Non-conventional therapeutic technique to replace CRISPR bacteria from biofilm by inducible lysogen

Figures & data

Table 1. Nomenclature.

Parameters	Description	Dimensions
$\overrightarrow{L}$	Density of lysogens with CRISPR system	cells cm^−2
$\overrightarrow{C}$	Density of bacteria with CRISPR-immunity/spacer	cells cm^−2
$\overrightarrow{V}$	Density of phage	phage cm^−2
t	Time	h
K	Carrying capacity of bacteria in biofilm	cells cm^−2
α	Prophage induction	h^−1
r	Bacterial growth rate	h^−1
b	Burst size	phage cell^−1
β	Adsorption rate	cm^2 phage^−1 h^−1
γ	Phage loss rate	h^−1
${\mathrm{\Phi }}_0$	Planktonic bacteria forming biofilm	cells cm^−2 h^−1
$M_{VL}$	Matrix of per capita infection rate of lysogens	–
$M_{VC}$	Matrix of per capita infection rate of CRISPR-immune bacteria	–
$M_L$	Probability Matrix for lysogens inducing prophage	–
$M_V$	Probability Matrix for producing free virions from infected bacteria	–
$p_L$	Probability of lysogenization	–
$p_F$	Probability of CRISPR failure	–
$p_S$	Probability of gaining spacer	–
$p_I$	Probability of infection by lower index viral strains (see text)	–
$p_i$	Probability of induction	–

Table 2. Baseline values for the parameters used in this study.

Parameter	Value	References
K	$5\times {10}^6$ cells cm^−2	[5, 15, 36]
r	1 h^−1	[1]
b	150 phage cell^−1	[1]
α	10^−5 h^−1	[10, 30]
η	0.01 h^−1	[4, 15]
β	$6\times {10}^{-9}$ cm^2 phage^−1 h^−1	[1]
γ	0.1 h^−1	[1]
${\mathrm{\Phi }}_0$	2.78 cells cm^−2 h^−1	[4, 15]
$p_F$	10^−5	[11]
$p_L$	0.01	[4]
$p_S$	10^−5	[11]
$p_I$	1	–
$p_i$	1/i	–

Figure 1. Classes of three populations are presented in the above figure: phage populations (dashed lines), bacterial populations with CRISPR/Cas systems (dash-dot lines) and lysogens (solid lines). The therapeutic procedure time is set$t=90\mathrm{h}$. The lysogens are inserted into the biofilm with up to three prophages either simultaneously or successively. The possible sequences of the number of prophages harboured in the lysogens are $\{1,2\},\{1,1,1\},\{2,1\}$ and $\left\{3\right\}$, as described in the text. The colours of the curves are used to distinct each class of lysogen and CRISPR bacteria and are associated as described in the legend: $L_{000}$ and $C_{000}$ is for grey, $L_{001}$ and $C_{001}$ is for blue, $L_{010}$ and $C_{010}$ is for green, $L_{011}$ and $C_{011}$ is for cyan, $L_{100}$ and $C_{100}$ is for red, $L_{101}$ and $C_{101}$ is for pink, $L_{110}$ and $C_{110}$ is for cream, $L_{111}$ and $C_{111}$ is for black. Colours for phage populations are blue for $V_1$, cyan for $V_2$ and black for $V_3$. Coloured areas under curves of therapeutic lysogens represent the insertion phase whereas light grey areas under curves correspond to the competence phase.

Figure 2. Populations of CRISPR bacteria at the final time of therapeutic procedure of 60, 90 and 180 h. The insertion phase is varied between 0 and 15 h for each therapeutic sequence. Four possible combinations of prophages are considered: (a) one prophage followed by two simultaneously, denoted by $\{1,2\}$, (b) three prophages each introduced separately, denoted $\{1,1,1\}$, (c) two prophages simultaneously followed by one separately, $\{2,1\}$, and finally (d) introduction of a lysogen containing all three prophage sequences, $\left\{3\right\}$. Therapeutic procedures for 60, 90 and 180 h are represented by blue circles, black solid lines and red + signs, respectively.

