

CPU in sec | 173 | 586 | 4208 | 4371 | 16,657 | 72,839 |

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Karageorgos, F.F.; Kiparissides, C.
Prediction of Viscoelastic Properties of Enzymatically Crosslinkable Tyramine–Modified Hyaluronic Acid Solutions Using a Dynamic Monte Carlo Kinetic Approach. *Int. J. Mol. Sci.* **2021**, *22*, 7317.
https://doi.org/10.3390/ijms22147317

**AMA Style**

Karageorgos FF, Kiparissides C.
Prediction of Viscoelastic Properties of Enzymatically Crosslinkable Tyramine–Modified Hyaluronic Acid Solutions Using a Dynamic Monte Carlo Kinetic Approach. *International Journal of Molecular Sciences*. 2021; 22(14):7317.
https://doi.org/10.3390/ijms22147317

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Karageorgos, Filippos F., and Costas Kiparissides.
2021. "Prediction of Viscoelastic Properties of Enzymatically Crosslinkable Tyramine–Modified Hyaluronic Acid Solutions Using a Dynamic Monte Carlo Kinetic Approach" *International Journal of Molecular Sciences* 22, no. 14: 7317.
https://doi.org/10.3390/ijms22147317