# MODELS

Due to the pandemic situation of this iGEM edition, we were not able to do as many lab experiments as we would have liked to do, hence we decided to focus on the modeling and built a strong theoretical framework that could inform the experiments. There are two stages of our project that could be modeled: the dynamics of phage infection and the effects of the transformed bacteria on the cancer cells.

We focused on the former and we attempted to answer the three following questions: what is the phage population dynamics? How much of the protein of interest is produced (and how fast)? How do the infection and the protein production impact the bacterial population dynamics? This also allowed us to compare the different phages that could be used, or to decide if we employ phagemids.

We developed a compartmental model of the phage infection inspired by currently popular SIR-like models [1], the Kuang/Beretta [2] model, and the numerous other phage populations modelizations that followed. We call this model SIφ: bacteria are divided into two populations: susceptible (S) bacteria can be infected by the free phages (φ), and infected (I) cells can thus produce the protein of interest. We extended compartmental models by explicitly considering the cellular physiology and the interplay between the expression of an exogenous protein and cell growth, i.e. the fact that infected bacteria producing the (oncolytic) protein of interest may show reduced growth and thus carry a disadvantage with respect to the susceptible population.

We chose a bacterial population that grows following a logistic curve (there is a maximum size N^{*} of the bacterial population), and we introduced a phenomenological description of the metabolic cost of gene expression: as the growth rate of infected bacteria is supposed to decrease with increasing the rate of synthesis of the exogenous protein, we assumed that the growth rate is inversely proportional to the total rate of exogenous protein synthesis. Although this is surely an approximation, the shape of this monotonous function does not impact qualitatively the results.

The first type of infection that we wanted to simulate was **chronic infection** like phage M13: phages infect bacteria, they reproduce in them and can exit continuously without killing their host.

Parameters | Meaning | Value (if fixed) | Unit |
---|---|---|---|

I | Number of infected bacteria | / | cells |

S | Number of susceptible bacteria | / | cells |

p | Number of protein of interest produces | / | proteins |

φ | Number of free phages | / | phages |

φ_{i} |
Average number of phages in a cell | 413 [3] | phages |

N^{*} |
Load capacity | / | cells |

k_{in} |
Phage infection rate constant | 5.4e-9 [3] | bacteria^{-1}.phage^{-1}.h^{-1} |

k_{out} |
Cell exit rate of phage | 413 [3] | h^{-1} |

δ | Bacteria death rate | 0,01 | h^{-1} |

λ_{max} |
Maximal bacteria growth rate | 0.77 [4] | h^{-1} |

λ_{i} |
Infected bacteria growth rate | / | h^{-1} |

λ_{s} |
Susceptible bacteria growth rate | / | h^{-1} |

δ_{φ} |
Phage decay rate | 0.074 [3] | h^{-1} |

j_{φ} |
Phage production rate per phage | 413 [3] | phage.h^{-1} |

j_{p} |
Protein production | 413 [3] | protein.h^{-1} |

A | Impact of the number of phages in cell (and protein production) on cell growth | / |

`(dS) / dt = lambda_(s) S - k_(i n) varphi S - delta S`

`(d varphi) / dt = k_(out) I - delta_(varphi) varphi - k_(i n) varphi S`

`(d p) / dt = varphi_(i) j_(p) I`

`lambda_(i) = lambda_(max) / (1 + A ( j_(p) +j_(varphi) ) varphi_(i) ) frac(N^star - (S + I))(N^star)`

`lambda_(s) = lambda_(max) frac(N^star - (S + I))(N^star)`

The second infection strategy that we simulate is the strictly **lytic strategy**: phage infect bacteria, reproduce in them but instead of exiting continuously like previously there is a burst size. When there are enough phages in a bacteria they produce lysine and kill their host going out of it.

Parameters | Meaning | Value (if fixed) | Unit |
---|---|---|---|

I | Number of infected bacteria | / | cells |

S | Number of susceptible bacteria | / | cells |

p | Number of proteins of interest produced | / | proteins |

φ | Number of free phages | / | phages |

φ_{i} |
Average number of phages in a cell | 115 [3] | phages |

N^{*} |
Load capacity | / | cells |

k_{in} |
Phage infection rate constant | 2.7e-8 [3] | bacteria^{-1}.phage^{-1}.h^{-1} |

k_{out} |
Cell exit rate of phage | 115 [3] | h^{-1} |

δ | Bacteria death rate | 0,01 | h^{-1} |

τ | Latency time before cell death after phage infection | 0.7 [3] | h |

λ_{max} |
Maximal bacteria growth rate | 0.77 [5] | h^{-1} |

λ_{i} |
Infected bacteria growth rate | / | h^{-1} |

λ_{s} |
Susceptible bacteria growth rate | / | h^{-1} |

δ_{φ} |
Phage decay rate | 0.072 [3] | h^{-1} |

j_{φ} |
Phage production rate per phage | 115 [3] | phage.h^{-1} |

j_{p} |
Protein production | 115 [1] | protein.h^{-1} |

A | Impact of the number of phages in cell (and protein production) on cell growth | / |

