While developing PREDCEL in the lab, we simultaneously developed it in silico so that both sides could benefit from each other. One of the most important parameter of phage assisted directed evolution experiments like PREDCEL and PACE is the phage titer itself. If the phage titer drops washout can occur and the experiment has to be restarted with the disadvantage of loosing library complexity. If the phage titer increases too much, the multiplicity of infection (MOI), that means the amount of phage relative the the amount of E. coli rises too. If fore example the MOI is 10 and an E. coli can only be infected by one phage, nine out of ten phages will not infect an E. coli and thus will not evolve, but still make up most of the phage population.

Modelling concentrations in one Lagoon

Here the concentrations \(c\) of uninfected E. coli, infected E. coli and phage producing E. coli as well as the M13 phage are modelled. They are denoted with the subscripts \(_{u}\), \(_{i}\), \(_{p}\) and \(_{P}\). If the whole E. coli population is referred to, \(c_{E}\) is used. If an arbitrary E. coli population is meant, the subscript \(_{e}\) is used. The phage concentration \(c_{P}\) refers to the free phage only, phage that are contained in an E. coli they infected are not included. The used parameters include the time \(t\), the affinity of phage for E. coli \(k\), the duration between infection of an E. coli and the first phage leaving the E. coli \(t_{P}\). The three different E. coli populations each have a generation time \(t\) that is denoted with their subscript. The fitness of a phage population is \(f\). Each term describing the change of an E. coli concentration contains its growth, \(g_{e}\). The growth rate of an E. coli population can be modelled by exponential growth or by logistic growth. Especially, when long durations per lagoon are modelled, the logistic growth model is more exact. [source]. In the exponential case the growth rate \(g_{e}\) is modelled as $$ g_{e} (t_{e}) = c_{e} \cdot \frac{log(2)}{t_{e} } $$ Note that the growth rate in the model increases over time, while in the modelled culture, the nutrient concentration decreases. That makes the logistic model more plausible, it models \(g_{e}\) as $$ g_{e} (t_{e}, \: c_{e}(t), \: c_{c}) = \frac{c_{c} - c_{e} (t)}{c_{c} } \cdot \frac{log(2)}{t_{e} } $$ In this case the learning rate decreases as the current concentration \(c_{e}\) approaches the maximum capacity for E. coli in the given setup \(c_{c}\). With this model \(c_{e} \leq c_{c}\) is true for any point in time. Change of concentration of uninfected E. coli, \(\frac{\partial c_{u} }{\partial t} \: [cfu/min]\) $$ \frac{\partial c_{u} }{\partial t}(t) = g_{u} (t_{u}, \: c_{u}(t), \: c_{c}) - k \cdot c_{u}(t) \cdot c_{p}(t) $$ In addition to the growth term, the concentration of uninfected E. coli is described by a term for infection that takes into account the concentration of uninfected E. coli and the concentration of free phage and reduces the conentration of uninfected E. coli. Change of concentration of uninfected E. coli, \(\frac{\partial c_{i} }{\partial t} \: [cfu/min]\) $$ \frac{\partial c_{i} }{\partial t}(t) = \begin{cases} g_{i} (t_{i}, \: c_{i}(t), \:c_{c}) + k \cdot c_{i}(t) \cdot c_{p}(t) - c_{i}(t - t_{P}), \quad \text{for} \: t > t_{P} \\ g_{i} (t_{i}, \: c_{i}(t), \: c_{c}) + k \cdot c_{i}(t) \cdot c_{p}(t), \quad \text{otherwise} \end{cases} $$ Until \(t > t_{P}\) the concentration of infected E. coli increases by growth and infection of previouly uninfected E. coli. When \(t > t_{P}\), a third term describing that infected E. coli turn into phage-producing E. coli is subtracted. Change of concentration of phage producing E. coli, \(\frac{\partial c_{p} }{\partial t} \: [cfu/min]\) $$ \frac{\partial c_{p} }{\partial t}(t) = \begin{cases} g_{p} (t_{p}, \: c_{p}(t), \: c_{c}) - c_{i}(t - t_{P}), \quad \text{for} \: t > t_{P} \\ g_{p} (t_{p}, \: c_{p}(t), \: c_{c}), \quad \text{otherwise} \end{cases} $$ The population of phage producing E. coli only increases by growth until \(t > t_{P}\). When infected E. coli drop their first phage they turn into producing E. coli as described by the second term. Change of concentration of M13 phage, \(\frac{\partial c_{P} }{\partial t} \: [cpu/min]\) $$ \frac{\partial c_{P} }{\partial t}(t) = c_{P}(t) \cdot \mu_{max} \cdot f - k \cdot c_{u}(t)\cdot c_{P}(t) $$ The phage concentration is only increased by phage that leave phage-producing E. coli, which happens at a rate of \(f \cdot \mu_{max}\) per time unit, with f being the fitness, a value between 0 and 1, equal to the share of the wildtype M13 phages fitness and \(\mu_{max}\) being the wildtype phages production rate. We assume that the only negative influence on the free phage titer is phage infecting E. coli, which depends on both the phage titer \(c_{P}\) and the titer of uninfected E. coli, \(c_{i}\). The fitness \(f\) is assumed to be constant during the time spent in one lagoon, it is assumed that all phages have the same fitness.