@@ -473,48 +473,38 @@ At first glance, this may imply that the _V. natriegens_ model is more sensitive
### Implications of Diffusion Modelling
Another important aspect of the mathematical model was gaining some insight into the distribution of our proteins of interest - **\*\***Staygold**\*\*** and **\*\***IL-10**\*\***, inside a potential bioreactor.
Another important aspect of the mathematical model was gaining some insight into the distribution of our proteins of interest - ******Staygold****** and ******IL-10******, inside a potential bioreactor.
Diffusion modelling is particularly useful in Scale-Up and Scale-Down studies; by modelling diffusion, we are able to predict the concentration of our proteins of interest as they change over time and space, which in turn provides the basis for calculations in transition from lab scale to industrial scale bioprocesses. Furthermore, it would also guide an industrial bioreactor’s geometry, identify potential regions of protein aggregation and optimize resource utilization.
Diffusion modelling is particularly useful in Scale-Up and Scale-Down studies; by modelling diffusion, we are able to predict the concentration of our proteins of interest as they change over time and space, which in turn provides the basis for calculations in transition from lab scale to industrial scale bioprocesses. Furthermore, it would also guide an industrial bioreactor’s geometry, identify potential regions of protein aggregation and optimize resource utilization.
### Setting up the Model
For this investigation, we modelled a stir rod with radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg" /></span> in the relative center of the bioreactor. A region of bacterial abundance surrounds it between radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg" /></span> and <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/b.svg" /></span>, which will be considered the \***\*\*\*\*\*\*\***source\***\*\*\*\*\*\*\*** of our two proteins of interest (**\*\***\*\*\*\***\*\***Staygold**\*\***\*\*\*\***\*\*** and \***\*\*\*\*\***IL-10\***\*\*\*\*\***). Finally, the surface of the interior surface of the bioreactor, would be where the proteins would exit the system from through a filter, acting as a **\*\***\*\***\*\***target/sink**\*\***\*\***\*\***. Our goal is the model the diffusion of **Staygold** and **IL-10** from the source to the sink, by generating a concentration profile on MATLAB, and investigating the effects of varying cell radius, diffusion coefficient and basal absorption rate.
For this investigation, we modelled a stir rod with radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg" /></span> in the relative center of the bioreactor. A region of bacterial abundance surrounds it between radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg" /></span> and <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/b.svg" /></span>, which will be considered the ************source************ of our two proteins of interest (****************Staygold**************** and **********IL-10**********). Finally, the surface of the interior surface of the bioreactor, would be where the proteins would exit the system from through a filter, acting as a **************target/sink**************. Our goal is the model the diffusion of **Staygold** and **IL-10** from the source to the sink, by generating a concentration profile on MATLAB, and investigating the effects of varying cell radius, diffusion coefficient and basal absorption rate.
caption="Graphical Representation of the Bioreactor System"
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<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled.png" num="13" caption="Graphical Representation of the Bioreactor System" alternate/>
_Figure 1_ shows a diagram of the system of interest, where the system is being modelled in cylindrical coordinates. <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg" /></span> is defined as the radius of the stir rod, while <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/b.svg" /></span> is defined as the radius of the bacterial abundance. Both these regions are being modelled as concentric cylinders. Following the log phase of growth, the transformed cells are likely to begin ejecting excess the split inteins into its surroundings at steady-state. Hence, we may set **Ci** as the initial concentration of the protein obtained at the surface of the stir rod, and consider it to be a perfectly reflective boundary (******\*\*\*\*******\*\*\*\*******\*\*\*\*******boundary condition******\*\*\*\*******\*\*\*\*******\*\*\*\*******). For this design of the bioreactor, we are assuming a concentric filter at radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/c.svg" /></span> allowing for perfect absorption of the protein of interest, making the concentration of the proteins in this region 0 (********\*\*********\*\*********\*\*********boundary condition********\*\*********\*\*********\*\*********).
*Figure 1* shows a diagram of the system of interest, where the system is being modelled in cylindrical coordinates. <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg" /></span> is defined as the radius of the stir rod, while <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/b.svg" /></span> is defined as the radius of the bacterial abundance. Both these regions are being modelled as concentric cylinders. Following the log phase of growth, the transformed cells are likely to begin ejecting excess the split inteins into its surroundings at steady-state. Hence, we may set **Ci** as the initial concentration of the protein obtained at the surface of the stir rod, and consider it to be a perfectly reflective boundary (************************************boundary condition************************************). For this design of the bioreactor, we are assuming a concentric filter at radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/c.svg" /></span> allowing for perfect absorption of the protein of interest, making the concentration of the proteins in this region 0 (**************************************boundary condition**************************************).
Having set up our boundary conditions, we may now make some general assumptions:
- Considering the region of bacterial abundance of the bioreactor to be the solitary region producing our proteins of interest.
- Assuming a constant, perfect efficiency filter at radius <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/c.svg" /></span> that has no protein aggregation.
