Difference between revisions of "Team:KU Leuven/Modeling/Internal"
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| − | <p>Proteins, mRNA and other metabolites have a turnover rate. They are degraded over time. The degradation rate will be described as proportional to the amount of biomolecules. The coefficient of proportionality d is the degradation constant. Not every molecule has the same degradation rate, since some molecules are more stable than others. We can influence the stability of the molecules. For example, in cell B it is important that there is a fast switch between conditions and a fast turnover of CheZ and RFP is necessary. This is why we add a LVA-tag to these proteins. This tag destabilizes the protein and makes them degradade faster. The degradation rates used in the model are put in the next table: <p> | + | <p>Proteins, mRNA and other metabolites have a turnover rate. They are degraded over time. The degradation rate will be described as proportional to the amount of biomolecules. The coefficient of proportionality d is the degradation constant. Not every molecule has the same degradation rate, since some molecules are more stable than others. We can influence the stability of the molecules. For example, in cell B it is important that there is a fast switch between conditions and a fast turnover of CheZ and RFP is necessary. This is why we add a LVA-tag to these proteins. This tag destabilizes the protein and makes them degradade faster. For TransaminaseB we choose a very high degradation rate, because we did not include degradation terms for the TransaminaseB bound to substrate. The degradation rates used in the model are put in the next table: <p> |
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Revision as of 17:06, 16 September 2015
Internal Model
1. Introduction
We can think of many relevant questions in implementing a new circuit: how sensitive is the system, how much will it produce and will it affect the growth? As such, it is important to model the effect of the new circuits on the bacteria. This will be done in the Internal Model. We will use two approaches. First we will use a bottom-up approach. This involves building a detailed kinetic model with rate laws. We will use Simbiology and ODE's to study the sensitivity and dynamic processes inside the cell. This is the bottom-up approach. Afterwards, a top-down model, Flux Balance Analysis (FBA), will be used to study the steady-state values for production flux and growth rate. This part is executed by the iGEM Team of Toulouse as part of a collaboration and can be found here
2. Simbiology and ODE
In the next section we will describe our Simbiology model. Simbiology allows us to calculate systems of ODE's and to visualize the system in a diagram. It also has options to make scans for different parameters, which allows us to study the effect of the specified parameter. We will focus on the production of leucine, Ag43 and AHL in cell A and the changing behavior of cell B due to changing AHL concentration. In this perspective, we will make two models in Simbiology: one for cell A and cell B. First we will describe how we made the model and searched for the parameters. Afterwards we check the robustness of the model with a parameter analysis and we do scans to check for the effects of molecular noise.
3. Quest for parameters
We can divide the different processes that are being executed in the cells in 7 classes: transcription, translation, DNA binding, complexation and dimerization, protein production kinetics, degradation and diffusion. We went on to search the necessary parameters and descriptions for each of these categories. To start making our model we have to chose a unit. We choose to use molecules as unit because many constants are expressed in this unit and it allows us to drop the dillution terms connected to cell growth. We will also work with a deterministic model instead of a stochastic model. A stochastic model will show us the molecular noise, but we will check this with parameter scans.
The next step is to make some assumptions:
- The effects of cell division can be neglected
- The substrate pool can not be depleted and the concentration (or amount of molecules) of substrate in the cell is constant
- The exterior of the cell contains no leucine at t=0 and is perfectly mixed
- Diffusion happens independent of cell movement and has a constant rate
3.1 Transcription
Transcription is the first step of gene expression. It involves the binding of RNA polymerase to the promoter region and the formation of mRNA. The transcription rate is dependent on a number of conditions like the promoter strength and the used strain. Our system has both constitutive promoters, which are always active, and inducable promoters, which can be activated or repressed.
3.2 Translation
The next step in gene expression is the translation of the synthesized mRNA to a protein. This involves the binding of a ribosome complex to the RBS site of the mRNA, elongation and termination. The initiation step is the rate-limiting step of the three. The translation rate is thus tied to the thermodynamics of RBS-ribosome interaction but other sequences (5’-UTR, Shine-Delgarno,repeats, internal startcodons) and possible secondary structures (hairpins) also play a role (Chiam Yu Ng, Salis). (Fong) Salis et al have developed an algorithm to predict the translation initiation rate based upon thermodynamic calculations. This algorithm is used in the Ribosome Binding Site Calculator (https://salislab.net/software/ , Borujeni & Salis). We used this Calculator to predict the translation rates of our mRNAs.
