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Stoer, J. Introduction to Numerical Analysis.

Strang, G. Introduction to Linear Algebra. An Analysis of the Finite Element Method. Wavelets and Filter Banks. Strikwerda, J.

Trefethen, L. Unpublished lecture notes, Bau, III. Numerical Linear Algebra.

Optimal control of partial differential equations and nonlinear optimization

Trottenberg, U. Oosterlee, and A. Burlington, MA: Academic Press, Van der Vorst, H. Van Loan, C.

Differential equation - Wikipedia

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Bathe, K. Finite Element Procedures. Multigrid Methods and Applications. Mallat, S. A Wavelet Tour of Signal Processing. Whitham, G. Linear and Nonlinear Waves. Need help getting started? Don't show me this again Welcome! For a detailed discussion of characteristics of Western blot data we refer to [ 55 ]. These signals converge on the level of Raf kinase, which they phosphorylate.

ERK induces downstream signaling and can down-regulate the Raf activity [ 49 ]. The latter establishes a negative feedback loop [ 52 , 57 ]. The pathway is illustrated in Fig. The model considers the six reactions:. The upstream signaling is summarized in the time-dependent rate constant k 1,max t with the flexible parameterization. Experimental studies proved an inhibition of Raf phosphorylation by pERK [ 52 ].

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This feedback is however context-dependent [ 49 ]. Using mass conservation and reformulations explained in detail in Additional file 1 : Section 3 we arrive at the ODE model. The model for the relative phosphorylation levels does not depend explicitly on the total abundances [Raf] 0 , [MEK] 0 and [ERK] 0 but only on products and ratios of these parameters with other parameters, e. Defining these products and ratios as new parameters eliminates non-identifiabilities and reduces the number of parameters.

The measurement noise is assumed to be normally distributed and its variance is estimated from the experimental data. As all parameters are non-negative, a log-parameterization is used for parameter estimation [ 15 ]. The states of the reformulated model are between 0 and 1. Details regarding parameters and initial conditions are provided in the Additional file 1 : Table S2.

In addition to the kinetic, scaling and noise parameters, the initial conditions of the models for H1 and H2 are unknown. These analytical expressions are provided in the Additional file 1 : Equation 10 , We inferred the model parameters and initial conditions from the Western blot data using ML estimation. The optimization problem is solved using multi-start local optimization.

Newton's Method