Chromate Model

Aim

The aim of modeling the chromate detection pathway is to compare the original biobrick system with the modified system which we use in the lab. Sensitivity analysis will also allow for optimisation of the processes in the experimental side of the project. Although most of the kinetic rate parameters were estimated, the general activity of the systems can be understood.

Results

In the lab a modified pathway is being used to insert into e.coli to detect chromate. In the modified pathway two plasmids are involved shown below:

In the absence of chromate, \(C_{V}\), the \(ChrB\) protein forms a dimer, \(D\), which then binds the pChr promoter, forming a closed promoter, \(P_{c}\). This promoter cannot function so no green fluorescent protein, \(GFP\), is produced. In the presence of chromate, the ChrB dimer instead bind to chromate forming a complex, \([C_{V} \cdot D]\). This allows the open promoter, \(P_{o}\) to function and produce \(GFP\). The reactions can be described as chemical equations, where the parameters \(k\) are the kinetic rates:

$$ \large{ \ce{ ->[k_{1}] ChrB ->[k_{2}]}\\ \ce{ ChrB + ChrB <=>[k_{3}][k_{-3}] D ->[k_{5}]}\\ \ce{ D + \text{\(C_{V}\)} <=>[k_{4}][k_{-4}] [D \cdot C_{V}]}\\ \ce{ D + P_{o} <=>[k_{6}][k_{-6}] P_{c} }\\ \ce{ P_{o} ->[k_{7}] GFP + P_{o} }\\ \ce{GFP ->[k_{8}] }} $$

This system was written as a system of ODE's and simulations were run with chromate added at specific times to investigate change in concentration over time. The steady state in the presence of chromate was calculated as the \(GFP\) reached a steady level and did not change once there. Once chromate was removed from the system the \(GFP\) decreased again as expected. The steady state analysis showed that there was a 18927.20323% increase of \(GFP\) concentration from the initial concentration.

We wish to compare the modified pathway with the original pathway, where only one plasmid is used:

The pChr promoter, when open, \(P_{o}\), produces \(ChrB\) protein and \(GFP\). In the absence of chromate, \(C_{V}\), the \(ChrB\) protein forms a dimer, \(D\), which then binds the promoter, forming a closed promoter, \(P_{c}\). This promoter cannot function so no green fluorescent protein, \(GFP\), is produced. In the presence of chromate, the ChrB dimer instead bind to chromate forming a complex, \([C_{V} \cdot D]\). This allows the open promoter, to function and produce \(GFP\) and \(ChrB\). The \(ChrB\) will then in turn dimerise and bind to chromate, if any is left, or inhibit the promoter. The reactions can be described as chemical equations, where the parameters \(k\) are the kinetic rates. Note that only the highlighted reactions differ from our modified pathway.

$$ \large{ \ce{ ChrB ->[k_{2}]}\\ \ce{ ChrB + ChrB <=>[k_{3}][k_{-3}] D ->[k_{5}]}\\ \ce{ D + C_{V} <=>[k_{4}][k_{-4}] [D \cdot C_{V}]}\\ \ce{ D + P_{o} <=>[k_{6}][k_{-6}] P_{c} }\\ \ce{ P_{o} ->[k_{7}] GFP + ChrB + P_{o} }\\ \ce{GFP ->[k_{8}] }} $$

This system was written as a system of ODE's, (1) and (3), simulations were run with chromate added at specific times to investigate change in concentration over time.

The steady state in the presence of chromate was calculated as the \(GFP\) concentration reached a steady level and did not change once there. Once chromate was removed from the system the \(GFP\) decreased again as expected. The steady state analysis showed that there was a 20396.68% increase of \(GFP\) concentration from the initial concentration. Note that this value is slightly more than that of the modified pathway. This suggests that the modified pathway will give a lower output than that of the original pathway. The lab had some concerns over the reliability of the original pathway, as it was shown to be prone to false positives due to leakiness. Leakiness, in this case, refers to when \(GFP\) is produced even when the promoter is closed, which will give gFP production even in the absence of chromate. To investigate this further a leakiness term was introduced to both the modified and original pathway models. The simulations were then ran again using ode23 and compared.

