Modeling

With Interactive Modelling iGEM Heidelberg provides a comprehensive set of tools that not only help to facilitate the implementation of PACE but also give an intuitive understanding of underlying mechanisms. To control highly complex processes such as PACE or PALE in a near-ideal way enables to exploit as much of it's potential as possible. The most important parameters were determined and examined with ODE systems, solved analytically or numerically, [stochastic and distributional] models. As far as possible the models are available online to make them accessible to anyone interested. When useful, a [tool for comparison of experimental data and the model] is available. In addition the Interactive modelling helps to monitor parameters that cannot be easily be interpreted from raw data, such as [] and combines different parameters to make useful statements about an experiment.

Modelling concentrations over multiple Lagoons

When transfer from one volume to the next is performed, new lagoon can be modelled with starting values calculated from the last lagoons end values. For each concentration from the previous lagoon \(c_{t}\), the concentration in the next lagoon \(c_{t+1}\) is calculated as $$ c_{t+1} = \frac{v_{t} }{v_{l} } \cdot c_{t} $$ with \(v_{l}\), the volume of a lagoon and \(v_{t}\), the volume that is transferred. If the transfered volume is spinned down before it is added to the new lagoon, the initial value for \(c_{P}\) is calculated this way. The initial concentration of uninfected E. coli is set to the initial cell density. Initial concentrations of infected and phage-producing E. coli are set to zero, because before the transfer, no phages are present in the new lagoon.

If the transfer volume is not spinned down, the concentration of infected and phage-producing E. coli are calculated, using the above formula. The initial concentration of uninfected E. coli is the calculated the same way, but the initial cell density is added.

In directed evolution the fitness should increase over time. A linear increase in fitness between to given values was implemented to show this. The problem with this approach is its basic assumption being that all phage-producing E. coli are infected by phages with the same fitness.

To make the model more plausible, a distribution of fitness was introduced. For a set of discrete fitness values each fitness values share of the phage-producing E. coli population is calculated.

That changes the equation for the change in the concentration of phage-producing E. coli to $$ \frac{\partial c_{P} (t)}{\partial t} = -k \cdot c_{u}(t) \cdot c_{P} (t) + \sum_{i = 0}^N f_{i} \cdot s_{i} \cdot \mu \cdot c_{p} (t) $$ The calculation is for \(N\) different fitness values \(f_{i}\) and their share of the total phage-producing E. coli population \(s_{i}\).

Numeric solutions

The problem described above is a system of four differential equations, of which two ( \(\frac{\partial c_{i} }{\partial t} \:, \: \frac{\partial c_{p} }{\partial t}\) ) are so called delayed differential equations. They contain a term that needs to be evaluated at a timepoint in the past \(t - t_{P}\). A custom script was used to solve the problem numerically, using the explicit Euler method.[Source!]

The basic idea is that from a point in time with all values and all derivatives values given, the next point in time can be calculated by assuming a linear progress between the two points. $$ f(t_{n+1}) = f(t_{n}) + (t_{n+1} - t_{n}) \cdot f'(t_{n}) $$ This is performed for \(c_{u}(t)\), \(c_{i}(t)\), \(c_{p}(t)\) and \(c_{P}(t)\) rotatory, to always have the needed values from \(t_{n}\) ready for \(t_{n+1}\).

To explore, how unprecise parameters and noise influence the outcome of the model, a mode was implemented, that adds gaussian noise to all parameters. It uses the function \(n\) that makes a value \(v\) noisy with a random parameter \(r\). $$ n(v) = \big(1 - 2r\big) \cdot \sigma_{G} \cdot \sigma_{v} \cdot v, \quad r \in (0, 1) $$ Here, \(\sigma_{G}\) is a factor that is the same for all \(v\), \(\sigma_{v}\) is specific for \(v\). This way, it is possible to have one parameter being noisier than another, while being able to tune the noise globally. [Results]

<code class="language-python">vl</code>

<code class="language-python">tl</code>

<code class="language-python">ceu0</code>

<code class="language-python">cp0</code>

<code class="language-python">epochs</code>

<code class="language-python">tsteps</code>

<code class="language-python">min_cp</code>

<code class="language-python">max_cp</code>

Fig: 1a Numeric solution calculated with explicit Euler approach

Logarithmic plot of the concentrations of all E. coli populations cE, uninfected E. coli ceu, infected E. coli cei, phage-producing E. coli cep and M13 phage cP

Fig: 1b Numeric solution calculated with explicit Euler approach

Non-logarithmic plot of the derivatives of concentrations of all E. coli populations cE, uninfected E. coli ceu, infected E. coli cei, phage-producing E. coli cep and M13 phage cP

Fig: 1c First derivative of concentrations calculated with explicit Euler approach

Logarithmic plot of the concentrations of all E. coli populations cE, uninfected E. coli ceu, infected E. coli cei, phage-producing E. coli cep and M13 phage cP

Fig: 2 Numeric solution for a range of values for \(t_{l}\) and for \(v_{t}\)

All combinations of setups for the two ranges were calculated. The number of epochs plotted is counted until either the phage titer is less than a minimal threshold (orange) or larger than a maximum threshold (blue)