Team:Valencia UPV/Model
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MODELING
OVERVIEW
MOTIVATION
In ChatterPlant we aim to understand and program our synbio device according to the necessities of certain situations.
Modeling our gene circuits provides us with deep insight and prediction capability of the biological processes taking place in ChatterPlant.
Furthermore, mathematical models in synthetic biology contribute not only to generate empirically contrastable hypothesis, but also to manage resources efficiently, skipping unnecessary experiments imposed by trial-error approaches.
WHAT WE ARE MODELING
The SynBio-based design integrated in ChatterPlant is composed by two gene circuits. In order to set ChatterPlant as a new sustainable and efficient agriculture system, we analyzed both their single performance and their interaction with several factors (e.g. the cell medium, environment and ChatterBox).
Our model comprises of:
- HUMAN-PLANT: Optogenetic circuit. . How to tune the synbio circuit parameters to get the desired plant response to red light stimulus? How long has to remain the LEDs system switched ON in order to get a certain protein amount?
- PLANT-HUMAN: AND gate. How much GP3 is necessary to let the color be expressed? Which is the optimal proportion between recombinases and GP3?
Check our Optogenetic circuit model
In our in silico experiments, we analysed empirical data and used optimization algorithms in order to set the optimal conditions which ensure a smooth bidirectional communication between plants and humans.
HOW WE ARE MODELING
In ChatterPlant we analyze the dynamic behavior of our biological system considering the biochemical species involved in a certain set of reactions. According to the degree of approximation to capture the dynamic behavior, we can differentiate two approaches:
- Deterministic. Deterministic models do not take into account the natural randomness of the reactions. For each chemical species, the amount of molecules transformed within reactions only depends on the initial amount of molecules, reaction rates and stoichiometry relations.
- Stochastic. Inherent noise due to random events plays a relevant role in the dynamics. As a deterministic model does not capture noise, we use stochastic linear differential equations.
Check our Deterministic Optogenetic Model
Check our Deterministic Recombinase-GP3 Model
MODELING SOFTWARE MODULES
We start building the genetic circuits from basic modules, coupling them to generate the mathematical model of the whole system. As UPV_iGEM is an interdisciplinary team, most of the models generated in ChatterPlant are included in the modeling software tool and are represented by modules in an artistic graphic interface, for the purpose of introducing researchers to a more realistic conception of the engineering in biology, meanly, SynBio.
HUMAN-PLANT: OPTOGENETIC CIRCUIT
Two constitutive modules express the E-PIF6 and PhyB-VP64 fusion proteins that regulate the output expression.
Constitutive modules representation of the fusion proteins E-PIF6 and PhyB-VP64
E-PIF6 binds to the promoter’s operator. When red light (660 nm wavelength) LEDs are switched on, PhyB changes its conformation (PhyB*) and binds to PIF6. Consequently, the transcription of the desired protein starts because of the RNAp recruitment by VP64.
Expression regulated by the transcriptional factors
Far red light (740 nm wavelength) reverts PhyB* to its natural conformation (PhyB). This change stops de transcriptional activity of the third optogenetic circuit’s module.
Switch off
DETERMINISTIC
REACTIONS
Now we take into account the principal reactions in each module representing them both graphic design and formal reactions.
E-PIF6 expression
PhyB-VP64 expression
Regulated expression
Constitutive module (A=E-PIF6)
Constitutive module (B=PhyB-VP64)
Regulated module
ASSUMPTIONS
Considerations in the model:
- The cRNAp constant considers that the cell has the sufficient free RNAp in excess to be utilized by all the active genes that are transcribing simultaneously in the cell, including the gene of interest. Under this conception, the free RNAp vary in an almost unappreciable way in time, so can be defined as the CRNApFree constant and consequently the sum of the RNAp linked to the DNA and the free RNAp as the cRNAp constant.
- The RNAp binding-unbinding reactions to the promoter are much faster than the elongation and degradation reactions, so can be considered in the equilibrium state.
- Transcription reaction is faster than translation reaction, so can be considered in the equilibrium state.
- The conformation change is instantaneous.
FINAL EQUATIONS
After a mathematical development (explained in the attached pdf), we obtained the following equations, which define the constitutive and regulated expression respectively (where sub P is a generalization to name the protein)
Constitutive expression production is directly proportional to the translation rate (kP), to the transcription effective rate (kmPe), to the gene copy number (cnP) and inverse proportional to the mRNA degradation rate (dmP). The protein degradation is defined by the protein degradation rate in the cellular medium.
Regulated expression is proportional to the translation rate (kP), to the transcription rate (kmP), to the gene copy number (cnP) and inverse proportional to the mRNA degradation rate (dmP).
Alphas and betas are defined by rates (see supplementary). When red light is off (y2=0), the expression becomes as:
The basal expression depends on the E-PIF6 production. With the obtained model we discovered that the leakage can be reduced with a strong promoter on E-PIF6.
