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Revision as of 03:29, 18 October 2018
TALE
The first system studied utilizes the TALE repressor and specifically the TALEsp1 or TALEsp2 sequence.
The kinetic reactions that make up the system are described by mass action kinetics for the transcription, translation and degradation of molecules, and by Hill kinetics described as mass action kinetics in two possible states for binding and dissociating the TALE repressor at the binding site of promoter P.
The TALE gene is replicated at a constant rate from each plasmid c and then translated into the TALE protein. This protein then binds to the promoter of sfGFP (G), whose gene is on the same plasmid as TALE. It is associated with dissociation constant kD = koff / kon and cooperativity n and produces the final output of the gene of interest, which in our case is sfGFP. This system has been studied and proved experimentally [1]. Our work on this model aims to better study its parameters and their characterization, as well as to reply to wet lab questions about the system and its possible extensions.
We started by describing the kinetic reactions that the model works with:
As the model follows the IFFL's network motif, what we expect is the stabilization of sfGFP at a specific level, regardless of changes in the number of plasmids (copy number, c). The analytical solution of the differential equations resulting from the above kinetic reactions at steady state, associates the concentration of sfGFP protein with the copy number through the following ratio:
where n is the cooperativity between the TALE repressor and the operator site of DNA. So, easily, we can conclude that in order to isolate the output from the plurality of plasmids, n should be equal to 1. After studying and approximating the system parameters, which are described in detail in the section of parameter estimations, we ended up with the final featured model, which we used to get results and insights for the system.
Specifically, we run the simulation of differential equations for a copy number of 5 to 100. For every copy number, the system was simulated for 1000 min and we noticed, that the steady state was reached after approximately 400 min.
System evaluation
From figure 2, we can conclude that the system responds to our expectations, but it seems necessary to use a metric to evaluate its performance. So to interpret the performance, we will measure the error by dividing the result of the difference between final (copy number = 100) and initial (copy number = 5) sfGFP, with the initial sfGFP.
The E of the system in figure 1 was proven experimentally [1] to be equal to 0.52. This information was used in the robustness analysis section of the parameter estimation.
Cooperativity comparisons
The binding of TALE to the DNA is done through Hill's kinetics as mentioned above and cooperativity n shows how strong the repress is.
We noticed that as we increase n, the steepness of the curve increases. Then we wanted to prove via the model, that n must be 1, in order for the system to function correctly. It is worth noting that the values of all the parameters for this experiment will not be used. Although TALE has been found to have n = 1, different cooperativity may refer to different kinds of repressor and their association with different promoters. The assumption is that different cooperativities n will be due to a change of the promoter and thus the parameters that will change are kon koff and the maximum transcription rate of sfGFP, aGmax.
We got the results from 3 robustness analyses, for n = 1, n = 2, and n = 0.5, using 10,000 different sets of parameters in each. The parameters of the model to be varied are the kon,koff of the TALE-DNA binding and maximum translation rate aG. The range obtained was based on what was used in the sensitivity analysis and is 0.001 to 10. Through this procedure we calculated the average sfGFP production for different copy numbers. Finally, we calculated the Standard Error for each copy number with 95% confidence intervals as shown in Figure 3.
We can easily notice, that if n is not 1 the system does not stabilize. We also calculated the Error for every parameter set.
| Cooperativity coefficient, n | Mean Error of parameter sets |
| 0.5 | 4.03 |
| 1 | 0.39 |
| 2 | 11.35 |
Parameter Estimation
Sensitivity analysis
After the initial modeling of chemical reactions of the system, and having set a range of values for each kind of parameter, based on the type of biological component and the nature of the reaction, we proceeded to sensitivity analysis for all parameters. Our goal is to characterize the parameters and apply this approach from a simple model to a more complex one such as the dcas-sgRNA system we studied later on.
Initially studying the bibliography and previous iGEM teams, we were able to gather several values for the rates of transcription, translation and degradation of the components, and we set the range in which each parameter ranges. Next, we created the samples within the defined range using Saltelli's sampling scheme from SALib Library [3]. In particular, we used for each variable N = 40,000 samples translated into 880,000 parameter sets, for the calculation of first and second order Sobol indices through the relation: sets=N(2D +2), where D is the number of parameters. At first, we had a smaller sample size and we noticed that several sensitivity indices values were negative, or with a confidence interval greater than 50% of their value, so we increased sample size to get more accurate values, which we judged based on how small their confidence interval is.
