Análisis económico

determined for each member of the population,pj using the expression below:

R2j = PN

i=1(Yi−Fi(pj))

N . (3.41)

The number of data points is denoted N, Yi and Fi(pj) denote the i-th elements of the experimental data and model curve vectors respectively. The elements of the population were ranked according to these values; the 200 with the highestR2_{values are eliminated. The process} is then repeated; 1000 generations were used.

Confidence intervals were estimated manually for these parameters using the statistical data provided by COPASI in Microsoft Excel (Microsoft Ltd, Reading, UK): assuming that error was normally distribution. The basic technique was as follows. For a given distribution of error,

X, confidence intervals were estimated for each parameter using the following equation:

v±=v±X(p)·se(v). (3.42)

Wherev is the estimated parameter value,pis a probability such that 1−2pgives the required confidence level andse(v) is the standard error (or standard deviation) associated withv. The value denoted by X(p) is the value at which P(X > x) =pfor the standardised distribution

X.

3.5

Simulation of reaction species concentrations

The concentrations of reaction species were predicted by numerical integration of a model system. The trapezoidal integration method was implemented manually in Microsoft Excel

and was used to eliminate background production from experimental data. Simulation of time courses for a single reaction were undertaken in Maple. Prediction of more complex pathway

dynamics was undertaken usingCOPASI. These three methods are described in detail below. Experimental observations of pathway dynamics were obtained using the assays described in Section 2.4. Over long time courses background production of the observed species could signif- icantly affect the measured data. Rates of background production at the start and end points of the time course were estimated from experimental data. These data were combined with models of background production to extrapolate rates for all points in the time course. The resulting curve was integrated numerically using a manual implementation of the trapezoidal integration method (Equation (3.43)) inMicrosoft Excel. The iterative function describing this method is given below:

xn+1=xn₊tn+1−tn 2 (f(x

n+1_{, t}n+1_{) +f(x}n_{, t}n_{)), (3.43)}

time steps are enumerated by superscripts; t, x, andf are time, concentration of species, and rate of change of species concentration respectively. This method is rather crude; being prone to accumulation of error and not suitable for stiff systems. However it is adequate to integrate the relatively low curvature functions used to model background production in the models used. The superior algorithms implemented in Maple or COPASI could have been used. However aligning the resulting time courses to the experimental data would have been a non-trivial problem and as such the simpler solution, a direct implementation in Microsoft Excel, was preferred.

Time courses for a single reaction were generated inMaple using thedsolvefunction. This uses an algorithm similar to that used byCOPASIwhich is described below. Maplewas used for this case since the subsequent analysis desired was best implemented in this program. Specifically

Maple was used to plot rates of accumulation of certain reaction species, corresponding to experimental data, and initial rates against initial concentrations of substrates, simulating the initial rate kinetic characterisations described in Sections 2.4 and 5. In order to create equivalent

3.5. SIMULATION OF REACTION SPECIES CONCENTRATIONS 69 plots from data simulated usingCOPASI it would have been necessary to export the data to an alternative program.

The differential equations describing model pathways were numerically solved using the time course functions inCOPASI [159]. This program has a number of features which make it ideal for modelling of metabolic pathways. Models of individual reactions can be selected from a list and populated with relevant parameters. Incorporation of additional reactions to an existing model is relatively simple. Finally it is possible to use stochastic or deterministic simulations, or a hybrid of both.

Time course simulations were deterministic and were produced using a variant of the Livermore Stiff ODE solver [176]. Given that the lowest particle numbers considered were of the order of 1011 deterministic simulations are appropriate. Options for the algorithm were left as de- fault. Thus the algorithm was allowed to use mass conservation laws; the relative and absolute tolerances were 10−6 _{and 10}−12 _{respectively, significantly higher than can reasonably be ex-} pected experimentally; and the maximal internal steps before the next reporting time was 105_. The algorithm incorporates both stiff, backward differentiation formula (BDF), and non-stiff, Adams-Moulton, integration methods.

The Adams-Moulton and BDF methods of orderqare given by the following expressions:

Yn=Yn−1+hn q−1 X j=0 βjfn−j, (3.44) Yn= q X j=1 αjYn−j+hnβ0fn, (3.45)

respectively. The interval required in divided into internal, not necessarily regular, increments denoted by the length N vectorξ. The step size is denoted hn(= ξ_n−ξ_n−1), the difference between the n-th and (n−1)-th elements of ξ. The estimated solution and the derivative of this solution are denoted byYandf respectively. Subscripts denote the position in the vector

corresponding to the mesh point at the same position in ξ. Thus fn is the derivative of Y at ξ_n. The coefficients {αj} and {βj} are associated with particular orders of these methods. Equations (3.44) and (3.45) are implicit, they require solutions at mesh points which have not yet been simulated, the unknown values are predicted using explicit methods and then corrected iteratively. This approach provides excellent accuracy, that is the difference between estimated and actual values at a given point is small, and stability, that is the accumulated difference between estimated and actual values grows slowly [177].

In the implementation inCOPASI, the maximal order the Adams method will attempt before switching to the BDF method is 12; the maximal order the BDF method will attempt before switching to smaller internal step sizes is 5.