Team:TorontoMaRSDiscovery/Modeling

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Home The Team The Project Parts Submitted to the Registry Modeling Notebook


Contents

Summary

The objective for this years modelling work is to lay the foundations for future efforts. A model has been constructed using approximate parameter values gathered from online tools and approaches for more the more detailed modelling of eCFP and Encapsulin pathways have been outlined.

Model

The construct that needs to be modelled is shown below. Our submission this year may/may not have all of the parts assebled.

BioBrick.png

The construct is represented by the reactions shown below File:Uft mod Rxns.xls. The reaction framework was mainly generated using the SynBioSS Designer (a really useful modelling tool for iGEM teams), which also provides "default" parameter values. These parameter values have been gathered from various literature sources, and should be taken as approximations on the appropriate order of magnitude. All units were taken to be (1 / molarity^n-1 * s), where n is the order of the reaction. For example, for the reaction 2 lacI --> lacI2, n = 2, thus the rate constant has the units 1/molality*s.

The pathways for the expressed proteins, eCFPt and Encapsulin, need to be modelled in more detail. For example, in this basic model eCFPt is degraded via a first order reaction and Encapsulin makes a dimer complex. However, in reality fluorescent proteins have been shown to degreade via Micheales-menton kinetics and Encapsulin is thought to make a complex of about 60 monomers. These issues will be addressed in future work.