STOCHASTIC
An overview about the theory behind this type of models is addressed at the Stochastic chapter of our downloadable Modeling detailed explanation. Basically, instead of using deterministic reaction rates, we use probabilistic reaction rates. These rates define the probability per time unit of one chemical changes to other or disappear.
In general terms, the next equation represents the production of proteins under the assumption of a white Gaussian noise, i.e. null mean and unitary variance:
SIMULATIONS AND CONCLUSIONS
Simulations were performed using our own developed scripts in Matlab 2016b.
Figure 1. mRNA expression in constitutive module
In this figure (figure 1), we can see the variation associated to the expression of mRNA for constitutive proteins in our system due to the Gaussian noise added to the expression. Due to the oscillating production of mRNA we obtain a similar variation on the production of the protein that depends on it (figure 2).
Figure 2. EPIF6 constitutive expression
Figure 3. mRNA expression in regulated module
Figure 4. Inactivated PhyB-VP64 complex
Figure 5. Activated PhyB-VP64 complex
In figures 3, 4 and 5, we realized about the behavior of our model when we put the red light on (regulated expression). All the inactive chimeric protein (PhyB-VP64) changes its conformation instantaneously because of its dynamic is fast enough to assume it
Figure 6. C protein regulated expression
Figure 7. C protein constitutive expression
We can confirm visually the dependency between the production of our target protein (C) and the presence of both activated PhyB-VP64 and EPIF6 (figures 5 and 6). In this case, we simulate a light stimulus in minute 100 and with a length of 20 min. It should be noted that apparently our optogenetic construction has a filter behavior. We can see that the noise variation in RNA messenger is not showed on the production of our target protein (C).
OPTIMIZATION OF PARAMETERS
For the optimization of our optogenetic model we have used luciferase assay results obtained in lab. Our optimization software is based on genetic algorithms.
In order to have an easily comprehension of changes in our model when we change parameters value, we decided to simplify the model taking into account some assumptions.
Figure 8. Final equation of optogenetic circuit
Figure 9. Simplification of our model in two steps
Then we proved that these assumptions didn’t eliminate simulation capability of our model and we ensure that we didn’t lose information about the system behavior.
Finally, we obtained these results:
Figure 10. Here we can see how the algorithm defined the Pareto front with different errors in the constitutive expression (darkness) and in the regulated one (red light presence)
In the image, lines represent the model fitted to the points (*), which are results obtained in this experiment. In this case, parameters correspond to the set which minimizes the red light model relative error. The election of any other set of parameters is possible and responds to different subjective criteria, which means that there is not a unique optimal solution.
Figure 11. Representation of experimental data (points) and adjusted model (lines)
Therefore, we got a model able to predict the expression of a target protein in our optogenetic construction. The following table contains estimated values of parameters according to different importance criteria.
Table 1. Table with different parameters values for extreme and middle points of Pareto Front represented above
IN SILICO EXPERIMENTS
We started simulating for a red-light pulse of 20 minutes.
Figure 12. Optogenetic circuit dynamics in a standar simulation
E-PIF6 is being expressed constitutively as well as PhyB-VP64. When red light is ON, as we considered an instantaneous conformation change, the active PhyB-VP64 equals the current concentration from the inactive form and continues increasing because of its constitutive expression. Therefore, the desired output protein starts expressing. When red light turns OFF, the active PhyB-VP64 disappears according to its degradation rate, while the inactive form continues expressing. Since the switch is off, the output protein expression stops (we have to consider both basal and active form degrading as they are transcriptional factors) and fall down by its own degradation rate. Firstly, we studied how to optimize energetic resources in our hardware,. We simulated the optogenetic circuit’s dynamic with different values of light pulse time. We assumed that LEDs are in their maximum power, since experiments were performed in these conditions, and our model is characterized using these data.
Figure 13. Optogenetic circuit dynamics with different light pulse length
When the light pulse time is longer, the protein from optogenetic circuit remains during more time. The difference between 1 minute and 30-70 minutes is small, nonetheless, compared with the range of 400-700 minutes, the difference is evident. Therefore, it the controller’s decision according to the specific needs, how long the LEDs system needs to remain in ON state. Finally, we knew that ChatterPlant performance does not only rely on ChatterBox setting. It also depends on the underlying genetic design. We simulated different gene copy numbers, both for E-PIF6 and PhyB-VP64.
Figure 14. Optogenetic circuit dynamics with different gen copy number
As can be seen in the graph above, results show that from 70 to 95 gene copy number the output expression saturates, being almost identical in these last scenarios. Consequently, in order to maintain genetic efficiency and avoid metabolic overload, the variation of gene copy number should be in the range from 1 to 70.
CONCLUSIONS
- Human plant communication is feasible and has been confirmed by our in vivo and in silico results.