3.3 Complexation and dimerization
Before the proteins can bind the promoter region, they first have to make complexes. cI and penI form homodimers, while LuxR first forms a heterodimer with AHL and afterwards forms a homodimer with another LuxR/AHL dimer. This next part will describe the kinetics of such complexation. We only need the parameters of LuxR and cI, because the Hill function found for penI already was adapted for the protein in monomer form.
| Biomolecule | Complexation rate | Source |
|---|---|---|
| LuxR/AHL association | $1{\cdot}10^{-5}$ 1/(molecule*s) | Goryachev et al. (2005) |
| LuxR/AHL dissociation | 0.00333 1/s | Goryachev et al. (2005) |
| LuxRdimer association | $1{\cdot}10^{-5}$ 1/(molecule*s) | Goryachev et al. (2005) |
| LuxRdimer dissociation | 0.01 1/s | Goryachev et al. (2005) |
| cI dimer association | 0.00147 1/(molecule*s) | iGEM Aberdeen 2009 |
| cI dimer dissociation | 0.01 1/s | Bernstein et al. (2002) |
3.4 Protein production kinetics
Our models comprise two product forming enzymes: LuxI and Transaminase B.
LuxI is the enzyme responsible for AHL production and Transaminase B is responsible for Leucine formation.
For LuxI kinetics Schaefer et al. (1996) found a value of 1 molecule/minute for the Vmax. Since we assume that LuxI is fully saturated with substrate, we take this value as the synthesis rate of AHL.
Transaminase B forms leucine from aKIC and glutamate. In this reaction glutamate is converted to aKG. The reaction kinetics is of the ping-pong bi bi form. This model describes a mechanism in which the binding of substrates and release of products is ordered. The enzyme shuttles between a free and a substrate-modified intermediate state.
Ping-Pong Bi-Bi equations
\begin{align} \frac{{\large d}{TB}}{d t}= & \beta_{TB} {\cdot} {m_{ilvE}} - {kf}_{1}{\cdot}{TB}{\cdot}{Glu} + {kf}_{-1}{\cdot}{[TB-GLU]} - {kr}_1{\cdot}{Leucine}_{in}{\cdot}{[TB-GLU]} \\\\ & + {kr}_{-1}{\cdot}{[TB-Leu]} + {kcat2}{\cdot}{[{TBNH}_2-aKIC]} + {kcat4}{\cdot}{[{TBNH}_2-aKG]} - d_{TB}{\cdot}{TB} \end{align} $$\frac{{\large d}{[TB-GLU]}}{d t}= -{kcat1}{\cdot}{[TB-GLU]} + {kf}_{1}{\cdot}{TB}{\cdot}{Glu} - {kf}_{-1}{\cdot}{[TB-GLU]} $$ \begin{align} \frac{{\large d}{[{TBNH}_2]}}{d t}= & {kcat1}{\cdot}{[TB-GLU]} + {kcat3}{\cdot}{[TB-Leu]}- {kf}_{2}{\cdot}{{TBNH}_2}{\cdot}{aKIC} + {kf}_{-2}{\cdot}{[{TBNH}_2-aKIC]} \\\\ & - {kr}_2{\cdot}{{TBNH}_2}{\cdot}{aKG} +{kr}_{2}{\cdot}{[{TBNH}_2-aKG]} \end{align} $$\frac{{\large d}{[{TBNH}_2-aKIC]}}{d t}= -{kcat2}{\cdot}{[{TBNH}_2-aKIC]} + {kf}_{2}{\cdot}{{TBNH}_2}{\cdot}{aKIC} - {kf}_{-2}{\cdot}{[{TBNH}_2-aKIC]} $$ $$\frac{{\large d}{[TB-Leu]}}{d t}= {kr}_1{\cdot}{Leucine}_{in}{\cdot}{[TB-GLU]} - {kr}_{-1}{\cdot}{[TB-Leu]} - {kcat3}{\cdot}{[TB-Leu]} $$ $$\frac{{\large d}{[{TBNH}_2-aKG]}}{d t}= {kr}_2{\cdot}{{TBNH}_2}{\cdot}{aKG} - {kr}_{2}{\cdot}{[{TBNH}_2-aKG]} - {kcat4}{\cdot}{[{TBNH}_2-aKG]}$$
Yang et al. (2005) studied these reactions and found a method to estimate the constants. Their constants of this reversible Ping-Pong Bi-Bi reactions are put in the next table :
| Constant | Value |
|---|---|
| ${kf}_{1}$ | 0.0098 |
| ${kf}_{-1}$ | 3300 |
| ${kf}_{2}$ | 0.0882 |
| ${kf}_{-2}$ | 5940 |
| ${kr}_{1}$ | 0.0031184 |
| ${kr}_{-1}$ | 4620 |
| ${kr}_{2}$ | 0.0041161 |
| ${kr}_{-2}$ | 3465 |
| $kcat1$ | 33.33 |
| $kcat2$ | 60 |
| $kcat3$ | 46.67 |
| $kcat4$ | 35 |
3.5 Degradation
