Team:ETH Zurich/core/model

Infection Dynamics Model

Rapid Phage Selection in Liquid Media

A major bottleneck in phage therapy is the labor-intensive screening procedure needed to identify the most infectious phage that can be used to treat a pathogen of interest. In order to be applicable to the clinics, experts made us aware that it is crucial to be able to rapidly select the best variants from our phage library. Selection in liquid bacterial culture leads to the amplification of infectious phage variants. However, our model shows that small differences in starting conditions such as initial concentrations and first time of amplification of the infectious phages may lead to the shadowing of the most infectious phage variant (Fig. 1). In order to overcome this problem, long selection periods with constant optimal bacterial concentrations are carried out with our bioreactor.
<figure class="figure-center"> <img src="T--ETH_Zurich--model-phage-selection.jpg" alt="phage selection"> <figcaption> Figure 1: Phage variant selection in liquid bacterial culture. In liquid bacterial culture, amplification of phages takes place. After a short selection period, slightly higher starting conditions lead to a higher count of infectious phages compared to a better phage variant with lower starting conditions. A longer selection period gives the best variants time to outcompete the rest.</figcaption> </figure> <p>

The main challenges we face during the reactor’s operation are the determination of host, phage and nutrient concentrations over time. We opted for a deterministic model in order to better understand the reactor’s dynamics, as we are interested in the average population behaviour and we don't expect stochastic effects to take place with such high bacteria and phage counts. The model predicts future concentrations of all entities inside the reactor. This allows for the real time tuning of various parameters of the bioreactor such that optimal bacterial concentrations for phage selection are maintained. 

       <figure class="figure-center">
         <img src="T--ETH_Zurich--reactor_general.png" alt="reactor">
         <figcaption> Figure 2: Overview of the Reactor. The reactor consists of two compartments: the host compartment where the bacterial host is cultured in exponential phase with constant OD and the main compartment where the selection of the best variant from the phage library takes place.</figcaption>
       </figure>


The reactor consists of two compartments. In the host compartment, the host bacterium is grown. Its concentration is kept constant in order to assure that the host is growing in exponential phase which allows for optimal phage amplification. The host can be pumped into the main compartment, where it is brought into contact with the phage library and where the selection takes place. A constant influx of fresh host and outflux of old host and amplified phages assures two critical points. Firstly, after a sufficient amount of time, only the best phage variant will remain in the reactor. Secondly, because old hosts are constantly replaced with fresh hosts, the selection pressure that favors mutations in the host bacterium is eliminated.
We model the dynamics in the reactor for two different cases: formation of phages in bacteria (recombineering) or in cell-free systems (yeast and in vitro), which, while similar, need two different models. Here, we broadly describe the models, explaining how they differ for each approach. A more thorough and detailed version of these models can be downloaded <a class="a-link" href="https://static.igem.org/mediawiki/2019/c/c6/T--ETH_Zurich--model-equations.pdf">here.</a>


Ensuring Constant Exponential Bacterial Growth in the Host Compartment

The host compartment of the reactor provides fresh bacteria to the main reactor. A constant bacterial concentration ensures exponential growth, which is crucial for effective amplification of the phages. Bacterial growth is limited by two factors: the exhaustion of nutrients and the accumulation of toxins. Control of the flux that replaces old medium with fresh medium in the host compartment of the bioreactor allows to eliminate growth exhaustion due to the accumualtion of toxins and for the active control of available nutrient concentrations. Therefore, bacterial per-cell growth rate \(g\) [\(h^{-1}\)] is modeled as a function of nutrient concentration \(s\) [\(g/L\)] by the Monod equation [1]<a style="color: #ffffff; text-decoration:none;" href="#biblio-model">Monod J. The Growth of Bacterial Cultures. Annu Rev Microbiol. 1949; 3:371–394.</a>:

         \begin{equation}
         g(s) = \frac{g_{max} \cdot s}{ K_s + s}\\
         \end{equation}

The per-cell growth rate \(g\) is determined by two constants: \(g_{max}\) [\(h^{-1}\)], the maximal per-cell growth rate and \(K_s\) [\(g/L\)], the nutrient concentration at half-saturation of \(g_{max}\). The parameters values are listed in table 1. This function shows a linear relation between the nutrient concentration and the growth rate at low nutrient concentrations, and develops into a zero order function at high nutrient concentration where the growth rate saturates at the \(g_{max}\) value (Fig. 3).


