Team:Heidelberg/SandboxMarita/Model/Mutagenesis Induction


Modeling
Mutagenesis Induction
Successful in vivo directed evolution by PREDCEL and PACE requires the thorough consideration of experimental para phage propagation times, culture dilution rates and inducer/inhibitor concentrations. We employed extensivmeters, e.g. phage propagation times, culture dilution rates and inducer/inhibitor concentrations. We employed extensive ODEblabla bland phage propagation required to tes and inducer/inhibitor concentrations. We employed extensive ODEblabla bland phage propagation required to express the required mutagenic polymerases. Finally, provide a web-based, fully intera phage propagation times, cultureexpress the and inducer/inhibitor concentrations. We employed extensive ODEblabla bland phage propagation required to express the required mutagenic polymerases. Finally, provide a web-based, fully intera phage propagation times, cultureon times, culture dilutployed extensivctive modeling platform, not only extensively employed by our wet lab, but highly informs future iGEM team phage propagation times, culture dilution rates and inducer/inhibitensivs building on our work.
An interactive webtool implementing the model described below is available.
Mutagenesis plasmids are crucial to enable rapid mutation that makes continuous evolution possible in short time scales. The mutagenesis plasmids we used have a \(P_{BAD}\) promotor that is arabinose inducible but suppressed by glucoseRN159. Consequently controlling the glucose concentration is important in order to have a strong induction of the mutagenesis plasmids that results in a high mutation rate, which leads to a larger covered sequence space. This model uses ordinaray differential equations (ODE) to model both the glucose and the E. coli concentration, assuming both are independend of each other. This is plausible because the medium used in the experiments contained other carbon sources than glucose. The glucose consumption rate per E. coli is assumed to be independent of the glucose concentration.
The arabinose concentration is easier to tune since higher arabinose concentrations induce the mutagenesis plasmids stronger and have no known negative side effects. This shifts the focus of the induction modeling to the glucose concentration.

Theory

Modeling the glucose concentration in a PACE experiments lagoon is based on a set of steady state equations while the model for PREDCEL has to take the development over time into account. The glucose concentration in the turbidostat \(c_{G_{T} }\) is increased with the incoming medium with a flow rate of \(\Phi\) and a glucose concentration of \(c_{G_{M} }\). It is decreased by with the medium that leaves the turbidostat with the same flow rate, but a glucose concentration of \(c_{G_{T} }\). Additionally E. coli with a concentration of \(c_{E}\) take up glucose at a rate of \(q\).
$$ \frac{\partial c_{G_{T} }(t)}{\partial t} = \Phi_{T} \cdot c_{G_{M} } - \Phi_{T} \cdot c_{G_{T} } - c_{E} \cdot q $$
In the case of a PACE experiment we can assume a dynamic equilibrium. That means that once a steady state is reached the concentrations stay the same and those concentrations are modeled here. First of all glucose is consumed in the turbidostat:
$$ \frac{\partial c_{G_{T} }(t)}{\partial t} = 0 $$
$$ \Leftrightarrow c_{G_{T} } = c_{G_{M} } - \frac{c_{E} \cdot q}{\Phi_{T} } $$
$$ \Leftrightarrow c_{G_{M} } (c_{G_{T} }) = c_{G_{T} } + \frac{c_{E} \cdot q}{\Phi_{T} } $$
When a lagoon with Volume \(V_{L}\) and a flowrate of \(\Phi_{L}\) is supplied by the turbidostat the glucose consumption in that lagoon can be modeled the same way. Because the E. coli titer, glucose concentration and flow rate into the lagoon are constant, a second steady state equilibrium can be assumed:
$$ c_{G_{L} } = c_{G_{T} } - \frac{c_{E} \cdot q}{\Phi_{L} } $$
In the context of PACE mutagenesis plasmids are induced in the lagoons which stops growth of E. coli, hence the E. coli titer is assumed to be the same as in the turbidostat. This simplifies the combination of both equations and results in the full equation
$$ c_{G_{L} } = c_{G_{M} } - \frac{c_{E} \cdot q}{\Phi_{T} } - \frac{c_{E} \cdot q}{\Phi_{L} } $$
$$ \Leftrightarrow c_{G_{M} } (c_{G_{L} }) = c_{G_{L} } + \frac{c_{E} \cdot q}{\Phi_{T} } + \frac{c_{E} \cdot q}{\Phi_{L} } $$
According to this the glucose concentration in the medium \(c_{G_{M} }\)has to be chosen so that it is the sum of the inteded glucose concentration in the lagoon \(c_{G_{L} }\) and the amount of glucose that is consumed in both the turbidostat with flow rate \(\Phi_{T}\) and the lagoon with flow rate \(\Phi_{L}\).

