Attila Gábor

Robust and efficient parameter estimation in dynamic models of biological systems

BackgroundDynamic modelling provides a systematic framework to understand function in biological systems. Parameter estimation in nonlinear dynamic models remains a very challenging inverse problem due to its nonconvexity and ill-conditioning. Associated issues like overfitting and local solutions are usually not properly addressed in the systems biology literature despite their importance.Here we present a method for robust and efficient parameter estimation which uses two main strategies to surmount the aforementioned difficulties: (i) efficient global optimization to deal with nonconvexity, and (ii) proper regularization methods to handle ill-conditioning. In the case of regularization, we present a detailed critical comparison of methods and guidelines for properly tuning them. Further, we show how regularized estimations ensure the best trade-offs between bias and variance, reducing overfitting, and allowing the incorporation of prior knowledge in a systematic way.ResultsWe illustrate the performance of the presented method with seven case studies of different nature and increasing complexity, considering several scenarios of data availability, measurement noise and prior knowledge. We show how our method ensures improved estimations with faster and more stable convergence. We also show how the calibrated models are more generalizable. Finally, we give a set of simple guidelines to apply this strategy to a wide variety of calibration problems.ConclusionsHere we provide a parameter estimation strategy which combines efficient global optimization with a regularization scheme. This method is able to calibrate dynamic models in an efficient and robust way, effectively fighting overfitting and allowing the incorporation of prior information.

Model complexity reduction of chemical reaction networks using mixed-integer quadratic programming

The model complexity reduction problem of large chemical reaction networks under isobaric and isothermal conditions is considered. With a given detailed kinetic mechanism and measured data of the key species over a finite time horizon, the complexity reduction is formulated in the form of a mixed-integer quadratic optimization problem where the objective function is derived from the parametric sensitivity matrix. The proposed method sequentially eliminates reactions from the mechanism and simultaneously tunes the remaining parameters until the pre-specified tolerance limit in the species concentration space is reached. The computational efficiency and numerical stability of the optimization are improved by a pre-reduction step followed by suitable scaling and initial conditioning of the Hessian involved. The proposed complexity reduction method is illustrated using three well-known case studies taken from reaction kinetics literature.

Reaction network realizations of rational biochemical systems and their structural properties

In this paper, a frequently used representation of mass-action type reaction networks is extended to a more general system class where the reaction rates are in rational function form. An algorithm is given to compute a possible reaction graph from the kinetic differential equations. However, this structure is generally non-unique, as it is illustrated through the phenomenon of dynamical equivalence, when different reaction network structures correspond to exactly the same dynamics. It is shown that under some technical assumptions, the so-called dense realization containing the maximal number of reactions, forms a super-structure in the sense that the reaction graph of any dynamically equivalent reaction network is the sub-graph of the dense realization. Additionally, optimization based methods are given to find dynamically equivalent realizations with preferred properties, such as dense realizations or sparse realizations. The introduced concepts are illustrated by examples.

Modeling and identification of a nuclear reactor with temperature effects and xenon poisoning

This paper presents the modeling and identification procedure for a VVER-type pressurized water reactor. The modeling goal is to produce a mathematical description in nonlinear state-space form that is suitable for control-oriented model analysis and preliminary controller design experiments. The proposed model takes temperature effects and xenon poisoning into consideration and thus it is an extension of formerly published simpler model structures. Real transient measurement data from the plant has been used for the identification that is based on standard prediction error minimization. It is shown that the model is fairly well identifiable and the newly inserted model components significantly improve the quality of fit between the measured and computed model outputs. Furthermore, the estimated parameter values fall into physically meaningful ranges.

Linear conjugacy in biochemical reaction networks with rational reaction rates

In this paper we show that the model form of a wide class of kinetic systems with rational terms in the reaction rates is invariant under a positive linear diagonal transformation. Thus, the concept of linear conjugacy defined originally for mass action systems is extended to rational biochemical models. The generalized Kirchhoff matrix and the kinetic weighting matrix of the linearly conjugate models are given as functions of the computed transformation parameters. It is shown through the illustrative examples that the dense realization of a linearly conjugate rational model may contain more reactions than that of a dynamically equivalent one due to the additional degrees of freedom introduced by the linear transformation. The proposed matrix-based representation is suitable for the computational search of preferred graph structures corresponding to linearly conjugate realizations of rational kinetic models.

Modeling and identification of a nuclear reactor with temperature effects and Xenon poisoning

This paper presents the modeling and identification procedure for a VVER-type pressurized water reactor. The modeling goal is to produce a mathematical description in nonlinear state-space form that is suitable for control-oriented model analysis and preliminary controller design experiments. The proposed model takes temperature effects and Xenon poisoning into consideration and thus it is an extension of formerly published simpler model structures. Real transient measurement data from the plant has been used for the identification that is based on standard prediction error minimization. It is shown that the model is fairly well identifiable and the newly inserted model components significantly improve the quality of fit between the measured and computed model outputs. Furthermore, the estimated parameter values fall into physically meaningful ranges.

Improved Parameter Estimation in Kinetic Models: Selection and Tuning of Regularization Methods

Kinetic models are being increasingly used as a systematic framework to understand function in biological systems. Calibration of these nonlinear dynamic models remains challenging due to the nonconvexity and ill-conditioning of the associated inverse problems. Nonconvexity can be dealt with suitable global optimization. Here, we focus on simultaneously dealing with ill-conditioning by making use of proper regularization methods. Regularized calibrations ensure the best trade-offs between bias and variance, thus reducing over-fitting. We present a critical comparison of several methods, and guidelines for properly tuning them. The performance of this procedure and its advantages are illustrated with a well known benchmark problem considering several scenarios of data availability and measurement noise.

On the Verification and Correction of Large-Scale Kinetic Models in Systems Biology

In this paper we consider the problem of verification of large dynamic models of biological systems. We present syntactical criteria based on biochemical kinetics to ensure the plausibility of a model and the positivity of its solution. These criteria include the positivity of the rate functions, their kinetic type dependence on the reactant species concentrations, and the absence of the negative cross-effects that together guarantee the nonnegativity of the dynamics. Further, the stoichiometric matrix of the truncated reaction system is checked against conservation using its algebraic properties. Algorithmic procedures are then proposed for checking these criteria with emphasis on good scaling up properties. In addition to these verification procedures, we also provide, for certain typical errors, model correcting methods. The capabilities and usefulness of these procedures are illustrated on biochemical models taken from the Biomodels database. In particular, a set of 11 kinetic models related with E. coli are checked, finding two with deficiencies. Correcting actions for these models are proposed.

Final Comments by the Authors

E-mail: fazekas@scl.sztaki.hu Figure 2 contains a simulation of the nuclear reactor with input output linearisation, where τ is set to 20min. The response to a step change of the set-pointN is shown. One can see from the top diagram that N has the desired linear behaviour and reaches the new steady state after about 2 hours. However, the internal states are far from stationary at that time, as is illustrated by the lowest diagram (note the different time scales in the diagrams). The zero dynamics is stable but slow. It needs a very long time to settle after a step change. This may be explained by the large time constants 1/λI and 1/λX of the iodine and the xenon concentration, which are between 10 h and 13 h.