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<h1 class="top">Information Transfer in Metabolic
Pathways:<br> Effects of Irreversible Steps in Computer
Models</h1>
<p class=navigation>PAGE 2 OF 4: <a href="irrev.htm">Back
to page 1</a> / <a href="irrev3.htm">Forward to page
3</a></p>
<p class="header">This page contains part 2 of the full
text of the following paper: Athel Cornish-Bowden and María
Luz Cárdenas (2001) "Information Transfer in Metabolic
Pathways: Effects of Irreversible Steps in Computer Models"
<cite><abbr title="European Journal of Biochemistry">Eur. J.
Biochem.</abbr></cite> <strong>268,</strong>
6616–6624.</p>
<h2 class=framed>RESULTS</h2>
<div class="centred"><p><img src="images/irrfig1.gif" width="500" height="163" alt="Linear pathway with an irreversible step"></div>
<p class="legend">Fig. 1. Linear pathway with an irreversible step. The kinetic equations for the component enzymes are listed in Table 1.
<h3 class=framed>Linear pathway with feedback</h3>
<p class="legend">Table 1. Kinetic parameters for the model illustrated in Fig. 1.
The symbols refer to those in eqn. (1) of the text, unless otherwise noted.
Metabolite concentrations are shown by lower-case italic letters that
correspond to the metabolite names, i.e. <i>s</i><sub>1</sub> is the concentration of S<sub>1</sub>, etc.
When numerical values are given for concentrations (i.e. for <i>x</i><sub>0</sub> and <i>x</i><sub>6</sub>) these
were treated as independent of the system, i.e. as constants; where no values
are given they were treated as dependent variables and calculated by the simulation
program. In all cases units are arbitrary (but consistent), but concentrations
can be taken to be in mM if this is considered helpful for understanding.<br><br>
<div class="centred"><table cellpadding="6">
<tr><td>Enzyme</td><td><i>a</i></td><td><i>p</i></td><td><i>K</i><sub>eq</sub></td><td><i>V</i></td><td><i>K</i><sub>m</sub></td><td><i>K</i><sub>p</sub></td></tr>
<tr><td>E<sub>1</sub><sup>a</sup></td><td><i>x</i><sub>0</sub> = 10</td><td><i>s</i><sub>1</sub></td><td>5</td><td>1.5</td><td>6</td><td>1.3</td></tr>
<tr><td>E<sub>2</sub><sup>b</sup></td><td></td><td></td><td>5</td><td>120</td><td></td><td></td></tr>
<tr><td>E<sub>3a</sub><sup>c</sup></td><td><i>s</i><sub>2</sub></td><td></td><td>10</td><td>25</td><td>4</td><td></td></tr>
<tr><td>E<sub>3b</sub></td><td><i>s</i><sub>3b</sub></td><td><i>s</i><sub>3a</sub></td><td>0.2</td><td>500</td><td>4</td><td>0.6</td></tr>
<tr><td>E<sub>4</sub><sup>d</sup></td><td><i>s</i><sub>3a</sub></td><td><i>s</i><sub>4</sub></td><td>1000</td><td>20</td><td>0.5</td><td>1.6</td></tr>
<tr><td>E<sub>5</sub></td><td><i>s</i><sub>4</sub></td><td><i>s</i><sub>5</sub></td><td>8</td><td>15</td><td>6</td><td>3</td></tr>
<tr><td>E<sub>6</sub><sup>e</sup></td><td><i>s</i><sub>5</sub></td><td><i>x</i><sub>6</sub> = 0</td><td>∞</td><td>0 to 2.4</td><td>2</td><td>∞</td></tr>
</table></div>
<p class="legend"><sup>a</sup>When E<sub>1</sub> was assumed to be feedback-inhibited by S<sub>5</sub> a term
(<i>s</i><sub>5</sub>/<i>K</i><sub>fb</sub>)<sup>2</sup> was added to the denominator of the rate expression.
