Note that the diffusivity of the enzyme in hydrogel (

D_{E}^{gel}) negatively scales with hydrogel crosslinking density and positively correlates to hydrogel mesh size (

ξ). As enzyme infiltrates the hydrogel, it catalyzes tyrosine residues into DOPA dimers, resulting in an increase in hydrogel crosslinking density (i.e., smaller mesh size) that may lead to lower diffusivity. Hence,

D_{E}^{gel} may be dependent on the location of the enzyme in the hydrogel. In this mathematical model, we assume that enzyme diffusion proceeds bi-directionally from the top and bottom (i.e., along the

x-axis) of the thin hydrogel with a thickness of

h (i.e., neglecting diffusion from the gel edge). Initially, the entire hydrogel (i.e., 0 <

x <

h) is free of enzyme, which gives the following initial condition:

we also assume that enzyme concentration in the solution remains unchanged (

C_{E0}), which yields the following boundary conditions:

since enzyme diffusion proceeds bi-directionally, at any given time there is no flux at the center of the hydrogel (i.e.,

x =

h/2), which yields the following boundary condition:

As mentioned earlier, diffusivity of any solute in a crosslinked hydrogel is affected by gel mesh size. Hence, it is necessary to determine whether the additional DOPA dimer crosslinks significantly affects

Tyr_{ase} transport in the stiffened gel. In this regard, the Lustig–Peppas estimation of solute diffusivity in a highly swollen gel can be used to establish a correlation between

Tyr_{ase} diffusivity and hydrogel mesh size [

35].

where

D_{E}^{sol} is the diffusivity of enzyme in solution (5.05 × 10

^{−10} m

^{2}/s for

Tyr_{ase}),

R_{E} is the hydrodynamic radius of the enzyme (4.5 nm for

Tyr_{ase}) [

51], and

Y is the critical volume required for a successful translational movement of the substrate relative to the average free volume of a water molecule (1 for PEG-based gels) [

35]. Note that this equation is used only when gel mesh size is larger than hydrodynamic radius of the soluble molecule (i.e.,

R_{E}/

ξ < 1) because no diffusion is possible when R

_{E} is larger than

ξ. Clearly, when

R_{E} approaches

ξ,

D_{E}^{gel} becomes much smaller than

D_{E}^{solution}. Therefore, it is critical to determine the relative sizes of

R_{E} and

ξ even for highly swollen PEG-based hydrogels (i.e.,

Q > 10). To obtain mesh size of a swollen hydrogel, we first obtained mass swelling ratio (

q) by measuring the mass (

m) of swollen and dried gels:

where the volumetric swelling ratio (

Q) of the hydrogel is determined using mass swelling ratio (

q) and the density (

ρ) of PEG (1.087 g/cm

^{3} at 37 °C) and water (0.994 g/cm

^{3} at 37 °C):

once

Q is obtained, gel mesh size (

ξ) can be derived using the following equation:

where

$\ell $ is the average bond length (1.47 Å) in the backbone of an ethylene glycol subunit (i.e., –CH

_{2}–CH

_{2}–O–), 3 is the total number of bonds in a PEG repeat subunit,

C_{n} is the Flory characteristic ratio (4 for PEG-based hydrogel), and

$\overline{{M}_{n}}$ is the number for average molecular weight of PEG8NB (20 kDa) [

1]. To obtain the average molecular weight between crosslinks (

$\overline{{M}_{c}}$) within a step-growth hydrogel, the following equation can be used:

where

MW_{A} and

MW_{B} represent molecular weights of the two macromer crosslinkers PEG8NB and CYGGGYC, respectively.

f_{A} and

f_{B} are the functionality of the macromer and peptide crosslinker (i.e., 8 and 2). Numerical model predictions were programmed and executed via Grapher (Arizona Software) using a spatial step size (Δ

x) of 1 µm and a temporal step size (Δ

t) of 1 s.