SENSITIVITY RATIOS: EXACT RESULTS

J. Tóth1,2*
1Budapest University of Technology and Economics, Department of Mathematical Analysis, Hungary;
 2Eötvös Loránd University, Laboratory for Chemical Kinetics, Hungary
jtoth@math.bme.hu

Recent numerical calculations on combustion and cell metabolism models suggest that ratios of sensitivities may show certain regularities and these relations might have deep meaning and consequences on the given chemical (combustion) or biological (cell cycle/metabolism) phenomena [1,3]. Here we give exact sufficient, necessary, and sufficient and necessary conditions for these regularities (local similarity, scaling relation, and global similarity) to hold. We also investigate the question, what happens if the conditions are only approximately fulfilled.
Consider the parameter dependent initial value problem 
 
with Let us suppose, for the sake of simplicity, that the complete solution to all initial value problems are defined on the whole real line, and let the derivative of the ith coordinate function with respect to the kth parameter be denoted by  These sensitivities are known to obey [2] the initial value problem 

Parameters  and  are said to be locally similar with respect to the state variables and  with the similarity function , if for all     and also  holds. A scaling relation is said to hold with respect to the state variables and  if for i,j and for all k and for all   is true. The parameters  and  are said to be globally similar with respect to the state variable with the similarity number and they are uniformly globally similar, if the similarity number does not depend on i.  
The necessary and sufficient condition for the scaling relation to hold is the existence of functions such that holds for all                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
If then 
 with 
The necessary and sufficient condition for uniform global similarity of the parameters   and  with the similarity parameter  is that  holds for all  This also implies the explicit form of the right hand side: all of its coordinate functions do not depend on the parameters  and separately, only on their linear combination  
If the coordinate functions of the right hand side do not depend on the parameters  and separately, only on their linear combination  up to an additive error then
   
If   then 
The figure shows in the case of the simple Volterra-Lotka model that approximate fulfilment of uniform global similarity implies approximate constancy of the right hand side.

Acknowledgement:
The present work has partially been supported by the National Scientific Foundation, Hungary (T047132). Useful discussions  with Mr. Z. Ruzsa are also acknowledged.

References

[1] Turányi T, Lovrics A, Csikász-Nagy A, Zsély I Gy, Zádor J, Novák B: Analysis of a budding yeast cell cycle model using the shapes of local sensitivity functions, Lecture at the present conference (2007)
 
[2] Turányi T: Sensitivity analysis of complex kinetic systems: Tools and applications. J Math Chem 5, 203–248 (1990) 

[3] Lovrics A, Csikász-Nagy A, Zsély I Gy, Zádor J, Turányi T, Novák B: Time scale and dimension analysis of a budding yeast cell cycle model. BMC Bioinformatics 7, 494 (2006)