Understanding Oscillatory Biological Systems using Dynamic Optimization Techniques
A. Katharina Wilkins, Paul I. Barton & Bruce Tidor
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 . In other systems, oscillations are used to convey function in more subtle ways. An example are the cAMP transients and Ca2+ spikes in spinal neurons, as described in . The system chosen for the present study is the circadian clock system found in many different organisms, from unicellular ones  to mammals .
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, or 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 . A temperature compensation mechanism exists which cancels the influence of temperature fluctuations on this intrinsic period . A gating mechanism actively regulates when external light input is processed .
Several mathematical models of different levels of complexity have been proposed recently to describe different circadian clock systems [4, 5, 6]. These models will be used in this research project to analyze and compare network performance and design, using methods from dynamic optimization theory.
The Use of Global Dynamic Optimization Methods for Biological Network Analysis
In addition to their predictive value, the aforementioned models can be employed to extract more general principles of biological function and 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 the 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.
The methods used in this project are mainly drawn from existing methods in optimization theory. Global dynamic optimization methods, in particular, 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)  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.
Open Questions and Strategies
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; the clock is being read. However, these activities divert - at least momentarily, if not spatially - the players away from the central clock mechanism, resulting in fluctuating levels of proteins, and/or in altered physical properties such as binding constants. The molecular clock is apparently robust with respect to those variations.
Most of the interesting properties of circadian systems are directly related to their oscillatory behavior. It is therefor a main goal in this project to develop a tool set to allow to analyze typical characteristics of oscillation using dynamic optimization methods. Which parameter(s) - and therefor which physical processes - are important to set the period, amplitude, phase? Which reaction rates contribute to dampening or rhythm generation (forcing)?
Such a tool set requires capturing the characteristics that are studied in form of objective and constraint functions for the optimization. Furthermore, gradient information is necessary for the optimization algorithms to work efficiently. Sensitivity information of the trajectories with respect to the parameters is needed to compute these gradients. Sensitivity analysis of oscillatory systems is still an area of active research [9, 10] and the present work will employ some of those results and strive to develop more.
 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.
 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.
 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