Workshop on Systems Biology, Thu, 08 Nov, 2007
Speaker: Katharina Wilkins
Abstract
A comprehensive analysis is presented of the different types of sensitivities of limit cycle oscillators with respect to parameters. This sensitivity analysis is, compared to the sensitivity analysis of other dynamical systems, complicated by several factors. First, from any initial condition within the region of attraction, the periodic orbit is reached only in asymptotic fashion, but it is of interest to study its exact properties. Furthermore, it is desired to analyze derived properties that are implicit in the mathematical description of the limit cycle, such as amplitude, period and phase of oscillation, and their parametric dependencies.
The general framework of this work is based on the solution of a boundary value problem consisting of n equations ensuring the periodicity of the solution, and the (n+1)st equation being a phase locking condition (PLC) which defines a reference for time zero. Differentiation of this system of equations permits exact calculation of period and state sensitivities. Furthermore, a decomposition of the state sensitivities into a sum of three terms each with a specific contribution (period, amplitudes, phases) to the overall sensitivity is discussed in detail.
The period sensitivity of the system lends itself to answering several questions that are currently discussed in circadian biology: Where in the network is the period control located and how is period control achieved? Why is the period stable with respect to fluctuations in protein levels, or external temperature? How can the clock be "read" without disturbing its own accuracy?
The new decomposition is also the starting point for the definition and computation of different types of phase sensitivities. First, a sensitivity of the phase set by the PLC with respect to the parameters is discussed. This quantity is then used to define a particular type of sensitivity, which is unique to oscillatory systems, and is termed a "peak-to-peak" sensitivity. It describes how an infinitesimal change in parameters would cause a change in the time between two peaks or troughs of different state variables. A simple equation is derived for the exact computation of these sensitivities and the results are compared to the proportional sensitivity of the peak-to-peak distance with respect to overall period change (i.e., stretching or compressing of the oscillation). In doing so, information about the flexibility of the system is gained, in the sense of whether the time span between specific events (e.g., peaks) can be varied independently of the overall period.
This quantity is of interest in the context of circadian biology; while the total length of the day is constant at 24 hours, the length of the sunlit day, or the time between dawn and dusk, undergoes seasonal variation. It can be hypothesized that the clock mechanism might have the capability of adjusting to this variation by varying the peak-to-peak distances of pertinent state variables without varying the period of oscillation. This hypothesis is evaluated using peak-to-peak sensitivity analysis.
The mathematical methods developed here were applied to the most detailed ordinary differential equation model of the mammalian circadian clock currently published, which consists of 73 state variables and 231 parameters. It is shown how sensitivity analysis can be used to relate network structure to network functionality, and that the network level computational analysis complements the detailed molecular knowledge derived from "wet" experiments and furthers our overall understanding of the inner workings of the biological clock.
URL: www.ricam.oeaw.ac.at/specsem/ssqbm/participants/abstracts/index.php
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