- The ******\*\*******\*\*******\*\*******Staygold******\*\*******\*\*******\*\******* and ******\*\*******\*\*******\*\*******IL-10******\*\*******\*\*******\*\******* are assumed to be moving purely through passive diffusion between the source (region of bacterial abundance) and the sink (filter).
- No ******\*\*******\*\*******\*\*******Staygold******\*\*******\*\*******\*\******* and ******\*\*******\*\*******\*\*******IL-10******\*\*******\*\*******\*\******* is being lost to internal bacterial metabolic processes.
- No ******\*\*******\*\*******\*\*******Staygold******\*\*******\*\*******\*\******* and ******\*\*******\*\*******\*\*******IL-10******\*\*******\*\*******\*\******* is being lost to protein degradation.
- The ******************************Staygold****************************** and ******************************IL-10****************************** are assumed to be moving purely through passive diffusion between the source (region of bacterial abundance) and the sink (filter).
- No ******************************Staygold****************************** and ******************************IL-10****************************** is being lost to internal bacterial metabolic processes.
- No ******************************Staygold****************************** and ******************************IL-10****************************** is being lost to protein degradation.
- The host bacteria’s endogenous activity is being ignored.
With these assumptions in place, we are ready to begin deriving a concentration profile for our system.
With these assumptions in place, we are ready to begin deriving a concentration profile for our system.
#### Deriving the Steady-State Concentration Profile of Our Proteins in our Diffusion System
We begin with the multi-dimensional diffusion equation at steady state (in cylindrical coordinates). The <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/s-r.svg"/></span> function incorporates the bacterial region’s basal production of ******\*\*******\*\*******\*\*******Staygold******\*\*******\*\*******\*\******* and ******\*\*******\*\*******\*\*******IL-10******\*\*******\*\*******\*\*******.
We begin with the multi-dimensional diffusion equation at steady state (in cylindrical coordinates). The <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/s-r.svg"/></span> function incorporates the bacterial region’s basal production of ******************************Staygold****************************** and ******************************IL-10******************************.
We will let <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg"/></span> be the location of the outer edge of the stir rod. <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/b.svg"/></span> will be the edge of the region of bacterial abundance, and <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/c.svg"/></span> will be the inner surface of the protein filter. This is summarized below, in \***\*\*\*\***Figure 2\***\*\*\*\***:
We will let <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/a.svg"/></span> be the location of the outer edge of the stir rod. <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/b.svg"/></span> will be the edge of the region of bacterial abundance, and <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/c.svg"/></span> will be the inner surface of the protein filter. This is summarized below, in *********Figure 2*********:
<Figure
num="14"
caption="Cylindrical view of the system of interest. The boundary conditions have been incorporated into the system."
<Figure num="14" caption="Cylindrical view of the system of interest. The boundary conditions have been incorporated into the system." alternate src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-1.png"/>
#### Finding the Steady-State Concentration of ACC Molecules
We can integrate this function over the volume of interest to find the total number of protein molecules (******\*\*******\*\*******\*\*******Staygold******\*\*******\*\*******\*\******* and ******\*\*******\*\*******\*\*******IL-10)******\*\*******\*\*******\*\*******, <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/u.svg"/></span>, between the cell nucleus and the rhizosphere. Below, <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/h.svg"/></span> denotes the height of the cell.
We can integrate this function over the volume of interest to find the total number of protein molecules (******************************Staygold****************************** and ******************************IL-10)******************************, <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/u.svg"/></span>, between the cell nucleus and the rhizosphere. Below, <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/h.svg"/></span> denotes the height of the cell.
We can divide <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/math31.svg"/></span> by the volume (<span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/u.svg"/></span>) of interest to obtain the average concentration of protein (******\*\*******\*\*******\*\*******Staygold******\*\*******\*\*******\*\******* and ******\*\*******\*\*******\*\*******IL-10)******\*\*******\*\*******\*\******* throughout the region.
We can divide <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/math31.svg"/></span> by the volume (<span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/u.svg"/></span>) of interest to obtain the average concentration of protein (******************************Staygold****************************** and ******************************IL-10)****************************** throughout the region.
Concentration profile of the <strong>StayGold</strong>. The diffusion
constant for the protein was calculated using GROMACS from the protein
modelling sub team, and the plots were generated in MATLAB. Interestingly,
we see the concavity of the function switch from concave up to concave
down at the edge of bacterial abundance, which is where the basal rate of
production turns off.
</>
}
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<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-2.png" num="15" caption={<>Concentration profile of the <strong>StayGold</strong>. The diffusion constant for the protein was calculated using GROMACS from the protein modelling sub team, and the plots were generated in MATLAB. Interestingly, we see the concavity of the function switch from concave up to concave down at the edge of bacterial abundance, which is where the basal rate of production turns off.</>}/>
Concentration profile of the <strong>IL-10</strong> proteins. The
diffusion constant for the protein was calculated using GROMACS from the
protein modelling sub team, and the plots were generated in MATLAB.
