Modelling the spatiotemporal complexity of interactions between pathogenic bacteria and a phage with a temperature-dependent life cycle switch

Abstract

We apply mathematical modelling to explore bacteria-phage interaction mediated by condition-dependent lysogeny, where the type of the phage infection cycle (lytic or lysogenic) is determined by the ambient temperature. In a natural environment, daily and seasonal variations of the temperature cause a frequent switch between the two infection scenarios, making the bacteria-phage interaction with condition-dependent lysogeny highly complex. As a case study, we explore the natural control of the pathogenic bacteria Burkholderia pseudomallei by its dominant phage. B. pseudomallei is the causative agent of melioidosis, which is among the most fatal diseases in Southeast Asia and across the world. We assess the spatial aspect of B. pseudomallei-phage interactions in soil, which has been so far overlooked in the literature, using the reaction-diffusion PDE-based framework with external forcing through daily and seasonal parameter variation. Through extensive computer simulations for realistic biological parameters, we obtain results suggesting that phages may regulate B. pseudomallei numbers across seasons in endemic areas, and that the abundance of highly pathogenic phage-free bacteria shows a clear annual cycle. The model predicts particularly dangerous soil layers characterised by high pathogen densities. Our findings can potentially help refine melioidosis prevention and monitoring practices.

Figure 1

Schematic diagram explaining the two types of infection cycles of the phage (lytic and lysogenic) under the scenario of temperature-dependent lysogeny. At hot temperatures ($T>35$ °C), most phages follow a lytic cycle by killing the pathogen after the infection (shown by red dashed lines). At cooler temperatures ($T<35$ °C), infection by the phage mostly occurs via a lysogenic cycle where bacterial cells become phage-associated (shown by blue lines). However, an increase in the ambient temperature causes the bacteria-associated phages to enter the lytic state and lyse their hosts (shown by the red solid line).

Figure 2

(a) Temperature dependence of the adsorption constants $K_i$ ($i=1,2$) of the phage (measured in ${\text{ml}}^{-1}{\text{day}}^{-1}$). (b) Growth rates of susceptible $\alpha (T)$ and lysogenic $\overline{\alpha }(T)$ bacteria and the transition rate ${\lambda }_1(T)$ from the lysogenic cycle to the lytic cycle (measured in ${\text{day}}^{-1}$). The corresponding analytical expressions for the temperature dependence are given by (3)–(5).

Figure 3

Vertical distribution of the temperature across the soil (a) and daily temperature variation at a fixed depth of the soil (b) for the first day of April in a typical field in Nakhon Phanom province in Thailand predicted by the heat equation (2) using historical surface temperature data for the period of 2013–2016.

Figure 4

(a,b) Daily and seasonal temporal dynamics of bacteria and phage numbers within the upper 20 cm of the soil predicted by the model for Nakhon Phanom province in Thailand. Model parameters are taken from Table 1 as default values. The unit of the densities of bacteria and phages are cell/ml and phage/ml, respectively.

Figure 5

Vertical distribution of infected bacteria predicted by the model (a): in lytic $I_1$, (b): in lysogenic $I_2$, (c): phage P and (d): susceptible bacteria S in the soil across the day of April 1st predicted by the model calculated for Nakhon Phanom province. Model parameters are taken from Table 1 as default values. Note that the curves in (d) overlap for depths h>20cm. The unit of the densities of bacteria and phages are cell/ml and phage/ml, respectively.

Figure 6

Daily average densities of susceptible bacteria S within the upper 20 cm of the soil calculated for different values of carrying capacity $C_{\text{surf}}$ (Nakhon Phanom province): The corresponding values of $C_{\text{surf}}$ (measured in cell/ml) are provided in figures: (a) $C_{\text{surf}}=1\times {10}^6\text{cell}/\text{ml}$, $C_{\text{surf}}=2\times {10}^6\text{cell}/\text{ml}$, $C_{\text{surf}}=3\times {10}^6\text{cell}/\text{ml}$. (b) highly enriched environment, $C_{\text{surf}}=5\times {10}^6\text{cell}/\text{ml}$, $C_{\text{surf}}=1\times {10}^7\text{cell}/\text{ml}$, $C_{\text{surf}}=5\times {10}^7\text{cell}/\text{ml}$. The unit of the density of S is cell/ml.

Figure 7

Influence of the carrying capacity on the vertical distribution of susceptible bacteria S in the soil predicted by the model calculated for Nakhon Phanom province. The left panel shows vertical distributions in the top 60 cm whereas the right panel presents zooms of the same profiles near the surface. The corresponding values of $C_{\text{surf}}$ (measured in cell/ml) are provided in figures. The graphs show the spatial distributions predicted for April 1st. The unit of the density of S is cell/ml.

Figure 8

Bifurcation diagrams showing dynamical regimes in the model (Nakhon Phanom province) depending on the parameters $\mu$ (the morality rate of phages)-$C_{\text{surf}}$ (carrying capacity on the surface) and K (overall phage adsorption rate) and b (phage burst size). The classification of regimes A–C is the following. Regime “A” corresponds to periodic daily variations of species densities; for regime “B” species shows irregular oscillations; regime “C” signifies the extinction of phages. Other model parameters are set at default values.