Figure 2. Dynamical error as the difference between two conditional expectations.
To illustrate, we consider a 2-stage model of gene expression with the input, 
, equal to the current rate of transcription, and the signal of interest 
. We model 
as a 2-state Markov chain and show simulated trajectories of the protein output, 
, corresponding to four different input trajectories, 
. These input trajectories (or histories) all end at time 
in the state 
(not shown) and differ according to their times of entry into that state (labelled 
on the time axis; 
is off figure). 
(black lines) is the average value of 
at time 
given a particular history of the input 
: the random deviation of 
around this average is the mechanistic error 
(shown at time 
for the first realisation of 
). 
is the average or mean value of 
given that the trajectory of 
ends in the state 
at time 
. 
(red line) can be obtained by averaging the values of 
over all histories of 
ending in 
. The mean is less than the mode of the distribution for 
because of the distribution's long tail. 
, not shown, is obtained analogously. The dynamical error, 
, is the difference between 
and 
and is shown here for the first trajectory, 
. Fig. 3B shows data from an identical simulation model (all rate parameters here as detailed in Fig. 3B).