Bulletin of Mathematical Biology

, Volume 79, Issue 7, pp 1539–1563 | Cite as

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation

  • Brian Ingalls
  • Maya Mincheva
  • Marc R. Roussel
Original Article


A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.


Sensitivity analysis Periodic solutions Delay-differential equations Gene expression RNA interference 



This research was supported by the Natural Sciences and Engineering Research Council of Canada.


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Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA
  3. 3.Alberta RNA Research and Training Institute, Department of Chemistry and BiochemistryUniversity of LethbridgeLethbridgeCanada

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