       <figure class="figure-center">
         <img src="T--ETH_Zurich--model-growth-rate.png" alt="growth rate">
         <figcaption> Figure 3: Model of the Growth Rate of Bacteria According to the Monod equation. The growth rate is linearly dependent on the nutrient concentration for low concentrations and saturates for high nutrient concentrations at the maximal growth rate.</figcaption>
       </figure>



In order to ensure a constant bacterial concentration in the host compartment of the reactor, a system of ordinary differential equations is used to describe the concentrations of host and nutrients over time [2]<a style="color: #ffffff; text-decoration:none;" href="#biblio-model">Allen, R. J., & Waclaw, B. (2019). Bacterial growth: a statistical physicist's guide. Reports on progress in physics. Physical Society (Great Britain), 82(1), 016601. doi:10.1088/1361-6633/aae546</a>. All concentrations are written with capital letters. Explicit dependency on time, for both concentrations and parameters, is removed for better readability.
\begin{equation} \frac{\partial N}{\partial t} = - \gamma \cdot g(N) \cdot H + s_0 \cdot in - out \cdot N \\ \end{equation} The nutrient concentration \(N\) is depleted by the bacterial consumption of nutrients. This can be described as the product of:

  • the amount of nutrients needed to produce one bacterium \(\gamma\) [\(ug/baterium\)]
  • the number of newly forming bacteria \(g(N) \cdot H\)

Additionally, the nutrient concentration \(N\) is decreased by the outflux \(out\) of old medium with nutrient concentration \(N\) and increased by the influx \(in\) of fresh medium with nutrient concentration \(s_0\) into the compartment.
\begin{equation} \frac{\partial H}{\partial t} = g(N) \cdot H - \delta_{H} \cdot H - out \cdot H \\ \end{equation} The host concentration \(H\) increases with bacterial growth (dependent on nutrient concentrations according to the Monod function) and decreases with bacterial death \(\delta_H\) and outflux \(out\) of the reactor.
A solution that describes the steady-state concentrations can be found by setting this system of differential equations to zero. 


       \begin{equation*}
       0 = - \gamma \cdot g \cdot H + s_0 \cdot in - N \cdot out
       \end{equation*}

       \begin{equation*}
       0 = g \cdot H - \delta_{H} \cdot H - out \cdot H\\
       \end{equation*}

       \begin{equation}
       N^\star = \frac{K_s \cdot out}{g_{max} - out}
       \end{equation}

       \begin{equation}
       H^\star = \frac{s_0 - N^\star}{\gamma}
       \end{equation}

Since the host compartment is needed to provide fresh bacteria to the main compartment, it is crucial to assure that the provided bacteria are in a state that is optimal for infection. Our wet lab experiments showed us that that if the bacterial concentration was too high, phages couldn't lysed the bacteria. Through our experimental data, we determined the optimal host concentration for the best infection and maximum yield of phages. Fixing the steady state concentration of host to the determined value of \(2.4 \cdot 10^{8}\) \( bacteria/mL\) (corresponding to an OD600 of 0.3) allows to chose the parameter values for the influx to and outflux from the host compartment.


We implemented this model in python in order to simulate our reactor and be able to predict the future concentration of each entity. This allowed us to vary the temperature amongst other parameters to make sure our system would behave as expected and as desired. While running our experiments, we monitor the OD in the compartments in real time and thanks to the prediction function of our model, we could be able to modify the temperature and reduce fluctuations in host concentration. At the time of the wiki freeze, we are still implementing this feature.



