Scaling and Continuum Percolation Model for Enzyme-Catalyzed Gel Degradation

Enzyme-catalyzed gel degradation is inherently controlled by diffusion of enzymes in the gel. We report kinetics measurements on the gelatin-thermolysin system, varying solvent viscosity as well as gel and enzyme concentrations. Scaling relations and reduced variables are proposed which are shown to account for the experimental results. Finally, we argue that the nontrivial experimental dependence on enzyme concentration for the degradation time demonstrates that enzyme random walk is self-attracting, leading to a continuum percolation model for gel degradation.

Figure 1

Experimental setup for determination of the gel-sol transition time $t_c$: while the sample is still a solid gel, the red ball stays at the top of the tube (tubes 1, 4–6 from the left in the inset); at $t_c$ the ball falls down (tube 3).

Figure 2

Mean degradation time $t_c$ vs thermolysin concentration [$E$] for different gel volume fractions ${\varphi }_{\mathrm{gel}}$. Lines are best fits $t_c=t_c(1)[E{]}^{-1/b}$.

Figure 3

Result of the power law fits to the data: $t_c([E])=t_c(1)[E{]}^{-1/b}$ vs ${\varphi }_{\mathrm{gel}}$. The solid line and the horizontal dotted line correspond to $(t_c/\mathrm{h})=(48\pm 4)({\varphi }_{\mathrm{gel}}/\%{)}^{2.50\pm 0.05}([E]/\mathrm{nM}{)}^{-1.46\pm 0.07}$. The nonhorizontal dashed line is a guide for the eyes.

Figure 4

The effect of glycerol on $t_c$ for ${\varphi }_{\mathrm{gel}}=7.5\%$. Bottom: $t_r$ vs $[E{]}_r$. Top: $t_r\times [E{]}^{1/b}$ vs $[E{]}_r$. $t_r$ is either $t_c$ or $t_c/{\eta }_r$, depending on symbols, with ${\eta }_r$ the relative solvent viscosity. $[E{]}_r$ is either [$E$] or $[E]A_r$, depending on symbols, with $A_r$ the relative enzyme activity.

Figure 5

Same data as in Fig. 2 using the reduced variables of Eq. (2) with ${\xi }_0=a{\varphi }_{\mathrm{gel}}^{-1}$ (${\nu }_F=1/2$). Adimensional quantities are obtained with $a=0.7\mathrm{nm}$ [19] and $D_0=83{\mathrm{nm}}^2/\mu \mathrm{s}$ [3].