When a PREDCEL experiment is carried out the steady state assumption does not apply as the glucose in a flask is consumed over time, resulting in end concentrations that differ from the initial ones.
The glucose concentration in a flask \(c_{G_{F} }\) only changes by consumption since no glucose is added during PREDCEL. The consumption of glucose until a given timepoint equals to the integral over the E. coli concentration \(c_{E}\) multiplicated with the glucose consumption factor \(q\). That is the amount of glucose that is consumed by a given amount of E. coli during a given duration.
$$ \frac{\partial c_{G_{F} }(t)}{\partial t} = q \cdot \int_{t_{0} }^{t} c_{E}(t) \: dt $$
At any point in time the current glucose concentration is simply the difference between the intial concentration and the glucose that was already consumed.
$$ c_{G_{F} }(t) = c_{G_{F} }(t_{0}) - q \cdot \int_{t_{0} }^{t} c_{E}(t) \: dt $$
For the calculation of \(c_{E}\) both exponential growth and logistic growth were assumed in two different models. Especially for high E. coli concentrations the logistic growth model should be more precise since it takes into account that there is an upper limit for the E. coli concentration.
Exponential growth is defined as
$$ \frac{\partial c_{E}(t)}{\partial t} = \frac{ln(2)}{t_{E} } \cdot c_{E}(t) $$
Introducing this term into the above equation gives
$$ c_{G_{F} }(t)= c_{G_{F} }(t_{0}) -q \cdot \int_{t_{0} }^{t} c_{E}(t_{0}) \cdot exp\left(\frac{ln(2) \cdot t}{t_{E} }\right) dt $$
$$ c_{G_{F} }(t)= c_{G_{F} }(t_{0}) -q \cdot \int_{t_{0} }^{t} c_{E}(t_{0}) \cdot exp\left(\frac{ln(2) \cdot t}{t_{E} }\right) dt $$
$$ = c_{G_{F} }(t_{0}) - q \cdot c_{E}(t_{0}) \cdot t_{E} \cdot \left(exp\left(\frac{ln(2) \cdot t}{t_{E} }\right) - exp\left(\frac{ln(2) \cdot t_{0} }{t_{E} }\right)\right) $$
This results in the glucose starting concentration \(c_{G_{F} }(t_{0})\) needed to reach a concentration of \(c_{G_{f} }(t)\) afer a duration of \(t\) being calculated by
$$ c_{G_{F} }(t_{0}) = c_{G_{F} }(t) + q \cdot c_{E}(t_{0}) \cdot t_{E} \cdot \left(exp\left(\frac{ln(2) \cdot t}{t_{E} }\right) - exp\left(\frac{ln(2) \cdot t_{0} }{t_{E} }\right)\right) $$

