Tài liệu Báo cáo khoa học: Optimal observability of sustained stochastic competitive inhibition oscillations at organellar volumes pptx

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

and Biochemistry, University of Lethbridge,
Lethbridge, Alberta, T1K 3M4, Canada
Tel: +1 403 329 2326
Fax: +1 403 329 2057
E-mail:
Website: />Note
The mathematical model described here has
been submitted to the Online Cellular Sys-
tems Modelling Database and can be
accessed free of charge at chem.
sun.ac.za/database/davis/index.html
*Present address
Centre for Nonlinear Dynamics in Physiology
and Medicine, McGill University, Montre
´
al,
Que
´
bec, Canada
(Received 19 August 2005, revised 12
October 2005, accepted 31 October 2005)
doi:10.1111/j.1742-4658.2005.05043.x
When molecules are present in small numbers, such as is frequently the
case in cells, the usual assumptions leading to differential rate equations
are invalid and it is necessary to use a stochastic description which takes
into account the randomness of reactive encounters in solution. We display
a very simple biochemical model, ordinary competitive inhibition with sub-
strate inflow, which is only capable of damped oscillations in the deter-
ministic mass-action rate equation limit, but which displays sustained
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-
teristics: first, we look for fluctuations away from a

steady state of reasonable amplitude which appear to
have a periodic character. Second, if we have enough
data, we look for a peak in the power spectral density
(PSD, the frequency spectrum of the data, derived
from its Fourier transform [20]). If we adopt this
operational definition of sustained oscillations, the
ingredients required for stochastic oscillations may be
observed in a very simple biochemical model, namely
the competitive inhibition (CI) mechanism with sub-
strate influx:
À!
k
0
S; ð1Þ
E þS
!

k
1
k
À1
C !
k
À2
E þP; ð2Þ
E þI
!

k
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-
ated, an uninteresting runaway condition results in

which the substrate accumulates without limit.) At the
parameters used in this study (given in Experimental
procedures), the deterministic system displays damped
oscillations with a natural frequency of f
0
¼
0.00272 Hz which decay to undetectable levels in five
or six cycles [21].
The situation is quite different in the stochastic
version of this model, simulated using Gillespie’s
algorithm [26,27]. The usual differential equation des-
cription assumes that concentrations are continuous
variables. Of course, since concentration is N ⁄ V, and
N is a discrete variable, this is not the case. In fact, in
the small-number regime, the random nature of react-
ive encounters becomes significant. Gillespie’s algo-
rithm generates realizations of the random process
which would result in a well-mixed reaction. Figure 1
shows the number of substrate molecules (N
S
) vs. time
0
500
1000
1500
2000
2500
3000
0 2000 4000 6000 8000 10000 12000 14000
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
we will see later, this is not the case in this class of

models, creating a dilemma for the modeller who
wants to describe the behaviour of a single cell.
The fluctuations appear to have a strong periodic
component, an impression confirmed by the peak in
the PSD (Fig. 2). The decrease in intensity with
increasing frequency seen at low frequencies is charac-
teristic of noise and would be observed in any stochas-
tic simulation of a chemical system as a consequence
of the temporal autocorrelation of chemical fluctua-
tions [28]. Note that the maximum in the PSD appears
just below the natural frequency of the deterministic
system, indicated by the arrow. This redshift, which is
consistently observed, can be understood as resulting
from the addition of the noise spectrum to that of the
oscillatory relaxation of the system. These two spectra
are of course not independent since they both originate
in the stochastic kinetics of the CI mechanism. It is
thus interesting that they behave as if they were inde-
pendent components which could be simply added to
give the overall frequency response of the system.
Due to the conservation of enzyme and inhibitor in
this model, there are only three free variables, which
can be taken to be the numbers of S, I and C mole-
cules for instance. However, as is the case in the deter-
ministic system, the existence of fast and slow
processes in this model at the parameters of interest
implies that the stochastic attractor, i.e. the distribu-
tion of points in the three-dimensional N
S
· 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
½S½E