- To optimize plant response to red light, maintaining the same gene copy number for PhyB-VP64 and E-PIF6, translation rates for PhyB-VP64 need to be 8-fold higher than E-PIF's.
- Given an importance criteria, the model could be used to tune the LEDs in order to obtain a desired amount of protein and economize energy resources.
PLANT-HUMAN: AND GATE
A constitutive module expresses the integrase-recombinase PhiC31 which recombines the specific attachment sites of the reporter in BP state to LR state, keeping this state because of the constitutive expression.
When the dispenser puts on dexamethasone to the plant, the integrase-excisionase gp3 (also called recombination directionality factor - RDF) expresses through the transitory effect of the dexamethasone. Gp3 with the presence of PhiC31 recombines the specific attachment sites to BP state, then the inducible promoter is addressed to produce the transcription product of the specific color under the respective stress inductor that switch on the expression in the reporter assembly.
For more information, visit the biological design.
DETERMINISTIC
REACTIONS
For this part of Chatterplant we have made two models to obtain information about the performance of the PhiC31 recombinase and Gp3. At first, we decided to produce a simple model to understand how PhiC31 works and how does the reporter assembly perform acts with the presence of PhiC31. We considered these reactions:
Graphic description of the behavior of our PhiC31 simple model with reaction parameters
In the image we can see that PhiC31 (green hexagon) forms dimers and then each dimer joint with our register assembly. In this case, the experiments in lab used luciferase so our model is inspired on them. When PhiC31 is connected with our register, it acts and recombines the sites. Then PhiC31 leaves the register dimer by dimer. We would have expression of our target protein (luciferase) when the sites are recombined (LR).To continue, we show the formal reactions that correspond to the previous image:
Reactions of PhiC31 – register assembly
Next, we show the diagram related to the action of PhiC31 and GP3 at the same time, also we have to consider the reactions that we explain before.
Graphic representation of GP3 actuation
As we can appreciate, a simple model about the GP3 actuation was considered without take into account the interaction between free GP3 and free PhiC31 in intracellular media. Even though we did that assumption, our model provides us with the enough information for our case of study. In order to check this, we compared the behavior of our model with a very complex model (The mechanism of C31 integrase directionality: experimental analysis and computational modelling - Alexandra Pokhilko, Jia Zhao et al., 2016).
GP3 formal reactions
MODEL AND CONSIDERATIONS
In the literature, there are different approaches to model the recombinase-excisionase action. We started modeling theoretically basic supposed reactions to have a general knowledge as a first contact point. Then, studying the experimental results, we could better understand which reactions can be happening inside ChatterPlant.
For the integrase-recombinase action:
- Monomeric PhiC31 doesn’t bind to the reporter, only dimers of PhiC31.
- The recombination performance is irreversible.
For the integrase-excisionase action:
- Excisionase doesn’t dimerize in the cellular medium.
- Excisionase doesn’t form complexes with PhiC31 in the cellular medium.
- We consider cooperativity in the excisionase binding to the reporter.
- The recombination performance is irreversible.
Sensitivity analysis: To continue we are going to show the behavior of our model when we change the proportion between
IN SILICO EXPERIMENTS
Sensitivity analysis: To continue we are going to show the behavior of our model when we change the proportion between
Sensitivity analysis: The biosensor circuit is studied by the PhiC31 – Gp3 performance on the registrer assembly. From the considerations in the reactions, we simulate the dynamic behavior for different concentrations of integrase-recombinase as initial value. The initial value of the registrer in BxP state is 0.1 microM, while the Gp3 is under the expression of a constitutive module.
Figure 15. Temporal evolution of chemical species
As integrase-recombinase increases, the LR state increases, while BP state decreases, because of the recombinase performance on the reporter. As a result, starts the expression of the protein A in the reporter since LR is the on state. As can be observed on the evolution of A, for a greater concentration of recombinase, the production has a delay because of the Gp3 that is pretending to reverse de LR state to de BP state. In this case, finally 100% of the register is in LR state.
CONCLUSIONS
It is crucial to select the concentration of PhiC31 and GP3 and its proportion to obtain a good behavior of our genetic device.We have studied the importance of unproductive complex formed by PhiC31 ans GP3. In the future, we consider that it is important to study the expression of our target protein with a dexamethasone stimulus.
MODELING CONCLUSIONS
We have obtained a general vision of our genetic system, understanding it at all and creating strategies to work with it. We also have optimized our parameter value in the optogenetic circuit, having a model that can predict the protein production depending on the light stimuli.
Taking into account that the experimental data are from plants, it usually is a point in favour because of its difficult extraction and consequently, a difficult optimization and characterization. With the recombinase model, we have obtained a model of a system that nowadays is un study and we have been able to understand how PhiC31 and GP3 works. Also, we think that it would be useful to future studies in this field.
