Balanced truncation

Liebermeister, W., Baur, U., Klipp, E., Biochemical network models simplified by balanced truncation, FEBS J. 2005, 272 4034-4043. 

Biosimulation has a dominant role to play in systems biology. In this chapter, we briefly outline two approaches to systems biology and the role that mathematical models has to play in them. Our focus is on kinetic models, and silicon cell models in particular. Silicon cell models are kinetic models that are firmly based on experiment. They allow for a test of our knowledge and identify gaps and the discovery of unanticipated behavior of molecular mechanisms. These models are very complicated to analyze because of the high level of molecular-mechanistic detail included in them. To facilitate their analysis and understanding of their behavior, model reduction is an important tool for the analysis of silicon cell models. We present balanced truncation as one method to perform model reduction and apply it to a silicon cell model of glycolysis in Saccharomyces cerevisiae. 

This chapter addresses how silicon cell models can be used in biosimulation for systems biology. We first describe the process of model building, as well as its purpose and how it fits in systems biology. Then we compare the use of silicon cell models with the use of the less-detailed core models. We briefly discuss various simulation methods used to model phenomena involving diffusion and/or stochas-ticity as well as methods for model analysis. Finally we discuss balanced truncation as a method for model reduction. This method is illustrated by applying it to a silicon cell model of yeast glycolysis. 

In mathematical system theory, the subject of model reduction has been studied for about 30 years. The focus is on model reduction of linear systems, in particular methods based on singular value decomposition. One of the best known of these methods is balanced truncation. It is used extensively for various engineering purposes, such as electronic chip design and the reduction of models of aerospace structures. This method does not require the type of a priori information about the system mentioned above. Only recently has it been tried out on biochemical systems [105, 106]. 

The aim of the project reported here is to develop system reduction methods for large biochemical systems, including silicon cell models. Here we present our first approach using balanced truncation. The plan is to develop reduction methods custom-made for biochemical systems. To use balanced truncation is a natural first step towards the development of finer methods, since it is a basic method in system theory, and many other methods are variants of this. 

We now introduce some concepts that will be needed for the later description of the method of balanced truncation. In Section 15.2 kinetic systems were introduced of the form ... 

In order to use balanced truncation the system must be a linear system. Since most biological systems are highly nonlinear, they have to be linearized before balanced truncation can be implemented. Linearization is the procedure of constructing a linear system of which the state variables are approximations of the deviations of the state variables in the original system from some selected state, for example the steady state. The approximation will be good when the state variables are close to the steady state, but less good when far away from this state. 

Truncation is the exclusion of some of the last state variables of the balanced system, which are the state variables of least importance for the input-output behavior. This results in a system of lower order than the original system, with an input-output behavior similar to that of the linearized, but not exactly the same. In the coming section we will describe the mathematical procedure of balanced truncation, but for more details we refer to Chapter 7 in the book by Athanasios [107]. 

Balanced Truncation in Action Reduction of a Silicon Cell Model of Glycolysis in Yeast... 

To compare the original system with the reduced system, the pyruvate flow in the original, the linearized, and the reduced model was simulated. Starting at steady state, with the constant input u(t) = 50 mM, 10% and 50% step increases in input glucose concentration were applied, resulting in increases in the flow of pyruvate by only 0.3% and 1.5% respectively (Fig. 15.2). Notwithstanding the small increase in flux, the balanced truncation introduced a 10% overestimation of the increase in pyruvate flux. 

Nonzero elements DEL(i) request finite-difference estimates of df/dOi and dPHl/dOi for those parameters, and are updated often to balance truncation and rounding errors. 

A combined Balanced Truncation and Multi-Parametric Programming approach for Linear Model Predictive Control... 

We present a novel approach to Model Predictive Control problems, which combines a model reduction scheme coupled with parametric programming. Balanced Truncation is used to first reduce the size of the original Model Predictive Control formulation, while multi-parametric programming is employed to derive the parametric control laws offline. The theoretical developments are presented with an example problem. 

Keywords MFC, Multi-Parametric Programming, Balanced Truncation. 

In this paper we present a Model Reduction technique incorporated with multi-parametric programming and control, namely Balanced Truncation (T3T). The use of Balanced Truncation eliminates a number of states of dynamic linear systems, while a bound on the maximum error obtained for the output vector can be established. This then allows for the derivation of (approximate) linear parametric controllers, which can be tested and validated (against the original high-fidelity model) off-line. These theoretical developments are presented next. 

Balanced Truncation in Multi-parametric programming and control... 

In Eq. (1) we present the mathematical formulation of the MPC problem we aim to solve. Given Eq. (1), we first seek to use balanced truncation to reduce the size of the model, and then solve the reduced control problem via our multi-parametric programming and control methodologies. The derived parametric controller can then be validated against the original, full space model. 

Where the initial state x(t) corresponds to the vector of parameters in the multi-parametric programming framework. Balanced truncation is then applied to Eq. (1). We work with the dynamic system (xt+k+i t = Axt+k t + But+kl yt+k t = Cxt+k t) and seek to find a transformation T such that the transformed system is balanced. Following the procedure as described in [5], we describe the dynamic system in an equivalent balanced form ... 

Note that Eq. (4) is not exactly equivalent to either Eq. (1) or Eq. (3) information on the dynamics is lost during the balanced truncation step. There is an inherenf error in the calculation of the output vector y even though a feasible solution may be obtained from the reduced problem, this may actually lead to constraint violations of the original problem. We consider here two ways to deal with this problem (i) neglect the output boimds and keep only the input bounds (ii) update the output bounds in order to ensure feasibility of all output solutions. These are presented next. 

We have presented a systematic procedure to derive parametric controllers based on (i) reduction of the original MFC model by the use of Balanced Truncation, and (ii) application of our multi-parametric programming and control toolbox [7]. 

Dones, 1., Skogestad, S., and Preisig, H.A. (2011) Application of balanced truncation to nonlinear systems. Industrial eL Engineering Chemistry Research, 50,10093-10101. 

Shape optimization of microfluidic structures is a challenging problem, where MOR is strongly desired to reduce the computational complexity during iterations. Utilization of reduced order models for shape optimization in microfluidic devices has been explored recently. Antil et al. [15] combined the POD and the balanced truncation MOR methods for shape optimization of capillary barriers in a network of microchannels. Ooi [9] developed a computationally efficient SVM surrogate model for optimization of a bioMEM microfluidic weir to enhance particle trapping. 

H.K. Fathy and J.L. Stein Fundamental concordances between balanced truncation and activity-based model reduction. 2nd International Conference on Integrated Modeling and Analysis in Apphed Control and Automation. Marseille, France (2005)... 

This work investigates the use of reduced order models of reactive absorption processes. Orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE) are compared. All three methods are able to accurately describe the steady state behaviour, but they predict different dynamics. In particular, the OC dynamic models show large unrealistic oscillations. Balanced truncation, residualization and optimal Hankel singular value approximation are applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy. 

Linearize the open-loop dynamic model of the process at the two fixed points in Table 4. Then perform model reduction [38,39] to derive two reduced linear 4-state models. From the Jacobian of the full state model, balanced truncation was used to reduce the model order. The... 

Vasilyev D, Rewienski M, White J (2006) Macromodel generation for bioMEMS components using a stabilized balanced truncation plus trajectory piecewise-linear approach. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25(2) 285-293... 

Summarizing the above considerations for symmetry-adapted and electrostatically balanced truncation of the infinite lattice sums, the following requirements must be satisfied in actual calculations (using again the convention that indices 0, q, Qx, qi refer to cells, and r, s, u, v and a, refer to orbitals and atoms, respectively). ... 

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