<sup>b</sup>The rate for E<sub>2</sub> was calculated from eqn. (2) of the text, with <i>K</i><sub>eq</sub> = 5,
<i>V</i> = 120, <i>a</i><sub>1</sub> ≡ <i>s</i><sub>1</sub>, <i>a</i><sub>2</sub> ≡ [ATP] = 8 (constant),
<i>p</i><sub>1</sub> = <i>s</i><sub>2</sub>, <i>p</i><sub>2</sub> ≡ [ADP] = 2 (constant), <i>K</i><sub>m</sub>1 = 7, <i>K</i><sub>m</sub>2 = 5,
<i>K</i><sub>p</sub>1 = 3, <i>K</i><sub>p</sub>2 = 0.8. Note that as the concentrations of ATP and ADP are fixed these
metabolites were treated as external to the model. They could not be treated as components of a conservation relationship
because the model as defined does not conserve the sum of their concentrations.
<sup>c</sup>The rate for E<sub>3a</sub> was calculated from a modified form of eqn. (1) in which <i>p</i> in the disequilibrium term
represents <i>s</i><sub>3a</sub><i>s</i><sub>3b</sub> and the denominator is written as a product of three terms
(1 + <i>s</i><sub>2</sub>/<i>K</i><sub>m</sub>)(1 + <i>s</i><sub>3a</sub>/<i>K</i><sub>p</sub>1)(1 + <i>s</i><sub>3b</sub>/<i>K</i><sub>p</sub>2),
with <i>K</i><sub>m</sub> = 4, <i>K</i><sub>p</sub>1 = 9 and <i>K</i><sub>p</sub>2 = 7.
<sup>d</sup>In some simulations the equilibrium constant of E<sub>4</sub> was treated as infinite (i.e. the disequilibrium term in the
numerator was treated as unity), and in some <i>K</i><sub>p</sub>, the Michaelis constant constant of the product, was also treated as infinite,
i.e. the term in <i>s</i><sub>4</sub> was omitted from the denominator. <sup>e</sup>Variation in demand for the end-product S<sub>5</sub>
was simulated by varying the value of the limiting rate <i>V</i> for E<sub>6</sub> in the range 0 to 2.4.
<p>The pathway shown in Fig. 1 is more elaborate than the
simple one examined earlier [19], and incorporates more of
the complications found in real metabolic pathways, but it
has the same essential character as a linear biosynthetic
pathway with a single end-product S<sub>5</sub>, which is consumed by
a demand pathway represented in vestigial form by the final
step catalysed by E<sub>6</sub>. To examine the importance of allowing
for the reversibility of nearly irreversible steps the
reaction catalysed by E<sub>4</sub> has an equilibrium constant of
1000 in favour of the forward direction, which was treated
as infinite in some simulations (i.e. the back reaction was
ignored). Feedback inhibition of E<sub>1</sub> by S<sub>5</sub> was likewise
considered in some simulations and not in others. Ordinary
product inhibition of E<sub>4</sub> by S<sub>4</sub> is shown explicitly in the
Figure because although such inhibition is an inevitable
property of any reversible enzyme-catalysed reaction it is
sometimes ignored in steps considered to be irreversible.
Details of the kinetic equations are given in Table 1.
Variation in metabolic demand was simulated by varying the
limiting rate of E<sub>6</sub>. Inclusion of ATP and ADP in the model
at constant concentrations is in a sense superfluous, as
their concentrations could, of course, be subsumed in the
rate constants of a simpler model. Nonetheless, we include
them to emphasize that the sort of conclusions we reach
here do not depend on the use of oversimplified models in
which every enzyme has just one substrate and just one
product.