`(dS) / dt = lambda_(s) S - k_(i n) varphi S - delta S`

`(d varphi) / dt = k_(out) I - delta_(varphi) varphi - k_(i n) varphi S`

`(d p) / dt = varphi_(i) j_(p) I`

`lambda_(i) = lambda_(max) / (1 + A ( j_(p) +j_(varphi) ) phi_(i) ) frac(N^star - (S + I))(N^star)`

`lambda_(s) = lambda_(max) frac(N^star - (S + I))(N^star)`

## NUMERICAL RESOLUTION OF EQUATIONS

The resolution was coded in python language using the scipy package. The function finally applied was solve_ivp with the LSODA method. This is a function already implemented in python in order to numerically integrate a system of ordinary differential equations given initial values.

We are interested in the steady-state values of the populations S and I and in the production of the protein p, at different values of the metabolic cost (represented by the parameter A) or of the protein synthesis rate (jp).

We fixed the load capacity to an arbitrary value N^{*}(2 x 10^{6}), the initial number of susceptible bacteria (S0 = 10^{3}, I0 = 1), and of phages injected (1e4). The death rate of bacteria is also fixed to be small but non zero (otherwise the population will always reach the saturating value ^{*}); we could not find any value in the literature for bacteria in a tumor microbiome, and we suspect that it will strongly depend on the type of bacteria and the type of tumor.

### Chronic Model

We start analyzing the **chronic** model. For the parameter values that we could find in the literature (see table), if the protein burden is neglected (A=0) the saturation value ^{*} is quickly reached and the total population stays infected. This is reflected by a constant production rate of the protein of interest (the derivative in p(t) is constant as it is proportional to I).

Instead, if A is small (A = 0.005 in Fig. 4), S reaches a lower steady state because of the tradeoff between gene expression and cellular growth. As a consequence, the exogenous protein is expressed at a lower rate.

Finally, if the cost of gene expression is too high (A=1), the infection kills the population as death outruns growth (fig 5). Thus, the protein of interest will be produced for a limited time-window.

Similar results can be obtained if, instead of varying the metabolic cost A, we vary the production rate of the protein of interest j_{φ}.

**The model thus informs us that promoters and RBSs**, which determine the expression levels, will **have an impact on the dynamics of the bacterial population**. According to the desired output one might want to choose strong or weak promoters/RBSs. In some cases we would like to have a long and continuous production of the desired protein, in other cases, we could opt for a shorter and more controlled production of the oncolytic proteins.

Although we mainly focused on the analysis of the chronic model, we observed an interesting oscillatory behavior in the population dynamics of S and I in the **lytic model**. The cost of gene expression still has a quantitative impact on the populations S and I (see figs 6.Left/Middle) but we now observed a strong dependence on the load capacity N^{*} (fig 6.Right).

### Lytic Model

In the lytic infection, thus we noticed two types of dynamics depending on the values of N*and S0. We observed a regime with oscillations, stabilizing at long times to a steady-state value depending on the metabolic cost A. If the population is small, those oscillations are repressed and the system more rapidly finds the steady-state.

## CONCLUSION/DISCUSSION

The results allow us to say that in both lytic and chronic strategies, the metabolic cost of the infection by phages has an impact on the bacteria's survival. That means that its impact also interferes in the competition between the two populations of infected and susceptible and should not be neglected. The choice of the RBS and promoter is extremely important for the success of the Phagent therapy. We need it to last in the time and we saw that the infected bacteria all die if the metabolic cost is too high. For the lytic strategy, the size of the target bacteria population needs to be taken into account in order to reach a steady-state.

Thanks to the modelization we have been able to study the behavior of phages and bacteria whereas we had no time to do it in the lab. However, our models are still to be discussed on many points. First of all the bacteria death is almost neglected which is not representative, even more in the chronic model as it means the bacteria cannot die in this model, they reach the load capacity and it is over. Moreover, we work on an average number of phages per cell which is not really representative as each bacteria does not count in reality the same number of phages. It can depend on its localization for example between bacteria that are between cancer cells and bacteria that are in the tumor cells. Finally the form of the metabolic cost is not optimal: another equation more elaborated could be used in order to see and study more the impact of the phage infection on the bacteria metabolism, growth and death.

##### References

[1] “Comportemental models in epidemiology” (last edition 2020), the 25th of October 2020.

[2] E. Beretta, Y. Kuang. Modeling and analysis of a marine bacteriophage infection. Math Bioscience. (1998)

[3] Marianne De Paepe et al. Viruses’ Life History: Towards a Mechanistic Basis of a Trade-Off between Survival and Reproduction among Phages. PLoS Biol (July 2006)

[4] Konrad Krysiak-Baltyn et al. Computational Modelling of Large Scale Phage Production Using a Two-Stage Batch Process. Pharmaceuticals. (April 2018)