Team:Heidelberg/Model/Medium Consumption
Modeling
Optimizations of PACE to reduce medium consumption
Inspired by this years' iGEM Goes Green initiative launched by the iGEM Team Dresden, we designed a model that has provided us and will provide the iGEM community with a versatile tool to minimize the amount of medium used in the process of operating Phage-Assisted Continuous Evolution (PACE) to develop enhanced proteins. With the help of this model, it is possible for everyone to calculate the appropriate amount of medium that is needed to operate a successful implementation by tuning all parameters. We also provide information about PACE applications conducted by others in the past to compare the values to.
Medium consumption is critical when it comes to the energy needed for an experiment especially because the medium has to be autoclaved, which is highly energy consuming. In a turbidostat, the cell density is held constant by adjusting the medium influx to the cell density. That means the growth of the cells is not affected, instead for a one new cell, emerging from proliferation, one is put to the waste. In a Chemostat, the cell density is controlled by adjusting the influx of an essential nutrient, which limits the growth of the culture to a certain cell density. This may cause the host cells to be less efficient in producing the proteins needed for PACE and replicating the phage genome. Additionally, there is a constant efflux from the chemostat to the lagoons, that compensates the growth of the cells.
iGEM Heidelberg used a turbidostat in all PACE experiments because we assumed that PACE works better when the E. coli cells are allowed to grow at their maximum rate under the given conditions. So their ability to produce phage as fast as possible is not impaired by any deprivation.
The flow rate through a turbidostat is described by $$ \frac{\partial V_{M} }{\partial t} = \Phi_{T} = \frac{log(2)}{t_{E} } \cdot V_{T} $$ The flow rate \(\Phi_{T}\) in has to be equal to the growth term normalized with the turbidostat size. Note that the total volume of the turbidostat is constant because the flow rate is the same for incoming and outgoing medium.
In case of fluctuations in the generation time of the E. coli cells it is crucial to have a buffer so that the turbidostat is not diluted when the culture grows slower. A buffer of 50 %, means \(b\) is set to \(1.5\). For a turbidostat, the volume can be calculated from the flow using $$ V_{T} = \Phi_{T} \cdot \frac{t_{E} }{log(2)} $$ For chemostats only the needed flow rate \(\Phi_{T}\) is calculated, the volume needs to be chosen so that the flow rate can be reached.
RN158 over 40 mlRN31 to 100 mlRN63 .
There are a variety of possible ways to estimate the ideal size of the lagoons, one is presented here. It is based on the sequence length and mutation rate. Alternatively, to adjust the size of the lagoons, it is possible to adjust the total duration of the experiment. But as that increases energy consumption for heating and stirring in addition to medium consumption, we decided to focus on the lagoon-size.
The size of the phage population \(N_{P}\) per lagoon is
$$
N_{P} = c_{P} \cdot V_{L}
$$
The total sequence length \(L_{T}\) is
$$
L_{T} = N_{P} \cdot L_{S}
$$
when \(L{S}\) is the length of one sequence. The number of mutations that occur during one generation \(N_{M}\) is
$$
N_{M} = L_{T} \cdot r_{M}
$$
Here \(r_{M}\) is the mutation rate. It is reported to be \(5.3 \cdot 10^{-7}\), or when increased by an induced mutagenesis plasmids \(5 \cdot 10^{-5}\) RN63 .
The number of possible n-fold mutants of a sequence with length sl can be calculated by
$$
N_{n} = 3^{n} \cdot r_{M} \cdot \frac{L_{S}!}{(L_{S} - n)!}
$$
as there are three possibilities for each basepair to be exchanged to and with each additional mutation there is one possible position less. The number of n-fold mutants that can occur in a lagoon can be calculated using
$$
M_{n} = N_{P} \cdot (r_{M} \cdot L_{S})^n
$$
therefore, the required lagoon volume is
$$
V_{L} = \frac{N_{n} \cdot t}{M_{n} }
$$
With a theoretical coverage factor of the n-fold mutants of \(t\).
Interactive Medium Consumption Model
Medium consumption model
PACE usually consumes an extraordinary amount of medium per experiment. This is due to the need for a continuous supply of host cells with a constant cell density. This can be achieved either by using a turbidostat or a chemostat. Here we provide a tool to both the community and ourselves to calculate medium consumption based on different tunable parameters of PACE. We also want to gain an understanding of how we can reduce the amount of medium needed for an experiment.Medium consumption is critical when it comes to the energy needed for an experiment especially because the medium has to be autoclaved, which is highly energy consuming. In a turbidostat, the cell density is held constant by adjusting the medium influx to the cell density. That means the growth of the cells is not affected, instead for a one new cell, emerging from proliferation, one is put to the waste. In a Chemostat, the cell density is controlled by adjusting the influx of an essential nutrient, which limits the growth of the culture to a certain cell density. This may cause the host cells to be less efficient in producing the proteins needed for PACE and replicating the phage genome. Additionally, there is a constant efflux from the chemostat to the lagoons, that compensates the growth of the cells.
iGEM Heidelberg used a turbidostat in all PACE experiments because we assumed that PACE works better when the E. coli cells are allowed to grow at their maximum rate under the given conditions. So their ability to produce phage as fast as possible is not impaired by any deprivation.
