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

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 equ

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, incorporatmislead-ing

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: roussel@uleth.ca

Website: http://people.uleth.ca/
roussel

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 http://jjj.biochem.

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.

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:

!k0

Eþ S
!k1

k 1

C!k2

Eþ I
!k3

k 3

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 cohercoher-ence 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 f0¼ 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 (NS) 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.

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 NS· NI· NC

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:

½E0¼ ½E þ ½C þ ½H ð4Þ and

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 ¼ k1½S ½E0 [C]  ½I0 [I]

 kð 1þ k2Þ[C]  0;

which leads to

½C ½S ½E 0 ½I0þ ½I

½S þ KM

;

where KM¼ (k-1+ k-2)⁄ k1 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:

NC¼NS NEðtotalÞ NIðtotalÞþ NI

NSþ VKM

where NE(total) and NI(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 NC 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

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.

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 NS and NI, 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 NS · NI 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 NSto 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 NI 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 NS· NI plane Note the difference in scales between the NS and NI axes The horizontal segments thus represent relatively long

0 1 2 3 4 5 6

N I

104ρ

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 NS¼ 832

and N I ¼ 752 The histogram bins used in this calculation are 10

units wide in the NSdimension, and 1 unit wide in NI.

-10

-5

0

5

10

2000 4000 6000 8000 10000 12000 14000

t/s

240

280

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.

sequences of reaction events without any change in the

total amount of active enzyme (from Eqn (4),

NE+ NC¼ NE(total)) NH, the latter being completely

determined according to Eqn (5) by NI and by the

total amount of inhibitor) Note that fluctuations

tak-ing NIaway from its steady-state value are essential to

this escape process, since fluctuations in NS 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 rotarota-tion, 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⁄ c0, where c0¼ k0V 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⁄ c0seconds 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: k3⁄ k1

and k-3⁄ (k–1+ k–2) must both be small, and the ratio

of [I]0 to [E]0, or equivalently of NI(total) to NE(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

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

N I

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 (NS¼ 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.

determine which inhibitors of a given reaction may

lead to oscillatory behaviour Inhibitor strength is

nor-mally described by the dissociation constant KI¼

k–3⁄ k3 Moreover, if we fix KI and vary, say, k-3, then

k3 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 KIand 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 KI), but not too tightly In

the limit as KIfi 0, tight-binding inhibitors become

nearly irreversible and, of course, oscillations are then

impossible For the parameters of this study, KM¼

1.1· 1015moleculesÆL)1 Note that the highest qualities

are obtained when KIis of a similar size to or smaller

than KM

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 Þ ¼ ffiffiffiffiffiffiffiffiffi

Pðf Þ

p

where S is the mean signal strength, in our case the mean number of substrate molecules We compute observabilities both at the natural frequency f0, and at the frequency of the peak in the power spectrum, fp The former is a fixed frequency while the latter is vari-able, approaching f0as Vfi 1

In Figs 9 and 10, we show the observability at the natural frequency, respectively, as a function of k-3 at fixed KIand as a function of KIat 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

log10(K I/molecules L-1)

(k-3

-1 )

Q

-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:

k0¼ 5 · 10 16 LÆmolecules)1Æs)1, k1¼ 10)15 moleculesÆL)1Æs)1,

k-1¼ 0.1 s)1, k-2¼ 1 s)1, [E]0¼ 10 17 moleculesÆL)1, [I]0¼ 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

f/Hz

0 1000 2000 3000

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.

a little higher Qualitatively, the behaviour is similar to

that observed in the deterministic model: the

observa-bility increases as we decrease k-3and displays a

maxi-mum at values of KIsimilar 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

NS ¼ 0 axis The latter is important: when NS 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 V1⁄ 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

(f0

log10(K I/molecules L-1)

Fig 10 Observability as a function of KIfor the stochastic model.

The parameters are set as in Fig 8, with V ¼ 5 · 10)15L and k 3 ¼

0.001 s)1.

0

20

40

60

80

100

120

140

(f0

log10(k-3/s-1) Fig 9 Observability at the natural frequency as a function of k-3for

the stochastic model The parameters are set as in Fig 8, with

V ¼ 5 · 10)15L and K I ¼ 10 15

moleculesÆL)1.

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

t/s

f/Hz

A

N S

0.6

0.4

0.2

0

1600

1200

800

400

0

t/s

f/Hz

B

N S

0.6

0.4

0.2

0

1200

800

400

0

t/s

f/Hz

C

N S

500

400

300

200

100

0

180000

170000

160000

150000

140000

t/s

f/Hz

D

N S

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)15L Arrows in the upper panels indicate the natural frequency of the system.

0 5 10 15 20 25 30 35 40 45

(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.

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
V, will overwhelm the oscillatory

signal at normal assay volumes

For the parameters used in this study, the observabili-ties O(fp) and O(f0) 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

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: k0¼ 5 · 1016

mole-culesÆL)1s)1 (8· 10)8 molÆL)1Æs)1), k1¼ 10)15

LÆmole-cule)1Æs)1 (6· 108 LÆmol)1Æs)1), k–1¼ 0.1 s)1, k–2¼ 1 s)1,

k3¼ 10)18 LÆmolecule)1Æs)1 (6· 105 LÆmol)1Æs)1), k–3¼

0.001 s)1 The total concentrations of enzyme ([E]0) and of

inhibitor ([I]0) were [E]0¼ 1017

moleculesÆL)1 (1.7· 10)7 molÆL)1) and [I]0¼ 2 · 1017

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:

c0¼ Vk0; for the first-order rate constants (k-1, k-2and k-3),

c-i¼ k-i; and for the second-order rate constants (k1and k3),

ci¼ ki⁄ V Similarly, the total numbers of enzyme and

inhibitor molecules were calculated by NE(total)¼ V[E]0and

NI(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 k0 k1[E][S]þ k1[C];

d[C]

dt ¼ k1[E][S] kð 1þ k2Þ[C];

d[H]

dt ¼ k3[E][I] k3[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

½Css¼ k0=k2;

½Hss¼ ½I0A þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2þ 4k2

2k3k3½I0 p

2k2k3

; and

½Sss¼ k0ðk1þ k2Þ

k1 k2 ½E0 ½H]ss

 k0

with A¼ k3[k)2([E]0) [I]0)) k0] + k)2k)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 Jij¼ ¶vi/¶cj, where c¼ ([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]:

Q¼ = kð Þ 2p< kð Þ

: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 (NS) 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)5Hz, 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 fp was defined as the frequency of the absolute maximum in the PSD in a window centered on f0 of width 0.2f0 (i.e 10% to either side of f0) 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