Team:Bielefeld-CeBiTec/Modeling/CFPS

Modeling

Termination of protein synthesis

Transcription and translation are a common element of biomathematical models, but to describe cell-free protein synthesis, the special conditions have to be taken into account. Due to the dilution of the cell extract and a partial loss of activity, transcription and translation rates in vitro are typically one to two orders of magnitude lower than in vivo {Karzbrun 2011 #10}. The most important difference, however, are the limited resources in a batch-mode CFPS. Our experimental data show that the fluorescence signal reaches a plateau after approximatly ?? minutes. This can have various reasons, for example depletion of energy resources or amino acids {Jewett 2004 #32}, degradation of ribosomes {Stögbauer #22}, accumulation of inhibitory byproducts {Kim 1999 #33} or an unfavorable pH shift {Kim 2008 #34}. Although the reasons can be manifold, several publications agree that translation is the limiting step, whereas transcription proceeds when protein synthesis has long stopped {Iskakova 2006 #35}{Stögbauer #22}. However, it is an important question whether translation stops because resources have been consumed by the translation reaction or due to a process that is independent of translation, such as the degradation of ribosomes. We performed several experiments to investigate this issue. The first was a spiking experiment to find out if sfGFP production can be reconstituted by adding fresh components once the plateau has been reached, but neither amino acids nor cell extract (??) had a significant effect. Next, we tested various plasmid concentrations in the CFPS reaction. If the reason for the termination of translation was the consumption of resources, one would expect that the final protein amount is the same regardless of the plasmid concentration. However, we observed a dependence of the protein amount on the plasmid concentration. Furthermore, the plateau was reached at approximately the same time, which indicated that the termination of protein synthesis is caused by a process that is not directly driven by translation. These results were underpinned by an experiment in which we added the plasmid after the other reaction components had already been incubated at 37 °C for a certain period of time. The longer the delay was, the less fluorescence was observed. This clearly showed that a process which proceeds independently of translation is responsible for the stop of protein synthesis. Consequently, we decided to include a species named "TL resources" that catalyzes the translation reaction and degrades over time {Stögbauer #22}. This virtual species comprises all components which are necessary for translation and its degradation could be caused by enzymatic activity as well as a pH shift. We found that the degradation can be best described by Michaelis Menten kinetics, which means that it is linear at first and slows down once the TL resources are mostly degraded.

$$\frac{d[\mathit{TL}\mathit{resources}]}{\mathit{dt}}=-\text{v\_deg\_TL}·\frac{[\mathit{TL}\mathit{resources}]}{\text{K\_deg\_TL}+[\mathit{TL}\mathit{resources}]}$$

Transcription and translation

When fitting parameters to our data, we noticed that it was not possible to obtain good fits for both high and low plasmid concentrations using mass action or Michaelis Menten kinetics for transcription and translation. However, we obtained good results when using Hill kinetics for both reactions. The Hill coefficients were empirically determined to be 2 for transcription and 3 for translation. Hill coefficients greater than one are generally interpreted as indicators of positive cooperativity. Although this might not be as intuitive as in the case of allosteric enzymes, it is not unreasonable to think of protein synthesis as a cooperative process. For example, when one ribosome is bound to an mRNA, it prevents the formation of secondary strucutes and thus facilitates the binding of a second ribosome to the same transcript {Iskakova 2006 #35}{Underwood 2005 #36}.
For the mRNA, a degradation constant of 0.08 min^-1 was chosen in accordance with an mRNA lifetime of 12 min, which has been measured in an E. coli cell extract {Karzbrun 2011 #10}. We found that it yielded better results to include an mRNA degradation term, even though our optimized CFPS contains an RNase E inhibitor. Furthermore, the degradation term prevents an unrealistic, unlimited increase of the mRNA concentration.

$$\frac{d[\mathit{reporter}\mathit{mRNA}]}{\mathit{dt}}=-\mathit{vTX}1·\frac{{[\mathit{operator}]}^2}{\mathit{KTX}{1}^2+{[\mathit{operator}]}^2}-\text{deg\_mRNA1}·[\mathit{reporter}\mathit{mRNA}]$$ $$\frac{d[\mathit{inactive}\mathit{reporter}]}{\mathit{dt}}=-\mathit{kTL}1·[\mathit{TL}\mathit{resources}]·\frac{{[\mathit{reporter}\mathit{mRNA}]}^3}{\mathit{KTL}{1}^3+{[\mathit{reporter}\mathit{mRNA}]}^3}-\mathit{kmat}·[\mathit{inactive}\mathit{reporter}]$$

Maturation of sfGFP

After its translation, sfGFP requires a maturation step in which the chromophore is converted into its active form. This leads to a delay until fluorescence can be measured and was thus incorporated into our model. According to the literature, this maturation takes approximately 5 minutes {Stögbauer #22}. GFP is a very stable protein and even in very long measurements, we observed only a very slight decrease of fluorescence intensity. Therefore, we did not include degradation terms for the reporter protein.

$$\frac{d[\mathit{reporter}]}{\mathit{dt}}=\mathit{kmat}·[\mathit{inactive}\mathit{reporter}]$$

Fitting

Since "TL resources" is a virtual species, we could only determine its initial concentration and its degradation rate by fitting to our data. We also optimized the Michaelis Menten constant of the translation to describe our data more accurately. To do this, we used the data sets which we had obtained by testing various plasmid concentrations and a delayed plasmid addition. In the first step, we estimated the sfGFP concentrations by comparing the fluorescence intensity to isolated GFP of known concentration. We then fitted three parameters and one concentration to the delayed plasmid addition data set using Matlab´s nlinfit (nonlinear least-squares problems) algorithm. The obtained Michaelis Menten constant of the TL resources´degradation was used for the model, while the other three values were fitted again to a portion of the data for various plasmid concentrations. We used this sequential fitting approach due to the day-to-day variations between experiments, which would otherwise compromise the goodness of fit. Furthermore, it seemed more important for the biosensor model that alterations of the plasmid concentration are described accurately, because a repressor effectively reduces the concentration of plasmids that can be transcribed.

Parameter	Description	Value	Source
vTX1	reporter transcription rate constant	18.2 nM min-1	{Stögbauer #22}
KTX1	Michaelis Menten constant for reporter transcription	8.5 nM	{Stögbauer #22}
deg_mRNA1	reporter mRNA degradation constant	0.08 min-1	{Karzbrun 2011 #10}
kTL1	reporter translation rate constant	0.0076 min-1	{Stögbauer #22}
TL resources	initial concentration of TL resources	1520 nM	Fitting
KTL1	Michaelis-Menten constant for translation	29.9 nM	Fitting
v_deg_TL	translation resources degradation rate constant	13.5 nM min-1	Fitting
K_deg_TL	Michaelis menten constant for degradation of translation resources	53.2 nM	Fitting
kmat	reporter maturation rate constant	0.2 min-1	{Stögbauer #22}

Validation

To check if the model performs equally well on unseen data as on the training data set, we compared the prediction of the model for two plasmid concentrations which had not been part of the fitting data set to the experimental data. As can be seen from the figure, the model predicts the sfGFP expression for 7 nM plasmid very well. For 1 nM, the model predicts a lower expression than we observed, but the difference appeared to be acceptable to us.

Conclusion