One of the central problems in the emerging field of Systems Biology is the analysis and functional classification of large complex biochemical reaction networks. Such networks are increasingly being scrutinized and their individual components painstakingly investigated in detail. However, a general methodology for inferring dynamical and functional behavior from the detailed network description is still sorely lacking. We propose methodologies by which large components of such networks can be replaced by components of much smaller state dimension that have similar functionality. We term this problem Functional Model Reduction to emphasize distinctions with traditional model reduction techniques. The enabling ideas behind this methodology consist of understanding how dynamical systems that are designed for prescribed functions (such as logical or hybrid operations) can be implemented with dynamical networks constrained to have specific types of building blocks. We investigate it in the specific context of building blocks that are available from basic biochemical kinetics. Enabled with this analysis, we pose the problem of carrying out this analysis in reverse, that is, given networks with specific types of building blocks, we ask what type of functional behavior they represent, and whether it is possible to mirror that behavior with dynamical system of much lower dimension. Our goal is not to develop a general nonlinear model reduction technique, but rather one that is particularly tailored to the differential equations that result from biochemical kinetics. Some novel aspects of systems theory will need to be developed such as realizations with prespecified network components as well as functional objectives for model reduction.
Intellectual Merit
Uncovering and classification of function from the detailed description of biochemical reaction networks is a central problem in systems biology and dynamical systems theory. The proposed work will contribute techniques that are particularly tailored to the dynamical network that arise from biochemical kinetics. A new paradigm for model reduction based on network function will be developed.
Broader Impact
The broader impacts of this work include the application of the model reduction techniques developed in this project to a high order complex model of ischemic stroke that is being developed, which will make possible new understanding of this disease and new treatments for it. The multi-disciplinary nature of this work will ensure that graduate students from dynamical systems and control and those from the life sciences will develop new skill sets from the other disciplines and will help create graduates who are comfortable working at the boundary of their disciplines.
Intellectual Merit: This research initiated an ambitious program of functional model reduction of large networks. We investigated this model reduction problem and its reverse, a realization theory for networks with a certain function built from a limited palette of simple dynamical components. We obtained partial but promising results using the tools of differential algebra and Groebner bases. These results are amongst few in the literature that provide a constructive realization theory with nonlinear components. We studied limits of performance of reaction networks based on limitations of optimal controllers caused by right-half-plane zeros of the dynamics, relating the network chain and feedback structure to those zeros, and simple formulae for the ultimate limitations of performance were obtained. The work on networks lead us to investigate related phenomena of performance limitations due to network structure. In several works which analyze vehicular formations and general consensus-like dynamical networks, we were able to derive asymptotic (in network size) lower bounds for the best achievable performance. This work provided a clear answer to the question of why the so-called vehicular platoons problem is inherently difficult for large systems, in that it is spatially a one dimensional problem. These results made strong contact with the problems of harmonic solids and order-disorder transitions, and our work exhibited the dimensionality dependence of the limits of performance, in addition to local node dynamics. This result is one of few in the distributed control literature that directly connect limits of performance statements with network topology, a significant question in this field, and one of the central questions in the current proposal. A recent paper has been nominated for the Axelby Outstanding Paper Award of the IEEE Transactions on Automatic Control, arguably the most prestigious theoretical publication award given by the controls community. Broader Impact: Functional model reduction was developed with biochemical kinetic networks as the prototype problem. Though our results are still somewhat limited, they are a step towards a full understanding of the function of large biochemical networks (through a model reduction approach). The reverse problem of realization from a limited palette of simple dynamical components was partially inspired by synthetic biology, and our work may yield useful design techniques for this important field. Analytical calculations of best achievable performance limits in networked dynamical systems were carried out for a biological system (autocatalysis), as well as vehicular formations and consensus algorithms. The latter problem making contact with and borrowing from fundamental results in statistical physics. Thus our work during this project exhibited significant cross-fertilization between areas in control engineering, physics and biology. Such interdisciplinary research and training of graduate students is often desired, and we believe we have demonstrated it with considerable success during this project. One female postdoctoral researcher trained and supported by this project, Stacy Patterson, has gone on to a promising academic career at the interface of Control Theory and Computer Science, thus increasing the participation women in STEM fields.