Figure 3. Three sets of bar plots for the population of CRISPR bacteria in the biofilm against each therapeutic sequence with 90-h procedural time. For the first set of therapeutic sequences, first four bars on the left, lysogen are inserted in the biofilm at the start of the first growth period of each element of the therapeutic lysogen sequence (new class of lysogen). For the second set of therapeutic sequences, middle four bars, lysogen are inserted for 1 h at the start of the first growth period of each element of the therapeutic sequence and 4 h at the end of the second growth period of the same element of therapeutic sequence. Similarly, for third set of sequences (last four bars), lysogen are inserted for 1 h in the start of the first growth period of each element of the therapeutic sequence and 4 h at the start of second growth period of the same element of therapeutic sequence. Each set contains four possible sequences namely lysogen with one prophage followed by two simultaneously, denoted by $\{1,2\}$ (black colour), three prophages each introduced separately, denoted $\{1,1,1\}$ (blue colour), two prophages simultaneously followed by one separately, $\{2,1\}$ (cyan colour), and all prophages simultaneously, $\left\{3\right\}$ (red colour).

Figure 4. Populations of CRISPR bacteria (C) against biofilm formation rate ${\mathrm{\Phi }}_0$ (subplots (a) and (c)) and adsorption rate β (subplots (b) and (d)) for 90-h therapeutic procedure. Four possible combinations of prophages are presented in each subplot, namely lysogen with one prophage followed by two simultaneously, denoted by $\{1,2\}$ (solid line), three prophages each introduced separately, denoted $\{1,1,1\}$ (dashed line), two prophages simultaneously followed by one separately, $\{2,1\}$ (dotted line), and all prophages simultaneously, $\left\{3\right\}$ (dash-dotted line).

Figure 5. Parameter space defined between the biofilm formation rate ${\mathrm{\Phi }}_0$ and the adsorption rate β for 90-h therapeutic procedure. Colour bar on the right shows the colours pattern for the population density of CRISPR bacteria. Dark blue is the least population density, i.e. 0.01, while bright yellow is the population density of CRISPR bacteria near carrying capacity. Biofilm formation rate ${\mathrm{\Phi }}_0$ is varied from 0.001 to 3 cells/cm²/h and adsorption rate is varied between ${10}^{-10}$ and ${10}^{-7}$. Four possible combinations of prophages are considered: (a) one prophage followed by two simultaneously, denoted by $\{1,2\}$ (solid line), (b) three prophages each introduced separately, denoted $\{1,1,1\}$ (dashed line), (c) two prophages simultaneously followed by one separately, $\{2,1\}$ (dotted line), and finally (d) introduction of a lysogen containing all three prophage sequences, $\left\{3\right\}$ (dash-dotted line).

Figure 6. Parameter space between the biofilm formation rate ${\mathrm{\Phi }}_0$ and the probability of gaining spacer $p_s$ for 90-h therapeutic procedure. Colour bar on the right shows the colour pattern for the population density of CRISPR bacteria. Dark blue is the least population density, i.e. 0.0005, while bright yellow is the population density of CRISPR bacteria near carrying capacity $5x{10}^6$. Four possible combinations of prophages are considered: (a) one prophage followed by two simultaneously, denoted by $\{1,2\}$ (solid line), (b) three prophages each introduced separately, denoted $\{1,1,1\}$ (dashed line), (c) two prophages simultaneously followed by one separately, $\{2,1\}$ (dotted line), and finally (d) introduction of a lysogen containing all three prophage sequences, $\left\{3\right\}$ (dash-dotted line).

Figure 7. Populations of CRISPR-immune bacteria at the final time after simultaneous insertion of lysogens with 1, 2, 3, 4 and 5 prophages. Each bar represents either population density of lysogen L, bacteria with CRISPR/Cas system C or phage V. The insertion phase is 5 h while the rest of the time is for the competence phase until the populations show a steady state for long enough time.

Figure A.1. Populations of CRISPR-immune bacteria (C) against probability of lysogeny $p_L$ at the final time t=180 h. Lysogen are introduced with one, two or three prophages at a time. Four possible combinations of prophages are considered for each procedure, namely one prophage followed by two simultaneously, denoted by $\{1,2\}$ (solid line), three prophages each introduced separately, denoted $\{1,1,1\}$ (dashed line), two prophages simultaneously followed by one separately, $\{2,1\}$ (dotted line), and finally the introduction of a lysogen containing all three prophage sequences, $\left\{3\right\}$ (dash-dotted line).

