Geometrically regulating evolutionary dynamics in biofilms

The theoretical understanding of evolutionary dynamics in spatially structured populations often relies on nonspatial models. Biofilms are among such populations where a more accurate understanding is of theoretical interest and can reveal new solutions to existing challenges. Here, we studied how the geometry of the environment affects the evolutionary dynamics of expanding populations, using the Eden model. Our results show that fluctuations of subpopulations during range expansion in two- and three-dimensional environments are not Brownian. Furthermore, we found that the substrate's geometry interferes with the evolutionary dynamics of populations that grow upon it. Inspired by these findings, we propose a periodically wedged pattern on surfaces prone to develop biofilms. On such patterned surfaces, natural selection becomes less effective and beneficial mutants would have a harder time establishing. Additionally, this modification accelerates genetic drift and leads to less diverse biofilms. Both interventions are highly desired for biofilms.

Figure 1

(a) $2+1\mathrm{D}$ setting by considering a $w\times l\times l$ environment with $l=200$ and $w=16$ as the environment in which populations grow. We apply temporal evolution until $95\%$ of the environment become occupied by cells. A plane consisting of one line at each $yz$ plane separates two populations and we study the geometry of these lines. (b) For each line, we analyze the displacement $X$ and winding angle $\theta$. (c) The statistics of displacement of these lines shows $\langle \mathrm{\Delta }{X}^2\rangle \sim t^{\alpha }$ (inset). For $1+1\mathrm{D}$ ($w=1$) we have $\alpha =1.31\pm 0.01$ and extrapolation suggests to have $\alpha (1/w=0)=1.02$. (d) For the statistics of the winding angle vs $t$ we have $\langle {\theta }^2\rangle =a+\beta ln\left(t\right)$ with $\beta =0.32\pm 0.01$ until $\langle {\theta }^2\rangle$ saturates to ${\langle {\theta }^2\rangle }_{\text{sat}}$ (inset). ${\langle {\theta }^2\rangle }_{\text{sat}}$ increases vs $w$ as $w^{0.33\pm 0.02}$. (e) To check if such a variation in $w$ has a dominant effect on the domain wall fluctuations or if it is just a crossover effect, we compared $\langle \mathrm{\Delta }{X}^2\rangle$ for environments with different sizes. This analysis shows that the behavior is irrelevant of system size.

Figure 2

(a) Normalized extinction probability vs normalized $t$ for different system sizes ($l$) in a $1+1\mathrm{D}$ environment ($w=1$). All sizes show a log-normal distribution with $\sigma =0.57\pm 0.01$. Inset: The averaged extinction time grows vs system size as $\overline{t}\sim l^{1.42\pm 0.01}$. These dynamics is significantly different from that of Brownian motion which has been used to analyze extinction during range expansion so far [45]. (b) The normalized averaged extinction time vs $t$ for $w=30$ and different values of $l$. Extinction times still have a log-normal distribution with $\sigma =0.54\pm 0.03$. Inset: The averaged extinction time grows vs system size as $\overline{t}\sim l^{1.35\pm 0.02}$.

Figure 3

(a) $2+1\mathrm{D}$ wedge setting as shown with angles of ${\varphi }_l$. (b) $\alpha$ vs ${\varphi }_l$ for $1+1\mathrm{D}$ and $2+1\mathrm{D}$. As we increase ${\varphi }_l, \alpha$ monotonically decreases. (c) $\beta$ vs ${\varphi }_l$. $\beta$ decreases vs ${\varphi }_l$. Inset: ${\langle {\theta }^2\rangle }_{\text{sat}}$ vs ${\varphi }_l$ for $1+1\mathrm{D}$ and $2+1\mathrm{D}$. These results together reveal that the initial condition can have a longstanding effect on the geometry of the domain lines.

Figure 4

(a) A $yz$ cut of a growth on a periodic wedged initial condition with an angle of ${\varphi }_l$ and a periodic length of $l^{\prime }$. Instead of initially separated populations, we consider two well-mixed populations on a periodic wedge initial condition. (b) Heterozygosity, defined as $H\left(t\right)=n_A\times n_B$, vs $t$ for different values of $l^{\prime }$ with ${\varphi }_l=\pi /4$ and $w=100$ (the qualitative behavior is independent of $w$). Inset: $H\left(t\right)$ vs $t$ for different values of ${\varphi }_l$ and $l^{\prime }=40$. Thus, the initial condition can interfere with the fluctuations of subpopulations and one can accelerate genetic drift through engineering the geometry of the substrate. (c) $H$ vs $t$ where for $R_A=1.05\times R_B$ with ${\varphi }_l=\pi /4$ and $w=100$ (the qualitative behavior is independent of $w$). Inset: $H\left(t\right)$ vs $t$ for different values of ${\varphi }_l$ and $l^{\prime }=40$. Due to a lower fitness advantage, (B) has a lower chance to survive and goes extinct over time. However, as we increase $l^{\prime }$, (B) gets more of a chance to survive. Changing the initial configuration leads to a long-lasting effect on the diversity of expanding populations and promotes their coexistence.

Figure 5

(a) A single member of (A) emerges in a 2D environment which already is occupied by (B). Due to mutations, (A) can have a different duplication rate as $R_A=R_B(1+\mathrm{\Delta }R)$, where $\mathrm{\Delta }R<0, \mathrm{\Delta }R=0$, and $\mathrm{\Delta }R>0$ represent deleterious, passenger, and driver mutations, respectively. Such a difference in the duplication rate affects the probability of fixation for (A). As we increase $\mathrm{\Delta }R$, the fixation probability increases. (b) For beneficial mutants, living on a 2D periodically wedged surface with ${\varphi }_l=\pi /4$ delays the fixation process, depending on the value of $l^{\prime }$. This result reveals that the initial geometry of the substrate directly regulates the fixation probability of beneficial mutants. (c) and (d) show a similar process happens in 3D environments as well.