Reaction Rate Constant
Constitutive Expression
--> lacl4 1E-10
--> tetR2 1E-10
Multimerization
2 lacI --> lacI2 1.00E+09
lacI2 --> 2 lacI 10
2 lacI2 --> lacI4 1.00E+09
lacI4 --> 2 lacI2 10
2 tetR --> tetR2 1.00E+09
tetR2 --> 2 tetR 10
2 Enc --> Enc2 100
Encapsulation
Enc2 + eCFPt --> Enc2:eCFPt 100
Transcription
RNAp + BBa_R0010 + LacI_binding_site --> RNAp:BBa_R0010:LacI_binding_site 10000000
RNAp:BBa_R0010:LacI_binding_site --> RNAp + BBa_R0010 + LacI_binding_site 0.057
RNAp:BBa_R0010:LacI_binding_site --> RNAp:BBa_R0010:LacI_binding_site* 0.1
RNAp:BBa_R0010:LacI_binding_site* --> RNAp:DNA_eCFPt + BBa_R0010 + LacI_binding_site 30
RNAp:DNA_eCFPt --> RNAp + mRNA_eCFPt 0.035
RNAp + BBa_R0040 + TetR_1 + TetR_2 --> RNAp:BBa_R0040:TetR_1:TetR_2 10000000
RNAp:BBa_R0040:TetR_1:TetR_2 --> RNAp + BBa_R0040 + TetR_1 + TetR_2 0.057
RNAp:BBa_R0040:TetR_1:TetR_2 --> RNAp:BBa_R0040:TetR_1:TetR_2* 0.1
RNAp:BBa_R0040:TetR_1:TetR_2* --> RNAp:DNA_Enc + BBa_R0040 + TetR_1 + TetR_2 30
RNAp:DNA_Enc --> RNAp + mRNA_Enc 0.0375
Translation
rib + mRNA_eCFPt --> rib:mRNA_eCFPt 100000
rib:mRNA_eCFPt --> rib:mRNA_eCFPt_1 + mRNA_eCFPt 33
rib:mRNA_eCFPt_1 --> rib + eCFPt 0.1154
rib + mRNA_Enc --> rib:mRNA_Enc 100000
rib:mRNA_Enc --> rib:mRNA_Enc_1 + mRNA_Enc 33
rib:mRNA_Enc_1 --> rib + Enc 0.1234
Protein-DNA
lacI4 + nsDNA --> lacI4:nsDNA 1000
lacI4:nsDNA --> lacI4 + nsDNA 1.6225
lacI4 + LacI_binding_site --> lacI4:LacI_binding_site 1.00E+09
lacI4:LacI_binding_site --> lacI4 + LacI_binding_site 0.005
tetR2 + nsDNA --> tetR2:nsDNA 1000
tetR2:nsDNA --> tetR2 + nsDNA 1.6225
tetR2 + TetR_1 --> tetR2:TetR_1 1.00E+09
tetR2:TetR_1 --> tetR2 + TetR_1 0.005
tetR2 + TetR_2 --> tetR2:TetR_2 1.00E+09
tetR2:TetR_2 --> tetR2 + TetR_2 0.005
Protein - Effector - DNA
lacI4 + IPTG --> lacI4:IPTG 50000000
lacI4:IPTG --> lacI4 + IPTG 0.1
lacI4:IPTG + LacI_binding_site --> lacI4:IPTG:LacI_binding_site 1.00E+09
lacI4:IPTG:LacI_binding_site --> lacI4:IPTG + LacI_binding_site 0.7
lacI4:LacI_binding_site + IPTG --> lacI4:IPTG:LacI_binding_site 1000000
lacI4:IPTG:LacI_binding_site --> lacI4:LacI_binding_site + IPTG 0.4
lacI4:IPTG + nsDNA --> lacI4:IPTG:nsDNA 1000
lacI4:IPTG:nsDNA --> lacI4:IPTG + nsDNA 1.6225
lacI4:nsDNA + IPTG --> lacI4:IPTG:nsDNA 1000
lacI4:IPTG:nsDNA --> lacI4:nsDNA + IPTG 1.6225
tetR2 + aTc --> tetR2:aTc 50000000
tetR2:aTc --> tetR2 + aTc 0.1
tetR2:aTc + TetR_1 --> tetR2:aTc:TetR_1 1.00E+09
tetR2:aTc:TetR_1 --> tetR2:aTc + TetR_1 0.7
tetR2:TetR_1 + aTc --> tetR2:aTc:TetR_1 1000000
tetR2:aTc:TetR_1 --> tetR2:TetR_1 + aTc 0.4
tetR2:aTc + TetR_2 --> tetR2:aTc:TetR_2 1.00E+09
tetR2:aTc:TetR_2 --> tetR2:aTc + TetR_2 0.7
tetR2:TetR_2 + aTc --> tetR2:aTc:TetR_2 1000000
tetR2:aTc:TetR_2 --> tetR2:TetR_2 + aTc 0.4
tetR2:aTc + nsDNA --> tetR2:aTc:nsDNA 1000
tetR2:aTc:nsDNA --> tetR2:aTc + nsDNA 1.6225
tetR2:nsDNA + aTc --> tetR2:aTc:nsDNA 1000
tetR2:aTc:nsDNA --> tetR2:nsDNA + aTc 1.6225
Degradation
lacI4 --> 0.000289
tetR2 --> 0.000289
mRNA_eCFPt --> 0.0015
mRNA_Enc --> 0.0015
eCFPt --> 0.000289
Enc2 --> 0.000289
Enc2:eCFPt --> 0.000289
Dilution
lacI4:nsDNA --> nsDNA 0.000193
tetR2:nsDNA --> nsDNA 0.000193
lacI4:IPTG --> IPTG 0.000289
tetR2:aTc --> aTc 0.000289
lacI4:IPTG:nsDNA --> IPTG + nsDNA 0.000193
tetR2:aTc:nsDNA --> aTc + nsDNA 0.000193

Simulations

All simulations were carried out using the SimBiology Matlab Toolbox, which was freely available to iGEM teams. The File:SimBiology project file used for the following simulations can be accessed File:Here.

Accumulation of Proteins and Encapsulin-eCFP complex

Uft mod pic1.png Uft mod pic2.png

It can be observed that the expressed proteins and Encapsulin-eCFP complex have some sort of oscillatory behavior, and that the later inverses the former. The Encapsulin and eCFPt proteins concentrations build up while the complex concentation stays low, and then the complex starts to form while the reservoirs of Encapsulin and eCFPt are depleted.

Addition of IPTG and aTc effectors

Uft mod pic3.png Uft mod pic4.png

IPTG and aTc effectors bind lacI4 and tetR2 respectively, preventing them from inhibiting transcription. Predictably, when IPTG and aTc are added to the system (initial amounts changed from 0 --> 100), we observe higher peaks in protein expresssion and complex formation.

Sensitivity Analysis

It is interesting to note which parameters have the greatest effect on the dynamics of the system. Since the parameters in our model are approximations from literature sources, the most sensitive parameters would be leading candidates to be experimentally determined.

