Model order reduction of CFFLs

Abstract

Biochemical reactions play a crucial role and tell us many about the behavior of the biological regulation processes . We will apply several methods of order reduction to describe the overall dynamics in a more compact way. For modelling a set of biochemical reactions is rewritten as first order differential equations. This set of first order differential equations defines the state space model. This rewriting is based on mass-action kinetics and Michaelis-Menten (MM) theory. With the Stoichiometry matrix the conservation laws and flux distribution in steady state can respectively be deduced. The systems we are looking at can be distinguished in many different biochemical regulatory networks . Examples of these networks can be found in gene expression, protein production and/or hormone production. The system that includes all given biochemical reactions of the network can be seen as a Coherent Feedforward Loop or CFFL. In synthetic biology these CFFLs are studied to gain insight into the desired production/expression: think about medicine production, agriculture and manufacturing. In biochemistry mostly a set of biochemical reactions can be given as a family or combination of CFFLs. To realize this a so called AND-gate or toehold switch is used.

First of all we use conservation laws to reduce the system in order equal to the dimension of the left nullspace. Afterwards we have the option to reduce the system even more by applying the Quasi Steady State Approach in a given network like the CFFL. This method suggests that some species concentrations will reach its steady states much sooner than other species concentrations (if we look at slow timescale). Therefore it is assumed that some species already have their steady state at the beginning of the experiment. This is the so called classical QSSA. Another way to reduce the system order is by applying the Kron reduction order method. This method assumes a complexes network that reduces the complexes and thus the number of species. Here the concept of complex balancedness will determine whether the steady states for both models will be the same. Eventually we will also deal with alternative modelling where the cycles and feedback mechanisms will be replaced by more simple ones. Then afterwards mass-action kinetics along with classical QSSA can be applied. To get an optimal reduction order model the way in which parameters within the model are estimated can be discussed by optimization techniques. Furthermore we will see how the system can be transformed if we also have to do with in-and outflows. It actually means that we will need to add an extra term . One term will be in matrix-vector form while the other method merely uses vector-scalar notation. We will also look at the relation between these two forms. A future challenge would be to make an auto based system that directly converts the given system into its reduced order form. Here the best reduction order model will be selected automatically and applied in the best determined
sequence.