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Revision as of 22:24, 17 October 2018
dcas9 - sgRNA Model
Figure 1: Schematic representation of the system
After the study of TALE system we moved on to the study of the model, which uses as repressor the complex dcas9-sgRNA, whose schematic representation is shown in Figure 1.
This system, consists of three subsystems.The main subsystem, is the one that replaces the system of TALE and has the same functionality. The only difference is that now the repressor consists of the complex dcas9-sgRNA and for that reason, instead of the transcription and translation of the repressor, we just have a binding of the two molecules. The other two subsystems which were studied are about the creation of those two molecules. The dcas subsystem works with extra repression for its production, which is controlled via Doxycycline.In contrast, the sgRNA subsystem operates by activation through Rhamnose-RhaS complex which is also adjustable.
For each one of those subsystems, we have studied separately their characteristics and the parameters which affects more the output of every subsystem. In order to achieve that, we used the knowledge that we gained from studying the TALE system. Along the way, we realized possible problems and gave solutions to them, which we then shared with Wet lab, so as to integrate them in system design.
dcas9 subsystem
We started by studying the dcas subsystem. The dcas is repressed by the tetR, because in large quantities can be toxic for the cell. Also, by adding doses of Doxycycline, which in turn creates complex with tetR, we managed to decrease the repression that happens in dcas to the desired level.
As shown in Figure 1, the dcas9 and the tetR are on a different plasmid than the rest of the system. This means, that the change in copy number that we examine does not affect the plasmids which are responsible for the production of the dcas9 and the tetR. After the completion of the modeling process, an analysis was held to consider other options that Wet lab proposed, regarding the different expression sites of dcas.
For the mathematical representation of the system, we took into consideration the following assumptions:
- Inducer dynamic Doxycycline: We assume that as soon as Doxycycline is injected into the solution, is stabilised in a steady state inside the cell and we do not take into account influx and outflux rates.The concentration of Dox inside the cell is set to 8ng/ml, that is 2.4 mM. When the complex [TetR: Dox] is degraded with the same rate as tetR, we assume that Dox can be free inside the cell again.
- tetR dimerization, binding to Doxycycline and binding rate to tetO2 operator sites: A tetR dimer is sufficient for specific binding to the operator sequence. However, a tetR monomer is unable to bind and exert any repressive effect. Thus, we assume that the repressor dimer binds to the operator (denoted as species O) with a one-to-one stoichiometry and also assume that a dimer binds one to one with Doxycycline. These assumptions are valid, as monomers forming a dimer and dimer forming a complex with Doxycycline are very fast interactions and do not affect other simulation components. As a consequence, we halved the translation rate of tetR and the concentration of Doxycycline that was injected into the solution.
- We expressed the Hill equation as mass action kinetics for the two different states as we did in TALE system.
| Parameters | Biological meaning |
| cn | psb3k3 plasmid |
| at | transcription rate of mtetR |
| bt | translation rate of tetR |
| dmt | degradation rate of mtetR |
| dt | degradation of tetR |
| kdt | binding rate of tetR to Doxycycline |
| kdt- | unbinding rate of tetR to Doxycycline |
| kon2 | binding of tetR to promoter site |
| koff2 | unbindng rate of tetR to promoter site |
| Tmax | maximum transcription rate of dcas9 |
| Tmin | minimum transcription rate of dcas9 |
| bd | translation rate of dcas9 |
| dmd | degradation of mdcas9 |
| dd | degradation rate of dcas9 |
| n | cooperativity tetR to pTet promoter |
Parameter estimation
For parameter values, first estimates were made through bibliography and methods which were used also for TALE, such as RBS Calculator.
The parameters for which it was deemed necessary to do further analysis, as bibliographic references proved to be quite uncertain, are the binding and unbinding rates between doxycycline and tetR and also those of tetR with DNA, in the two position of pTet promoter. So the sensitivity analysis, was applied to these parameters.
As can be seen in Figure 2, the kon2 between tetR and DNA shows approximately 0.6 first order index and taking into account also higher order interactions, this is the most influential parameter of the system. The toxicity of dcas is an important limiting factor which we should consider.Therefore, its concentration must be under control via the tetR repression and the Doxycycline dose.
Taking into consideration all the above, we decided to examine the system behaviour with different sets of parameters. Specifically, we used all the parameters that was used in sensitivity analysis and we created 5000 different sets. The values of the kinetics ranged from 0.001 to 10.