Table A1. Four strategies of therapeutic procedures are defined, namely constant phases, mixed phases, semi-mixed phases and separate phases.

	$\{1,2\}$	$\{1,1,1\}$	$\{2,1\}$	$\left\{3\right\}$
Constant phases	L${}_{001}$: I${}_P$,C${}_P$	L${}_{001}$: I${}_P$,C${}_P$	L${}_{011}$: I${}_P$,2C${}_P$	L${}_{111}$: I${}_P$,3C${}_P$
	L${}_{110}$: I${}_P$,2C${}_P$	L${}_{010}$: I${}_P$,C${}_P$	L${}_{100}$: I${}_P$,C${}_P$
		L${}_{100}$: I${}_P$,C${}_P$
Mixed phases	L${}_{001}$: I${}_P$,C${}_P$	L${}_{001}$: I${}_P$,C${}_P$	L${}_{011}$: I${}_P$,C${}_P$,I${}_P$,C${}_P$	L${}_{111}$: I${}_P$,C${}_P$,I${}_P$,
	L${}_{110}$: I${}_P$,C${}_P$,I${}_P$,C${}_P$	L${}_{010}$: I${}_P$,C${}_P$	L${}_{100}$: I${}_P$,C${}_P$	C${}_P$,I${}_P$,C${}_P$
		L${}_{100}$: I${}_P$,C${}_P$
Semi-mixed phases	L${}_{001}$: I${}_P$,C${}_P$	L${}_{001}$: I${}_P$,C${}_P$	L${}_{011}$: 2I${}_P$,2C${}_P$	L${}_{111}$: 3I${}_P$,3C${}_P$
	L${}_{110}$: 2I${}_P$,2C${}_P$	L${}_{010}$: I${}_P$,C${}_P$	L${}_{100}$: I${}_P$,C${}_P$
		L${}_{100}$: I${}_P$,C${}_P$
Separate phases	L${}_{001}$: I${}_P$	L${}_{001}$: I${}_P$	L${}_{011}$: 2I${}_P$	L${}_{111}$: 3I${}_P$,3C${}_P$
	L${}_{110}$: 2I${}_P$,3C${}_P$	L${}_{010}$: I${}_P$	L${}_{100}$: I${}_P$,3C${}_P$
		L${}_{100}$: I${}_P$,3C${}_P$

Notes: The therapeutic sequences are shown for three number of prophages in the table. I${}_P$ and C${}_P$ stand for the insertion phase and the competence phase. The symbol L${}_{ijk}$ represents lysogen with prophage index i, j and k (= 0 or 1). 0 means the absence of prophage, whereas 1 represents the presence of prophage.

Figure A.2. Colour combinations for the therapeutic procedure. Blue, cyan and black colours are used for the population of the first, second and third phage, respectively. The bacterial populations are based on the combinations of associated phages as follows. $L_{000}$ and $C_{000}$: Gray; $L_{001}$ and $C_{001}$: Blue; $L_{010}$ and $C_{010}$: green; $L_{011}$ and $C_{011}$: Cyan; $L_{100}$ and $C_{100}$: Red; $L_{101}$ and $C_{101}$: Pink;$L_{110}$ and $C_{110}$: Rose; $L_{111}$ and $C_{111}$: Black.

Figure A.3. Populations of bacteria with CRISPR/Cas systems (dash-dot lines), lysogens (solid lines) and phage (dash lines) during the mixed phases therapeutic procedure. The simulation times are $t=60$ h ($\mathrm{Ip}=5$ and $\mathrm{Cp}=15$), 180 h ($\mathrm{Ip}=15$ and $\mathrm{Cp}=45$) and 300 h ($\mathrm{Ip}=25$ and $\mathrm{Cp}=75$). The lysogens are inserted into the biofilm with up to three prophages either simultaneously or successively. The possible sequences of the number of prophages harboured in the lysogens are $\{1,2\},\{1,1,1\},\{2,1\}$ and $\left\{3\right\}$, as described in the text.

