Optimal observability of sustained stochastic competitive
inhibition oscillations at organellar volumes
Kevin L. Davis* and Marc R. Roussel
Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, Alberta, Canada
When a system contains only a small number of work-
ing units, whether these be molecules in a chemical sys-
tem or individuals in a biological population, random
changes in the number of individuals of a population
play an important dynamical role. Living cells in par-
ticular often have biochemical components which are
present in very small numbers. In these cases, the usual
deterministic differential equations may give mislead-
ing results and a stochastic description, incorporating
the essentially random nature of individual reaction
events, is required. If we are interested in biochemical
kinetics, a mesoscopic description, i.e. one which does
not take into account the microscopic details of the
positions and internal states of the molecules involved,
is often sufficient. This is the level of description adop-
ted in this study.
Noise can have a variety of effects in nonlinear
systems [1–4]. In some cases, these effects are more
quantitative than qualitative [5–9]. In others, new
behaviours are observed when either internal [10,11]
or externally imposed noise is considered. It is now
relatively well known that external noise can excite
Keywords
stochastic kinetics; enzyme inhibition;
oscillation; stochastic resonance
Correspondence
M.R. Roussel, Department of Chemistry

oscillations in stochastic simulations. We define an observability parameter,
which is essentially just the ratio of the amplitude of the oscillations to the
mean value of the concentration. A maximum in the observability is seen
as the volume is varied, a phenomenon we name system-size observability
resonance by analogy with other types of stochastic resonance. For the
parameters of this study, the maximum in the observability occurs at vol-
umes similar to those of bacterial cells or of eukaryotic organelles.
Abbreviations
CI, competitive inhibition; PSD, power spectral density; SSA, steady-state approximation.
84 FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS
oscillatory modes in systems which, in the absence of
noise, would decay to equilibrium [4,12–17]. It is also
accepted that internal noise due to stochastic kinetics,
whether in small chemical systems or in ecological mod-
els, can enhance oscillatory motion [18,19]. There is
often an optimal level of noise at which the periodic
character is most evident, a phenomenon known as
stochastic coherence. Stochastic coherence has mostly
been studied in systems which are close to a Hopf bifur-
cation leading to sustained oscillations [14,15,18,19], or
which are excitable [4,14]. However, neither of these fea-
tures is necessary. In one recent study closely related to
our own, internal noise was shown to induce bistability
in a system which otherwise would have a unique steady
state. Fluctuations in molecule numbers then also
induced random transitions between the two states, and
thus an oscillatory mode appeared in the dynamics due
exclusively to the internal noise [17].
Due to the presence of noise, oscillatory behaviour
is often recognized experimentally by a pair of charac-

3
k
À3
H: ð3Þ
In the deterministic limit, this model displays damped
oscillations when reaction (3) is slow but not thermo-
dynamically disfavored [21]. In the small-number
regime however, the concentrations undergo fluctua-
tions of large amplitude with a characteristic period,
i.e. sustained oscillations. Moreover, we find that there
is an optimal volume at which these oscillations should
be most clearly observable, this volume coinciding with
typical volumes of organelles or bacteria. This observa-
tion is related to, but distinct from, system size coher-
ence resonance, a type of stochastic coherence found
in mesoscopic chemical or biochemical systems in
which the noise level is controlled by the system size
[22–25]. Specifically, we find that the signal-to-noise
ratio, a classical measure of oscillatory coherence [15],
increases monotonically with system size, but that the
amplitude of the oscillations relative to the baseline of
the oscillations, a ratio we call the observability, goes
through a maximum as a function of system size.
Results
Mechanism of stochastic oscillations
As mentioned above, in the deterministic (mass-action
differential equation) limit, the CI mechanism with
substrate influx always has a stable steady state unless
the rate of substrate (S) influx exceeds the enzyme’s
(E) turnover capacity. (If the latter condition is viol-

