# 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 drylab and built a strong model. We divided our project in three main parts:

1. Phage construction
2. Population dynamics in the tumor microbiome submitted to phage infection
3. Effect of the transformed bacteria on the cancer cell

We focused on the second point and tried to answer those three questions: what is the phage population dynamics ? What amount of protein of interest is produced? How does 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 were inspired by the Kuang/Beretta model and the numerous other phage population modelisations that followed.

In the differents models we used, we chose a bacterial population with a logistic growth (there is a maximum bacterial population N*) and a metabolic cost for the infected bacteria: their growth rate was divided by its increase in protein synthesis.

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

Table 1: Meaning, value and unit of parameters of the chronic model
Parameters Meaning Value (if fixed) Unit
I Number of infected bacteria / cells
B Number of infected bacteria / cells
p Number of proteins of interest produced / proteins
φ Number of free phages / phages
φi Average number of phages in a cell 413 [1] phages
kin Phage infection rate 5.4e-9 [1] bacteria-1.phage-1.h-1
kout Cell exit rate of phage 413 [1] phage.h-1
δ Bacteria death rate 0 (neglicted) cell.h-1
λmax Maximal bacteria growth rate 0.77 [2] bacteria.h-1
λi Infected bacteria growth rate / bacteria.h-1
λs Susceptible bacteria growth rate / bacteria.h-1
δφ Phage decay rate 0.074 [1] phage.h-1
jφ Phage production rate per phage 413 [1] phage.h-1
jp Protein production 413 [1] protein.h-1
A Impact of the number of phages in cell (and protein production) on cell growth /
frac(dI)(dt) = lambda_(i) I + k_(i n) varphi S - delta I
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_(s) = lambda_(max) / (1 + A ( j_(p) +j_(varphi) ) varphi_(i) ) frac(Nstar - (S + I))(Nstar)
lambda_(i) = lambda_(max) frac(Nstar - (S + I))(Nstar)

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.

Table 2: Meaning, value and unit of parameters of the lytic model
Parameters Meaning Value (if fixed) Unit
I Number of infected bacteria / cells
B Number of infected bacteria / cells
p Number of proteins of interest produced / proteins
φ Number of free phages / phages
φi Average number of phages in a cell 115 [1] phages
kin Phage infection rate 2.7e-8 [1] bacteria-1.phage-1.h-1
kout Cell exit rate of phage 115 [1] phage.h-1
δ Bacteria death rate 0 (neglicted) cell.h-1
τ Latency time before cell death after phage infection 0.7 [1] h-1
λmax Maximal bacteria growth rate 0.77 [2] bacteria.h-1
λi Infected bacteria growth rate / bacteria.h-1
λs Susceptible bacteria growth rate / bacteria.h-1
δφ Phage decay rate 0.072 [1] phage.h-1
jφ Phage production rate per phage 115 [1] phage.h-1
jp Protein production 115 [1] protein.h-1
A Impact of the number of phages in cell (and protein production) on cell growth /
frac(dI)(dt) = lambda_(i) I + k_(i n) varphi S - delta I
dS / dt = lambda_(s) S - k_(i n) varphi S - delta S - I/tau
d phi / dt = k_(out) I - delta_(varphi) varphi - k_(i n) varphi S
d p / dt = varphi_(i) j_(p) I
lambda_(s) = lambda_(max) / (1 + A ( j_(p) +j_(varphi) ) phi_(i) ) frac(Nstar - (S + I))(Nstar)
lambda_(i) = lambda_(max) frac(Nstar - (S + I))(Nstar)

[1] 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)
[2] Konrad Krysiak-Baltyn et al. Computational Modelling of Large Scale Phage Production Using a Two-Stage Batch Process. Pharmaceuticals. (April 2018)
[3] Corinne Rieul et al. Effect of bacteriophage M13 infection on phosphorylation of dnaK protein and others Escherichia coli proteins. European journal of biochemistry (June 1987)