From the figure, it can be seen that both pathways will have the same response to the same level of leakiness. This indicates that the modified, two plasmid, system is just as reliable as the original, one plasmid system, giving us confidence in our two plasmid system.

Sensitivity analysis was used to investigate \(GFP\) production with varying levels of chromate added to the system.

For both systems it could be seen that for higher and lower values of chromate added, the final concentration of \(GFP\) will be the same. The only difference is the time it takes for this steady level to be obtained. The higher the initial concentration of chromate, the faster it will reach the steady level, described earlier. Also note the modified pathway can achieve the optimal \(GFP\) concentration in less time than the original. This information was passed to lab for future decisions.

Method: Deriving Equations

Using the law of mass action the modified pathway reactions can be written as a system of ordinary differential equations:

$$ \large{ \begin{eqnarray} \frac{dChrB}{dt}&=&k_{1}-k_{2}ChrB+k_{-3}D-k_{3}ChrB^{2}, \nonumber \\ \frac{dD}{dt}&=&k_{3}ChrB^{2}-k_{-3}D-k_{5}D-k_{6}DP_{o}+k_{-6}P_{c}-k_{4}DC_{V}+k_{-4}[C_{V}\cdot D], \nonumber \\ \frac{dC_{V}}{dt}&=&k_{-4}[C_{V} \cdot D]-k_{4}DC_{V}, \nonumber \\ \frac{d[C_{V} \cdot D]}{dt}&=&k_{4}DC_{V}-k_{-4}[C_{V} \cdot D], \\ \frac{dP_{c}}{dt}&=&k_{6}P_{o}D-k_{-6}P_{c}, \nonumber \\ \frac{dP_{o}}{dt}&=&k_{-6}P_{c}-k_{6}P_{o}D, \nonumber \\ \frac{dGFP}{dt}&=&k_{7}P_{o}-k_{8}GFP, \nonumber \end{eqnarray}} $$

with assumed initial conditions:

$$ \large{ \begin{eqnarray} ChrB(0)&=&1, \nonumber \\ D(0)&=&1, \nonumber \\ C_{V}(0)&=&1, \nonumber \\ \text{[} C_{V} \cdot D \text{ ]} (0)&=&0, \\ P_{c}(0)&=&1, \nonumber \\ P_{o}(0)&=&1, \nonumber \\ GFP(0)&=&0.01. \nonumber \end{eqnarray}} $$

Using the law of mass action the original pathway reactions can be written as a system of ordinary differential equations, where the highlighted is the only equation that differs from the modified equations.

$$ \large{ \begin{eqnarray} \frac{dChrB}{dt}&=&k_{7}P_{o}-k_{2}ChrB+k_{-3}D-k_{3}ChrB^{2}, \nonumber \\ \frac{dD}{dt}&=&k_{3}ChrB^{2}-k_{-3}D-k_{5}D-k_{6}DP_{o}+k_{-6}P_{c}-k_{4}DC_{V}+k_{-4}[C_{V}\cdot D], \nonumber \\ \frac{dC_{V}}{dt}&=&k_{-4}[C_{V} \cdot D]-k_{4}DC_{V}, \nonumber \\ \frac{d[C_{V} \cdot D]}{dt}&=&k_{4}DC_{V}-k_{-4}[C_{V} \cdot D], \\ \frac{dP_{c}}{dt}&=&k_{6}P_{o}D-k_{-6}P_{c}, \nonumber \\ \frac{dP_{o}}{dt}&=&k_{-6}P_{c}-k_{6}P_{o}D, \nonumber \\ \frac{dGFP}{dt}&=&k_{7}P_{o}-k_{8}GFP, \nonumber \end{eqnarray}} $$

with the same initial conditions as the modified system. Both sets of equations were solved using MATLAB's ode23 solver (Bogacki, 1989).