Differences in the Models used for Production of Phages in Bacteria or In Vitro


For the recombineering approach, we only need to grow one host, the target pathogen. The yeast and in vitro approaches both make phages in a cell-free transcription-translation system. For these we need two different bacterial strains: the target pathogen as well as a host that can be infected by the wild type phage. Modified phages produced in vitro still possess the wild type T7 tail fiber because during in vitro assembly a phage genome can assemble with proteins expressed by different neighboring genomes. To exert a control on which tail fiber each genome is packaged with, we perform in vitro assembly with an excess of original tail fibers. This leads to the formation of phages with the modified genome packaged with the original tail fiber. After the infection of original host, the modified phages will finally be correctly assembled with the corresponding tail fiber proteins. Growing the two hosts independently in different reactors will allow us to better control their ratio in the main reactor. 


       <figure class="figure-center">
         <img src="T--ETH_Zurich--model-table.png" alt="table values">
         <figcaption> Table 1: Different parameters used in the model.</figcaption>
       </figure>


Prey-Predator Equations


The goal of the second and main component of the reactor is to infect the target pathogen with a variant from the phage library. Infectious phage variants will amplify exponentially and evolve to infect even more rapidly. A Lotka-Volterra prey-predator equation is used to model the interaction of phages with bacteria. The infection rate depends on the adsorption rate \(\alpha\), as well as the concentrations of bacteria and phages. The adsorption rate \(\alpha\) describes how likely it is that a phage is adsorbed to a bacterium per unit of time, taking into account the diffusion rate, the size of the bacterium and the affinity to the cell-surface receptor. The infection rate is modelled by mass action kinetics. Again, concentrations are written in capital letters and time dependency will be omitted.

       \begin{equation}
       infection\ rate = \alpha \cdot H \cdot P \\
       \end{equation}

In order to model the nutrient and host concentrations in the main compartment, ODEs similar to the ones used for the host compartment were established. Small adjustments for the influx and outflux as well as for the influence of the additional entities were made. Note that the host concentration increases due to the influx from the host compartment with a constant host concentration \(s_A\), and decreases due to the interaction with phages. For the nutrient concentration, the assumption was made that an infected host, whose machinery has been hijacked by a phage, consumes the same amount of nutrients as an uninfected host. The limiting factor for nutrient consumption is likely to be the same as for an uninfected host, namely how fast the bacterium can transport nutrient through its membrane. 

       \begin{equation}
       \frac{\partial H}{\partial t} = g(N) \cdot H - \delta_{H} \cdot H\\
       - \alpha \cdot H \cdot P \\
       + in_{HN} \cdot H_{reactorA} - out \cdot H
       \end{equation}
       \begin{equation}
       \frac{\partial N}{\partial t} = - \gamma \cdot g(N) \cdot (H +H_{OO} + H_{NN}) 
       + s_0 \cdot in_{LB} + s_A \cdot in_{HN} - N \cdot out\\
       \end{equation}


In addition, delay differential equations were developed in order to model the novel entities present in the main reactor, specifically the infected hosts and free phages [3]<a style="color: #ffffff; text-decoration:none;" href="#biblio-model">Stopar, D. (2008). Modeling bacteriophage population growth. In S. T. Abedon (Ed.), Bacteriophage Ecology: Population Growth, Evolution, and Impact of Bacterial Viruses, Advances in Molecular and Cellular Microbiology (pp. 389–414). chapter, Cambridge: Cambridge University Press.</a>. The time delay is necessary to account for the time span between the infection and the release of progeny phages. A lysis time \(l\) of 13.3 minutes for bacteriophage T7 was used [4]<a style="color: #ffffff; text-decoration:none;" href="#biblio-model">Heineman, R. H., & Bull, J. J. (2007). Testing optimality with experimental evolution: lysis time in a bacteriophage. Evolution; international journal of organic evolution, 61(7), 1695–1709. doi:10.1111/j.1558-5646.2007.00132.x</a>. This is congruent with our experiments.
The concentration of infected host increases at the rate of infection and decreases as the host gets lysed by the phages. The decrease is modeled by substracting the amount of infected host at one lysis time ago. In order to model the number of each type of phages, we also need to make a distinction between hosts infected by the wild type T7 (\(H_{OO}\)) and those infected by new phages (\(H_{NN}\)). The former will create wild type phages (\(P_{OO}\)) while the latter will create new phages (\(P_{NN}\)). The term \(e^{-l \cdot out}\) accounts for the loss of infected host due to the outflux. 

