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.

Comparing the plots for 20 % and 100 % fitness show that a higher fitness increases the final phage titer and decreases the amount of uninfected E. coli earlier.

Modeling 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 modeled. 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 \(f\).

Bacterial Growth

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 modeled by exponential growth or by logistic growth. Especially, when long durations per lagoon are modeled, the logistic growth model is more exact. [source]. In the exponential case the growth rate \(g_{e}\) is modeled 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 modeled 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.

Transitions between uninfected, infected and phage-producing

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.

Phage propagation

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, \(c_{i}\). When the fitness is assumed to be the same for all phages, it is modeled to be constant during the time in one lagoon.

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 python script was used to solve the problem numerically, using the explicit Euler method. 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 \(t_{n+1}\).

Fitness distributions

If a fitness distribution is assumed, the fitness does not necessarily stay constant in one lagoon. The distribution is initialized with 100 % of the phages having the starting fitness. First changes in the fitness distribution occur after the first E. coli start to release phages.
The fitness distribution is modeled to change by mutation and selection. The selection is implemented in the following equation: $$ s_{i}(t_{n + 1}) = f_{i}(t_{n}) \cdot s_{i}(t_{n}) $$ \(f_{i}\) is one of \(N\) fitness values, \(s_{i}\) is the share of phages with that fitness value relative to the total phage population. Since the fitness is interpreted as a percentage of the wildtype fitness in this model, the new shares can be calculated als the product of the fitness value \(f_{i}\) and its old share \(s_{i}(t_{n})\). The resulting values are then normalized by divsion with $$ \sum_{i = 0}^{N} f_{i}(t_{n + 1}) $$ in order to always have shares that sum up to 1.
The mutation is implemented as a function that manipulates the fitness shares and applied to the fitness shares after selection. Here, \(r_{M}\) is the mutation rate, that means the amount of the fitness share \(i\) that changed during the mutation. \(w\) is the width of a fitness bin: $$ w = \frac{1}{N - 1} $$ The new shares are calculated as $$ s_{i}(t_{n + 1}) = \big(1 - r_{M}\big) \cdot s_{i}(t_{n}) + r_{M} \sum_{i = 0}^{N} \int_{f_{i} - \frac{1}{2} \cdot w}^{f_{i} + \frac{1}{2} \cdot w} \mathcal{N}_{\mu=f_{i}, \: \sigma}(x) \: dx $$ Special cases:
For \(f_{0}\) the integral is changed to $$ \int_{- \infty }^{f_{0} + \frac{1}{2} \cdot w} \mathcal{N}_{\mu=f_{i}, \: \sigma}(x) \: dx $$ For \(f_{0}\) the integral is changed to $$ \int_{f_{N} - \frac{1}{2} \cdot w}^{\infty} \mathcal{N}_{\mu=f_{i}, \: \sigma}(x) \: dx $$ to have the integrals over the normal distribution add up to 1. The normal distribution \(\mathcal{N}\) is used to model that many mutations have only small effects on the fitness, while some have a larger effect. \(\sigma\) is a parameter of the model that influences how much the new scores differ from the previous ones. In this model it is theoretically possible that mutation turns any given fitness into any other fitness, however small values for can \(\sigma\) prevent this.

When the the concentrations are modeled over a longer period of time the fitness of the wildtype is reached. To more accurate model evolution in which the wildtype fitness is never reached, \(\mu_{max}\) can be set to a smaller value. This value will be reached by most of the phages, if enough time is given. Finally almost no phages with the lowest fitness can be found in the lagoon.

