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Summary

Biological oscillations are found ubiquitously in cells and are widely variable, with periods varying from milliseconds to months, and scales involving subcellular components to large groups of organisms. Interestingly, independent oscillators from different cells often show synchronization that is not the consequence of an external regulator. What is the underlying design principle of such synchronized oscillations, and can modeling show that the complex consequences arise from simple molecular or other interactions between oscillators? When biological oscillators are coupled with each other, we found that synchronization is induced when they are connected together through a positive feedback loop. Increasing the coupling strength of two independent oscillators shows a threshold beyond which synchronization occurs within a few cycles, and a second threshold where oscillation stops. The positive feedback loop can be composed of either double-positive (PP) or double-negative (NN) interactions between a node of each of the two oscillating networks. The different coupling structures have contrasting characteristics. In particular, PP coupling is advantageous with respect to stability of period and amplitude, when local oscillators are coupled with a short time delay, whereas NN coupling is advantageous for a long time delay. In addition, PP coupling results in more robust synchronized oscillations with respect to amplitude excursions but not period, with applied noise disturbances compared to NN coupling. However, PP coupling can induce a large fluctuation in the amplitude and period of the resulting synchronized oscillation depending on the coupling strength, whereas NN coupling ensures almost constant amplitude and period irrespective of the coupling strength. Intriguingly, we have also observed that artificial evolution of random digital oscillator circuits also follows this design principle. We conclude that a different coupling strategy might have been selected according to different evolutionary requirements.

Introduction

Biological oscillations are observed in all organisms at widely varying temporal and spatial scales. Oscillations play an important role in maintaining homeostasis and delivering encoded information. Many cellular processes have circadian rhythms, whereas oscillators with time constants from milliseconds to minutes with profound biological consequences include the heart beat and neural oscillations, the glycolytic system of muscle or yeast cells, and widespread calcium ion, inositol 1,4,5-trisphosphate and calmodulin oscillations (Goldbeter, 1996; Kearns et al., 2006; Matsu-ura et al., 2006).

Many biological oscillators show synchronized oscillations, not because of the presence of an invariant central or controlling clock, but because the independent oscillating systems (normally including an internal negative feedback; Fig. 1A) are coupled. Examples where biological data show the presence of pairs of coupled oscillators are given in Table 1. For most of the cases in Table 1, the coupling between biological oscillators is made through communication via signaling messengers such as hormones and neurotransmitters (Fig. 1A). For instance, Dictyostelium cellular oscillators are coupled by cyclic adenosine monophosphate (cAMP), and neuronal oscillators are coupled by neurotransmitters such as glutamate (for excitatory coupling) and γ-aminobutyric acid (GABA) (for inhibitory coupling). We found that synchronization between two homogeneous biological oscillators is most widely induced by a positive feedback loop that couples the local oscillators. There are two kinds of a positive feedback loop, depending on the regulation type of the interaction constituting the feedback: a double-positive feedback loop (PP) in which both links activate (positive) interactions (Fig. 1B), and a double-negative feedback loop (NN) in which both links inhibit (negative) interactions (Fig. 1C). Therefore, coupling of oscillators tends to involve two negative feedback loops and one positive feedback loop (which can have either PP or NN coupling structure). There have been studies on the dynamics of coupled feedback loops (Hasty et al., 2001; Brandman et al., 2005; Locke et al., 2006; Matsu-ura et al., 2006; Kim, D. et al., 2007; Kim, J.-R. et al., 2008; Tsai et al., 2008; Shin et al., 2009) and on the synchronization of biological oscillators (Bier et al., 2000; Takamatsu et al., 2000; McMillen et al., 2002; Gonze et al., 2005; Fukuda et al., 2007; Li and Wang, 2007; Yu et al., 2007; Meng et al., 2008; Morelli et al., 2009), but these structures and their consequences have not been modeled in a general context. Kearns and colleagues (Kearns et al., 2006) show that the coupling of two positive oscillatory signals in an antiphase relationship can produce stable activity in response to stimulation, with computational simulations suggesting that the relative strength of two feedback mechanisms and their temporal relationship to each other might account for cell-type-specific dynamic regulation. To unravel the most general design principles of synchronized oscillations, we focused on the positive feedback that couples two local oscillators. In general, positive feedback can amplify signals (Hasty et al., 2000), reduce a response speed, induce hysteresis (Becskei et al., 2001; Ferrell, 2002; Isaacs et al., 2003; Kim, J.-R. et al., 2008) and realize a toggle switch (Gardner et al., 2000; Hasty et al., 2000; Tyson et al., 2003; Kobayashi et al., 2004), but its role in coupling local independent oscillators and inducing synchronized oscillations has not been investigated yet.

Table 1.

Various examples of synchronized oscillations and the corresponding coupled oscillators

In this paper, we explore interesting and important features of the PP and NN feedback that couples two oscillators and enables synchronized oscillation. Through mathematical modeling and extensive computational simulations in which the strength of feedbacks coupling the two oscillators (coupling strength) and other parameters (including delay in feedback) were varied, we reveal that the two types of coupling structures have their own characteristic features in synchronization. In particular, we aimed to analyze the differences of the two coupling types with respect to the time delay between two oscillators, to the period and amplitude of the resulting synchronized oscillations, and to noise robustness. Specifically, we have shown that PP coupling is more advantageous when local oscillators are connected with a short communication time delay, whereas NN coupling is advantageous for a long communication time delay. We also found that this design principle holds for various examples of synchronized oscillations (Table 1). We have also investigated whether artificial evolution of random coupling of digital oscillators follows such a design principle for synchronized oscillations and, intriguingly, find that this is indeed the case.