       \begin{equation}
       \frac{\partial H_{OO}}{\partial t} = \alpha \cdot H \cdot P
       - H_{OO}(t-l) \cdot e^{-l \cdot out} -out(t) \cdot H_{OO}
       \end{equation}

       \begin{equation}
       \frac{\partial H_{NN}}{\partial t} = \alpha \cdot H \cdot P
       - H_{NN}(t-l) \cdot e^{-l \cdot out} - out(t) \cdot H_{NN}
       \end{equation}

Free phage concentration is increased though host lysis which is proportional to the amount of infected cells at time \(t\ - l\) and the burst size \(\beta\) (number of phages produced when the bacterium lyses). It decreases when a phage is adsorbed to a bacterium. The influx of the library \(in_{LIB}\) also increases the concentration of free phages by a factor \(R\) representing the fraction of each phage type in the library. 

       \begin{equation}
       \frac{\partial P_{OO}}{\partial t} = \beta \cdot H_{OO}(t-l) \cdot e^{-l \cdot out}
       - \alpha \cdot H_T \cdot P_{OO}
       - out \cdot P_{OO} - \delta_P \cdot P_{OO} + in_{LIB} \cdot R_{POO}
       \end{equation}

       \begin{equation}
       \frac{\partial P_{NN}}{\partial t} = \beta \cdot H_{NN}(t-l) \cdot e^{-l \cdot out}
       - \alpha \cdot H_T \cdot P_{NN}
       - out \cdot P_{NN} - \delta_P \cdot P_{NN} + in_{LIB} \cdot R_{PNN}
       \end{equation}

Simultaneous infection of a single bacterium by multiple phages is possible, but will not lead to majors changes. The only effect it has is the additional loss of one or more phages by absorption. The burst size or lysis time are limited by the translation step, not the transcription, so the addition of a phage genome will not lead to a faster or higher multiplication. We model this by keeping the infection rate dependent on the concentration of all host (\(H_T\)), whether infected or not. 


       <figure class="figure-center">
         <img src="T--ETH_Zurich--model-graphs-host-infected-phage.png" alt="graph host infected phage">
         <figcaption> Figure 4: Graphs showing the modeled concentrations of various entities in our reactor.The leftmost graph shows the host concentration, it goes up then drops down due to the phages' infection. That's where the infected host concentration starts to increase (middle graph) and then decreases as bacteria get lysed. One lysis period after the infection, we can see on the rightmost graph that the phages concentration increase rapidely. At the end of the simulation, all bacteria have been lysed and the phage concentration is very high.</figcaption>
       </figure>



Difference between Recombineering and In Vitro

For the recombineering approach, we only have two types of phages, the wild type and newly recombined phages, possessing a new genome as well as new tail fibers. With phages produced in vitro (yeast approach), things are a bit more complicated as this production can create mixed phages whose genome doesn't match their tail fibers.


Bibliography

[1] Monod J. The Growth of Bacterial Cultures. Annu Rev Microbiol. 1949; 3:371–394.
[2] Allen, R. J., & Waclaw, B. (2019). Bacterial growth: a statistical physicist's guide. Reports on progress in physics. Physical Society (Great Britain), 82(1), 016601. doi:10.1088/1361-6633/aae546
[3] Stopar, D. (2008). Modeling bacteriophage population growth. In S. T. Abedon (Ed.), Bacteriophage Ecology: Population Growth, Evolution, and Impact of Bacterial Viruses, Advances in Molecular and Cellular Microbiology (pp. 389–414). chapter, Cambridge: Cambridge University Press.
[4] Heineman, R. H., & Bull, J. J. (2007). Testing optimality with experimental evolution: lysis time in a bacteriophage. Evolution; international journal of organic evolution, 61(7), 1695–1709. doi:10.1111/j.1558-5646.2007.00132.x
[5] Shao X, Mugler A, Kim J, Jeong HJ, Levin BR, Nemenman I (2017) Growth of bacteria in 3-d colonies. PLoS Comput Biol 13(7): e1005679.


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