# Circadian Rhythms**Team:** 8
**School:** ALBUQUERQUE ACADEMY
**Area of Science:** Biochemistry
**Interim:** Definition of problem:
A typical biochemical oscillator in all biological systems, called the Circadian Rhythms, describes an organism’s sleep-wake cycle through sustained oscillations produced by the reactions of period proteins. Robustness of such oscillations is critical to the function of a biological system. A robust system will produce a persistent periodic signal even in the presence of environmental disturbance, while a less robust system may collapse under these perturbations.
This project/program will simulate the Circadian Rhythm using the mathematical model proposed by Albert Goldbeter (1995). In each simulation, an environmental disturbance is added to a selected model parameter. From these simulations, I will investigate how these disturbances affect the periodicity of the signals and elucidate under what conditions the system collapses, that is, losses its biological functions.
Solving the problem computationally:
The Circadian Rhythm model is represented by five nonlinear ordinary differential equations (ODEs):
1. dM / dt = vs*KI^4 / (KI^4 + PI^4) – vm*M / (Km + M)
2. dP0 / dt = ks*M – V1*P0 / (K1 + P0) + V2*P1 / (K2 + P1)
3. dP1 / dt = V1*P0 / (K1 + P0) – V2*P1 / (K2 + P1) – V3*P1 / (K3 + P1) + V4*P2 / (K4 + P2)
4. dP2 / dt = V3*P1 / (K3 + P1) – V4*P2 / (K4 + P2) – k1*P2 + k2*PN – vd*P2 / (Kd + P2)
5. dPN / dt = k1*P2 - k2*PN
These equations are solved numerically using the finite difference method, for example:
dx/dt = vx
(x[i] - x[i – 1])/Δt = vx[i - 1]
x[i] = vx[i – 1]Δt + x[i – 1].
Equ. 5 becomes
PN[i] = (k1*P2[i - 1] - k2*PN[i - 1]) Δt + PN[i - 1]
An environmental disturbance is simulated with a random generator, for example, for v in the above equation:
x[i] = (v + ε(t))x[i – 1]Δt + x[i – 1]
Where ε(t) is the a random term used for Monte-Carlo simulations.
A graphical interface allows the user to input model parameters and select the amplitude and time duration of the disturbance. A Java applet is used to graphically display the calculated results.
Progress:
The basic code has been completed. I am now in the process of code revisions and simulations. The preliminary simulations show that the program can successfully solve and display Circadian Rhythm oscillations. By adding environmental disturbances, the system does collapse under certain conditions.
Results expected:
From model simulations, I expect to get better understandings of the robustness of a biochemical oscillator for a given set of variables. This includes finding what magnitudes of disturbance the system can withstand before collapsing.
Mentor: Yifeng Wang
Sources:
Fall C.P., Marland E.S., Wagner J.M., & Tyson J.J. (eds.). 2002. Computational Cell Biology. Springer-Verlag New York. 20: 250-54.
Goldbeter, A. 1996. Biochemical Oscillators and Cellular Rhythms: The molecular bases of periodic and chaotic behavior. Cambridge University Press. 459-90.
Goldbeter, A. 1995. A model for Circadian Oscillations in the Drosophila Period Protein (PER). Proceedings: Biological Sciences, 261: 319-24
Goldbeter A. & Gonze D. 2000. Entrainment vs. Chaos in a Model for a Circadian Oscillator Driven by Light-Dark Cycles. Journal of Statistical Physics, 101: 649-63.
Tyson J.J., Hong C.I., Thron C.D., & Novak B. 1999. A Simple Model of Circadian Rhythms Based on Dimerization and Proteolysis of PER and TIM. Biophysical Journal, 77: 2411-2417.
**Team Members:**
Michael Wang
**Sponsoring Teacher:** Jim Mims
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