<div class="centred"><p><img src="images/irrf2ab.gif" width="638" height="299" alt="Full model (feedback, reversible)">
<p><img src="images/irrf2cd.gif" width="636" height="299" alt="Modified model (feedback, irreversible)">
<p><img src="images/irrf2ef.gif" width="637" height="299" alt="Modified model (no feedback, reversible)">
<p><img src="images/irrf2gh.gif" width="643" height="299" alt="Modified model (no feedback, irreversible +)">
<p><img src="images/irrf2ij.gif" width="644" height="338" alt="Modified model (no feedback, irreversible -)">
</div>
<p class="legend">Fig. 2. Simulation of the model in Fig. 1. In each row the left-hand panel (acegi)
shows flux control coefficients and (in the inset) the flux through E<sub>6</sub> as a function of the demand
expressed by the value of <i>V</i> and the right-hand panel (bdfhj) shows the metabolite concentrations
in the same range of con-ditions. The different rows represent different versions of the model,
and in each case a schematic version of Fig. 1 is added where there is room for it to give a quick
indication of the properties considered, which were as follows: (ab) full model as illustrated in
Fig. 1, with feedback inhibition of E<sub>1</sub> by S<sub>5</sub> and E<sub>4</sub> treated as reversible; (cd) E<sub>4</sub> treated as
irreversible; (ef) E<sub>4</sub> reversible but no feedback inhibition of E<sub>1</sub>; (gh) no feedback inhibition of E<sub>1</sub>,
and E<sub>4</sub> irreversible, but still inhibited by its product S<sub>4</sub>; (ij) no feedback inhibition of E<sub>1</sub>, E<sub>4</sub>
irreversible and no effect of S<sub>4</sub> on E<sub>4</sub>. Note that a more expanded scale for metabolite concentrations
is used in the top two rows (bd) in order to make the much smaller variations in concentrations that
occur in the presence of feedback inhibition more easily visible.
<p>As seen in Fig. 2a, flux control in the complete model
is concentrated in E<sub>6</sub> at low demand; as the demand
increases it changes smoothly to majority control by E<sub>1</sub>,
with a small contribution from E<sub>2</sub>. This means that at low
demand the synthetic flux is essentially equal to the
demand, falling significantly below it only when the demand
starts to exceed what the supply can support (inset to Fig.
2a). This satisfactory degree of flux control is achieved
without requiring enormous changes in metabolite
concentrations (Fig. 2b), even though the feedback
inhibition incorporated in the model is weak compared with
what occurs in many real systems. The general implications
of this sort of behaviour for the regulatory design of
pathways in living organisms have been discussed previously
[24, 25], and we shall not consider them here.
<p>As long as there is a feedback loop the small degree of
reversibility in E<sub>4</sub> is irrelevant. Treating this reaction
as strictly irreversible has no perceptible effect on the
flux control coefficients (Fig. 2c), on the flux (inset to
Fig. 2c), or on the metabolite concentrations (Fig. 2d). It
is also irrelevant in the absence of a feedback loop if the
irreversible enzyme is subject to ordinary product
inhibition, as may be seen by comparing Fig. 2ef with the
virtually indistinguishable Fig. 2gh. Thus reversibility as
such is no more important in the absence of a feedback loop
than it is in the presence of one.
<p>The major change in the model of Fig. 1 comes on
eliminating not only the feedback loop and the
reversibility of step 4 but also the inhibitory action of
S<sub>4</sub> on E<sub>4</sub> (Fig. 2ij). The flux and flux control coefficients
then do not change at all at high levels of demand (Fig.
2i), but at low demand there is no steady state at all; the
concentrations of the late metabolites rise even more
steeply than in Fig. 2f as the demand falls close to the
critical level (Fig. 2j), and their uncontrolled increases
at this level explains the loss of steady state. In simple
terms it occurs because the supply steps produce S<sub>5</sub> faster
than the limiting rate (or "maximum velocity" in the
terminology often used) of the demand pathway.
<p>These results are consistent with previous observations
(e.g. [24]) that apparently similar flux behaviour in
different models may well conceal substantial differences
in the behaviour of the metabolite concentrations. They
suggest that if one is forced by lack of experimental
information or by the need to avoid unnecessarily
complicated rate equations to use models that are simpler
than reality it will normally be better to ignore
reversibility in near-irreversible steps than to ignore
feedback loops that go around such steps.
<p>The fact that product inhibition must always be possible
at high product concentrations does not exclude more
complicated behaviour at low concentrations, such as
product activation in nitrite reductase [26], but such
behaviour is rarely reported and is probably rare in
nature, so it is unlikely to be an important consideration
in the design of metabolic models; by contrast, ordinary
product inhibition is not only theoretically necessary but
is also frequently detected experimentally, and must
certainly be an important consideration in the design of
metabolic models.