Figure A.4. Populations of bacteria with CRISPR/Cas systems (dash-dot lines), lysogens (solid lines) and phage (dash lines) during the semi-mixed phases therapeutic procedure. The simulation times are t=60 h ($\mathrm{Ip}=5$ and $\mathrm{Cp}=15$), 180 h ($\mathrm{Ip}=15$ and $\mathrm{Cp}=45$) and 300 h ($\mathrm{Ip}=25$ and $\mathrm{Cp}=75$). The lysogens are inserted into the biofilm with up to three prophages either simultaneously or successively. The possible sequences of the number of prophages harboured in the lysogens are $\{1,2\},\{1,1,1\},\{2,1\}$ and$\left\{3\right\}$, as described in the text.

Figure A.5. Populations of bacteria with CRISPR/Cas systems (dash-dot lines), lysogens (solid lines) and phage (dash lines) during the separate phase therapeutic procedure. The simulation times are t=60 h ($\mathrm{Ip}=5$ and $\mathrm{Cp}=15$), 180 h ($\mathrm{Ip}=15$ and$\mathrm{Cp}=45$) and 300 h ($\mathrm{Ip}=25$ and $\mathrm{Cp}=75$). The lysogens are inserted into the biofilm with up to three prophages either simultaneously or successively. The possible sequences of the number of prophages harboured in the lysogens are $\{1,2\},\{1,1,1\},\{2,1\}$ and $\left\{3\right\}$, as described in the text.

Figure A.6. Populations of bacteria with CRISPR/Cas systems (dash-dot lines), lysogens (solid lines) and phage (dash lines) during the constant phase therapeutic procedure. The simulation times are t=60 h ($\mathrm{Ip}=5$ and $\mathrm{Cp}=15$), 180 h ($\mathrm{Ip}=15$ and$\mathrm{Cp}=45$) and 300 h ($\mathrm{Ip}=25$ and $\mathrm{Cp}=75$). The lysogens are inserted into the biofilm with up to three prophages either simultaneously or successively. The possible sequences of the number of prophages harboured in the lysogens are $\{1,2\},\{1,1,1\},\{2,1\}$ and $\left\{3\right\}$, as described in the text.

M. Ballyk, D. Jones, and H.L. Smith, The biofilm model of Freter: A review, in Structured Population Models in Biology and Epidemiology, P. Magal and S. Ruan, eds., Springer, Berlin, 2008, pp. 265–302.

 Google Scholar

R. Freter, H. Brickner, J. Fekete, M.M. Vickerman, and K.E. Carey, Survival and implantation of Escherichia coli in the intestinal tract, Inf. Immun. 39 (1983), pp. 686–703.

 PubMed Web of Science ®Google Scholar

A. Mašić and H.J. Eberl, Persistence in a single species CSTR model with suspended flocs and wall attached biofilms, Bull. Math. Biol. 74 (2012), pp. 1001–1026. doi:10.1007/s11538-011-9707-8.

 PubMed Web of Science ®Google Scholar

S.T. Abedon, Bacteriophage Ecology, Advances in Molecular and Cellular Microbiology, Cambridge University Press, Cambridge, UK, 2008.

 Google Scholar

S. Chibani-Chennoufi, A. Bruttin, M.L. Dillmann, and H. Brüssow, Phage-host interaction: An ecological perspective, J. Bact. 186 (2004), pp. 3677–3686. doi:10.1128/JB.186.12.3677-3686.2004.

 PubMed Web of Science ®Google Scholar

E. Kutter and H. Brussow, Phage ecology, in Bacteriophages: Biology and Applications, E. Kutter and A. Sulakvelidze, eds., CRC Press, Boca Raton, FL, 2004.

 Google Scholar

Q. Ali and L.M. Wahl, Mathematical modelling of CRISPR-Cas system effects on biofilm formation, J. Biol. Dyn. 11 (2017), pp. 264–284. doi:10.1080/17513758.2017.1314025.

 PubMed Web of Science ®Google Scholar

L.M. Childs, N.L. Held, M.J. Young, R.J. Whitaker, and J.S. Weitz, Multiscale model of CRISPR-induced coevolutionary dynamics: Diversification at the interface of Lamarck and Darwin, Evolution 66 (2012), pp. 2015–2029. doi:10.1111/j.1558-5646.2012.01595.x.

 PubMed Web of Science ®Google Scholar