Present position: Temporary researcher
|Thesis title:||Observer Design and Parameter Identification for Biochemical Reaction Networks|
|Research area:||Control Systems|
The last decades have seen rapid advancements in the understanding of organisms and their biological behaviour on a sub-cellular and cellular level. A huge demand of research efforts in life science and in medical applications has been required, since many years, to focus on the cellular, sub-cellular and gene level. These advancements are part-wise driven by developments in proteomics, genomics, and measurement technologies. This has led to a significant increase in knowledge, data, and information available on the sub-cellular and cellular level. It has become clear that the identification of single components (such as genes or proteins) and the understanding of their taxonomy and teleonomy do not lead to a complete picture and understanding of the biological processes involved. Instead, a holistic understanding requires a systems approach including mathematical modelling and analysis.
This awareness has led to the research direction of systems biology. It aims to analyzing the organization and control of biological systems. They are, from a control theory perspective, dynamical systems. Systems and control theory is then considered as a major framework which can lead to many important advances in biology in the next years.
It is remarkable, nowadays, that the challenges in biological science and particular in systems biology are several. It is equally remarkable that many research efforts have been in the last decades fruitfully addressed towards the solution of many open problems in this framework. These efforts contribute, on many different directions, to complete the ''big picture'' of understanding on biological systems, their organization and dynamics. In particular understanding in systems biology is constituted by understanding of biological systems on four main different levels: systems structure, systems dynamics, control mechanisms actuated by cells, cell control. Understanding on these four levels would lead to a breakthrough for diagnostics and therapy of diseases. Scientific contributions to systems biology are several and cover different areas, facing the overcoming problems through diverse approaches. In biological system modelling the major bottlenecks are the problems related to the quantification and the estimation (from data) of the variables representing the state of the studied systems and parameter estimation.
The estimation of the values assumed by the state variables in a biological system is important in many fields of application (i.e. real time monitoring of the levels of species taking part of the reaction network, better understanding of processes and so on). Such problems have been mathematically addressed to in the past, for small classes of chemical reaction networks (such as zero deficiency mass action networks).
On the other hand, significant achievements in parameter estimation can be carried out by two complementary ways: advances in measurement techniques and advances in parameter identification techniques. As for the latter, efforts are devoted in literature towards the solution of this problem. In general algorithms for parameter identification do not guarantee identifiability and are mainly funded on linearization-based numerical optimization techniques. A solution is not guaranteed and no general criteria for identifiability have been formalized.
2. Methodology and results
In this thesis an important class of biochemical network models is considered. Namely, our investigation is addressed towards mass action network models. These models are formalized according to the mass action law.
The thesis is organized in three parts. Part I mainly deals with the definition of the class of systems investigated throughout the thesis and with the analysis of its properties. Part II is mainly devoted to the problem of state estimation. Part III deals with the problem of parameter identification.
In Part I a special form of the general model is obtained and some properties are inferred by the system structure. First we review the main properties of the models of this class. In particular, much attention is paid in literature to a subclass of these systems, namely the zero deficiency network class. The main results on the subject are discussed. Then some novel advanced issues on the mass action networks class are investigated. For instance we work out a special factorization of the system (considering inputs and outputs) which will be addressed to throughout the thesis. We include in the model degradation and formation reactions. Finally some novel properties (which are of main importance in the following chapters) of these systems are inferred.
As for Part II, we investigate the problem of state estimation for mass-action biochemical networks.
The questions arising, from a practical perspective in this frame are manly two:
Which sets of species do we have to measure so as to gain observability, according to experimental constraints?
How do I design an observer?
In control theory, the issue of nonlinear system observability has been thoroughly addressed to, but the application of general analysis and design techniques to models of biochemical reaction networks can result to be difficult, especially for high dimensional systems typically encountered in systems biology. In particular, the problem of the determination of the outputs in order to gain observability is very tricky, since the observability property is usually tested by computing the so called observability map. Besides being burdensome from the computational point of view, global invertibility of this map is a hard property to test. Furthermore general design methods do not apply successfully in some cases. In this thesis we propose a Luenberger observer and we prove that such observer can asymptotically provide the estimation of the system's state variables (for mass-action networks in general). In particular the main conditions required lead to the determination of a set of suitable output maps such that this observer is successfully applied. A wide set of examples of applications and discussions are finally provided.
In Part III we explore the issue of identifiability and identification for mass action biochemical reaction networks. The key questions we investigated in this frame are:
How do we characterize structural and practical identifiability for models of mass action biochemical reaction networks?
How to perform parameter identification for this class of models?
The approach we adopted towards answering these questions involves a model expansion. Taking the specific model of biochemical reaction networks into account, we derive sufficient and necessary conditions for local parameter identifiability based on a suitable system expansion. For instance, we work out the so-called expanded normal form. The key idea is that observability of this expanded form leads to parameter structural identifiability. The presented results lay a theoretically sound basis for the development of new identification methodologies for biochemical reaction networks. Then we investigate some possible solutions of this problem, ranging from well known nonlinear asymptotical observers to linearization-based nonlinear observers. We show that a simple high gain observer can be designed if a certain conservative condition on the available measurement is satisfied. Examples of application of the explored techniques are shown and a discussion on the obtained results is provided.