Self-organized homogenization of flow networks

TL;DR

It is shown that also artificial flow networks can self-organize toward homogeneous perfusion by the versatile adaption of controlled erosion, paving the way for a very versatile self-organized increase in the performance of porous media.

Abstract

From the vasculature of animals to the porous media making up batteries, the core task of flow networks is to transport solutes and perfuse all cells or media equally with resources. Yet, living flow networks have a key advantage over porous media: they are adaptive and self-organize their geometry for homogeneous perfusion throughout the network. Here, we show that also artificial flow networks can self-organize toward homogeneous perfusion by the versatile adaption of controlled erosion. Flowing a pulse of cleaving enzyme through a network patterned into an erodible hydrogel, with initial channels disparate in width, we observe a homogenization in channel resistances. Experimental observations are matched with numerical simulations of the diffusion-advection-sorption dynamics of an eroding enzyme within a network. Analyzing transport dynamics theoretically, we show that homogenization only occurs if the pulse of the eroding enzyme lasts longer than the time it takes any channel to equilibrate to the pulse concentration. The equilibration time scale derived analytically is in agreement with simulations. Lastly, we show both numerically and experimentally that erosion leads to the homogenization of complex networks containing loops. Erosion being an omnipresent reaction, our results pave the way for a very versatile self-organized increase in the performance of porous media.

Figures (5)

