Scale invariance in biological systems

In this dissertation we will discuss various techniques related to modeling and identification problems arising in complex biological networks, and demonstrate how control theory approaches can be used to validate mathematical models coming from exhaustive computational experiments or noisy experimental data. The methodology based on systematic exploration of the basic dynamic processes, feedback control loops, and signal processing mechanisms in complex networks or their parts provides powerful tools for guiding the reverse-engineering of networks, and allows one to design artificial systems that are capable of achieving various objectives. Adaptation is an essential property of many cellular systems and it means that the measured variables return to their basal levels after a transient response to a step increase in stimulus. By definition, neither the concepts of perfect nor approximate adaptation address the characteristics of the transient signaling which occurs prior to a return to steady state, which are physiologically relevant. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes exhibit an additional feature: scale invariance, meaning that transient behavior remains approximately the same when the background signal level is scaled. Recent interest in scale-invariance was triggered by a pair of papers published in 2009, in which scale-invariant behavior was experimentally observed in several highly conserved eukaryotic signaling pathways that play roles in embryonic patterning, stem cell homeostasis, cell division, and other central processes, and their misregulation results in diseases including several types of cancer. In this thesis we will review the biological phenomena of adaptation and scale invariance, and present the relevant mathematical results for several classes of systems that exhibit these properties. We will use a model from the literature which describes the class of enzyme networks, to prove the impossibility of perfect scale invariance, and develop the mechanism which gives rise to an approximate scale invariance. We will demonstrate results on a biological example of soil-living amoeba Dictyostelium discoideum. Additionally, it has been often remarked in the literature that certain systems whose output variables respond at a faster time scale than internal components, give rise to an approximate scale-invariant behavior. We will state a fundamental limitation of such a mechanism, showing that there is a minimal error that cannot be overcome, no matter how large the separation of time scales is. We will highlight the extensions and challenges in analyzing adaptation and scale-invariance in a stochastic setting. Finally, we will discuss the development of tools for the identification of time-varying parameters in nonhomogeneous Poisson processes, in applications where discrete measurements such as "spikes" or "tumbles" are observed from the behavior of free swimming bacteria in response to the nutrient (input) signals. The objective is to estimate the underlying rate of a nonhomogeneous Poisson process that describes these events, which can then be used to analyze transient behaviors of various species and postulate a plausible model. This work has been motivated by the novel experimental methods for assaying various chemotactic bacteria based on microfluidics devices, with the goal to analyze scale invariance property and model the behavior of different species using various inputs (nutrients).Ph.D.Includes bibliographical referencesby Maja Škatari