N
S
t/s
Fig. 1. Number of substrate molecules as a function of time from
a stochastic simulation of the CI mechanism at V ¼ 5 fL (red). The
blue line is the corresponding result obtained from the deterministic
differential equations. The model parameters are given in the
Experimental procedures.
K. L. Davis and M. R. Roussel Stochastic oscillations in competitive inhibition
FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS 85
for a typical realization of the CI stochastic process.
For comparison, the number of molecules of S compu-
ted from the usual deterministic rate equations is also
shown. Note that the stochastic oscillations continue
long after those predicted by the differential equations
have died away.
The comparison made in Fig. 1 is one of two we
could make between the deterministic and stochastic
systems. The other possibility would be to compare the
deterministic solution to the average behaviour of an
ensemble of identically prepared stochastic systems.
Due to phase diffusion in the stochastic system, the
average behaviour would display damped oscillations,
just like the deterministic system. However, in many
studies, one uses a set of deterministic differential
equations to represent the time evolution of chemicals
in a single cell. The comparison made in Fig. 1 then
becomes relevant. Classical theory suggests that the
behaviours of the deterministic and stochastic systems
should agree in the large-number limit of the latter. As

· N
I
· N
C
space after neglect of an initial transient, will be relat-
ively thin, staying near a surface which we can roughly
identify with a version of the classical steady-state
approximation (SSA). Specifically, oscillations appear
when the inhibitor system reacts slowly [21]. We
should therefore be able to apply the steady-state
approximation to [C]. Let us pursue this idea systemat-
ically. The mass conservation relations are:
½E
0
¼½Eþ½Cþ½Hð4Þ
and
½I
0
¼½Iþ½Hð5Þ
where [E]
0
and [I]
0
are, respectively, the total concen-
trations of enzyme and inhibitor. Using these mass
conservation relations, the SSA is then:
d½C
dt
¼k
1

is the usual Michaelis con-
stant of the enzyme. In the stochastic model, we track
numbers of molecules rather than concentrations.
Making this transformation, we get, finally:
N
C
¼
N
S
N
EðtotalÞ
À N
IðtotalÞ
þ N
I
ÀÁ
N
S
þ VK
M
; ð6Þ
where N
E(total)
and N
I(total)
are, respectively, the total
numbers of enzyme and inhibitor molecules in the
reaction volume V. Of course, the stochastic system
cannot exactly conform to the SSA as Eqn (6) will typ-
ically predict noninteger values for N

and N
I
, trajectories being essentially confined
to a thin region near the SSA. Note that all our simu-
lations were carried out with the full system. We used
the above result only to justify the use of two-dimen-
sional projections in our data analysis.
In Fig. 4, we show the probability of visiting states
in the N
S
· N
I
plane, the so-called invariant density,
which we obtain as a simple histogram of visitation
frequency from a long trajectory after removing a
transient. In excitable systems, stochastic oscillations
often take the form of stochastic limit cycles, in which
the oscillations follow a relatively well-defined path,
leading to a ring-like structure in the invariant density
[14]. In a model with noise-induced bistability such as
that of Samoilov and coworkers [17], the system will
linger near each equilibrium point for long periods of
time such that the density is expected to have two
maxima. Clearly neither of these scenarios applies here.
Rather, the density has a single peak near the deter-
ministic steady state. In fact, the density shown in
Fig. 4 is not obviously different from that of an ordin-
ary chemical system whose fluctuations around its
steady state are incoherent, leading to a noise spec-
trum. The density is therefore mainly a reflection of

first stays close to the steady state. In fact, the traject-
ory shown returns 18 times to the steady state in the
first 6 s of the evolution. However, fluctuations eventu-
ally bring the system to a state where N
I
is well below
its steady-state value, i.e. where a greater-than-steady-
state amount of the inactive enzyme form H has
accumulated. This allows S to accumulate, moving the
system to the right in the N
S
· N
I
plane. Note the
difference in scales between the N
S
and N
I
axes.
The horizontal segments thus represent relatively long
0
1
2
3
4
5
6
N
S
N

¼ 752. The histogram bins used in this calculation are 10
units wide in the N
S
dimension, and 1 unit wide in N
I
.
-10
-5
0
5
10
2000 4000 6000 8000 10000 12000 14000
d
t/s
240
280
4000 8000 12000
N
C
t/s
Fig. 3. Distance of the stochastic trajectory shown in Fig. 1 from
the SSA surface, Eqn (6). The trajectory mostly stays within four
molecules of the value predicted by the SSA, while the amplitude
of stochastic oscillations in N
C
(inset) is much larger.
K. L. Davis and M. R. Roussel Stochastic oscillations in competitive inhibition
FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS 87
sequences of reaction events without any change in the
total amount of active enzyme (from Eqn (4),