• Figure 1: Erosion by the MMP-1 enzyme homogenizes the resistances of parallel channels whose walls are made of the hydrogel PEG-NB, cross-linked with a cleavable peptide sequence. (a) Experimental setup: glass slide with six microfluidic chambers. Each chamber contains a different microfluidic network, whose hydrogel walls are made of PEG-NB cross-linked with a cleavable peptide sequence. Inset: four-channel network. (b) Time-lapse of the erosion of a four-channel network, whose hydraulic resistances are initially imbalanced. The images, taken at $t=0$ (top), $t=5min$ (middle), and $t=8min$ (bottom), show the hydrogel (light green) being eroded by the enzyme (red). Scale bar: 1. (c) Time-evolution of the widths $w_i(t)$ of the four channels. The injection of the enzyme upstream of the network occurs from $t=2$ min to $t=7$ min (light red patch). Points: experimental data. Lines: numerical simulations. (d) Homogenization of the channels' hydraulic resistances $R_i(t)$ as the channel walls are getting eroded: the hydraulic resistances $R_i(t)/R_4(t)$ normalized by the narrowest channel's resistance homogenize as they increase over time. Channels are numbered from the largest to the smallest width. Points: experimental data. Lines: numerical simulations. Inset: normalized standard deviation $\sigma(t)/\sigma(0)$ of the normalized hydraulic resistances $R_i(t)/R_4(t)$. Similarly to the hydraulic resistances, the flow rates also homogenize, see Supp. Fig. 11 supp.
• Figure 2: Wall erosion follows Michaelis-Menten kinetics. (a) Spatio-temporal evolution of the channel width $w$ of a single channel, see Movie 1 in the Supplemental Material supp, at a fixed position along the channel. The hydrogel (light green) is eroded by the enzyme (red, $c\geq1.5e-6\mol\per L$) diffusing into the hydrogel and back into the channel after the pulse passed. The hydrogel walls, i.e. the boundaries between the channel and the hydrogel, are highlighted in blue. Three pulses of enzyme of duration $t_e=5$ min are inter-spaced by 60 min of PBS flowing into the network. (b) Time-evolution of the channel width $w$ (blue) as pulses of the enzyme are flowed in. The concentration of the enzyme in the channel (red) is averaged along the cross-section of the channel in $y$ and along the channel in $x$, $\langle\bar{c}\rangle$. (c) Erosion rate $dw/dt$ of the hydrogel, as a function of the local enzyme concentration $c_w$ within 50 of the wall, follows Michaelis-Menten kinetics. Squares: raw data. Triangles: averaged data. Error bars show the standard deviation. Line: power-law fit of the data. Data over 10 experiments with 10 different channel geometries but keeping the PBS concentration and channel heights constant. The kinetic model is consistent over variations of height and buffer concentration in altered gels, see Supp. Fig. 2c supp.
• Figure 3: High absorption, low desorption and reducing Péclet number improve homogenization. (a) Normalized enzyme concentration averaged over the channel length $\left<\bar{c}(x,t)\right>$ for two parallel channels shows a slower rise in the narrow channel (green) relative to the wide channel (purple). Channels are of equal length $L=4.5mm$. The time scale to reach the upstream equilibrium concentration $t_\mathrm{eq}$ is extracted as the time-point of $\left<\bar{c}(x,t)\right>$ crossing 0.7. (b) Low values of $t_{\mathrm{eq}}$ in the narrow channel, $t_{\mathrm{eq}}^{\mathrm{nar}}$, correspond to better homogenization - the shorter it takes for the concentration in the narrow channel to reach the upstream concentration, the more the channel erodes, leading to better homogenization of the hydraulic resistances reflected by a greater decrease in $\sigma(t=60\s)/\sigma(0)$. Numerically determined $t_{\mathrm{eq}}$ (blue dots) obtained from (a) for different values of $w_{\mathrm{nar}}$ with a constant value for $w_{\mathrm{wide}}=$ 300 match the corresponding analytically calculated $t_{\mathrm{eq}}$ from Eq. \ref{['eq:t_eq']} (crosses). The simulation parameters are: diffusivity $D=30\square\um\per\s$, absorption rate $K_a=20\um\per\s$ and desorption rate $K_d=30\per\s$. (c) Homogenization $\sigma(t=60\s)/\sigma(0)$ (color coded) by erosion is best for high absorption, low desorption. Also, reducing the Péclet number increases homogenization for $\mathrm{Pe}>1$ considered here. Here, we only show the plane with $k_d=K_dw^2_{\rm nar}/D=4$ for clarity; see Supp. Fig. 13a supp and Movie 6 in the Supplementary Material for the full 3D phase plot supp. $D$, $K_a$ and $K_d$ varied for $w_{\mathrm{nar}}=$ 200 and $w_{\mathrm{wide}}=300\um$. For all plots, homogenization dynamics employ a channel height $H=1mm$, enzyme concentration $c_0=3e-5\mol\per L$, erosion constant $\chi_0=3.3\pm0.6e3\um\per\min\mol\tothe{-1/2}L\tothe{1/2}$, and inflow rate $Q=6\uL\per\min$. Note that the pulse duration $t_e$ was set to $t_e=1min$ for numerical efficiency. Increasing the pulse length increases the erosion, which results in a steeper drop in the homogenization metric $\sigma(t)/\sigma(0)$, indicating better homogenization, see Supp. Fig. 13c supp.
• Figure 4: Transition from heterogenization to homogenization at increasing erosion pulse length. (a) Two parallel channel design where parallel channels are now partitioned into three sections of varying width. The disparity in width in the entry zone of the parallel channels $w'_{\text{nar}}\ll w_{\text{wide}}$ creates a much longer time to equilibrate in the narrower channel than in the wider channel. The small difference in width in the center section $w_{\text{nar}}<w_{\text{wide}}$ reduces the required heterogeneity in channel width erosion for heterogenization, $\Delta w_{\text{nar}}/\Delta w_{\text{wide}}<w_{\text{nar}}/w_{\text{wide}}$. (b) The normalized standard deviation $\sigma(t)/\sigma(0)$ of the normalized hydraulic resistances $R_{\text{nar}}(t)/R_{\text{wide}}(t)$ of the center sections increases as the erosion pulse passing is shorter than 10 s but decreases at longer pulse length, much larger than the time to equilibrate in the narrow parallel channel $t_{\text{eq}}=0.66\s$. The simulation parameters are: inflow rate $Q=60\uL\per\min$, diffusivity $D=3\square\um\per\s$, absorption rate $K_a=20\um\per\s$ and desorption rate $K_d=e3\per\s$ and channel height $H=200\um$.
• Figure 5: Erosion homogenizes complex, looped networks, and works even better with bubbles. Erosion of a loopy, hexagonal network with an initial bimodal distribution of channel widths perfused without (a,b,c,d) and with (e,f,g,h) air bubbles into wider channels before enzyme pulse (see Movies 3 & 4 in the Supplemental Material supp). (a,e) Time-lapses of the erosion at $t=0$ (top), $t=7min$ (middle), and $t=60min$ (bottom) show the hydrogel (light green) getting eroded by the enzyme (red). With bubbles blocking wide channels, see dark channels, the enzyme does not reach and thus does not erode wide channels hydrogel walls. Scale bar: 1. (b,f) Increase of the channels' width due to the hydrogel erosion. The initially wide (resp. narrow) channels are depicted in blue (resp. red). Black lines indicate numerical simulations. (c,d,g,h) Distribution of the channels' hydraulic resistances $R(t)$ normalized by the average hydraulic resistance of the narrow channels $\overline{R}_\mathrm{nar}(t)$, shown at the start of the experiment ($t=0$, c & g) and after one enzyme pulse ($t=60min$, d & h). The value of the normalized standard deviation $\sigma(t)/\sigma(0)$ of the normalized hydraulic resistance $R(t)/\overline{R}_\mathrm{nar}(t)$ of all channels (wide and narrow) is annotated for each distribution.