# Analysis of Delays in Transcriptional Signaling Networks with Time-Varying Temperature-Dependent Rate Coefficients

### Marcella M. Gomez, Matthew R. Bennett and Richard M. Murray

Submitted, 2014 Winter q-bio Conference (5 Nov 2013)

This paper provides preliminary work in an aim to fundamentally understand the effects of temperature fluctuations in the dynamics of biological oscillators. Motivated by circadian rhythms, we are interested in understanding how time-varying temperatures might play a role in the properties of biochemical oscillators. This paper investigates time-dependent Arrhenius scaling of biochemical networks with delays. We assume these time-delays arise from a sequence of simpler reactions that can be modeled as an aggregate delay. We focus on a model system, the Goodwin oscillator, in which we use time-varying rate coefficients as a mechanism to understand the possible effects of temperature fluctuations. The emergence of delays from a sequence of reactions can be better understood through the Goodwin model. For a high order system and comparably high reaction rates, one can approximate the large sequence of reactions in the model with a delay, which can be interpreted as the time needed to go through the âqueueâ. Such types of delays can arise in the process of transcription for example. To study how these delays are affected by temperature fluctuations, we take the limit as the order of the system and the mean reaction rates approach infinity with a periodically time-varying rate coefficient. We show that the limit cycle of the Goodwin oscillator varies only in the limit when the oscillator frequency is much larger than the frequency of temperature oscillations. Otherwise, the instantaneous frequency of the oscillator is dominated by the mean value of the time-varying temperature.

- Conference Paper: http://www.cds.caltech.edu/~murray/preprints/gbm14-wqbio_s.pdf
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