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: 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. 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 Âà ; 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. 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We define an observability parameter, which is essentially just the ratio of the amplitude of the oscillations