0
À[C] À½I
0
À[I]
ÀÁÈÉ
À k
À1
þk
À2
ðÞ[C] %0;
which leads to
½C%
½S½E
0
À½I
0
þ½I
ÀÁ
½SþK
M
;
where K
M
¼ (k
-1
+ k
-2
) ⁄ k
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
C

. Nevertheless,
as seen in Fig. 3, the distance from the SSA surface is
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.002 0.004 0.006 0.008 0.01
P/10
9
f/Hz
Fig. 2. Smoothed power spectral density P as a function of fre-
quency f. The PSD at V ¼ 5 fL was computed from 501 realizations
of the stochastic process, as described in the Experimental proce-
dures. The arrow indicates the natural frequency of the correspond-
ing deterministic system.
Stochastic oscillations in competitive inhibition K. L. Davis and M. R. Roussel
86 FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS
generally much smaller than the size of the oscillatory
fluctuations. It follows that the behaviour of this
stochastic model can be understood with reference to
only two variables, which can conveniently be taken to
be N
S
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
the stability of the steady state and of the lack of the

sort of phase space structure which is normally associ-
ated with stochastic limit cycles. How is it then that
this system is able to oscillate?
The mechanism of these oscillations is identical to
that leading to damped oscillations in the deterministic
version of this system [21]: oscillations occur when the
inhibition process is much slower than the catalytic
removal of S. On the other hand, thermodynamics
favours the conversion of a substantial amount of
enzyme to the unproductive form H. When the con-
centration of S exceeds its steady-state value, the
enzyme–substrate complex C tends to accumulate,
leading to a depletion of H. The removal of S being
faster than the recovery of H, this in turn causes N
S
to
fall below its steady-state value. The concentration of
H thus recovers to a value somewhat in excess of its
steady-state value, which allows S to reaccumulate
beyond its steady-state value. In the deterministic
system, S does not reaccumulate to its original value,
and thus the oscillations are damped. In the stochastic
system on the other hand, fluctuations can move the
system away from the deterministic steady state. This
can occur when the system has relaxed into the vicinity
of the steady state, but fluctuations can also keep the
system from moving to the steady state.
Figure 5 shows a trajectory escaping from the steady
state and initiating an oscillation. The trajectory at
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
I

10
4
ρ
0 1000 2000 3000 4000
720
730
740
750
760
770
780
790
Fig. 4. Invariant density (q, histogram of visitation frequencies)
computed from a 201 270 s trajectory of the stochastic system,
leaving out a 1000 s transient, at V ¼ 5 fL. The steady-state con-
centrations for the deterministic version of this model are [S] ¼
1.67 · 10
17
moleculesÆL
)1
and [I] ¼ 1.50 · 10
17
moleculesÆL
)1
which, rounded to the nearest molecule, correspond to N
S
¼ 832
and N
I
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),
N

E
+ N
C
¼ N
E(total)
) N
H
, the latter being completely
determined according to Eqn (5) by N
I
and by the
total amount of inhibitor). Note that fluctuations tak-
ing N
I
away from its steady-state value are essential to
this escape process, since fluctuations in N
S
alone are
restored relatively rapidly by the kinetics.
While the system returns frequently to the vicinity of
the steady state (Fig. 4), most oscillatory cycles bypass
this point. Figure 6 shows an example of an oscillatory
cycle in which favourable fluctuations keep the system
away from the steady state. Note that the correspond-
ing deterministic trajectory contracts strongly toward
this point. Obviously, not every sequence of fluctua-
tions will tend to move the system away from the
steady state, but this occurs sufficiently often to lead
to the sustained oscillations seen, for instance, in
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
little if any role in the oscillations. To test this, we

modified the standard algorithm so that a molecule of
substrate was added exactly every 1 ⁄ c
0
seconds. The
results, shown in Fig. 7, are essentially identical to
those of the standard simulation algorithm. Thus we
may conclude that the kinetics of competition is really
responsible for the oscillations, with the kinetics of
delivery of the substrate playing at most a minor role.
Parameter dependence
The oscillatory mechanism in the deterministic and
stochastic models being similar, the conditions which
lead to oscillations are the same in both cases: k
3
⁄ k
1
and k
-3
⁄ (k
–1
+ k
–2
) must both be small, and the ratio
of [I]
0
to [E]
0
, or equivalently of N
I(total)
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
755

