Figure 1.
The dynamics of the protein output can result in a faithful representation of the current biological environment.
We consider a 2-stage model of gene expression [22]. The extracellular environment or input, , gives the current rate of transcription and the signal of interest
. We model
as either a 2-state Markov chain with equal switching rates between states (the states each have unconditional probability of
) (A&C); or as proportional to a Poissonian birth-death process for a transcriptional activator (B&D; proportionality constant of 0.025). The transformed signals
(in red, lower panels) are a perfect representation of
, although protein levels
(in blue) are not.
, the lifetime
of
equals 1 hr, and the translation rate
. Degradation rates of mRNA and protein are chosen to maximize the fidelity, Eq. 7. The units for
are chosen so that its variance equals one.
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).
Figure 3.
As the protein lifetime decreases, a trade-off between dynamical and mechanistic error determines fidelity.
We consider a 2-stage model of gene expression with the input, , equal to the current rate of transcription, and the signal of interest
. (A) The magnitude of the relative fidelity errors as a function of the protein degradation rate,
(from Eqs. 11, 12 and 13), using a logarithmic axis. (B–D) Simulated data with
as in Fig. 1A. The units for
are chosen so that its variance equals one in each case (hence
and
). Pie charts show the fractions of the protein variance due to the mechanistic (m) and dynamical (d) errors and to the transformed signal. The latter equals
. In B, the relative protein lifetime,
, is higher than optimal (
) and fidelity is 2.2; in C,
is optimal (
) and fidelity is 10.1; and in D,
is lower than optimal (
) and fidelity is 5.3. Dynamical error,
, is the difference between
(black) and the faithfully transformed signal
(red), and decreases from B to D, while mechanistic error increases. The lower row shows the magnitudes of the relative dynamical error (black) and relative mechanistic error (orange). All rate parameters are as in Fig. 1 A&C with
, unless otherwise stated.
Figure 4.
Increasing the strength of negative feedback decreases fidelity.
We consider a 2-stage model of gene expression with the signal of interest , and with
proportional to the level of a transcriptional activator. We simulate
as in Fig. 1A. Upper row compares the time course of the protein output (blue) to the faithfully transformed signal (red),
. Lower row shows the distributions for the output,
, that correspond to each of the two possible values of the input,
(low and high). Vertical lines indicate the means of the distributions. Pie charts show the fractions of the variance of each (conditional) distribution due to dynamical (d) and mechanistic (m) error, weighted by the probability of the input state: summing these gives the overall magnitude (variance) of the dynamical and mechanistic errors. (A) No feedback (
), fidelity equals 2.4. (B) Intermediate feedback (
), fidelity equals 2.0. (C) Strong feedback (
), fidelity equals 1.3. As the strength of feedback increases, the underlying state of the input is more difficult to infer (the conditional distributions overlap more) because increasing (relative) mechanistic error dominates the decreasing (relative) dynamical error. Note the decrease in the (relative) dynamical error when
is in its high state (yellow conditional distribution) because stronger negative feedback gives faster initiation of transcription. Transcription propensities are given by
, and all parameters except
are as in Fig. 3B.
Figure 5.
The fidelity of the collective response of a group of cells exceeds that of a single cell.
We consider a 2-stage model of gene expression with the signal of interest , and with
proportional to the level of a transcriptional activator and modeled as an Ornstein-Uhlenbeck process. The unconditional distribution of
is therefore Gaussian. Pie charts show fractions of the protein variance due to the mechanistic (m) and dynamical (d) errors and are computed using our Langevin method (SI). (A) For a single cell with negative autoregulation (
), fidelity is low and equal to 0.2, with a dominant mechanistic error. (B) For 100 identical and independent cells (given the input's history), with negative autoregulation (
): fidelity between
and the average protein output for the group is higher and equal to 3.5. All parameters as in Fig. 3B except
.