The data from the analysis (Figure 3), indicate that the mean value of dcas9 concentration is approximately 0.826 nM. This result appears to be problematic, as a concentration this low, would set an upper bound on the final repressor and by extension, sfGFP could not be independent from the copy number, in any case.
While trying to locate the problem, we noticed that, in order for the system to acquire a reasonable dcas concentration, we had to change its dissociation constant, .In Figure 3, kd’s maximum value is ,but scanning this parameter for a bigger range we found that if we increase it to ,we will finally get satisfying results.
After we shared this result with Wet lab, combined with the fact that the kd for which our system works, disagrees with those found in bibliography [4][5], we decided to make an experiment to validate the functionality of the dcas subsystem. [ ΠΛΗΡΟΦΟΡΙΕΣ ]. The results of the experiment, indicate that dcas is produced, in very good level, without causing any problem for the cell growth.
The reason that the concentration of dcas in the model depends that much by tetR’s ability to bind to DNA, we believe is due either to wrong configuration of tetR or to the modeling of the Doxycycline dose in the system. Having received the feedback from the experiment, we wanted to correct the model so as to bring the right result. We decided, to not change the modeling of the dose, but to adjust the concentration of dcas, by changing the aforementioned kd as seen in Fig.4. We ended up, that the value of the constant must be equal with as its concentration is greatly increased for other, bibliography based, values [2].
sgRNA subsystem
The sgRNA is produced by the promoter Pbad, only when he is activated by the complex Rhamnose-RhaS. In contrast, with the repressed function of dcas, the sgRNA is activated by that complex and that is induced by Rhamnose, which we add as a dose to the system.
As shown in Fig.1, the production of protein RhaS is done by the genome of the cell and therefore the concentration is independent from changes in its copy number of sfGFP. From that fact, we can understand that the complex which it creates with Rhamnose has this upper limit in each experiment for different copy number. The exact amount of complex and by extension the activation for the production of sgRNA, will directly depend from the Rhamnose that will be provided to the system.
For the mathematical representation of the system, considered the following assumptions:
- Inducer dynamic Rhamnose: Stabilization of Rhamnose as soon as it’s injected to the solution. As Rhamnose degradation rate is quite lower than this of rhaS, we assume that the degradation rate of the complex is the same as rhaS degrade. When rhaS degrades the free Rhamnose molecule is added to total Rhamnose concentration.
- RhaS dimer and binding rate to operator site of PrhaBAD:Like in tetR, we assume that Rhas is produced as a dimer because of it’s fast dimer binding. Because of its fast binding with L-Rhamnose we also assume, one to one stoichiometry. In the presence of L-rhamnose, its concentration is 0.02 g/ml or 1.2 mM, Expression from the rhaBAD promoter is log linear with n=1 [7] with respect to inducer concentration.
- Again, Hill equation was expressed as mass action kinetics equations.
| Parameters | Biological meaning |
| ah | transcription rate of mrhaS |
| bh | translation rate of rhaS |
| dmh | degradation rate of mrhaS |
| dh | degradation of rhaS |
| khr | binding rate of Rhas to Rhamnose |
| khr- | unbindng rate of Rhas to Rhamnose |
| kon1 | binding of [rhaS:Rham] to promoter site |
| koff1 | unbindng rate of[rhaS:Rham] to promoter site |
| hmax | maximum transcritpion rate of sgRNA |
| hmin | minimum transcription rate of sgRNA |
| ds | degradation of sgRNA |
| n | cooperativity of [rhaS:Rham] to PBAD promoter |
Parameter estimation
After using a bibliographic approach (see bibliography based estimation) to find the parameter values of the subsystem sgRNA, we ended up in the following uncertain parameters:
- Binding and unbinding rates of Rhamnose with RhaS.
- Binding and unbinding rates of Rhamnose:RhaS complex with Pbad promoter.
- Degradation rate of RhaS
We applied sensitivity analysis for these parameters, using the same rate ranges as dcas9 subsystem and the degradation variation is 0.001 to 1. The results are shown in Figure 5 where you can see that the most influential parameter is clearly the binding rate to Pbad promoter.
Binding and unbinding rates of RhaS to Rhamnose do not have any effect at all and we assume this is due to the very big concentration of Rhamnose in the system. (set to 1.2 mM) Τhe other two parameters have relatively high total order indices because of their interaction with kon (same differential equation).