Proteins, mRNA and other metabolites have a turnover rate. They are degraded over time. The degradation rate will be described as proportional to the amount of biomolecules. The coefficient of proportionality d is the degradation constant. Not every molecule has the same degradation rate, since some molecules are more stable than others. We can influence the stability of the molecules. For example, in cell B it is important that there is a fast switch between conditions and a fast turnover of CheZ and RFP is necessary. This is why we add a LVA-tag to these proteins. This tag destabilizes the protein and makes them degradade faster. For TransaminaseB we choose a very high degradation rate, because we did not include degradation terms for the TransaminaseB bound to substrate. The degradation rates used in the model are put in the next table:
3.6 Diffusion
Our model has 2 types of diffusion. Diffusion from the inside of the cell to the outside over the cell membrane and diffusion in the external medium. The diffusion over the cell membrane is more complicated because some proteins play a role in it and the membrane is not equally permeable for every molecule. For AHL, we found a value given in molecules/second. This unit seems strange because diffusion is usually used with units in concentration. It also leads to strange results since the amount of molecules is being leveled out so the inside amount equals the outside amount even though the concentrations are different. This is why we added a correction for the volume of the cell and the external volume to it.
4. System
4.1 Cell A
The designed circuit in Cell A is under control of a temperature sensitive cI repressor. Upon raising the temperature, cI will dissociate from the promoter and the circuit is activated. This leads to the initiation of the production of LuxR and LuxI. LuxI will consecutively produce AHL, which binds with LuxR. The newly formed complex will then activate the production of Leucine and Ag43. Leucine and AHL are also able to diffuse out of the cell into the medium. Ag43 is the adhesine which aids the aggregation of cells A, while Leucine and AHL are necessary to repel cells B.
We can extract the following ODE's from this circuit:
Cell A equations
Symbols:${}$ ${\alpha}$: transcription term, ${\beta}$: translation term, $d$: degradation term,
$D$: diffusion term, ${ K_d}$: dissociation constant, n: Hill coefficient, L: leak term
$$\frac{{\large d} m_{cI}}{d t} = \alpha_1 {\cdot} cI_{gene} - d_{mCI} {\cdot} m_{cI}$$ \begin{align} \frac{{\large d}{cI}}{d t} = \beta_{cI} {\cdot} {m_{cI}} -2 {\cdot} {k_{cI,dim}} {\cdot} {cI}^2 + 2 {\cdot} {k_{-cI,dim}}{\cdot} {[cI]_2} - d_{cI} {\cdot} {cI} \end{align}
We visualize these ODE's in the Simbiology Toolbox which results in the following diagram:
4.2 Cell B
The system of Cell B is also under control of the cI repressor and is activated similar as cell A. The activation by the temperature raise, leads to the production of LuxR. AHL of the medium can diffuse into the cell, binding LuxR and activating the next component of the circuit. This leads to the production of CheZ and PenI. CheZ is the protein responsible for cells to make a directed movement, governed by the repellent Leucine. PenI is a repressor which will shut down the last part of the circuit which was responsible for the production of RFP.
We can extract the following ODEs for Cell B from this sytem:
Cell B equations
Symbols: ${\alpha}$:transcription term, ${\beta}$:translation term, $d$:degradation term,
$D$:diffusion term, ${ K_d}$:dissociation constant, n:Hill coefficient, L:leak term
$$\frac{{\large d} m_{cI}}{d t} = \alpha_1 {\cdot} cI_{gene} - d_1 {\cdot} m_{cI}$$ $$\frac{{\large d}{cI}}{d t} = \beta_1 {\cdot} {cI} -2 {\cdot} {k_{cI,dim}} {\cdot} {cI}^2 + 2 {\cdot}{k_{-cI,dim}}{\cdot} {[cI]_2} - d_{cI} {\cdot} {cI} $$
We visualize these ODE's in the Simbiology toolbox. This gives us the following diagrams:
5. Results
Cell A graph of all, graph of Leucine, graph of AHL
Cell B graph of all, graph with induction and without induction
Sensitivity analysis
Conclusion and discussion
