Parameter Relationship Analysis

Overview and Motivation

During the early experimental phase of the model production, it was noticed that for some parameters the actual value did not matter too much, these 'sloppy' parameters could have a large range of values with minimal impact on the model predictions. Furthermore, some parameters were often coupled and whilst individually they are 'sloppy' some relationship between them is in fact not. This analysis is to look at and highlight these relationships.

The motivation for this analysis is it can tell you what parameters require extensive literature or experimental research and which can afford a higher final uncertainty.

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Methodology

1) Generate probability distributions for each kinetic parameter required from our collected data.

2) Simulate the model with different sets of kinetic values that are sampled from probability distributions. In our study, 5000 samples for each reaction were modelled, i.e. the model was simulated with 1000 different sets of kinetic values.

3) The data was assessed using a mean squared error

$$MSE = \frac{1}{n}{\sum_{i=1}^n(y_{i,experimental}-y_{i,model})^2}$$the top 10%, 11%-20% and the rest of model runs were recorded. The parameter sets which generated this data were then stored seperately.

4) For each combination of the parameters, the data was plotted with the different groupings coloured: green(top 10%) , yellow(11%-20%) and red(the rest) .

The results were normalised to 1 to make trends easier to spot since this analysis is about spotting correlations. Rather than quantively measuring any relationship.

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Results

Figure 1 shows K_{cat} vs K_{m} for HRP. Green indicates values that scored the top 10% when compared to experimental data. Yellow indicated 11%-20% and red indicated the remainder It shows that the best parameter sets are at a constant K_{m,HRP} but varying K_{cat,HRP}. Hence K_{cat,HRP} is a sloppy variable and K_{m,HRP} is not.

Figure 2 shows K_{cat} for HRP vs K_{cat} for GOx. Green indicates values that scored the top 10% when compared to experimental data. Yellow indicated 11%-20% and red indicated the remainder. It shows both are fairly sloppy parameters and no correlation between rate constants.

Figure 3 shows K_{m} for HRP vs K_{cat} for GOx. Green indicates values that scored the top 10% when compared to experimental data. Yellow indicated 11%-20% and red indicated the remainder. K_{cat,Gox} is sloppy and the value ofK_{m<,HRP/sub> is in the lower end of its PDF. }

Figure 4 shows K_{cat} for HRP vs K_{m} for GOx. Green indicates values that scored the top 10% when compared to experimental data. Yellow indicated 11%-20% and red indicated the remainder. K_{cat,HRP} is a sloppy parameter and K_{m,GOx} is at the lower end of its PDF.

Figure 5 shows K_{m} for HRP vs K_{m} for GOx. Green indicates values that scored the top 10% when compared to experimental data. Yellow indicated 11%-20% and red indicated the remainder. Both rate constants are on the lower end of their PDFs.

Figure 6 shows K_{m} vs K_{cat} for GOx. Green indicates values that scored the top 10% when compared to experimental data. Yellow indicated 11%-20% and red indicated the remainder. This is the most interesting parameter analysis. It's Hard to see (verified with model experimentation) but a very steep straight line of the best data points is shown. Hence K_{cat}/K_{m} is a constant value even though both are sloppy parameters as shown independently earlier.

Conclusions

From the graphs it is quite clear that some parameters have no relationship, as shown by the random distribution of green and yellow points amongst the red points. For other parameter combinations there are clear relationships shown by a band of green, bounded by bands of yellow amongst the red points. This validates our decision to provide a constraint in the suitable parameter values selected from certain PDFs.

This analysis was only undertaken using irreversible Michaelis-Menten kinetics. Further analysis should be performed using other more complicated systems, as it could for example in systems with multiple pathways to the same point this could indicate if one pathway is heavily limiting the overall network compared to another.

Some parameters were found to be sloppy some were not as such this analysis can guide what experiments you should do in the lab with respect to determining rate constants.

Finally this type of analysis can find relationships between your parameters. Such as in figure 6. This can lead to Constraints (halfway down linked page.) to further improve the quality of your data set.

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