In order to get more insights regarding the system behaviour, we applied a robustness analysis, shown in Figure 6. The result appeared as expected with sgRNA concentration increasing for higher copy number. The standard deviations show us that for little copy number the variation of the concentration is trivial but for higher copy numbers the variation increases accordingly. However, we can clearly tell that for most parameter sets the linearity of the system cannot be ruined.
Main system
The main system consists of the formation of the repressor complex and the production of sfGFP, which is being repressed by the complex. The complex binds to DNA slowly, as dcas9 has to find the PAM sequence.The dissociation is also very slow and approaches the dillution rate of the cell. The disruption percentage of RNAP, arises from sgRNA’s complementarity with DNA and the concentration of polymerase. We also assumed the degradation of the repressor complex, being the same rate as dcas9 degradation.
| Parameters | Biological meaning |
| c | inducible copy number ( low,medium,high) |
| kds | binding rate of dcas9:sgRNA |
| kds- | unbinding rate of dcas9:sgRNA |
| kon3 | binding of [dcas9:sgRNA] to promoter site |
| koff3 | unbinding of [dcas9:sgRNA] to promoter site |
| aGmax | maximum transcription rate of sfGFP |
| aGmin | minimum transcription rate of sfGFP |
| bG | translation of sgGFP |
| dmG | degradation of msfGFP |
| dG | degradation of sfGFP |
Parameter estimation
As previously done with the subsystems we studied bibliography to find out about system parameters and their ranges. After that, we applied sensitivity analysis for the association and dissociation constants. Interestingly, the results showed that the repressor’s kdson and kdsoff does not affect the system output. After changing some values of these parameters we found that this is true and the system appeared to be very robust regarding the repressor concentration. Due to the strong repression that it does, combined with its very small degradation rate, the repressor is binding to DNA and almost never dissociates, thus making small repression concentrations sufficient for the system to work. Specifically, using different values we were able to get exactly the same output error with many combinations of kdson and kdsoff.
The second thing we noticed that reinforces the previous statement is that the second order indices have a peak for the interaction of kon3 with kdsoff. That makes sense, because in the case of decreasing the repression strength, the system needs more repressor to operate the same.
Knowing from fact that the dcas9-sgRNA repression is very strong [1][2][3], we fixed the parameters accordingly (see Bibliography based parameter estimation) and finished model characterization. However, the final model produced an Error of approximately 22, which we will discuss in the next section.
dcas9-sgRNA Repressor analysis
In the previous section, we draw the conclusion from the results of sensitivity analysis, that the system is not sensitive regarding the formation of repressor, thus we can tell as an extension of this that the system is robust concerning the concentration of dcas and sgRNA.
On the other hand, we found out that the system appears to be generally unusable with an error of 22, that was hard to change. Scanning the most influential parameter, kon3, as evidenced by Figure 7, we were able to change the repressive power of the repressor complex and produce sfGFP concentration of different magnitudes, though the system Error (as described on the system evaluation section of TALE) did not show to be affected.
This could potentially mean, that the system would fail, even if we didn’t have set the binding affinities of the system to their right values with our characterization. That is, because even when trying to change the error by scanning different parameters and combinations of them, we weren’t able to correct the situation.
In order to find the root of this problem we had to think backwards. While plotting different component concentrations during simulations, we realised, that the repressor concentration was the same for different plasmid copy numbers and then we presented the problem mathematically, proving, that a constant repressor amount can not make the system’s output independent from the copy number. Below we present the proof of this statement.
Our goal is to minimize the error, between sfGFP expression for low (cmin) and high (cmax) copy numbers, find the conditions under which, the system is stabilized and to prove that this cannot be achieved with a constant repressor concentration over increasing copy number. For the gene of interest and the repressor we use the annotatuon G and R respectively.
In our case, the gene expression of hill equation, has cooperativity equal to one and the ratio of unbinding and binding rate is equal to kD (dissociation constant) expressed as
with
kD is the concentration of the repressor, in which half repression of G is achieved and we can assume, that in the steady state for every copy number, , thus (1) becomes
For simplification purposes let's consider that we have constitutive promoters for the expression of dcas9 and sgRNA. Dcas9 is expressed constantly by a medium value copy number, cn, while sgRNA's copy number increases in the interval [cmin, cmax], as it is located on the same plasmid as sfGFP is. Then, the analytical solution on the steady state for sgRNA and dcas9 concentration is
with cn = constant and
The dcas9 and sgRNA binding has one-to-one stoichiometry, then for each copy number when the reversible reaction reaches in equilibrium, the relationship between the reactants and the product R is
If for then dcas9 acts as the limiting reagent for the formation of the complex R, which by extension has a maximum concentration of acn. Specifically its concentration is given below
with Notice that R is independent of the copy number c.
