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,

. shows data from an identical simulation model (all rate parameters here as detailed in ).

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