From 2009.igem.org

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Summary

A kinetic model of our system has been constructed. Approximate parameter values were gathered from online resources to represent all processes up to the production of the eCFP and Encapsulin proteins. Current simulations predict the production of these two proteins at various initial concentrations of effector molecules. Furthermore, we have outlined approaches for the detailed modelling of the eCFP degredation/maturation and Encapsulin assembly pathways.

The initial simulations will be used to identify the critical concentration of Encapsulin mononomer required for microcompartment assembly. Subsequently, the detailed model of Encapsulin assembly will be integrated into the current model, and its parameters filled-in using parameter-scanning methods and experiments outlined in the relevant references. By comparing experimental measures of microcompartment formation with that predicted by the kinetic model, we aim to gain insight on the mechanisms of Encapsulin assembly.

Model

Our model is based on the following design (See Project Page for a full description of this system).

The reaction framework was mainly generated using the [http://synbioss.sourceforge.net/ SynBioSS Designer (a really useful modeling tool for iGEM teams)], which also provides default parameter values. These parameter values have been gathered from various literature [http://neptune.cems.umn.edu/designer/designer_defaults.pdf sources], and should be taken as approximations. 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 modeled 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.

The reactions can be accessed here:

Model (opens new page)

Model (download Excel file)

Simulations

All simulations were carried out using the SimBiology Matlab Toolbox, which was freely available to iGEM teams.

Accumulation of eCFP at Different Initial Concentrations of IPTG

The IPTG effector eventually loses effect, and the eCFP concentration approaches a maximum. This happens because the IPTG proteins eventually tie up all available LacI4 proteins, leaving the transcription and translation of eCFP to operate at its maxiumum rate.

Accumulation of Encapsulin monomer at Different Initial Concentrations of aTc

(NOTE: in order to observe Encapsulin monomer levels, the above simulations were carried out without any Encapsulin assembly reactions in place)

The production of Encapsulin occurs at a slower rate than eCFP because the BBa_R0040 promoter has two tetR binding sites, resulting in more repression and more off-pathway scenarios.

The Encapsulin concentration also seems assymptotic but at higher concentrations of aTc (relative to IPTG).

It is beneficial to have the Encapsulin protein under control of the BBa_R0040-tetR-aTc construct because at zero levels of aTc we can have essentially zero levels of Encapsulin (whereas even at zero levels of IPTG, there is still significantly more eCFP). This will enable us to probe extremely low concentrations of Encapsulin monomer when testing for optimal Encapsulin assembly conditions. The disadvantage is that if Encapsulin assembly requires a very high initial concentration of Encapsulin monomer, we would have to wait a long time to see microcompartments. If this later scenario is observed, it may be better to switch the eCFP and Encapsulin expression systems.

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.

(Currently, the SimBiology Toolbox reports an "OUT OF MEMORY" error when running a sensitivity analyis on our system).

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 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=95514&blobtype=pdf 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.

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 [http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=1302223&blobtype=pdf 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).

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.

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 shown on the right:

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.

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