<h3 class=framed>Branched pathway with feedback inhibition
and cross-activation</h3>
<div class="centred"><p><img src="images/irrfig3.gif" width="927" height="615" alt="Branched pathway"></div>
<p class="legend">Fig. 3. Branched pathway with feedback inhibition and cross-activation.
The model is shown in the central part of the Figure, and kinetic parameters are listed in Table 2.
Each of the panels (a) to (e) shows the response to changes in the demand in step 4a of the control
coefficients for flux through branch a; in each case the inset shows the corresponding changes in
the three fluxes over the same range of demand. In all cases the axes and scales are as shown
explicitly in Panel c. The grey curves linking the panels to the scheme in the centre of the Figure
show which special effects apply to each panel. In Panel (a) there were no special effects, i.e.
no reversibility in step 3a, no feedback inhibition of step 2a and no cross activation of step 2b.
Each of the other panels shows the effect of adding just one line of communication from S<sub>3a</sub> to E<sub>1</sub>.
Panel (b) shows the effect of allowing for the reverse reaction in step 3a (but no feedback or
activation of the other branch); Panel (c) shows the effect of feedback inhibition of E<sub>2a</sub> by S<sub>3a</sub>
(while keeping step 3a irreversible); Panel (d) shows the effect of moderate activation of E<sub>2b</sub> by S<sub>3a</sub>
(with no feedback inhibition of E<sub>2a</sub> or reversibility in step 3a); and Panel (e) shows the effect of
almost unlimited activation of E<sub>2b</sub> by S<sub>3a</sub> (again, without any other effects).
<p class="legend">Table 2. Kinetic parameters for the model illustrated in Fig. 3. The caption to Table 1 applies <i>mutatis mutandis</i> to this Table. <br><br>
<div class=centred><table cellpadding="6">
<tr><td>Enzyme</td><td><i>a</i></td><td><i>p</i></td><td><i>K</i><sub>eq</sub></td><td><i>V</i></td><td><i>K</i><sub>m</sub></td><td><i>K</i><sub>p</sub></td><td>Note</td></tr>
<tr><td>E<sub>1</sub></td><td><i>x</i><sub>0</sub> = 10</td><td><i>s</i><sub>1</sub></td><td>3</td><td>5</td><td>4</td><td>2</td><td>See note b</td></tr>
<tr><td>E<sub>2a</sub></td><td><i>s</i><sub>1</sub></td><td><i>s</i><sub>2a</sub></td><td>104.6</td><td>20</td><td>1.7</td><td>8</td><td>See notes c–d</td></tr>
<tr><td>E<sub>3a</sub></td><td><i>s</i><sub>2a</sub></td><td><i>s</i><sub>3a</sub></td><td>30000</td><td>150</td><td>0.05</td><td>0.1</td><td>See note e</td></tr>
<tr><td>E<sub>4a</sub></td><td><i>s</i><sub>3a</sub></td><td><i>x</i><sub>4a</sub> = 0</td><td>∞</td><td>0 to 2</td><td>1.5</td><td>∞</td><td>See note f</td></tr>
<tr><td>E<sub>2b</sub></td><td><i>s</i><sub>1</sub></td><td><i>s</i><sub>2b</sub></td><td>62.5</td><td>75</td><td>0.3</td><td>2</td><td>See note g</td></tr>
<tr><td>E<sub>3b</sub></td><td><i>s</i><sub>2b</sub></td><td><i>s</i><sub>3b</sub></td><td>4.67</td><td>50</td><td>2.5</td><td>3.5</td><td>See note d</td></tr>
<tr><td>E<sub>4b</sub></td><td><i>s</i><sub>3b</sub></td><td>x4b = 0</td><td>∞</td><td>25</td><td>3</td><td>∞</td></tr>
</table></div>
<p class="legend"><sup>a</sup>E<sub>1</sub> followed the reversible Hill equationÃ