Figure 1.
Schematic view of the translation process.
Table 1.
Typical translation parameters for E. coli.
Table 2.
Non-dimensional quantities introduced.
Table 3.
Parameters used for computational studies.
Figure 2.
Ribosome density for given kinetic rate constants and ribosome flux as a function of density.
(A) The loci of dimensionless termination () and initiation (
) rate constants that yield a steady state solution with a given density,
. (B) Ribosome flux (
) for steady state solutions as a function of ribosome density. For the two curves shown,
, the initiation rate constant is approximately in the ranges
and
, respectively. The remaining parameter values are
,
,
,
and
for
.
Figure 3.
Ribosome distribution as a function of time.
The ribosome distributions along the mRNA chain during induction, as predicted by the mechanistic model (Eqs. 10, dots), and the time delay model (Eqs. 21, open circles) at times (A) , (B)
, (C)
and (D)
. Parameter values are
,
,
,
,
,
and
for
. Distance along the chain is measured in ribosome lengths and the scale for the y-axis is different in each panel.
Figure 4.
Ribosome density and protein concentration as functions of time.
Numerical simulations, at high ribosome flux, of the ribosome density () and protein concentration (
) as functions of time; mechanistic model (Eqs. 10, dashed line), time-delay model (Eqs. 21, solid line). The initiation rates constants used are (A)
, (B)
and (C)
, respectively. The remaining parameter values are
,
,
,
,
and
for
.
Figure 5.
Steady state comparisons of the models.
Steady state density (top), protein concentration (middle) and protein production rate (bottom) as functions of the steady state density of the mechanistic model, . Mechanistic model (Eqs. 10) shown with the broken line, continuous line corresponds to results from delay model (Eqs. 21). Left panels: maximum flux (
) and
. Right panels: minimum flux (
) and
. For the abscissas, the initiation rate constant grows in the direction of increasing density. Protein concentration (middle panels) and protein production rate (bottom panels) shown at the representative time of
. Other parameters:
,
,
,
and
for
.
Figure 6.
Ribosome distribution as a function of time with a time-varying initiation rate.
Numerical simulation showing (dots) and
(open circles) starting from an empty mRNA chain at times (A)
, (B)
, (C)
and (D)
. Parameters are
,
,
,
,
,
and
for
. Note the change in the vertical scale in each panel. Distance along the chain is measured in ribosome lengths.
Figure 7.
Protein concentration and ribosome density as functions of time with a time-varying initiation rate.
(A) Ribosome density and (B) protein concentration as functions of time, mechanistic model shown with the dashed curve, approximate delay model shown with the continuous one. Parameters as in Figure 6.
Figure 8.
Maxima and minima of the mRNA concentration for different codon numbers.
The maxima and minima of the mRNA concentration are shown after transients have decayed. The result from the mechanistic model is shown with crosses, the time-delay model is shown with circles. The two models undergo a Hopf bifurcation near a codon number of , at this point the behavior changes from steady-state decay to oscillatory. Parameters are
,
,
,
,
for
,
,
,
,
and
.