760
765
770
775
0 500 1000 1500 2000
N
I
N
S
Fig. 5. Stochastic trajectory illustrating escape from the determinis-
tic steady state, marked by a dot. A trajectory segment was cho-
sen from a simulation at V ¼ 5 fL starting from the deterministic
steady state (N
S
¼ 832, N
I
¼ 752, N
C
¼ 250; all rounded to nearest
integer from exact result). Unlike our other simulations which were
sampled every second of simulation time, here every reaction
event was stored. The segment shown comprises 550 000 stoch-
astic simulation steps, covering a period of 688 s.
Stochastic oscillations in competitive inhibition K. L. Davis and M. R. Roussel
88 FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS
determine which inhibitors of a given reaction may
lead to oscillatory behaviour. Inhibitor strength is nor-
mally described by the dissociation constant K
I
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
and on k

-3
. The
longest lasting oscillations are found for inhibitors
which release the enzyme slowly, in accord with the the-
ory developed elsewhere [21,29]. Moreover, we note that
the inhibitor must bind the enzyme relatively tightly
(small dissociation constant K
I
), but not too tightly. In
the limit as K
I
fi 0, tight-binding inhibitors become
nearly irreversible and, of course, oscillations are then
impossible. For the parameters of this study, K
M
¼
1.1 · 10
15
moleculesÆL
)1
. Note that the highest qualities
are obtained when K
I
is of a similar size to or smaller
than K
M
.
In the stochastic system, persistent oscillations are
observed. We would nevertheless like to have a meas-
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
0

, and at
the frequency of the peak in the power spectrum, f
p
.
The former is a fixed frequency while the latter is vari-
able, approaching f
0
as V fi 1.
In Figs 9 and 10, we show the observability at the
natural frequency, respectively, as a function of k
-3
at
fixed K
I
and as a function of K
I
at fixed k
–3
. A plot of
the observability at the peak frequency looks nearly
identical, except that the values of the observability are
0
1
2
3
4
5
6
7
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:
k

0
¼ 5 · 10
16
LÆmolecules
)1
Æs
)1
, k
1
¼ 10
)15
moleculesÆL
)1
Æs
)1
,
k
-1
¼ 0.1 s
)1
, k
-2
¼ 1s
)1
,[E]
0
¼ 10
17
moleculesÆL
)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
0 5000 10000 15000

N
S
t/s
Fig. 7. PSD of the model with regular substrate influx at V ¼ 5 fL.
The arrow indicates the natural frequency. This Figure should be
compared to Fig. 2, which, except for the treatment of substrate
influx, was computed identically to this one. In the present case,
the PSD was computed from 191 stochastic trajectories. The inset
shows a typical trajectory of the model with regular substrate influx
which may be compared to the trajectory shown in Fig. 1 for the
fully stochastic model.
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 89
a little higher. Qualitatively, the behaviour is similar to
that observed in the deterministic model: the observa-
bility increases as we decrease k
-3
and displays a maxi-
mum at values of K
I
similar to those where the quality
of the deterministic model reaches a maximum. The
behaviour of the deterministic model can thus be used
as a guide to the behaviour of the stochastic model,
except that the damped oscillations of the former
become sustained oscillations in the latter.
Volume dependence
If we vary the volume while holding the concentrations
constant as we have in this study, then the mean
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
and in Fig. 1 is the same. At very small volumes, as