Fig. 1. Comparing the stochastic and deterministic tra-
jectories in Fig. 6, we also note that two display rota-
tion by a similar amount. The rate of rotation, and
thus the period of oscillation, originates in the inter-
play between the time scales for catalysis and inhibi-
tion, and is thus preserved in the stochastic model
(Fig. 2). Escape from the steady state, when it occurs,
lengthens the average cycle, which explains physically
why the frequency spectrum is redshifted relative to
the natural frequency. However, the relatively small
redshifts observed indicate that fluctuation-sustained
cycles such as seen in Fig. 6 dominate the dynamics
rather than escape events. The amplitudes, being dicta-
ted by the sequence of random fluctuations experi-
enced by the system, vary quite a bit from cycle to
cycle, as seen in Fig. 1.
In the usual stochastic simulation algorithm, all
elementary reactions, including the substrate influx
process (1), are treated stochastically (26,27). In other
words, molecules of S are added according to reaction
(1) at random times, with a Poisson distribution of
mean 1 ⁄ c
0
, where c
0
¼ k
0
V is the stochastic rate con-
stant for reaction (1). The above argument suggests
that the random arrival of substrate molecules plays

to N
E(total)
,
must not be too large [21]. The inhibitor subsystem
rate constants are of particular interest because they
735
740
745
750
755
760
765
200 400 600 800 1000 1200 1400 1600 1800
N
I
N
S
Fig. 6. Segment of a stochastic trajectory illustrating a typical oscil-
latory cycle. The steady state is marked by the dot, while the cross
represents the initial point and the diamond the final point of this
segment, which was drawn from the same simulation as that
shown in Fig. 5. The segment shown here includes 384 260 simu-
lation steps representing a time period of 480 s. The dotted curve
is the corresponding deterministic trajectory, run from the initial
point marked by the cross for the same duration.
730
735
740
745
750

¼
k
–3
⁄ k
3
. Moreover, if we fix K
I
and vary, say, k
-3
, then
k
3
will vary in proportion to the former rate constant,
which corresponds to a change in the time constant of
the inhibitor subsystem. In order to understand the
factors which lead to stochastic oscillations, we thus
start by considering an analysis of the deterministic
model which extends our earlier work [21] slightly.
Damped oscillations can be characterized by a meas-
ure of their persistence known as the quality, Q [29,30].
We define the quality so that the amplitude decreases by
a factor of e
)1 ⁄ Q
during one period of oscillation [29].
(See Experimental procedures for details.) A quality of
zero indicates a nonoscillatory state. Large qualities
mean that the oscillations persist longer, which in turn
means that they are more readily observable. In Fig. 8,
we show how the quality depends on K
I

ure of the observability of the oscillations. The signal-
to-noise ratio is typically used for this purpose in
studies of stochastic systems [15]. However, we have
not found this to be a particularly revealing measure
for this system.
In experiments, we have to contend both with the
internal noise and with the inevitable random measure-
ment errors generated by the detection electronics,
among other sources. The observational noise gener-
ally increases with the signal strength, i.e. with the
number of molecules under observation [31]. The
observability of the oscillations will thus depend critic-
ally on the amplitude of the oscillations relative to the
time-averaged number of molecules, which forms the
baseline for the oscillations. The value of the PSD at
frequency f, P(f), is proportional to the square of the
amplitude of the signal at that frequency. We therefore
define the observability of a frequency component of
the signal, O(f), by:
Oðf Þ¼
ffiffiffiffiffiffiffiffiffi
Pðf Þ
p
=

S; ð7Þ
where
S is the mean signal strength, in our case the
mean number of substrate molecules. We compute
observabilities both at the natural frequency f

8
log
10
(K
I
/molecules L
-1
)
log
10
(k
-3
/s
-1
)
Q
10 12 14 16 18
-6
-5
-4
-3
-2
-1
0
Fig. 8. Quality of oscillations of the deterministic model as a func-
tion of K
I
and k
-3
. The other parameters were fixed as follows:

