Systems Biology, Optimal Control Problems And Their Applications

In this thesis, we mainly deal with some optimal control problems in systems biology.We investigate four aspects of systems biology:neural system, cellular signal con-duction, gene expression modeling and estimation of systematic parametric sensitivity. In the article, the four aspects of the related problems are some of the most front and progressive research directions. Moreover, they are related to the theory of optimal con-trol. The problems involve approximate conversion of a stochastic control problem to a determinate control problem, the optimal estimation of the state and the optimal control problem with the state constraint. Presently, the application of the theory of optimal control in systems biology is still at its preliminary stage and we think it is an important and promising field.The thesis consists of seven chapters.In chapter 1, we will introduce the development history of systems biology and its recent research status. In the meanwhile, we will also present applications of control methods in systems biology and our work.Chapter 2 introduces some propaedeutics of reaction kinetics and optimal control problems.In chapter 3, we investigate the optimal control of neuronal spiking activity for classical Stein's model, Stein's model with reversal potentials with continuous random inputs, characterized by a positive parameter a and Stein's model with Poisson inputs. We solve the optimal control problems and obtain optimal ratesλ(t) for different kinds of models. The numerical simulations on variable parameters show that it is possible to make the interval of spikes the same as our expected time in the specific range of the values of parameters.In chapter 4, a mathematical model for the regulation of the lac operon are pre-sented to study diauxic growth and non-diauxic growth on glucose and lactose. The model takes into account all of the known regulatory mechanisms including catabolite repression, inducer exclusion, lactose hydrolysis to glucose, as well as the time delays inherent to transcription and translation. A new mechanism for catabolite repression is tested:cyclic adenosine monophosphate (cAMP) synthesis is correlated not only with the external glucose concentration but also with the internal glucose concentration. We have paid particular attention to the intracellular cAMP and theβ-galactosidase in the models. Numerical studies have been tested to show that agreement in manners be- tween the experiments and the models predictions. Changes in the extracellular glucose concentration that inhibited lactose transport could not extend or contract the diauxic growth period during growth in the presence of glucose and lactose.A challenge for any system is to filter the real signals from information-carrying signals obscured by noise. This task is particularly acute for the signal transduction path that mediates bacterial chemotaxis. And thereby we investigated the Dictyostelium discoideum system by using a mathematical model to describe the signal, noise, and system in chapter 5. We formulated and solved an optimal filtering problem to predict the process of ligand binding to the receptors. There was good agreement between the simulation results and experimental phenomena. Finally, we proposed the concept of the amplification factor of the noise to characterize the property of noise propagation.Chapter 6 mainly presents the application of the model reduction method in the lac-tose operon based on parametric estimation. Through the parametric sensitivity equa-tions, the 13 dimensional model is reduced to 3 dimensions by principal component analysis. The reduction not only cuts down the complexity of parametric estimation, but also does not greatly affect the degree of accuracy of the estimation.Chapter 7 presents optimal parametric sensitivity control designs for estimation of parameters in bioreactors. We investigate the law of optimal parametric sensitivity controls for the model of a single specie for microbial growth on a single substrate, the model of a single specie on multiple substrates and the model of multiple species for microbial growth on a single substrate. On the purpose of the estimation of the structural parameters the maximum specific growth rateμmax and the half-saturation coefficient K, we find out that an increase in the concentration of the system state results in the improvement of the exactness and the reliability of the parametric estimation through the theoretical analysis and numerical simulations.In chapter 8, we will give the conclusions. A future development prospect in the applications of the control theory in systems biology will be presented.