Method: Parameter Estimation.

Very little literature was available on the chromate pathways described. Therefore most of the values for the kinetic rates are assumptions.

$$ \large{ \begin{array}{|c|c|c|} \hline Parameter & Value & Units \\ \hline \hline k_{1} & 1 & mol \, l^{-1}\\ \hline k_{2} & 1 & t^{-1}\\ \hline k_{3} & 1 & mol^{-1} \, l \, t^{-1}\\ \hline k_{-3} & 1 & t^{-1}\\ \hline k_{4} & 1 & mol^{-1} \, l \, t^{-1}\\ \hline k_{-4} & 1 & t^{-1}\\ \hline k_{5} & 1 & t^{-1}\\ \hline k_{6} & 1 & mol^{-1} \, l \, t^{-1}\\ \hline k_{-6} & 1 & t^{-1}\\ \hline k_{7} & 1 & t^{-1}\\ \hline k_{8} & \frac{0.25}{60} & t^{-1}\\ \hline \end{array}} $$
Table 1: Parameter values.

Note that \(k_{8}\) is set as a more specific value. In both systems \(k_{8}\) represents the decay of \(GFP\), which has been studied in literature. From Halters research (2007), it was found that \(GFP\) decays at a rate of \(\frac{0.25}{60}\) \(t^{-1}\).

Although most of the kinetic rate parameters were estimated, the general activity of the systems can be understood

Method: Steady States

To calculate the steady states of both systems it is first noted that:

$$ \large{ \begin{eqnarray*} \frac{dC_{V}}{dt} + \frac{d[C_{V} \cdot D]}{dt} &=&0,\\ \frac{dP_{c}}{dt}+\frac{dP_{o}}{dt}&=&0, \end{eqnarray*}} $$

suggesting that:

$$ \large{ \begin{eqnarray*} C_{V}+ [C_{V} \cdot D] &=&constant,\\ P_{c}+ P_{o} &=&constant, \end{eqnarray*}} $$

therefore:

$$ \large{ \begin{eqnarray*} C_{V}+ [C_{V} \cdot D] &=&C_{V}(0)+[C_{V} \cdot D](0),\\ P_{c}+ P_{o} &=&P_{c}(0)+ P_{o}(0) . \end{eqnarray*}} $$

Substituting in the initial values then the equations become:

$$ \large{ \begin{eqnarray*} C_{V}+ [C_{V} \cdot D] &=&1,\\ P_{c}+ P_{o} &=&1, \end{eqnarray*}} $$

which can be re-arranged to give:

$$ \large{ \begin{eqnarray} [C_{V} \cdot D] &=&1-C_{V},\\ P_{c}&=&1-P_{o}. \end{eqnarray}} $$

Now using (4)and(5) the steady states of both the modified and original systems can be evaluated. Firstly consider the steady state of the modified system, begin by setting the left hand side of system (1) to zero and removing duplicate or inverse equations:

$$ \large{ \begin{eqnarray*} 0&=&k_{1}-k_{2}ChrB+k_{-3}D-k_{3}ChrB^{2}, \\ 0&=&k_{3}ChrB^{2}-k_{-3}D-k_{5}D-k_{6}DP_{o}+k_{-6}P_{c}-k_{4}DC_{V}+k_{-4}[C_{V}\cdot D], \\ 0&=&k_{-4}[C_{V} \cdot D]-k_{4}DC_{V}, \\ 0&=&k_{6}P_{o}D-k_{-6}P_{c}, \\ 0&=&k_{7}P_{o}-k_{8}GFP. \end{eqnarray*}} $$