we might expect, the trajectories don’t show any obvi-
ous regularities (Fig. 11A). As the volume increase,
two things happen: the regular component becomes
more evident and the trajectories move away from the
N
S
¼ 0 axis. The latter is important: when N
S
¼ 0,
reaction (2) cannot compete with (3), for obvious rea-
sons. Accordingly, the inactive form of the enzyme
tends to accumulate, resetting the system to a state
which is far from the steady state. Accordingly, the
system is constantly undergoing transient motion
toward the stochastic attractor rather than evolving in
this attractor and the periodicity cannot be fully
expressed. Once the volume becomes large enough that
excursions to zero are unlikely (Figs 11C and 1), the
periodic component of the motion begins to dominate
the PSD (Figs 2 and 11D). Note the vertical scales in
these figures: These oscillations occur in a mesoscopic
regime where there are quite a few molecules so that
the microscopic details of individual molecular encoun-
ters are of little importance, but where the internal
noise generated by the random occurrence times of
reactions is important.
In chemical systems, the level of internal noise
increases as V
1 ⁄ 2
, 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 parameters are set as in Fig. 8, with V ¼ 5 · 10

)15
L and k
3
¼
0.001 s
)1
.
0
20
40
60
80
100
120
140
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5
O(f
0
)
log
10
(k
-3
/s
-1
)
Fig. 9. Observability at the natural frequency as a function of k
-3
for
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,
which is similar to the volumes of bacteria [33] and

of some eukaryotic organelles [34]. We dub this new
phenomenon ‘system-size observability resonance’, by
analogy to other stochastic resonance phenomena, but
also to distinguish it from classical resonances in which
the signal-to-noise ratio passes through a maximum.
1.2
0.9
0.6
0.3
0
1200
900
600
300
0
100000 105000 110000 115000
0 0.002 0.004 0.006 0.008 0.01
t/s
f/Hz
A
N
S
P/10
9
0.6
0.4
0.2
0
1600
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
500

400
300
200
100
0
180000
170000
160000
150000
140000
100000 105000 110000 115000
0 0.002 0.004 0.006 0.008 0.01
t/s
f/Hz
D
N
S
P/10
9
Fig. 11. Sample stochastic trajectories (lower panel) and smoothed
PSDs (upper) at V ¼ (A) 2.1 · 10
)16
(B) 10
)15
(C) 1.4 · 10
)15
,and
(D) 10
)12
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
p

) vs. volume. We plot the peak
observability because it is experimentally more easily measured
than the observability at the natural frequency, O(f
0
). The observa-
bility at the natural frequency shows a similar trend, reaching its
maximum at a slightly larger volume.
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 91
Discussion
We have shown that sustained stochastic oscillations
with a well-defined frequency can be observed in a
very simple biochemical model. Unlike previous mod-
els which showed similar behaviour [4,14,15,18,19],
ours is neither excitable nor can it produce sustained
oscillations at nearby parameter values. Traditionally,
biochemical oscillations have been modelled using
mass-action differential equations with limit-cycle (sus-
tained oscillatory) behaviour. Our work shows that
models may produce oscillations under much weaker
conditions, provided the stochastic nature of reactive
events is taken into account. Our intent is not to con-
test the excellent modelling work which has been
carried out in the last few decades. Many robust bio-
chemical rhythms, such as the circadian clock, are
almost certainly of the limit-cycle variety [35,36],
although stochastic effects must be considered there
too [37–39]. However, it is worth keeping in mind in
light of our study that cellular rhythms may originate
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
grow roughly as N µ V, will overwhelm the oscillatory

