Team:Dundee/Modelling
DRY LAB
Introduction
BioSpray
The models for the BioSpray focus on each of the components of the BioSpray binding with their targets in the sample.
faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl.
The aim of a model describing the binding between haptoglobin and haemoglobin is to find the optimum concentration and binding rates that we require for visual detection of haemoglobin in the sample from the crime scene. The more complex formed the more likely it will be that the haemoglobin will be visually detected using the biospray.
Model FormationHaemoglobin is a tetramer, with two \(\alpha\) chains and two \(\beta\) chains. Haptoglobin binds to haemoglobin in two stages. Firstly the haptoglobin binds to the \(\alpha\) chains of the haemeoglobin only. This first reaction is reversible and the complex can dissociate. The haptoglobin then binds to the \(\beta\) chains of the haemoglobin to form an extremely strong complex. This reaction is not reversible. These reactions can be described by the scheme: $$ \ce{Hp + \alpha_{H}<=>[K_{a}][K_{d}] [Hp \cdot \alpha_{H}] ->[K_{i}] [Hp\cdot\alpha_{H}\cdot\beta_{H}]} $$ where \(Hp\) is the amount of free haptoglobin, \(\alpha_{H}\) is the amount of free haemoglobin, \([Hp\cdot\alpha_{H}]\) is the haptoglobin-haemoglobin-\(\alpha\)-chains complex and \([ Hp\cdot\alpha_{H}\cdot\beta_{H}]\) is the full haptoglobin-haemoglobin complex. \(K_{a}\), \(K_{i}\) are the forward rate reactions, and \(K_{d}\) is the reverse reaction rate.
Using the law of mass action (Guldeberg and Waage,1879) the scheme can be written as a system of ordinary differential equations (ODEs): $$ \begin{eqnarray} \frac{dHp}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H}\\ \frac{d \alpha_{H}}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \\ \frac{d[Hp \cdot \alpha_{H}]}{dt}&=& K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] - K_{i}[Hp \cdot \alpha_{H}]\\ \frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \end{eqnarray} $$ with initial conditions: $$ \begin{eqnarray} Hp(0)&=&4.17 \alpha_{H0} \\ \alpha_{H}(0)&=&\alpha_{H0}\\ \lbrack Hp \cdot \alpha_{H} \rbrack (0)&=&0\\ \lbrack Hp \cdot \alpha_{H} \cdot \beta_{H} \rbrack (0)&=&0 \end{eqnarray} $$
Parameter FindingThe parameters were estimated by considering the steady state of the system.Setting the left hand side of equations (1-4) to zero, from (1) and (3) the following equations are given: $$ \begin{eqnarray} K_{d} [Hp \cdot \alpha_{H}]&=&K_{a} Hp \alpha_{H} \\ K_{a} Hp \alpha_{H}&=&K_{d} [Hp \cdot \alpha_{H}] - K_{i} [Hp \cdot \alpha_{H}] \end{eqnarray} $$ Rearranging equation (9) gives: $$ \begin{equation} \frac{[Hp \cdot \alpha_{H}]}{Hp \alpha_{H}}=\frac{K_{a}}{K_{d}} \end{equation} $$ Considering the first binding reaction, it is found that the total amount of haptoglobin, $HpT$, will be equal to: $$ \begin{equation} HpT=Hp+[Hp \cdot \alpha_{H}] \end{equation} $$ Now using equations (11) and (12) it can be written that: $$ \begin{equation} \frac{Hp}{HpT}=\frac{1}{\frac{K_{a}}{K_{d}} \alpha_{H} + 1} \end{equation} $$ It is known that 4.17 haptoglobin per 1 haemoglobin is required for binding, and that haemoglobin and haptoglobin bind at a 1:1 ratio. Therefore the ratio of free haptoglobin to total haptoglobin will be: $$ \begin{equation} \frac{Hp}{HpT}=\frac{3.17}{4.17} \end{equation} $$ By substituting (14) into (13) the ratio between \(K_{a}\) and \(K_{d}\) can be found: $$ \begin{equation} \frac{K_{a}}{K_{d}}=\frac{100}{317} \end{equation} $$ For equation (10) we can use equations (13) and (14) to find the ratio between \(K_{i}\) and \(K_{d}\): $$ \begin{equation} \frac{K_{i}}{K_{d}}=\frac{83}{317} \end{equation} $$