Re-arranging these and substituting in values for the parameters from Table 1, (4) and (5) gives:

$$ \large{ \begin{eqnarray} 0&=&1-ChrB+D-ChrB^{2},\\ 0&=&ChrB^{2}-D-D-DP_{o}+1-P_{o}-DC_{V}+1-C_{V}, \label{eq:b} \\ 0&=&1-C_{V}-DC_{V},\nonumber \\ 0&=&P_{o}D-1-P_{o},\nonumber \\ 0&=&P_{o}-\frac{0.25}{60}GFP.\nonumber \end{eqnarray}} $$

Re-arranging the bottom three equations gives:

$$ \large{ \begin{eqnarray*} C_{V}&=&\frac{1}{1+D}, \\ P_{o}&=&\frac{1}{1+D}, \\ GFP&=&\frac{P_{o}}{\frac{0.25}{60}}, \end{eqnarray*}} $$

which can be substituted back into (5):

$$ \large{ \begin{eqnarray*} [C_{V} \cdot D]&=&1-\frac{1}{1+D}, \\ P_{c}&=&1-\frac{1}{1+D}. \end{eqnarray*}} $$

Now we can substitute these into (7):

$$ \large{ \begin{eqnarray*} 0&=&ChrB^{2}-2D,\\ \rightarrow D&=&\frac{1}{2} ChrB^{2}. \end{eqnarray*}} $$

Now substitute the value for \(D\) into (6):

$$ \large{ \begin{eqnarray*} 0&=&1-ChrB+\frac{1}{2} ChrB^{2}-ChrB^{2}, \\ \rightarrow 0&=&\frac{1}{2} ChrB^{2}+ChrB-1, \\ \rightarrow ChrB&=&-1 \pm \sqrt{3}. \end{eqnarray*} } $$

Since we are describing a biological system we take the positive value for \(ChrB\) and back substitute it to find the steady state of the modified system to be when:

$$ \large{ \begin{eqnarray*} ChrB&=&0.732050808076 \, mol\, l^{-1}, \\ D&=&0.2679491925\, mol\, l^{-1}, \\ C_{V}&=&0.7886751346\, mol\, l^{-1}, \\ \text{[} C_{V} \cdot D \text{ ]}&=&0.21132486454 \, mol\, l^{-1}, \\ P_{c}&=&0.21132486454 \, mol\, l^{-1}, \\ P_{o}&=&0.7886751346 \, mol\, l^{-1}, \\ GFP&=&189.2820323 \, mol\, l^{-1}. \end{eqnarray*}} $$

From this we can determine that there has been a 18927.20323% increase in \(GFP\) concentration throughout the reactions.

The steady state of the original system was calculated using the same method as above and was found to be:

$$ \large{ \begin{eqnarray*} ChrB&=&0.5943130164 \, mol\, l^{-1}, \\ D&=&0.1766039807\, mol\, l^{-1}, \\ C_{V}&=&0.8499036348\, mol\, l^{-1}, \\ \text{[} C_{V} \cdot D \text{ ]}&=&0.0.150096 \, mol\, l^{-1}, \\ P_{c}&=&0.150096 \, mol\, l^{-1}, \\ P_{o}&=&0.8499036348 \, mol\, l^{-1}, \\ GFP&=&203.9768 \, mol\, l^{-1}. \end{eqnarray*}} $$

From this we can determine that there has been a 20396.68 % increase in \(GFP\) concentration throughout the reactions.

References

• Bogacki, P., Shampine, L. F. (1989). A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.
• Halter, M., Tona, A., Bhadriraju, K., Plant, A. L., Elliott, J. T. (2007). Automated live cell imaging of green fluorescent protein degradation in individual fibroblasts. Cytometry Part A, 71(10), 827-834

Bone Incision Experiments

Objective

The aim is to represent the relationship between the applied force, sliding distance and wear volume when a stainless steel knife collides with bone. By calculating the force required to make the incision, this could potentially indicate the size or strength of a possible suspect, while the sliding distance would correspond to the length of the blade. This relationship is determined by the Archard Equation for wear¹:

W = kSP

where, W is the volume of wear, S is the sliding distance, P is the applied force and k the wear coefficient. This is intended to complement the chromium detector which would be used in a stabbing or decapitation case.