Interestingly, we see the concavity of the function switch from concave up
to concave down at the edge of bacterial abundance, which is where the
basal rate of production turns off.
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-3.png" num="16" caption={<>Concentration profile of the <strong>IL-10</strong> proteins. The diffusion constant for the protein was calculated using GROMACS from the protein modelling sub team, and the plots were generated in MATLAB. Interestingly, we see the concavity of the function switch from concave up to concave down at the edge of bacterial abundance, which is where the basal rate of production turns off.</>}/>
#### Exploring Concentration Response to Media Conditions by Varying _D_
#### Exploring Concentration Response to Media Conditions by Varying *D*
Since the value for the Diffusion Constant _D_ was obtained from a source with a high degree of uncertainty, the team found it useful to visualize the maximum and minimum deviations to our standard concentration profile due to the error within D. This result is presented below:
Since the value for the Diffusion Constant *D* was obtained from a source with a high degree of uncertainty, the team found it useful to visualize the maximum and minimum deviations to our standard concentration profile due to the error within D. This result is presented below:
Concentration profile of <strong>StayGold</strong> at the edges of their
uncertainty values, as predicted by GROMACS.{" "}
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-4.png" num="17" caption={<>Concentration profile of <strong>StayGold</strong> at the edges of their uncertainty values, as predicted by GROMACS. </>}/>
Concentration profile of <strong>IL-10</strong> at the edges of their
uncertainty values, as predicted by GROMACS.{" "}
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-5.png" num="18" caption={<>Concentration profile of <strong>IL-10</strong> at the edges of their uncertainty values, as predicted by GROMACS. </>}/>
We also conducted Sensitivity analysis of a varying Diffusion coefficient, plotting changes to the concentration profile by varying the GROMACS value by up to 50% in the positive and negative directions. The results are plotted below:
We also conducted Sensitivity analysis of a varying Diffusion coefficient, plotting changes to the concentration profile by varying the GROMACS value by up to 50% in the positive and negative directions. The results are plotted below:
Sensitivty analysis profiles of <strong>StayGold</strong> at the edges of
their uncertainty values, as predicted by GROMACS.{" "}
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-6.png" num="19" caption={<>Sensitivty analysis profiles of <strong>StayGold</strong> at the edges of their uncertainty values, as predicted by GROMACS. </>}/>
Sensitivty analysis profiles of <strong>IL-10</strong> at the edges of
their uncertainty values, as predicted by GROMACS.{" "}
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-7.png" num="20" caption={<>Sensitivty analysis profiles of <strong>IL-10</strong> at the edges of their uncertainty values, as predicted by GROMACS. </>}/>
#### Exploring Concentration Response to Filter Efficiency by Varying <span><img src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/c.svg" /></span>
Optimal filter efficiency is essential to the our model assumptions, preventing protein aggregation near the edge of the bioreactor at steady state, making the filter a perfect absorber. However, it is likely that under sustained use the filter would lose efficiency over time, and hence the team found it useful to model the weakening filter’s efficiency as an increase in distance to the protein filter in our diffusion model : the idea being that a weaker filter is analogous to a perfect sink existing further away from the region of bacterial abundance. The results from this analysis are plotted below:
Optimal filter efficiency is essential to the our model assumptions, preventing protein aggregation near the edge of the bioreactor at steady state, making the filter a perfect absorber. However, it is likely that under sustained use the filter would lose efficiency over time, and hence the team found it useful to model the weakening filter’s efficiency as an increase in distance to the protein filter in our diffusion model : the idea being that a weaker filter is analogous to a perfect sink existing further away from the region of bacterial abundance. The results from this analysis are plotted below:
Concentration profiles of <strong>StayGold</strong> with varying filter
distance. It appears that a inefficient filter produces higher levels of
protein near the stir rod and in the region of bacterial abundance at
steady state.
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-8.png" num="21" caption={<>Concentration profiles of <strong>StayGold</strong> with varying filter distance. It appears that a inefficient filter produces higher levels of protein near the stir rod and in the region of bacterial abundance at steady state.</>}/>
Concentration profiles of <strong>IL-10</strong> with varying filter
distance. It appears that a inefficient filter produces higher levels of
protein near the stir rod and in the region of bacterial abundance at
steady state.
</>
}
/>
<Figure src="https://static.igem.wiki/teams/4796/wiki/mathmodelling/untitled-9.png" num="22" caption={<>Concentration profiles of <strong>IL-10</strong> with varying filter distance. It appears that a inefficient filter produces higher levels of protein near the stir rod and in the region of bacterial abundance at steady state.</>}/>
### Conclusion
These findings provide a general idea of what the concentration profiles of our proteins of interest would look like inside a bioreactor. Follow up calculations could be to experimentally determine protein concentrations that negatively impact the bacterial populations producing the split inteins, before using these values as a upper bound to start constraining the model (for example, to calculate the optimal filter distance within the upper limit of the concentration profile).
