Team:UofC Calgary/Model
Model
Best Model Websites
https://2015.igem.org/Team:Waterloo/Modeling
https://2015.igem.org/Team:Oxford/Modeling
https://2015.igem.org/Team:Czech_Republic/Software
Look at “Modelling Story” in Google Drive for ideas
*Talk to Syed (for a modelling page) and Christine (for layout)
Table of Contents
- Introduction
- Growth Curves
- Diffusion Model
- Production Rate
- Degradation Rate
- Summary
- Improvements
Introduction
How long can a patch last on an astronaut?
How much radioactive peptide will be diffused through in the astronaut’s system?
When should the astronaut pop the extra media packet?
These were the types of questions we were able to answer by using a combination of lab experiments and mathematical models. These computative/analytical models / modelling / numerical techniques were used to evaluate how realistic our patch is, how beneficial it would be in space applications and where to make improvements, so that, one day, we can make our project a reality.
Growth Curves
Our first step was to determine the lifetime of our bacteria, Bacillus subtilis, as it would determine the nature of how our peptide would be produced how long the peptide would be produced for, and therefore, the lifetime of our patch. This was done through experimental growth curves:
Figure 1: Raw data of growth curves at different temperatures. Three replicates of growth at various temperatures was measured every hour over a period of 24 hours. Optical density was measured by spectrophotometry at a wavelength of 600 nm (OD 600). The most relevant growth curve is the one in orange, as this is the average temperature of the surface of the skin.
The growth curves revealed that B. subtilis, under expected conditions, underwent death phase after 16-18 hours. This established the process when the packets of super-rich media in our patch needed to be added to increase the bacteria’s lifetime. For ease of remembrance for the user, we decided that the packets should be popped every 12 hours.
To simulate our patch, another experiment was conducted in which a packet of super rich media was added to the bacteria every 12 hours. This was in triplicates as there are three packets in our patch. Through this, we determined that the bacteria’s lifetime in our patch is 60 hours (about 2.5 days). Since B. subtilis begins to release toxic material that may pass into the skin during its death phase (i.e. after 60 hours), we determined that the patch must be disposed of at this time.
The complete experiments can be found here (link).
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Diffusion Model
MATLAB was used to develop a diffusion model to numerically represent the diffusion of our peptide, Bowman–Birk protease inhibitor (BBI), from the patch, through the skin and into the blood. This diffusion model was developed with the hopes to answer questions, such as:
- Does the peptide reach a constant concentration in the blood while the patch is on the user?
- Literature values show that the minimum required amount of peptide needed for radioprotection is 10 micromolar (link). Does the concentration in the blood reach 10 micromolar or higher?
- How long does it take for the concentration in the blood to reach zero after the patch has been removed?
The following is a visual representation of what the system looks like.
Figure 2: A visual representation of the diffusion model system. Note that the main barriers in this diffusion system are the patch’s rate-controlling membrane and the skin. The skin was assumed to be one homogeneous layer (other factors, such as accounting differences in cellular structure of the skin, any absorption by cells and sweat ducts, enzymes, etc. were not considered).
To start developing our model, Fick’s first and second law were used:
J = -D dCdx (1)
dCdt = D d2Cdx2 (2)
Where:
J = concentration flux (molm2s)
D = diffusion coefficient (m2s)
C = concentration (molm3)
x = distance (m)
t = time (s)
We were able to develop three equations (one for each section - see Figure 2) which were used to determine the concentration in the patch, C0, the concentration in the adhesive, C1, and the concentration in the blood, C2. These equations were developed on the basis that the change in concentration over time for a block is defined as the flux of peptide entering the block minus the flux of peptide leaving the block. To ensure that the units match, the flux variables were multiplied by the cross-sectional area and divided by the respective volume of the section.