Figure 4, shows First and Total order sensitivity indices from the first sensitivity analysis we applied. This analysis contains all the parameters of the system, and we used it as the initial criterion for the significance of each parameter. The ranges of parameters, that were defined after our study, are listed in the workflow sensitivity analysis section. Second order indices of this analysis were not quoted, because their values were so small that they make their investigation unnecessary.
Thus, we see that First-order indices have too low values, which we expected because of the complexity of the model. This shows, that each parameter alone does not affect the result. However, summarizing first and higher order interactions in Total-order indices, we observe the overall sensitivity of the model in changing each parameter. Parameter values with a small Total-order index can be considered as constant at any value within the range given to them without affecting the output variance. As can be seen, the variables that most affect output, with Total-order indices above 0.2 is the degradation of sfGFP, dG followed by the binding rate kon, aT, bT and bG. The high influence of degradation on sfGFP is due to the direct relationship to output of about 0.8 with confidence interval (0.17).
Bibliography based estimation
Transcription and translation rates
Transcription and translation rates of TALE and sfGFP were calculated by using the average transcription rate of 45 nt / s and an average translation rate of 15 aa / s, respectively [7]. Also, the RBS Calculator from Salis Lab [7] was used and the translation rates of mRNA in the system were also estimated. To obtain the mRNA translation rate from each sequence, we had to calculate the ratio as described by [6], based on total Gibbs free energy between the ribosome and mRNA. In order to find the translation rate of sfGFP, we found its bibliographic rate and based on it and the corresponding sequence, in relationship with our’s, starting at another codon the fold-change was calculated between them and hence the rate.
In order to approximate the translation rate, we put weight in both the RBS calculator results and the bibliography, which provided the average translation rate of sequences with only criterion, it’s length. In order to include both translation rates from the two approaches, we used a weight function with a weight factor for each.
| Product | Sequence length | Transcription rate /min | Translation rate(1)/min | Translation rate(2)/min | Translation rate/min |
| TALEsp1 | 2600 | 01.03 | 1.38 | 0.03 | 0.44 |
| TALEsp2 | 2600 | 01.03 | 1.38 | 0.18 | 0.55 |
| sfGFP | 714 | 3.78 | 05.04 | 03.05 | 3.65 |
Degradation rates
The degradation of the protein’s mRNA, was found to be an average of 0.2 [8] in E. coli cells. The degradation of TALE was evaluated through the range of reasonable values from protein degradation and as shown in figure 4, it does not affect the end result.
| Parameters | Biological Meaning | Value | Units | Bibliography |
| aT | Transcription of TALE | 1.03 | nM/min | [7], |
| bΤ | Translation of TALEsp2,sp1 | 0.55,0.44 | 1/min | [6],[7] |
| dmT | Degradation of mRNA TALE | 0.2 | 1/min | [8] |
| dT | Degradation of TALE | 0.047 | 1/min | estimated |
| n | Cooperativity | 1 | dimentionless | [1] |
| kon | Binding rate to Promoter | 100 | 1/nMmin | estimated |
| koff | Unbinding rate to Promoter | 0.1 | 1/min | estimated |
| aG | maximum production rate | 3.78 | nM/min | [6] |
| bG | Translation of sfGFP | 3.65 | 1/min | [6],[7] |
| dmG | Degradation of mRsfGFP | 0.2 | 1/min | [8] |
| dG | Degradation of sfGFP | 0.019 | 1/min | [9] |
The kon, koff and degradation values of Tale, dT were not found bibliographically, and after replacing the constant values of the other parameters in the model, we applied another Sensitivity to these three values.
As shown, the binding of the repressor to the promoter is a critical parameter for the final stabilization of sfGFP. Under these circumstances, it is necessary to estimate kD (= koff / kon). kD can be evaluated experimentally as the point of the curve that the repressor concentration leads to the expression of sfGFP in half of its repression. However, the kon and koff reaction rates are not available and can not be found from experimental data. Thus, different combinations of kon, koff can lead to the same kD, which can cause a different time of arrival of the system to a steady state. This does not affect our analysis as we do not study the speed of the system.
For these parameters, second order interactions appear to have interesting results.
We see that a major part of the Total influence of dT to the final output is coming from the value of the kon. This is understandable as the two parameters are in the same reaction and are therefore dependent. After the process of sensitivity analysis and parameter definition, the question we had, was whether we should set a value for dT. Finally, we have left it variable for the moment and its definition was made after the Robustness analysis.
Robustness analysis
At this point three parameters made the result of the model variable, kon, koff and dT. Wishing to approach the behavior of the model under different values of these parameters, we applied a Robustness analysis by keeping the average of the values of sfGFP per copy number for 10,000 parameter sets. The Robustness analysis process is explained in the workflow.