We can now replace R in (2) and get
Likewise if sgRNA is the limiting reagent, that is for Replacing the corresponding R to (2) we end up with the following relationship
proving that for R, which is scaling proportionally to copy number increments, expression of G is independent of the copy number variation.
Figure 9: On the y axis the concentrations of dcas9, sgRNA and by extension, the repressor’s are presented. On the x axis we have copy number variation. The concentration concerns the steady state of the system for every copy number.
From the graphical representation, we can easily tell, that sgRNA indeed surpasses the level of dcas concentration at copy number 50. Thus, after that -let’s name it- breaking point, the repressor is constant for the remaining copy numbers, with dcas9 acting as the restriction reagent. It is obvious that, we had to change the repressor amount so as to be increased for increasing copy number. This could be achieved by either increasing dcas9 concentration or decreasing sgRNA’s one. We chose the latter option, because, as we concluded in dcas subsystem section, the amount of dcas is already enough and if increased, due to its toxicity it could be a problem for the cell’s growth.
In Fig. 10 we present the fixation of copy number variation. For a kd of 10 raised to the power of 4 we can see that dcas9 becomes the restriction reagent for high copy numbers. It is important to mention, that changing the binding affinity is difficult as it requires to change also the promoter and different promoters have discrete amount of affinity. This can be solved by changing the Rhamnose dose for small changes needed. Though the amount of Rhamnose should be small to make a difference because this time the restriction reagent is RhaS and its concentration is very small. We were able to make the aforementioned kd value stay under dcas9 concentration barrier, by providing Rhamnose near the value of 150 nM or lower.
Figure 10: Changing the binding affinity of the Rhamnose:RhaS complex with the promoter of sgRNA.
Finally, as we found that the concentration of Repressor is not a problem for the system (main system section), we concluded, that we can have different magnitudes of sfGFP with very small errors, by optimizing sgRNA concentration.
dcas9 Expression Site
In this section we are going to examine the different options we had for the expression site of dcas9 in order to inform Wet lab, which is the better option for our system. The three options that the Wet lab was thinking were:
- dcas to be expressed on the same plasmid with sgRNA and sfGFP
- Double plasmid solution where dcas is expressed in the same plasmid as tetR.
- dcas to be expressed directly from the genome.
Figure 11: Dcas9 Concentration over different copy numbers for all three options
Figure 12: sfGFP concentration over different copy numbers for all three options
As we expected, the results show, that when dcas is expressed from the genome, sfGFP is greatly increased as copy number is increased, because it sets a very low upper limit for the repressor concentration, which is reached very soon even for low copy numbers.
When dcas9 is expressed within the same plasmid as sfGFP, there is no restriction reagent in the system and both sgRNA and dcas are increasing. The final sfGFP production and system error is exactly the same with the final option, which is also our main system. This can be interpreted as in the main system we concluded that the system is robust, regarding the repressor binding and concentration, due to the final strong repression, slow unbinding and also slow degradation rate of the repressor. However, in the case of double plasmid, we are able to monitor the expression of dcas, in contrast with the single plasmid were dcas is constantly increasing.
Taking the case that we do not use tetR for dcas repression, the only difference is that the expression of dcas would be even higher.
Also we thought of having tetR and dcas both expressed in the same plasmid as sfGFP, but the results were not good. Actually dcas9 concentration was decreasing while copy number were increased. This meant that for high copy numbers, tetR concentration scales fast, thus the repression on dcas is stronger.
Bibliography based parameter estimation
All the values for the parameters of this model are presented in the following table. The parameters for which the bibliography is estimated are the ones that were uncertain and where estimated within each subsystem section of parameter estimation.
Using the same methods as we used in TALE system found or calculated the transcription, translation and degradation rates of the proteins [8][9].
The minimum transcription rate of repressed promoters was assumed 0.0001 for all proteins, because.
For all mRNAs, we found that degradation rate is approximately 0.2 for E.coli cells [10]. Regarding protein degradation rates, they all have been found through different studies with the exception of rhaS, whose parameter has been fixed, because it did not create much variance to sgRNA expression, as shown in Fig. 5. The fixated value was small based on other protein degradation rates.