signal at normal assay volumes.
For the parameters used in this study, the observabili-
ties O(f
p
) and O(f
0
) both peak in the femtolitre range.
We note that we did not specifically optimize the param-
eters to obtain this result but that the volume at which
the observability peaks will vary with parameters and
from model to model. Nevertheless, this is a very inter-
esting result. Bacteria [33] and some eukaryotic organ-
elles [34] have volumes in this range. Accordingly, the
stochastic oscillations described in this contribution
may be observable in at least some biochemical settings.
We note that the competitive inhibition mechanism
studied here is but one representative of a class of bio-
chemical oscillators [29] first discovered by Sel’kov and
Nazarenko [46]. Although the mechanism is somewhat
different, the hydrolysis of benzoylcholine by butyrylcho-
linesterase has recently been shown to display damped
oscillations in macroscopic experiments [47], and would
therefore be a candidate for sustained stochastic oscilla-
tions of the sort described here in experiments carried
out on a microscopic scale. Butyrylcholinesterase can be
immobilized in biosilica without detectable loss of activ-
ity in a form suitable for use in microreactors [48]. A
microscopic analogue to the experiment described in one
of our earlier papers [29] could therefore be attempted,
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
92 FEBS Journal 273 (2006) 84–95 ª 2005 The Authors Journal compilation ª 2005 FEBS

macroscopic and mesoscopic pictures of chemical reac-
tions is clearly worthy of further investigation.
Experimental procedures
Simulations
All stochastic simulations reported here were carried out
using Gillespie’s algorithm [26,27], which generates realiza-
tions of a stochastic process consistent with the kinetics of
a well-mixed system. Except where noted otherwise, we
fixed our bulk rate constants as follows: k
0
¼ 5 · 10
16
mole-
culesÆL
)1
s
)1
(8 · 10
)8
molÆL
)1
Æs
)1
), k
1
¼ 10
)15
LÆmole-
cule
)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
Æs

)1
), k
–3
¼
0.001 s
)1
. The total concentrations of enzyme ([E]
0
) and of
inhibitor ([I]
0
) were [E]
0
¼ 10
17
moleculesÆL
)1
(1.7 · 10
)7
molÆL
)1
) and [I]
0
¼ 2 · 10
17
moleculesÆL
)1
(3.3 · 10
)7
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
inhibitor molecules were calculated by N

E(total)
¼ V[E]
0
and
N
I(total)
¼ V[I]
0
.
The deterministic simulation reported in Fig. 1 was car-
ried out using the simulation program xpp, version 5.85
[53]. The stiff integration method was used, with a step size
of 1 s. The rate equations are as follows [21]:
d[S]
dt
¼ k
0
À k
1
[E][S] þ k
À1
[C];
d[C]
dt
¼ k
1
[E][S] À k
À1
þ k
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
2

þ 4k
2
À2
k
3
k
À3
½I
0
p
2k
À2
k
3
;
and
½S
ss
¼
k
0
k
À1
þ k
À2
ðÞ
k
1
k
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
i

/¶c
j
, where c ¼ ([S],[C],[H]), and v ¼
(d[S] ⁄ dt,d[C] ⁄ dt,d[H] ⁄ dt). This 3 · 3 matrix is evaluated at
the steady state and its eigenvalues are computed. In the
oscillatory regime, J evaluated at the steady state has a pair
of complex conjugate eigenvalues which we denote by k
±
.
The real parts of these eigenvalues give the time scales for
relaxation, while their imaginary parts give the frequencies
[54–56]. We define the quality by [29]:
Q ¼
= k
Æ
ðÞ
2p< k
Æ
ðÞ








:E
Note that the quality is identically zero for nonoscillatory
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
maximum in the PSD in a window centered on f

0
of
width 0.2f
0
(i.e. 10% to either side of f
0
). This opera-
tional definition was sufficient to capture the peak due to
the oscillatory mode, excluding the low-frequency tail of
the 1 ⁄ f noise.
Acknowledgements
This work was supported by the Natural Sciences and
Engineering Research Council of Canada. Most of the
calculations were carried out using WestGrid resources
funded in part by the Canada Foundation for Innova-
tion, Alberta Innovation and Science, BC Advanced
Education, and the participating research institutions.
WestGrid equipment is provided by IBM, Hewlett
Packard and SGI.
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 93
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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 95