Logistic growth is defined as
$$ \frac{\partial c_{E}(t)}{\partial t} = \frac{ln(2)}{t_{E} } \cdot \frac{c_{c} - c_{E}(t)}{c_{E}(t)} $$
The analytical solution to this gives
$$ c_{E}(t) = \frac{c_{c} }{1 + \big(\frac{c_{c} }{c_{E} (t_{0})} - 1\big) \cdot exp\big(-\frac{ln(2)}{t_{E} } \cdot t\big)} $$
If logistic growth is assumed, the term for \(c_{E}(t)\) is substituted by expression above. Here \(c_{c}\) is the capacity, the maximum concentration of E. coli under the present conditions.
$$ c_{G_{F} } (t) = c_{G_{F} } (t_{0}) - q \int_{t_{0} }^{t} \frac{c_{c} }{1 + \big(\frac{c_{c} }{c_{E} (t_{0})} - 1\big) \cdot exp\big(-\frac{ln(2)}{t_{E} } \cdot t\big)} \: dt $$
$$ = c_{G_{F} }(t_{0}) - \frac{q \cdot t_{E} \cdot c_{c} }{ln(2)} \cdot \Bigg(ln\bigg(c_{c} + c_{E}(t_{0})\cdot\Big(exp\big(\frac{ln(2)}{t_{E} } \cdot t \big) - 1\Big)\bigg) - ln\big(c_{c} \big)\Bigg) $$
This results in the glucose starting concentration \(c_{G_{F} }(t_{0})\) needed to reach a concentration of \(c_{G_{f} }(t)\) after a duration of \(t\) being calculated by
$$ c_{G_{F} }(t_{0}) = c_{G_{F} }(t) + \frac{q \cdot t_{E} \cdot c_{c} }{ln(2)} \cdot \Bigg(ln\bigg(c_{c} + c_{E}(t_{0})\cdot\Big(exp\big(\frac{ln(2)}{t_{E} } \cdot t \big) - 1\Big)\bigg) - ln\big(c_{c} \big)\Bigg) $$

Further calculations for simplification of entering data:
The glucose concentration in grams of dryweight is needed in order to work with the literature value of \(q\). It can be calculated from the optical density at a wavelength of \(\lambda=600 \: nm\) as
$$ c_{E. coli_{DW} } = c_{E. coli_{OD600} } \cdot 0.36 $$
according to Milo et al.Milo2009. The value of \(q\) is
$$ q = 0.183 \: g_{Glucose} \: g_{DW}^{-1} \: h^{-1} = 0.5083 \: g_{Glucose} \: OD600^{-1} \: h^{-1} $$
according to Neubauer et al.Neubauer2001. Because turbidostats are operated at a constant cell density, the flow rate \(\Phi\) can be calculated from the generation time \(t_{E}\) since the qualification of a turbidostat is that its dilution rate \(\Phi_{T}\) equals the growth rate of the culture.
$$ \Phi_{T} = \frac{ln(2)}{t_{E} } $$

Table 1: Variables and Parameters used for the calculation of the glucose and E. coli concentrations. List of all paramters and variables used in the analytical solution of this model and in the interactive webtool that is provided.

Symbol Value and Unit Explanation
\(c_{G_{T} }\) [g/ml] or [mmol/ml] Glucose concentration in Turbidostat
\(c_{G_{M} }\) [g/ml] or [mmol/ml] Glucose concentration in medium
\(c_{G_{L} }\) [g/ml] or [mmol/ml] Glucose concentration in lagoon
\(t\) [min] Time
\(\Phi_{T}\) [ml/min] Flow rate through Turbidostat
\(\Phi_{L}\) [ml/min] Flow rate through Lagoon
\(c_{E}\) [cfu/ml] or OD600 E. coli concentration
\(q\) \([g_{glucose} \: g_{DW}^{-1} h^{-1}]\) Glucose consumption by E. coliNeubauer2001
\(t_{E}\) [min] E. coli generation time

Practice

Many values can be taken from literature but since phage infection has an effect on the performance of E. coli, the maximum capacity for E. coli in our setup had to be determined with our conditions. For one accessory plasmdis the optical density was determined with and without phage infection over more then ten hours. Both cultures reached the maximum density after 510 min with an OD600 of 6.09 for the phage free culture and 5.133 for the infected culture. In the context of PREDCEL these values may serve as an estimation of the maximum E. coli capacity.
E. coli titer measurement for estimation of the E. coli capacity
The experiment was carried out in triplicates, the mean is plottet. If the optical density was higher than 1, samples were diluted and measured. The accessory plasmid (AP_dark) has a phage shock promotor under which gene III of the phage is expressed. The culture volume was 20 ml.

References