dC0dt=production rate - J1 (3)
= production rate - (-Dm (C1-C0) xmx-areaVpatch)(4)
dC1dt= J1 -J2 (5)
= (-Dm (C1-C0) xmx-areaVadhesive) - (-Ds (C2-C1) xsx-areaVadhesive) (6)
dC2dt= J2 - degradation rate (7)
= (-Ds (C2-C1) xsx-areaVblood)- degradation rate(8)
Where the values are the following:
Table 1: Values Used In Equations (4), (6) and (8)
|
Parameter |
Value |
Where Value Is Obtained From |
|
Cross sectional area, x-area (m2) |
4.9 10-3 |
Patch is 7 cm by 7 cm |
|
Diffusion coefficient of membrane, Dm(m2s) |
? |
|
|
Thickness of membrane, xm (m) |
50.8 10-6 |
3M information sheet |
|
Volume of patch,Vpatch (m3) |
1 10-5 |
10 mL |
|
Diffusion coefficient of skin, Ds (m2s) |
1 10-10 |
Based on 1-slab model from analytical model from Nelly |
|
Thickness of skin, xs (m) |
0.001 |
Ask Tiffany? |
|
Volume of adhesive,Vadhesive (m3) |
100 10-6 |
Cross - sectional area thickness of adhesive (estimated to be 100 micrometers) |
|
Volume of blood,Vblood (m3) |
0.005 |
Volume of blood ranges from 4.7 L to 5 L (according to Google) |
*An important note is that we were not able to find literature values for the diffusion of our peptide, BBI, through our patch’s size controlling membrane and the skin. For this reason, we used literature values for ____, to represent the diffusion of BBI through the skin. For the diffusion coefficient through the size-controlling membrane, which works .
Although this does not produce data that exactly represents the diffusion process of BBI in our system, it does reveal the general pattern of how it would diffuse. This was important to analyze as well.
The production rate is the amount of peptide (BBI) produced by the bacteria (B. subtilis) in a given period of time.
The degradation rate is the amount of peptide lost in a given period of time through enzyme degradation in the liver and excretion through the kidneys. The equation to represent the degradation rate is determined to be:
degradation rate = kC2 (9)
Where:
k= decay constant (s-1)
To determine the decay constant, the exponential decay equation was used:
N(t) = N0e-kt (10)
Rearranging equation (10) for k:
k = ln (N(t)N0)-t (11)
Half-life conditions for the peptide were used in equation (11) to solve for k. The values used were the following:
Table 2: Values Used in Equation (11)
|
Parameter |
Value |
Where Value Is Obtained From |
|
Time, t ( s) |
16200 |
The half life of the peptide |
|
(Concentration at time t, N(t) Initial concentration, N0 ) |
0.5 |
It doesn’t matter what the initial concentration is… at the half life, this ratio is always 0.5? |
After solving for k, we found that its value was 4.2786863 10-5 s-1. Thus, equation (9) can be rewritten to be:
degradation rate = 4.2786863 10-5 s-1C2 (12)
Now that we have determined both the production and degradation rate, we could plug in these values, as well as table 1, into equations (4), (6) and (8):
(13)
(14)
(15)
These equations were then solved through MATLAB by using the following codes (link?/download code). The following graphs for the concentration in each section vs time for the different locations were determined:
Figure 3: Concentration of BBI in the patch vs time
Figure 4: Concentration of BBI in the adhesive vs time
Figure 5: Concentration of BBI in the blood vs time
From these graphs, we were able to make the following conclusions to our initial problems:
- The peptide does reach a constant concentration in the blood while the patch is on the user; from Figure 3, we can also determine exactly at what time this happens (from Figure 3, it shows this happens at ___. However, recall that this number does not represent when BBI reaches equilibrium in the system).
- We are also able to determine whether the peptide will reach 10 micromolar (the radioprotection minimum). From Figure 3, it shows that the peptides reaches a maximum of _____, and since this is greater than 10 micromolar, it means that the patch will provide radioprotection for the user.
- From Figure 3, we are also able to determine that it takes roughly ____ hours for the concentration of peptide in the blood to reach zero (or 10 micromolar?). Hence, we can determine that a new patch should be applied after ____ hours, when the peptide concentration has reached zero micromolar (or 10 micromolar?).
Some other interesting results from the model were:
- There was an exponential increase in the concentration of the peptide.
References (for Diffusion Model):
http://serc.carleton.edu/quantskills/methods/quantlit/expGandD.html (exponential decay equation)
________ (literature value for when bacteria starts creating its peptide)
Value for the patch (3m info sheet)
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Improvements???
E.g. change the diffusion coefficient and see what the result is and put in the graph
Summary of Results
Here is a summary of how the patch works, based on our results and conclusions from the lab experiments and computational models:
Timeline picture - Christine … caption it “Our patch’s story”
Figure 6: A visual representation of how the patch works.
- The patch can be applied for 60 hours (about 2.5 days)
- The packets need to be popped every 12 hours
- The steady state concentration of radioactive peptide in the blood is _____, which is. This is __inbetween? than the required amount for radioprotective effects in the human body. It reaches this steady state concentration at ____ hours.
- It takes ___ hours for the peptide concentration in the blood to reach zero
- A new patch can be applied after ____ hours