The flow rate through a turbidostat is described by $$ \frac{\partial V_{M} }{\partial t} = \Phi_{T} = \frac{log(2)}{t_{E} } \cdot V_{T} $$ The flow rate \(\Phi_{T}\) in has to be equal to the growth term normalized with the turbidostat size. Note that the total volume of the turbidostat is constant because the flow rate is the same for incoming and outgoing medium.
Table 1: Additional Variables and Parameters used in the numeric solution of the model. List of all paramters and variables used in the analytic solution of this model and in the corresponding interactive webtool
| Symbol | Value and Unit | Explanation |
|---|---|---|
| \(V_{T}\) | [ml] | Volume of Turbidostat |
| \(V_{M}\) | [ml] | Volume of Medium consumed |
| \(t_{E} \) | [min] | E. coli generation time |
| \(\Phi_{T}\) | [ml/h] | Flow rate through turbidostat |
| \(t_{max}\) | [min] | Duration of the experiment |
Minmal Turbidostat Volume
Larger turbidostats or chemostats with a larger flow needs more medium for the same duration than smaller ones. When working with the minimal required volume or flow you can save medium and thus also energy. The minimal flow that is required can be calculated using $$ \Phi_{T} = b \cdot V_{L} \cdot N_{L} \cdot \Phi_{L} $$ which is simply the product of the lagoon volume, count and flow rate and a safety factor \(b\).In case of fluctuations in the generation time of the E. coli cells it is crucial to have a buffer so that the turbidostat is not diluted when the culture grows slower. A buffer of 50 %, means \(b\) is set to \(1.5\). For a turbidostat, the volume can be calculated from the flow using $$ V_{T} = \Phi_{T} \cdot \frac{t_{E} }{log(2)} $$ For chemostats only the needed flow rate \(\Phi_{T}\) is calculated, the volume needs to be chosen so that the flow rate can be reached.
Table 2: Additional Variables and Parameters used for this calculation. List of all additional paramters and variables used in the analytic solution of this model and in the corresponding interactive webtool
| Symbol | Value and Unit | Explanation |
|---|---|---|
| \(V_{L}\) | [ml] | Volume of Lagoons |
| \(N_{L}\) | Number of Lagoons | |
| \(\Phi_{L}\) | [ml/min] | E. coli generation time |
| \(b\) | \(1.5\) | Buffer |
Minimal Lagoon Volume
Obviously smaller lagoons require smaller turbidostats or chemostats with a lower flow rate and are therefore saving medium. However, there is a lower limit to lagoon size, if the phage population is too small, the sequence space that can be covered is insufficient to find variants that are better than previous ones. Lagoon sizes used by other vary from 15 mlTable 3: Additional Variables and Parameters used for this calculation. List of all additional paramters and variables used in the analytic solution of this model and in the corresponding interactive webtool
| Symbol | Value and Unit | Explanation |
|---|---|---|
| \(N_{P}\) | [pfu] | Amount of phages per lagoon |
| \(c_{P}\) | [pfu/ml] | Phage concentration |
| \(L_{S}\) | [bp] | Sequence length in basepairs |
| \(L_{T}\) | [bp] | Total sequence length in basepairs in lagoon |
| \(N_{M}\) | Number of mutations | |
| \(r_{M}\) | [1/generation] | Number of mutated basepairs per basepair per generation |
| \(n\) | [bp] | Number of mutated basepairs |
| \(M_{n}\) | Number of real sequences with \(n\) mutations | |
| \(N_{n}\) | Number of possible sequences with \(n\) mutations | |
| \(t\) | Theortical coverage of double |
Practice
The PACE experiments we carried out ranged from 3 to 7 days with a turbidostat volume of about 1.2 l. We observed E. coli generation times of 30 to 40 min.
Comparison of iGEM Heidelbergs PACE experiments to publishe experiments
A setup representative for the experiments conducted by iGEM Heidelberg is predicted to consume about 160 l of medium which fits the actual amount of medium used. The setup included a turbidostat volume of 1.25 l, an E. coli generation time of 40 min and was calculated with our interactive webtool for 5 days.
Since the goal was to improve the method reproducibility was prioritized over minimizing the medium consumption and we always used lagoon volumes of 100 ml.
The turbidostat volume was then adjustet to the number of lagoons.
