A temperature compensated sloppy model
For many characteristics and behavior of organisms, it has been possible to discern a direct link between the DNA sequence(s) of one or more genes (the genotype) and its manifestation in the whole organism (the genotype) Gutenkunst and Sethna (2007) suggested it would be useful to consider (at least) two additional layers of organization of biological systems, the chemotype and the dynatype. For a system composed on a particular combination of proteins and small molecules in a defined environment, the dynatype might be the observed state over time of some of the components. Suppose a model which quantitatively describes how each component changes over time, including any relevant interaction between the components. The model would consist of a set of equations describing the rates at which a particular component change from one state to another, which would depend on the two states as well as the states of some of the other components. The behavior of the model will depend on the values of the variables (parameters) in each equation. For each protein component, the actual rate at which the transition occurs will depend, in part, on the nucleotide sequence of its corresponding gene (its genotype), and possibly, any post-translational modification(s). If the model actually is a good representation of a biological, the set of parameters that best approximated the behavior (the chemotype), might give some insight on what aspects of the DNA sequence are crucial to the observed functioning
Sethna and his colleagues (Guntenkunst et al. 2007a, 2007b, Daniels et al. 2008) analyzed such models for many biological systems to explain some type of behavior, The behavior to be fitted consisted of the concentrations of various components of the model that had been observed at various times. The aim was to vary all the parameters of model over a wide range and determine how close the predicted behavior of the model differed from the observed behavior.
For every set of parameters (θ1..θn) they ran a computer simulation of the model and recorded the difference between the predicted and observed values at time for which there was data and added up the square of these differences. So this was just a standard sum of squares estimation to judge the goodness of the fit.
They were interested in how much the goodness of fit would change if they varied the values of the parameters in the vicinity of parameter space that gave the best fit.
Daniels et al, (2008) used sloppy modeling techniques to apply a modified version of the model of van Zon et al. (2007) to the data on dephosphorylation of KaiC observed by Tomita et al. (2006), where the rate of dephosphorylation was similar over a range of temperatures.
The figure at right shows the model they used. They included in their model the rates in both directions between phosphorylation states, rather than assuming one or the other direction (depending on allosteric configuration) was so small it could be taken as 0. They also made no assumptions about the "flip" rates between configurations, while van Zon had argued that only f6 and b0 were important, and the others could be taken as 0.
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They assumed that the rates of phosphorylation and dephosphorylated of KaiC monomers did not depend on phosphorylation level of the heamer (other than the number of unphosphorylated or phosphorylated monomers). There were thus 18 rates to be adjusted to fit the observed data (14 for the rates of "flipping" between hexamer configurations with from 0-6 phosphates, and 2 different rates of phosphorylation, and 2 different rates of dephosphorylation, again depending on configuration.
They assumed that each reaction rate followed Arrhenius law, so the the ith rate is:
r(i) = a(i) X e^(-(E(i)/kT)) ,
where E(i) is the energy barrier, k is Boltzmann's constant, T is the temperature in Kelvin.
A trivial solution that is temperature-compensated would be to have all the E(i) small. They avoided this by limiting the E for the phosphorylation was near 13kT, which they