File:Uft mod sa1.png


Future Work

Improvements to the basic model presented above can be separated into three areas; 1) performing experiments to determine parameter values that specific to our system, 2) more detailed modeling of eCFPt behavior, 3) more detailed modeling of Encapsulin assembly.

The experiments required to determine the general transcription and translation parameters need to be further researched, so this will be skipped over for now. Here we will discuss the more interesting areas – the detailed modeling of eCFP and Encapsulin pathways. This will be accomplished by incorporating previous modeling efforts that are applicable to the eCFP and Encapsulin proteins.


Degredation of eCFPt

The expression and degradation of Green Fluorescent Protein (GFP) was modeled previously. This paper noted that GFP must mature before going into fluorescent phase, and GFP degradation was shown to follow M-M kinetics, as depicted below. This paper also outlined how the degradation parameters could be derived for any expression system using only simple fluorescence and optical density measurements.

Uft mod ecfp1.png


The GFP maturation and degradation parameters found in this paper were:

m = 0.0004279 1/s
Vmax = 2E-10 molality/s
Km = 9E-13 molality/s


Assuming eCFPt would behave similarly to eGFP, the following additional reactions would be included in the model:

Maturation
eCFPt --> F_eCFPT 0.0004279
Enc2:eCFPt --> Enc2:F_eCFPt 0.0004279
Encapsulation
Enc2 + F_eCFPt --> Enc2:F_eCFPt 100
Degradation
eCFPt --> Vmax = 2E-10
Km = 9E-13
F_eCFPt --> Vmax = 2E-10
Km = 9E-13


Assembly of Encapsulin microcompartments

Though Encapsulin microcompartment assembly is not well-studied, it has been theorized to resemble the assembly of viral capsids. Thus, we could look to the modeling of viral capsid assembly, a relatively well-studied procedure, as a representation of Encapsulin assembly.

Adam Zlotnick has published models of viral capsid assembly using several different approaches, the most recent of which is based on viral capsid assembly but is general enough to be applied to other examples of spherical polymerization.

Model consists of reactions from an initial monomer subunit leading to a final capsid product. The sequential reactions take into account DeltaG for contact (Kn'), and parameters describing path degeneracy (varies for different intermediates).

Uft mod encap1.png

Simulations from a preliminary incorporation of this approach are shown below. In this version of our basic model, Encapsulin assembles in a series of first order reactions. However, for simplicity, the rate constants are kept identical and do not take path degeneracy into account.


Uft mod enc1.png
Uft mod enc2.png
Uft mod enc3.png


Encapsulin Assembly
2 Enc --> Enc2 1.00E9
2 Enc2 --> Enc4 1.00E9
2 En4 --> Enc8 1.00E9
2 Enc8 --> Enc16 1.00E9
2 Enc16 --> Enc32 1.00E9
2 Enc16 --> Enc64 1.00E9








Encapsulin assembly intermediates are observed to quickly come to a steady state. The last intermediate Enc64 is the most depleted probably because of the formation of Enc64:eCFPt complex. Formation of Enc64:eCFPt then mirrors the production of eCFPt.


The Zlotnick model also describes two important features of spherical polymerization that we should also expect to observe in Encapsulin assembly, kinetic trapping and nucleation. Kinetic trap occurs when the deltaG or initial concentration of monomer subunit is too high, which results in over-initiation of assembly and quickly the reactions become starved for subunits. This is prevented by the inclusion of nucleation, where the initial reactions occur at a slower rate because the first few intermediates in a series of reactions are less stable than downstream products. Essentially, nucleation serves as a “slow first step” that regulates the assembly pathway.

It will be interesting to note how the translational dynamics of Encapsulin interplay with its assembly dynamics. Since our system does not start with any initial concentration of Encapsulin, there will be an optimum rate of production that essentially serves the role of nucleation – avoiding kinetic traps while still maintaining a maximum production of full Encapsulin assemblies.

The general model is presented as:

Uft mod encap2.png


Where u is the initial subunit concentration (Enc in our case), m is the "mth" intermediate species, and N is the number of monomers in the final assembly.

All of the included parameters can be calculated from the four basic parameters; the microscopic, per-contact equilibrium constant (KAcon), the nucleus size (nuc), the nucleation on-rate (fnuc), and the elongation on-rate (felong). The paper describes methods to aquire these parameters from kinetic experiments, and provides examples for N = 12 and 30 unit viral capsid assemblies.