Results

From experimental methods a value of 6.23x10^-9 was found for the wear coefficient. Using this the Archard Equation was modelled on MATLAB to produce a parabolic surface plot, relating the volume of incision to the length of the blade and the force of stabbing. As expected the volume will increase, as the force and sliding distance increase. The following graph was obtained.

Method

To use the Archard Equation, first a value for the wear coefficient, k, between bone and stainless steel needed to be determined experimentally. The linear relationship between the volume of wear and force exerted can be investigated and the gradient of the produced graph then found. The gradient will be equal to kS, where k is the wear coefficient and S is the known sliding distance. Therefore k can be calculated.

To do this, a piece of equipment called an Instron 4204 was used, this is a computer controlled electro-mechanical testing system capable of performing a variety of tests based on tension and compression². This was used alongside the Bluehill program. For our needs, this equipment is used to measure the force needed to continue compression at a set rate, and therefore the force needed to cut through the bone. This was done for speeds of 5mm/min, 10mm/min and 20mm/min on 2 pig ribs and 4 pig shoulder bones. A knife is clamped to the Instron machine and a bone placed on a stand below. The program is started and the knife is lowered at a set speed using the set up below:

The program is started and stopped manually, allowing the extension to also be determined manually. For this purpose a start and end point is marked on the knife, meaning the extension is a control factor throughout the experiment and kept at 32mm. The Bluehill program then produces a graph of the force against extension, so the maximum force exerted to cut through bone can be found and recorded. This is done for 25 different incisions and the force exerted for each is recorded
The next stage is to determine the volume of each incision. As the incisions are so small they could not be measured by hand. A Kodax dxs 4000 Pro System was used to X-ray the bones. This was used along side two programs; Carestream and Image J. Carestream allowed a high resolution image of the incision to be produced, by choosing the optimum focal length and field of view. The image is then edited and the clearest contrast chosen.

Figure 3: A example of the Images captured using X-Ray Imaging. The Incisions are barely visible and therefore could not be measured by hand.

The image is imported into Image J and the scale set. Now the width and depth of the incision can be measured using the Line tool then Analyse and Measure. The results are then saved in a table. Finally the length if the incision is measured by had using a Vernier Calliper.

Now the volume is calculated by assuming the shape of the incision as a prism. The simple formula;

W=(^hb⁄₂)l

where W is the volume, h is the depth, b the width and l the length, is used. In Excel, the volume can be plotted against the force for each incision. A linear trendline is added and the gradient is found, in this case a value of 2x10^-11, and is equal to the wear coefficient times the blade length. The wear coefficient is therefore found to be 6.23x10^-9.

Now the wear coefficient has been found the Archard Equation can be modelled in MATLAB.

Discussion

From the experimental process a value of 6.23x10^-9 was obtained. However the graph itself had a very low R² value. This value is the percentage of variance in the date which can be explained by the model. This low value means the graph could be considered inaccurate. As the values of volume measured are all relatively accurate, with all uncertainties in the range of 3x10^-12 to 7x10^-11. This would suggest that the anomalies are due to the method used to make incisions. Although the actual measurement of force was accurate, the methid of applying force may not have been. The knife was clamped onto the jaws of the Instron 4204, but still swivelled slightly when force was applied. A manual attempt was made to ensure the blade stayed straight. However, as this coefficient had never been previously calculated it is unknown how much of an effect this had on the overall result.

References

[1] Thompson, J.M. and Thompson, M.K. A Proposal for the Calculation of Wear, Proc. of the 2006 International ANSYS Conference and Exhibition
[2] Instron Load Frame Standard Operating Procedure.