Discussion Our analysis aims at giving a better understanding of the system behaviour. We achieved that by building the subsystems of the model and combining them, in an effort to characterize the model as best as possible. Methods such as sensitivity and robustness analysis helped us to analyze and see connections between parameters and focus on the more important aspects of the system. By using those, we were also able to detect problems that the system had. Finally, we gave feedback for the system design by providing the better option regarding the dcas9 expression site. Also, by making the system sgRNA optimized, after the analysis of the repressor that we made, we provided helpful insights regarding the relative concentrations of sgRNA and dcas9 that made the system work properly.
| Product | Sequence length | Transcription rate nt/min | Translation rate(1)/min | Translation rate(2)/min | translation rate /min |
| tetR | 624 | 4.33 | 5.76 | 0.13 | 1.82 |
| dcas9 | 4107 | 0.66-0.73 | 0.87 | 0.42 | 0.56 |
| rhaS | 837 | 3.22 | 4.3 | 0.02 | 1.3 |
| sfGFP | 714 | 3.78 | 05.04 | 03.05 | 3.65 |
| Parameters | Biological meaning | Value | Units | Bibliography |
| cn | medium psb3k3 plasmid | 15 | no dimensions | - |
| at | transcription rate of mtetR | 4.33 | 1/min | [9] |
| bt | translation rate of tetR | 1.819 | 1/min | [8][9] |
| dmt | degradation rate of mtetR | 0.2 | 1/min | [10] |
| dt | degradation of tetR | 0.017 | 1/min | [5] |
| kdt | binding rate of tetR to Doxycycline | 126 | 1/nM*min | [6] |
| kdt- | unbinding rate of tetR to Doxycycline | 0.12 | 1/min | [6] |
| kon2 | binding of tetR to promoter site | 0.0001 | 1/nM*min | estimated |
| koff2 | unbindng rate of tetR to promoter site | 10000 | 1/min | estimated |
| Tmax | maximum transcription rate of dcas9 | 0.73 | 1/min | [9] |
| Tmin | minimum transcription rate of dcas9 | 0.0001 | 1/min | - |
| bd | translation rate of dcas9 | 0.55 | 1/min | [8][9] |
| dmd | degradation of mdcas9 | 0.2 | 1/min | [10] |
| dd | degradation rate of dcas9 | 0.005 | 1/min | [3] |
| n | cooperativity tetR:DNA | 2 | no dimensions | [4][5] |
| ah | transcription rate of mrhaS | 3.22 | nM/min | [9] |
| bh | translation rate of rhaS | 1.304 | 1/min | [8][9] |
| dmh | degradation rate of mrhaS | 0.2 | 1/min | [10] |
| dh | degradation of rhaS | 0.06 | 1/min | estimated |
| khr | binding rate of Rhas to Rhamnose | 100 | 1/min*nM | estimated |
| khr- | unbindng rate of Rhas to Rhamnose | 100 | 1/min | estimated |
| kon1 | binding rate of [rhaS:Rham] to DNA site | 0.001 | 1/min*nM | estimated |
| koff1 | unbindng rate of [rhaS:Rham] to DNA site | 10 | 1/min | estimated |
| hmax | maximum transcritpion rate of sgRNA | 5 | nM/min | [3] |
| hmin | minimum transcription rate of sgRNA | 0.0001 | 1/min | - |
| ds | degradation of sgRNA | 0.18 | nM/min | [3] |
| n | cooperativity [rhaS:Rham]:DNA inducible copy number ( low,medium,high) | 1 | no dimensions | [7] |
| c | binding rate of [dcas9:sgRNA] | 5-100 | no dimensions | - |
| kds | unbinding rate of [dcas9:sgRNA] | 10 | 1/nM*min | [3] |
| kds- | binding of transcriptional repressor [dcas9:sgRNA] to promoter PG site | 0.001 | 1/min | [3] |
| kon3 | unbinding of transcriptional repressor | 0.693 | 1/nM*min | [3] |
| koff3 | [dcas9:sgRNA]to promoter site | 0.018 | 1/min | [3][2] |
| aGmax | maximum transcription rate of sfGFP | 3.78 | 1/min | [9] |
| aGmin | minimum transcription rate of sfGFP | 0.0001 | 1/min | - |
| bG | translation of sgGFP | 3.65 | 1/min | [8][9] |
| dmG | degradation of msfGFP | 0.2 | 1/min | [10] |
| dG | degradation of sfGFP | 0.0193 | 1/min | [11] |