Since most mutations decrease fitness in reality, a symmetric function like the normal distributions density function may not be the best choice. Consequently we also implemented a mutation function with a skew normal distribution \(\mathcal{S}\), that replaces \(\mathcal{N}\). It has an additional parameter \(\alpha\) that modulates the skewness. $$ s_{i}(t_{n + 1}) = \big(1 - r_{M}\big) \cdot s_{i}(t_{n}) + r_{M} \sum_{i = 0}^{N} \int_{f_{i} - \frac{1}{2} \cdot w}^{f_{i} + \frac{1}{2} \cdot w} \mathcal{S}_{\mu=f_{i}, \: \sigma, \: \alpha}(x) \: dx $$ Special cases:
For \(f_{0}\) the integral is changed to $$ \int_{- \infty }^{f_{0} + \frac{1}{2} \cdot w} \mathcal{S}_{\mu=f_{i}, \: \sigma, \: \alpha}(x) \: dx $$ For \(f_{0}\) the integral is changed to $$ \int_{f_{N} - \frac{1}{2} \cdot w}^{\infty} \mathcal{S}_{\mu=f_{i}, \: \sigma, \: \alpha}(x) \: dx $$ to have the integrals over the normal distribution add up to 1.
The equation for \(\frac{\partial c_{P} (t)}{\partial t}\) is changed into the following to incorporate the distributed fitness. $$ \frac{\partial c_{P} (t)}{\partial t} = -k \cdot c_{u}(t) \cdot c_{P} (t) + \sum_{i = 0}^N f_{i} \cdot s_{i} \cdot \mu \cdot c_{p} (t) $$

Other assumptions for the mutation function are possible, for example a function that randomly samples from a normal distribution but is biased towards mutations that decrease fitness.
Since this model of fitness and mutation is heavily abstracted it is most likely limeted to qualitative statements. Still it provides an intuitive understanding on how directed evolution can work and illustrates the role of selection and mutation. If the model without mutation (Fig. 3) is compared to one with mutation and selection (Fig. 6) role of mutation, diversificating the pool of phages can be seen. When comparing a model without selection (Fig. 9) to the one with mutation and selection (Fig. 6) the role of selection can be described as channeling the variants caused by mutation into a specific direction.

Effects of noise

Errors in the parameters of the model can be due discrepancies of the modeled experiments conditions and the conditions under which parameters were determined. To estimate how problematic the resulting errors of the model are, we compared the model to data that was obtained from our other experiments.
To make general statements on how sensitive the model reacts to noise in its parameters, a function was introduced that adds adjustable gaussian noise to all parameters. It was scaled relative to each parameters value. The function \(n(v)\) returns the noisy value of \(v\), the random argument \(r\) is sampled from a uniform distribution over \([0, 1]\). $$ n(v) = v + \big(1 - 2r\big) \cdot \sigma_{G} \cdot \sigma_{v} \cdot v, \quad r \in [0, 1] $$ \(\sigma_{G}\) is a global factor that is the same for all \(v\), \(\sigma_{v}\) is specific for \(v\). This makes it possible to have one parameter being noisier than another, while being able to tune the noise globally.
When different levels of noise are added to the same model (Fig. 10), the comparison of predicted phage titer shows almost no difference. While the predicted E. coli titers show some differences, the total E. coli titer is similar between the different versions. That can hint at errors in the E. coli concentrations cancelling each other out.
The model seems to be sufficiently robust against random errors. However systematic errors in the parameters can still have a large effect on predicted phage and E. coli titers.

Modeling concentrations over multiple Lagoons

PREDCEL is based on transferring phage from one lagoon into the next. This substitutes the constant supply with fresh medium, fresh E. coli and the constant dilution by a turbidostat. Because transferring decreases the phage by the relation of the transfer volume \(V_{t}\) to the lagoon volume \(v_{L}\), there is an increased risk of washout. This model is partly motivated by the goal to minimize that risk. The basic idea is to enable the user to react to a diminishig phage titer early enough while maintaining a reasonable level of monitoring.
In all following model versions, a distributional fitness, transfer of phage and E. coli and logistic growth are assumed. Due to their lower plausibility the alternatives are not shown, but are available as functions in the script.
When transfer from one volume to the next is performed, the new lagoon can be modeled 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 The calculation is for \(N\) different fitness values \(f_{i}\) and their share of the total phage-producing E. coli population \(s_{i}\).

Parameters used for the figures