)1
, [I]
0
¼ 2 · 10
17
moleculesÆL
)1
, and k
3
¼ k
-3
⁄ K
I
.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.002 0.004 0.006 0.008 0.01
P/10
9
f/Hz
0
1000
2000
3000

number of each type of molecule in the system is
proportional to V. Additionally, the amplitude of
random fluctuations in chemical systems scales as
ffiffiffiffi
N
p
and thus as
ffiffiffiffi
V
p
[28]. At small volumes (small numbers
of molecules), the PSD shows no peak near the natural
frequency and the spectrum is dominated by the con-
tribution from the internal noise (Fig. 11A). As we
increase the volume, the PSD develops a shoulder
(Fig. 11B) which develops into a distinguishable peak
(Fig. 11C). Increasing the volume further raises this
peak far above the noise level (Figs 2 and 11D). Note
also that the redshift (the difference in frequency
between the peak in the PSD and the natural fre-
quency) decreases as we increase the volume. As
explained earlier, this occurs because the PSD is a sum
of a noise spectrum and of the natural frequency
response of the system. At high noise levels (small V),
the spectrum is more noise-like, while at lower noise
levels (large V), the PSD is dominated by the system’s
natural frequency response.
It is interesting to note how the appearance of the
trajectories changes as we vary the volume. Note that
the time span in each of the lower panels of Fig. 11

, while the number of molecules of
course increases as V. Accordingly, the relative
strength of the internal noise goes as V
)1 ⁄ 2
, decreasing
with volume. It is thus tempting to look for system-
size coherence resonance [22–25] in this system, which
in the present case would be a type of stochastic coher-
ence [4,32] in which the signal-to-noise ratio [15] passes
10
15
20
25
30
35
40
12.5 13 13.5 14 14.5 15 15.5 16
O(f
0
)
log
10
(K
I
/molecules L
-1
)
Fig. 10. Observability as a function of K
I
for the stochastic model.

the stochastic model. The parameters are set as in Fig. 8, with
V ¼ 5 · 10
)15
LandK
I
¼ 10
15
moleculesÆL
)1
.
Stochastic oscillations in competitive inhibition K. L. Davis and M. R. Roussel
90 FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS
through a maximum as a function of system volume.
However, the signal-to-noise ratio just increases mono-
tonically as a function of volume (not shown). We can
understand this behaviour by reference to Figs 2 and
11: As the volume increases, the amplitude of the oscil-
lations goes up faster than that of the background
chemical noise. We thus do not observe conventional
system-size resonance. The observability does however,
show a resonance-like phenomenon [Fig. 12]. The
observability is low at small volumes because noise
dominates in this regime. It increases as the volume
increases and the oscillations become more distinct as
described above. Unlike the signal-to-noise ratio how-
ever, the observability eventually decreases because the
amplitude of the oscillations does not increase as fast
as the mean number of molecules at large V. There is
therefore an optimum system size which, for our
parameters, turns out to be in the femtolitre range,

1200
800
400
0
100000 105000 110000 115000
0 0.002 0.004 0.006 0.008 0.01
t/s
f/Hz
B
N
S
P/10
9
0.6
0.4
0.2
0
1200
800
400
0
100000 105000 110000 115000
0 0.002 0.004 0.006 0.008 0.01
t/s
f/Hz
C
N
S
P/10
9

L. Compare also Figs 1 and 2, which give analogous
results for V ¼ 5 · 10
)15
L. Arrows in the upper panels indicate the
natural frequency of the system.
0
5
10
15
20
25
30
35
40
45
10
-16
10
-15
10
-14
10
-13
10
-12
O(f
p
)
V/L
Fig. 12. Peak observability O(f

from reactions which, in the macroscopic mass-action
limit, produce only damped oscillations. Because of
the phase diffusion implied by the stochastic kinetics
of these processes, in the absence of external syn-
chronizing factors, these rhythms may appear to be
damped in population-level measurements which aver-
age over a large number of cells. We have not investi-
gated the effect of diffusible synchronizing agents on
these oscillations. If they can be synchronized between
cells, this might yield robust multicellular oscillators
which again would challenge our reflex to seek cellu-
lar limit-cycle oscillators to explain biochemical
rhythms.
The conditions under which oscillations are observed
roughly correspond to the case of slow, tight-binding
inhibitors [40–45]. Recall that the mass-action differen-
tial equation model only displays damped oscillations.
We usually expect the behaviour of a stochastic model
to tend toward the behaviour of the corresponding
mass-action system at large volumes. However, as
noted above, the amplitude of the oscillations actually
increases with system size in this case. Thus, the beha-
viour of the stochastic model never approaches that of
the mass-action model. We can only reconcile the
experimental behaviour of systems with slow, tight-
binding inhibitors, where oscillations have not to our
knowledge been observed, with that of our stochastic
model when we take into account the fact that the
observability of the oscillations tends toward zero at
large volumes. Observational noise, which we expect to

