Mathematical modelling of CRISPR-Cas system effects on biofilm formation

Figures & data

Table

Nomenclature
Parameters	Description	Dimensions
H	Density of wild-type (non-lysogenic) bacteria	cells cm^−2
$H_{\mathrm{L}}$	Density of lysogens	cells cm^−2
$C_{\mathrm{S}}$	Density of bacteria with CRISPR-immunity/spacer	cells cm^−2
$C_{\mathrm{L}}$	Density of lysogens with CRISPR system	cells cm^−2
V	Density of phage	phage cm^−2
t	Time	hr
K	Carrying capacity of bacteria in biofilm	cells cm^−2
α	Prophage induction	hr^−1
r	Bacterial growth rate	hr^−1
b	Burst size	phage cell^−1
β	Adsorption rate	cm^2 phage^−1 hr^−1
γ	Phage loss rate	hr^−1
θ	Adhesion rate per cell	hr^−1
η	Slouging off rate of non-lysogens	hr^−1
${\eta }_{\mathrm{L}}$	Sloughing off rate of lysogens	hr^−1
φ	Planktonic bacteria forming biofilm	cells cm^−2 hr^−1
$p_{\mathrm{L}}$	Probability of lysogenization	−
$p_{\mathrm{F}}$	Probability of CRISPR failure	−
$p_{\mathrm{D}}$	Probability of cell death	−

Figure 1. Bacterial populations considered in the models. H represents the population of wild-type bacteria, $H_{\mathrm{L}}$ represents the population of bacteria with prophage (lysogens), $C_{\mathrm{S}}$ is the population of bacteria with a CRISPR-Cas system, $C_{\mathrm{L}}$ are lysogens with a CRISPR-Cas system. The bacterial populations H and $H_{\mathrm{L}}$ are considered in Model 1, $C_S$ and $C_{\mathrm{L}}$ in Model 2 and $H_{\mathrm{L}}$ and $C_S$ in Model 3.

Table 1. Existence and stability conditions for the steady states in Model 1.

Steady State	Existence Criteria	Stability Criteria
TE${}_1$	–	$\eta >r$
PFE${}_1$	$\eta <r$	$\eta >r\left(1-{\displaystyle \frac{1}{R_{\beta }}}\right)$
LE${}_1$	$\eta <r-\alpha$	$\eta <r\left(1-{\displaystyle \frac{1}{R_{\beta }}}\right)-\alpha$
AEE${}_1$	$r\left(1-{\displaystyle \frac{1}{R_{\beta }}}\right)-\alpha <\eta <r\left(1-{\displaystyle \frac{1}{R_{\beta }}}\right)$	Evaluated numerically

Note: The abbreviations used in this table are $R_{\beta }=Kb\beta /\gamma$, TE${}_1$: Trivial equilibrium, PFE${}_1$: Phage free equilibrium, LE${}_1$: Lysogenic equilibrium, AEE${}_1$: all existing equilibrium.

Table 2. Existence and stability conditions of the steady states in Model 2.

Steady state	Existence criteria	Stability criteria
TE${}_2$	–	$\eta >r$
PFE${}_2$	$\eta <r$	$\eta >r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)$
LE${}_2$	$\eta <{\eta }_{\mathrm{L}}<r-\alpha$	$\eta >r-(r-(\alpha +{\eta }_{\mathrm{L}}\left)\right){\displaystyle \frac{R_{\beta }\alpha p_{\mathrm{F}}+r}{R_{\beta }\alpha p_{\mathrm{D}}+r}}$
CE${}_2$	$\eta <r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)$	$\eta <r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)\left(1-{\displaystyle \frac{p_{\mathrm{F}}}{p_{\mathrm{D}}}}\right)+{\displaystyle \frac{p_{\mathrm{F}}}{p_{\mathrm{D}}}}(\alpha +{\eta }_{\mathrm{L}})$
AEE${}_2$	$\eta <r-(r-(\alpha +{\eta }_{\mathrm{L}}\left)\right){\displaystyle \frac{R_{\beta }\alpha p_{\mathrm{F}}+r}{R_{\beta }\alpha p_{\mathrm{D}}+r}}$
	$\eta >r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)\left(1-{\displaystyle \frac{p_{\mathrm{F}}}{p_{\mathrm{D}}}}\right)+{\displaystyle \frac{p_{\mathrm{F}}}{p_{\mathrm{D}}}}(\alpha +{\eta }_{\mathrm{L}})$, $p_{\mathrm{F}}>p_{\mathrm{D}}$	Evaluated numerically

Note: The abbreviations used in this table are $R_{\beta }=Kb\beta /\gamma$, TE${}_2$: Trivial equilibrium, PFE${}_2$: Phage free equilibrium, LE${}_2$: Lysogenic equilibrium, CE${}_2$: CRISPR equilibrium and AEE${}_2$: All existing equilibrium.

Table 3. Existence and stability conditions of the steady states in Model 3.

