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Understanding Oscillatory Biological Systems using Dynamic Optimization Techniques

Oscillations in Biology

Numerous examples show that oscillatory systems play an important role in biology. Some biological processes are clearly periodic, e.g. the cell cycle [1]. In other systems, oscillations are used to convey function in more subtle ways. An example is the cAMP transients and Ca2+ spikes in spinal neurons, as described in reference [2]. The context chosen for the present study is the circadian clock system found in many different organisms, from unicellular ones [3] to mammals [4]. Circadian clocks are thought to enhance the fitness of organisms by improving their ability to adapt to extrinsic influences, specifically daily changes in environmental factors, such as light, temperature, and humidity. Clock-controlled genes facilitate the modulation of many physiological properties during the course of one day. In human beings, those properties include blood pressure, mental performance, and hormone levels. The fact that circadian clocks have been identified in a variety of species indicates the presence of a very broad evolutionary advantage. A number of fascinating phenomena are found in these systems: The intrinsic ~24 hour clock period is astonishingly robust [5]. A temperature compensation mechanism exists which cancels the influence of temperature fluctuations, and their associated changes in reaction rates, on this intrinsic period [6]. Moreover, a gating mechanism actively regulates when external light input is processed [7].

Several mathematical models of different levels of complexity have been proposed recently to describe different circadian clock systems [4-6]. These models are being used in this research project to analyze and compare network performance and design, using local methods such as sensitivity analysis followed by global methods from dynamic optimization theory.

Open Questions and Strategies

In addition to their predictive value, the aforementioned models can be employed to extract more general principles of biological function as well as to learn about nature's implementation. One strategy is to use tools and concepts from the engineering sciences, control, and systems theory in particular. In order to extract general principles from mathematical models of biological systems, there is a need to study both local and global dynamics of the networks. To determine the function of a gene product, or a pathway, it is necessary to study its long-range influence on the behavior of the entire system, ideally the entire cell. In more complex systems this task is not tractable by intuition-guided methods.

While we know which structures in molecular systems typically result in oscillation (feedback loops), we don't know much about how the 24-hour period is set and why it is so stable. It is known that the players of the circadian clock play roles in influencing the expression levels of clock-controlled genes; that is, the clock is read. However, these activities divert - at least momentarily, if not spatially - the players from the central clock mechanism, resulting in fluctuating levels of proteins, yet the molecular clock is apparently robust with respect to such variation.

Most of the interesting properties of circadian systems are directly related to their oscillatory behavior. It is therefore a main goal in this project to develop a tool set to allow the analysis of typical characteristics of oscillation using dynamic optimization methods. Which parameter(s) - and therefore which physical processes - are important for setting and adjusting the period, amplitude, and phase? Which reaction rates contribute to dampening or rhythm generation (i.e., forcing)?

Sensitivity Analysis of Oscillating Systems

Sensitivity analysis is a useful tool for the analysis of dynamic systems. It can be used to give local information on the impact of an infinitesimal parameter change on the behavior of the system, including derived functions of its output. As such, sensitivity analysis can be applied in model reduction, stability analysis, and in the analysis of biochemical pathways, to name but a few. However, sensitivity analysis of oscillating systems, in particular that of limit cycle oscillators, is more challenging than for other dynamic systems and is an area of ongoing active research [9. 10]. A significant part of this research project includes adapting existing methods to sensitivity analysis of oscillating systems, and developing new methods, in particular, for the sensitivity analysis of derived properties of oscillators, such as the period, amplitude, and different types of phase relationships. Sensitivity analysis can answer questions such as 'in the given parameterization, which parameters have a large influence on a particular network behavior?'. However, its results only describe the network behavior locally. In order to study the network on a global level, sensitivity information is used to provide gradient information needed to perform global dynamic optimization.

Global Dynamic Optimization Methods for Biological Network Analysis

Global dynamic optimization methods have an advantage over previously employed methods (sensitivity analysis, bifurcation analysis) in that they enable exhaustive analysis of dynamic systems in multidimensional parameter space. Global Dynamic Optimization (GDO) [8] is a powerful tool to impose a design goal on a model and find a set of parameters that realize the goal globally. Depending on whether there is a feasible parameter set, one can determine why the network cannot fulfill the function at all or analyze the strategies involved in performing such function. Is the system optimized by nature to best fulfill different functions, and is there a trade off between them? What would be the optimal tuning for either case? GDO can also be used to analyze robustness by trying to optimize the system away from the intended function, or in other words maximize constraint violations.

Summary

This project aims to develop and use mathematical tools for the analysis of oscillating biological networks. By using both local and global methods, relationships between network parameters, architecture, and its behavior and functionalities can be discovered and analyzed. The results of such analysis are helpful in understanding the biological requirements and design goals that might have led to the development of complex biological networks such as the circadian clock mechanism.

References:

[1] Andrea Ciliberto, Bela Novak and John J. Tyson. Mathematical model of the morphogenesis checkpoint in yeast. In JCB, 163:1243-1254,2003.

[2] Yuliya V. Gorbunova and Nicholas C. Spitzer. Dynamic Interactions of cyclic AMP transients and spontaneous Ca2+ spikes. In Nature, 418:93-96,2002.

[3] Irina Mihalescu, Weihong Hsing and Stanislas Leibler. Resilient circadian oscillator revealed in individual cyanobacteria. In Nature, 430:81-85, 2004.

[4] Daniel B. Forger and Charles S. Peskin. A detailed predicitve model of the mammalian circadian clock. In Proc. Natl. Acad. Sci. U.S.A. , 100:14806 - 14811, 2003.

[5] Hiroki R. Ueda, Masatoshi Hagiwara and Hiroaki Kitano. Robust Oscillations within the Interlocked Feedback Model of Drosophila Circadian Rhythm. In J. theor. Biol. 210:401-406, 2001.

[6] Jean-Christophe Leloup and Albert Goldbeter. Temperature Compensation of Circadian Rhtythms: Control of the Period in a Model for Circadian Oscillations of the PER protein in Drosophila. In Chronobiol. Int. , 14:511-520, 1997.

[7] Peter Ruoff, Merete Vinsjevik, Christian Monnerjahn and Ludger Rensing. The Goodwin Model: Simulating the Effect of Light Pulses on the Circadian Sporulation Rhythm of Neurospora Crassa. In J. theor. Biol. , 209:29-42, 2001.

[8] Adam B. Singer and Paul I. Barton. Global Solution of Optimization Problems with Parameter-Embedded Linear Dynamic Systems. In J. Opt. Th. App. , 121:613-646, 2004.

[9] Daniel E. Zak, Joerg Stelling and Francis J. Doyle III. Sensitivity Analysis of oscillatory (bio)chemical systems. In Comp. Chem. Eng. , 29:663 - 673, 2004.

[10] Brian P. Ingalls. Sensitivity Analysis of Autonomous Oscillations: application to biochemical systems. In Proceedings of the Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS) , July 2004.