viz. a flow-through system in which the substrate is con-
tinuously fed into the reaction chamber where the
enzyme is held. These would no doubt be very difficult
experiments, if they are feasible at all at this time, but
they promise to enhance our understanding of kinetics
on cellular and subcellular scales.
We developed a steady-state approximation (Eqn 6)
to justify our use of two-dimensional representations
of the stochastic trajectories. Steady-state approxima-
tions can also be used to accelerate stochastic simula-
tions [49,50]. The SSA typically works well in
stochastic systems in roughly the same cases as it does
in the deterministic mass-action limit [50]. The success
of the SSA, among other lines of evidence, suggests
that some of the structure of the deterministic system
is retained in the stochastic system. Thus, other tech-
niques used in biochemical modelling could be exten-
ded to the stochastic case. For instance, it is tempting
to try to replace the SSA by a higher-order approxi-
mation to the underlying slow manifold [51,52] in
those cases in which the simpler approximation gives
poor results.
Our study features both well-understood ideas and
some surprises with regard to the relationship between
deterministic and stochastic biochemical systems. The
nonconvergence of the stochastic simulations to the
deterministic result was a particular surprise, especially
given the extreme simplicity of the model in which this
observation was made. The relationship between the
Stochastic oscillations in competitive inhibition K. L. Davis and M. R. Roussel

)1
Æs
)1
(6 · 10
8
LÆmol
)1
Æs
)1
), k
–1
¼ 0.1 s
)1
,k
–2
¼ 1s
)1
,
k
3
¼ 10
)18
LÆmolecule
)1
Æs
)1
(6 · 10
5
LÆmol
)1

molÆL
)1
). The bulk rate constants are transformed to stoch-
astic rate constants for a simulation at a given volume V in
the Gillespie algorithm according to the following formulae:
c
0
¼ Vk
0
; for the first-order rate constants (k
-1
,k
-2
and k
-3
),
c
-i
¼ k
-i
; and for the second-order rate constants (k
1
and k
3
),
c
i
¼ k
i
⁄ V. Similarly, the total numbers of enzyme and

À2
ðÞ[C];
d[H]
dt
¼ k
3
[E][I] À k
À3
[H];
ð8Þ
with [E] and [I] calculated from the mass conservation rela-
tions (4) and (5).
Quality
We outline here the computations leading to Fig. 8. The
steady state of Eqns 8 is
½C
ss
¼ k
0
=k
À2
;
½H
ss
¼½I
0
À
ÀA þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A

À2
½E
0
À½H]
ss
ÀÁ
À k
0
ÂÃ
;
with A ¼ k
3
[k
)2
([E]
0
) [I]
0
) ) k
0
]+k
)2
k
)3
. The Jacobian
matrix, J, is the matrix whose elements are the partial
derivatives of the rates with respect to the concentra-
tions, i.e. J
ij
¼ ¶v

solutions.
Power spectral densities
At each volume, we ran a minimum of 50 simulations, each
covering 500 000 s of simulation time with a 100 000 s dis-
carded transient at a time resolution of 1 s. In important
regions, we used upward of 500 simulations. The PSD (the
frequency spectrum of a signal) was computed from the
time series of the number of substrate molecules (N
S
) for
each simulation individually [20], and the average PSD was
then computed. The main features of the PSD were found
to converge using 50 simulations in these calculations. In
those cases in which we used more simulations, the main
effect was to reduce the noise, but not to change the fre-
quency profile in any significant way. The PSDs were
further smoothed by summing nine consecutive points,
reducing the frequency resolution from 2.5 · 10
)6
to
2.25 · 10
)5
Hz, a procedure which was particularly import-
ant for those points where we used fewer simulations to
compute the PSD.
Signal-to-noise ratios and observabilities were computed
from the smoothed PSDs. In both cases, the peak fre-
quency f
p
was defined as the frequency of the absolute

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