Non-DimensionalisationTo implement equations (1-4) in a model of the haptoglobin and haemoglobin binding, non-dimensionalisation is used to simplify the equations.This was done using the substitutions: $$ \begin{eqnarray*} u=\frac{Hp}{[A]} \qquad v=\frac{\alpha_{H}}{[B]} \qquad w=\frac{[Hp \cdot \alpha_{H}]}{[C]}\\ x=\frac{[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{[D]} \qquad \tau=\frac{t}{[t]} \end{eqnarray*} $$ where \([A],[B],[C],[D],[t]\) are constant. Therefore the concentration of haptoglobin is now represented by u, the concentration of haemoglobin by v, the initial complex by w and the full complex by x. By setting new constants as: $$ \begin{eqnarray*} [A]=[B]=[C]=[D]=\alpha_{H0} \qquad [t]=\frac{1}{K_{d}} \\ \lambda=\frac{K_{a}}{K_{d}} \alpha_{H0} \qquad \gamma=\frac{K_{i}}{K_{d}}, \end{eqnarray*} $$ equations (1-4) become: $$ \begin{eqnarray} \frac{du}{dt}&=&w - \lambda uv\\ \frac{dv}{dt}&=&w - \lambda uv \\ \frac{dw}{dt}&=&\lambda uv - w - \gamma w\\ \frac{dx}{dt}&=&\gamma w \end{eqnarray} $$ with new initial conditions: $$ \begin{eqnarray} u(0)&=&4.17 \\ v(0)&=&1 \\ w(0)&=&0\\ x(0)&=&0 \end{eqnarray} $$ From literature it is known that 2.5 $\times$ 10$^{-5}$ g/cm$^{3}$ haemoglobin is found in blood plasma (Weatherby and Ferguson,2004). It is also known that the molecular weight of haemoglobin is 64458 g/mol. From this the following equation can be used: $$ \begin{equation*} \text{Mass (g)}=\text{Concentration (mol/L)} \times \text{Volume (L)}\times \text{Molecular Weight (g/mol)}. \end{equation*} $$ with the values; $$ \begin{eqnarray*} \text{Mass}&=& 2.5 \times 10^{-5} \text{ g}\\ \text{Volume}&=& 0.001 \text{ l}\\ \text{Molecular Weight}&=&64458 \text{ g/mol}, \end{eqnarray*} $$ to find that the initial concentration of hemoglobin in 1ml of blood is \(\alpha_{H0}=\) 0.3878494524 \(\mu M\) From equations (15) and (16) the values for both \(\lambda\) and \(\gamma\) are known.
Initial ResultsNumerical simulations of the non-dimensionalised system of ODEs were run using MATLAB's ode23 solver (Bogacki, 1989).
To find the value of the steady state of the system, the left hand side of equations (17-20) are set to zero: $$ \begin{eqnarray*} w&=&\lambda uv \\ w&=&\lambda uv \\ \lambda uv&=&w+ \gamma w \\ \gamma w&=&0. \end{eqnarray*} $$ Comparing this to the numerical simulation it is assumed at the steady state that \(v=w=0\) and \(u\) and \(x\) are positive numbers. The equations above cannot be solved to give values for \(u\) and \(x\) in the form shown. By looking at the data statistics for the plot the assumed values of the steady state are: $$ \begin{equation*} (u,v,w,x)=(3.171,0,0,0.9986) \end{equation*} $$ The steady state can be estimated using the non-dimensionalised system of equations, (17-20). It can be noted that: $$ \begin{eqnarray*} \frac{du}{d\tau}-\frac{dv}{d\tau}&=&0\\ \Rightarrow u-v&=&u_{0}-v_{0}\\ \Rightarrow v&=&u-(u_{0}-v_{0}) \end{eqnarray*} $$ This can then be substituted into equations (17),(18) and (20). $$ \begin{eqnarray} \frac{du}{d\tau}&=&w-\lambda u(u-(u_{0}-v_{0}))\\ \frac{dw}{d\tau}&=&\lambda u (u-(u_{0}-v_{0}))-w-\gamma w\\ \frac{dx}{d\tau}&=&\gamma w. \end{eqnarray} $$ It is also noted that: $$ \begin{eqnarray*} \frac{du}{d\tau}+\frac{dw}{d\tau}+\frac{dx}{d\tau}&=&0\\ \Rightarrow u+w+x&=&\text{constant}\\ \Rightarrow u+w+x&=&u_{0}+w_{0}+x_{0}\\ \Rightarrow w&=&u_{0}-u-x. \end{eqnarray*} $$ Substituting this into equations (25) and (26) yields: $$ \begin{eqnarray} \frac{du}{d\tau}&=&u_{0}-u-x-\lambda u(u-(u_{0}-v_{0}))\\ \frac{dx}{d\tau}&=&\gamma (u_{0}-u-x). \end{eqnarray} $$ Now the steady state can be investigated by setting the left hand side of equations (27) and (28) to be zero. $$ \begin{eqnarray*} u_{0}-u-x&=&\lambda u(u-(u_{0}-v_{0}))\\ \gamma (u_{0}-u-x)&=&0 \end{eqnarray*} $$ Re-arranging these gives that: $$ \begin{eqnarray*} u-(u_{0}-v_{0})&=&0\\ u_{0}-u&=&x. \end{eqnarray*} $$ Solving this equation gives that the steady state is found to be when: $$ \begin{equation*} (u,v,w,x)=(3.17,0,0,1). \end{equation*} $$ This result seems to be similar to, but less accurate than, that given by the data statistics of Figure 1.
Sensitivity AnalysisThe parameters \(\gamma\) and \(\lambda\) are estimated above, however we can assess the optimal value of these. Notice that the value of both \(\gamma\) and \(\lambda\) are governed by the binding rates of the system. By running the ode23 solver over one hundred different values for both parameters. The range of values has the mean as the estimated values from above. That is the max values, A and B, are twice the estimated values for \(\lambda\) and \(\gamma\) respectively. All other conditions were kept the same as previous analysis.
where the colour bar represents the concentration of the final complex and:
$$ \begin{eqnarray*} A&=&2 \times \frac{83}{317}, \\ B&=&2 \times \frac{100}{317} \times 0.3878494524. \end{eqnarray*} $$Conclusions that can drawn from Figure 2 will be discussed in the conclusions section.
ConclusionsFrom the model of haemoglobin and haptoglobin binding several conclusions are inferred. The binding reaction should be relatively quick, meaning that a visual change should be apparent after a short period of time. This is demonstrated in Figure 1. The optimal binding rates were investigated via sensitivity analysis, and it was discovered that \(\gamma\) and \(\lambda\) should be as high as possible for the optimal visualistion of the traces within the sample. The previously estimated values of both parameters is the equivalent to the centre of the figure 2. From the figure it can be seen that the optimal values for both parameters is as large as possible. This suggests that the association rates should be much greater than the dissociation rates, to ensure optimal binding. This could be done by modifying haptoglobin to become more sticky, and thus more likely to bind to haemoglobin and less like to dissociate from the complex. The estimated values for \(\gamma\) and \(\lambda\) represent the natural binding reaction, if these are increased then more efficient reactions will occur.
References- Bogacki, P., Shampine, L. F. (1989). A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.
- Guldberg, C. M., Waage, P. (1879). Concerning chemical affinity. Erdmanns Journal fr Practische Chemie, 127, 69-114.
- Weatherby, D., Ferguson, S. (2004). Blood Chemistry and CBC Analysis (Vol. 4). Weatherby and Associates, LLC.
Aim of experiment: ultricies nec tellus eu
Protocols Used:
faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl.
Aim of experiment: ultricies nec tellus eu
Protocols Used:
faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl.
Aim of experiment: ultricies nec tellus eu
Protocols Used:
faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl. Morbi felis lectus, ultricies nec tellus eu, faucibus eleifend lectus. In molestie augue leo, id imperdiet ante imperdiet at. Quisque ultrices neque sit amet felis tincidunt tristique at sed nisi. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Ut suscipit ut massa in ultricies. Nunc nec tincidunt nunc, ut consectetur lorem. Sed non justo nislo nislo nisl.