Steady state	Existence criteria	Stability criteria
TE${}_3$	–	$\eta >r$
PFE${}_3$	$\eta <r$	$\eta >r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)$
LE${}_3$	$\eta <r-\alpha$	$\eta <r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)-\alpha$
CE${}_3$	$\eta <r\left(1-{\displaystyle \frac{1}{R_{\beta }{p}_{\mathrm{F}}}}\right)$	$r\left(1-{\displaystyle \frac{1}{p_{\mathrm{F}}{R}_{\beta }}}\right)-\alpha <\eta <r\left(1-{\displaystyle \frac{1}{p_{\mathrm{F}}{R}_{\beta }}}\right)$
LE${}_{\phi }$	−	$\phi >{\displaystyle \frac{K\alpha (r-\eta )}{\eta }}$ (sufficient condition)
AEE${}_{\phi }$	$\phi >{\displaystyle \frac{\alpha \gamma K(r-\eta )(\gamma r-Kb\beta p_{\mathrm{F}}(r-\eta -\alpha \left)\right)}{(Kb\alpha \beta p_{\mathrm{F}}+\gamma r)(Kb\alpha \beta p_{\mathrm{F}}+\eta \gamma )}}$
	$\phi <K\alpha \left(1-{\displaystyle \frac{r}{\alpha +\eta }}\left(1-{\displaystyle \frac{\gamma }{Kb\beta p_{\mathrm{F}}}}\right)\right)$	Evaluated numerically

Notes: The abbreviations used in this table are $R_{\beta }=Kb\beta /\gamma$, TE${}_3$: Trivial equilibrium, PFE${}_3$: Phage free equilibrium, LE${}_3$ and LE${}_{\phi }$: Lysogenic equilibrium, CE${}_3$: CRISPR Equilibrium and AEE${}_3$ and AEE${}_{\phi }$: All existing equilibrium.

Table 4. Baseline values for the parameters used in this study.

Parameters	values	Source
K	$5\times {10}^6$ cells cm^−2	[6, 21, 40]
α	10^−5 hr^−1	[13, 36]
r	1 hr^−1	[1]
b	200 phage cell^−1	[1]
β	10^−7 cm^2 phage^−1 hr^−1	[1]
γ	0.05 hr^−1	[1]
η	0.1 hr^−1	[21]
${\eta }_{\mathrm{L}}$	$\sqrt{\eta }$ hr^−1	–
φ	2.78 cells cm^−2 hr^−1	see text
$p_{\mathrm{F}}$	10^−4	[14]
$p_{\mathrm{L}}$	0.05	–
$p_{\mathrm{D}}$	0.1	–

Figure 2. Dynamics of phage and bacterial populations inside the biofilm at baseline parametric values. H and $C_{\mathrm{S}}$ represent populations of wild-type bacteria and CRISPR-immune bacteria (dashed lines), $C_{\mathrm{L}}$ and $H_{\mathrm{L}}$ represent lysogens with and without CRISPR systems (dots) and V is the population of bacteriophage (solid lines). Model 1 approaches the lysogenic equilibrium LE${}_1$, Model 2 and Model 3 (case (a)) approach the phage-free equilibria (PFE${}_2$ and PFE${}_3$) whereas Model 3 (case (b)) approaches the all existing equilibrium (AEE${}_{\phi }$).

Figure 3. Population densities of bacteria and bacteriophage at stable (solid lines) and unstable (dashed lines) equilibrium states against adsorption rate constant β in the above models. Each colour represents a unique population while the line width is increased to visualize overlapping populations. Bacterial populations are H (green) and $H_{\mathrm{L}}$ (blue) in Model 1, $C_{\mathrm{S}}$ (black) and $C_{\mathrm{L}}$ (purple) in Model 2 and $C_{\mathrm{S}}$ and $H_{\mathrm{L}}$ in Model 3 whereas the phage population is represented by V (red) in all models.

Figure 4. (a) Population densities of bacteria and bacteriophage at stable (solid lines) and unstable (dashed lines) equilibrium states against phage loss rate constant γ in the Model 3 (case (b)). Bacterial populations $C_{\mathrm{S}}$ (black) and $H_{\mathrm{L}}$ (blue) and the phage population V (red) are shown. Panels (b), (c) and (d) show the critical value of φ necessary to ensure the stability of the lysogenic equilibrium, versus α, η and K. In each case, the parameter space above each curve corresponds to a stable lysogenic equilibrium (LE${}_{\phi }$) while the space under each curve corresponds to stability of the all existing equilibrium (AEE${}_{\phi }$).

Figure 5. Population dynamics of phage-bacteria interaction in Models 2, 3 (case (b)) and 1. Three Vertical lines are drawn to separate the simulations of the models into four panels. Model 2 is simulated for baseline parameters in the first panel, Model 3 (case (b)) is first simulated for baseline parameters and then simulated after an increase in φ in the second and third panels respectively. In the fourth panel, simulations of Model 1 are presented for baseline parameters (top two curves) and for varied parameters based on possible therapy, i.e. $\eta =0.45$, $\alpha =0.3$ and r=0.75, (bottom two curves with increased line-width). The dashed lines represent $C_{\mathrm{S}}$, dash-dot lines represent $C_{\mathrm{L}}$ in Model 2, and $H_{\mathrm{L}}$ in Model 1 and 3, whereas dots symbolize the population density of phage V.

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