Using Lie algebra to asses the parameters identifiability and to perform experimental design
Integration of life sciences & engineering
Integration of Life Sciences & Engineering - Poster (T5-P)
Keywords: system of biotransformations, Lie algebra, qualitative experimental design
Florin Paul Davidescu1, Henrik Madsen2, Sten Bay Jørgensen1
1CAPEC, Deptartment of Chemical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
2Informatics and Mathematical Modeling, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
Keywords: system of biotransformations, Lie algebra, qualitative experimental design,
The increasing interest in producing expensive fine chemicals in the pharmaceutical industry using biochemical synthesis motivates this paper. To develop a purely enzymatic synthesis of complex molecules from inexpensive substrates, large reaction networks are necessary. One way to achieve such a functional network is by using a System of Biotransformations (SBT). The SBT is based on a microorganism's metabolic network containing the synthesis path including cofactor regeneration reactions down to the desired product, which most often is an intermediate in the metabolic network. Expression of the enzymes catalyzing reactions from this intermediate are turned off prior to the extraction i.e the genes are knocked-out. The SBT is used as cell free extract in the production phase. In order to identify the bottlenecks, to describe them qualitatively and subsequently to optimize the productivity of the SBT a dynamic model with good long term predictions properties over the operating window is necessary. Since the SBT is a system of high dynamics and complexity, it may be modeled based upon physical knowledge. In order to validate such models it is however necessary to assess the parameter identifiability and to design experiments.
In addressing the identifiability problem one has to determine which input variables should be varied and which outputs to be measured during the experiments in order to be able to render all the model parameters identifiable. In a general setting, one can ask, when all the possible inputs are varied and the possible outputs measured which are the parameters that can be identified. Once the minimal set of inputs and outputs which render the maximum number of parameters identifiable has been chosen, that is, qualitative experimental design has been performed, the next step is to design experiments which aims at improving the statistical quality of the model parameter estimates. During this step the idea is to determine how to manipulate the inputs and to measure the outputs chosen during the previous phase, that is, to do quantitative experimental design.
The identifiability analysis method for qualitative experimental design used in this contribution aims at addressing the theoretical identifiability of the parameters and is based on generating series. First, the original state space description is converted into a series expansion based on the Lie derivatives where the model output (the measured states) can be expanded in generating series with respect to inputs and time around an initial time. The coefficients of the series are used to form systems of algebraic equations. At a minimum, a system of algebraic equations equal to the number of the parameters occurring in model needs to be formed using the coefficients of the series. The system of equations needs to be solved analytically. If a combination of equations can be solved uniquely for the parameter set, then the set is theoretically identifiable. For qualitative experimental design, more states and perturbation of additional input variables related to the model are included and the analysis repeated.
Presented Wednesday 19, 13:30 to 15:00, in session Integration of Life Sciences & Engineering - Poster (T5-P).