As stated at the beginning, our main goal was to characterize all the parameters of the model. In order to do this, we combined the data we obtained from robustness analysis with the proven TALE experimental error [1]. This error is 0.52 for TALEsp1 and 0.07 for TALEsp2.
The system seems to be very robust as the error is very low for almost all parameter values. To characterize our last parameters all we needed to do was, find the kon, koff and dT set with the aforementioned experimental errors and also with the corresponding transcription rate. Though, in order for the characterization to be valid we needed more data about sfGFP concentration from experiments. Using only the system error we could find many sets that were producing it. For different parameter sets, sfGFP expression was different too. With an extra scan for bigger values of kd, we found the same errors for very big amount of expressed sfGFP (Fig. 1). Finally, we decided to keep the small kd because of the strong repression that TALE does.
| Repressor | kon | koff | dT |
| TALEsp1 | 9.85 | 2.19 | 0.047 1/min |
| TALEsp2 | 1.86 | 3.85 | 0.047 1/min |
TALE tuning with IPTG-LacI
For the tuning of the final concentration of sfGFP, the system was studied when LacI was placed after the sfGFP promoter on the same plasmid, repressed by TALE. Our system is stabilized for specific binding characteristics of the TALE with the sfGFP promoter as discussed above.
When LacI binds to an operator site of the promoter of sfGFP, we have gene expression even if a repressor is bound to the promoter too. The ideal case with the incorporation of LacI would be to regulate the final amount of sfGFP in each copy number at a small deviation around it ‘s final concentration .
Although, when LacI binds to the promoter, it appears to reduce the possibility of TALE to also bind, due to steric interactions between the two, thus making the system’s behaviour undesirable.
Τhe competition mechanism between TALE and LacI on the binding sites of sfGFP’s promoter P can be described as the following diagram:
We will analyze the mechanism, in which, two ligands occupy different sites of the promoter and assume that both interact due to electrostatic interaction. Because their binding sites are close enough, we assume that due to the size of LacI, Repressor TALE will display weak DNA binding when it is to bind in the presence of LacI, increasing the dissociation constant KT of TALE. This means that the binding affinity of the enzyme will be reduced, but can be readjusted in the presence of IPTG in the system by reacting with the free LacI reducing the likelihood of it, binding to the promoter. The dissociation constant for binding to the free promoter site is KT for TALE binding and KL for LacI binding. Similarly, when Tale is bound to the promoter, the affinity of LacI with the corresponding operator site will be reduced. The new dissociation constants will be k times bigger and we assume that k is the same for both ligands since interference between them will cause the same difficulty for each.
The expression of the hill equation, as described for TALE, is transformed as, the maximum transcription rate function for sfGFP, multiplied by the competitive equation, in the presence of LacI and includes the concentrations of the TALE repressor, LacI inhibitor and dissociation constant KT, KL, kKT, kKL. Despite the fact, that LacI does not interfere with the same operator site on the promoter, the interaction has a competitive nature, because of the steric interactions. As a result, TALE affinity to the DNA decreases.
Τhe production rate of sfGFP expressed as
where kKT >> KT because of reduced affinity.
For the differential equations that were added to the model:
- We did not include IPTG as although in high concentrations it can reduce the concentration of LacI and smooth it out, it cannot neutralize this phenomenon to some degree. We have studied the case with only LacI regulating sfGFP expression.
- Repressor TALE also represses the production of LacI with a similar dissociation constant and cooperativity characteristics with the sfGFP promoter.
- The rates for LacI were found bibliographically and added to the TALE model.
Parameters
TALE rates are characterized above as well as dissociation constant KT = kd is equal to 106 nM.
| Parameters | Biological Meaning | Value | Units | Bibliography |
| aL | Transcription of LacI | 0.23 | nM/min | [11] |
| bL | Translation of Laci | 0.128 | 1/min | [12] |
| dmL | Degradation of mRNA LacI | 0.1824 | 1/min | [12] |
| dL | Degradation of LacI | 0.0231 | 1/min | [13] |
| KL | dissociation constant LacI to Promoter | 800 | nM | [13] |
| k | constant | 100 | - | Est. |
From Figure 9, the final values of the sfGFP concentration for each copy number in the steady state, in the presence of LacI, are constantly increasing and the repressor cannot hold the same fold change for different copy number. Also, the deviation from the balancing of sfGFP shown in the TALE system is two orders of magnitude larger and the control of its quantity with LacI cannot satisfy the desired adjustment levels around the steady state.
For a low copy number over time, it is observed that although the amount of TALE produced is the same, it cannot repress with the same power as its absence, since the affinity is kx times larger than natural binding.