Results and Discussion

The coupling strength of two local oscillators determines synchronization as well as synchronization time

Let us consider the case where two isolated oscillators oscillate independently with a 170° phase difference and they are then coupled through a positive feedback loop as shown in Fig. 1B,C. For a weak coupling strength (f), the two oscillators are not synchronized (left area of Fig. 1D and left panel of 1E). If not synchronized, they show a phase portrait moving around a limit cycle (red curve in Fig. 1E) located in the diagonal direction of the phase plane (f=0). On the other hand, synchronized oscillators show a limit cycle aligned in the direction of X1=X2 (yellow curve in Fig. 1E). As coupling strength passes a threshold of approximately 1.068, synchronization is induced and the time taken for synchronization rapidly decreases as coupling strength increases to 2 (Fig. 1D). If the coupling strength becomes much larger (f>3.3), then the limit cycle of the yellow curve changes into a stable steady state point (see supplementary material Movie 1) implying disappearance of the oscillation (yellow region in Fig. 1D). A larger coupling strength makes the positive feedback predominant and thereby suppresses the oscillatory behavior of two negative feedback oscillators. Together, these imply that a certain range of coupling strength (orange region in Fig. 1D) is required for synchronized oscillation and a specific coupling strength is related to the synchronization time (see supplementary material Figs S1-S4 for further details).

PP and NN couplings have contrasting properties depending on time delay in feedback between oscillators

Considering the dynamics of coupled oscillators, the coupling strength and the communication time delay between two oscillators are the two main parameters that determine the dynamics such as synchronization, and the period and amplitude of synchronized oscillations. We have considered a parameter region for synchronization of two oscillators having a phase difference of 170° (Fig. 2A for PP and Fig. 2D for NN). Fig. 2A shows that coupled oscillators of type PP can be synchronized when the time delay (which corresponds to τ in the mathematical model in Materials and Methods) between the oscillators is short compared to their periods of uncoupled oscillators. By contrast, coupled oscillators of type NN can be synchronized when the time delay between the oscillators is relatively long (Fig. 2D). In Fig. 2B, when two uncoupled oscillators with a phase difference of 170° are coupled by PP with a time delay of zero, two oscillators can be synchronized. This is because if the time delay is zero, the two oscillators coupled by PP simultaneously enhance each other as soon as they are coupled. If the time delay is, however, long (here, 0.5 cycles), the two oscillators cannot be synchronized (Fig. 2C). To explain this, let us consider two oscillators coupled at two nodes X1 and X2 as shown in Fig. 1A. Since the time delay between X1 and X2 is 0.5 cycles, the value of X1 at time 0 is positively influenced by the value of the other node X2 at time −0.5 cycles. Note that the decreasing pattern of X1 around time 0 is similar to that of X2 around −0.5 cycles. Since X1 is positively regulated by X2, the phase of X1 will not so much change by the coupling. The same also holds for the phase of X2. This means that the phase difference between X1 and X2 will still be kept around 180°, implying no synchronization. Hence, if the time delay is long, two oscillators cannot be synchronized. The synchronization characteristics of NN in Fig. 2E,F can also be explained similarly.

Fig. 1.

An example of the coupling structure of two biological oscillators. (A) The coupling is made through signaling messengers such as hormones, neurotransmitter, light, or sound. Coupling occurs at all scales from subcellular to inter-ecosystem, and might involve identical or non-identical oscillators, and physical or chemical couplers. (B) Two oscillators coupled by double-positive interactions (PP) with coupling strength f. (C) Two oscillators coupled by double-negative interactions (NN). (D) The synchronization time after two oscillators with a phase difference of 170° are coupled. (E) The phase portraits in the (X1, X2)-plane (upper figures) and temporal oscillation patterns (lower figures) of the coupled oscillators for various values of f (left to right) 0, 1.07, 1.5 and 2.5. The red curve denotes the limit cycle of the oscillator with f=0 (no coupling) and the yellow curve denotes the limit cycle of synchronized oscillators (f>1.068; for f=1.07, shown, synchronization is reached after some 300 cycles). As f increases, the phase portrait of the coupled oscillators moves from around the red curve towards the yellow curve (see supplementary material Movie 1). Violet lines denote X1; green lines denote X2 before (dotted lines) and after (solid lines) coupling.

Even homogeneous biological oscillators can exhibit different oscillatory patterns, such as different periods or amplitudes as well as noise. To address this issue, for each time delay, we have randomly perturbed all the parameter values of the coupled oscillators by 10% and examined the phase difference of two oscillators. The insets in Fig. 2A,D show the average phase difference obtained from 100 repetitions of the above procedure for each time delay. Since a smaller phase difference implies better synchronization, we find that coupled oscillators of type PP are well synchronized for a short time delay (see the inset in Fig. 2A) but coupled oscillators of type NN are well synchronized for a long time delay (see the inset in Fig. 2D). Taken together, we conclude that oscillators show increased synchronization behavior with PP coupling when they have a short communication time delay, whereas NN is better for a longer communication time delay (Fig. 2G).

PP and NN induce different features of the resulting synchronized oscillations

We have examined the period and amplitude of the resulting oscillations produced by the coupled oscillators for a wide range of parameter values (Fig. 3A,B). For PP, the coupled oscillator shows no oscillation when the coupling strength is larger than a certain threshold (f >3.3), regardless of the time delay (Fig. 3A, left) (zero amplitude in yellow region of Fig. 3A,B means no oscillation). In this case, the variance of the period of the resulting synchronized oscillations is very large as the coupling strength changes (the left inset in Fig. 3A). We note that the period of unsynchronized oscillations does not change much with respect to the variation of either coupling strength or time delay. Compared to PP, NN shows different features of the period (Fig. 3A, right). For NN, the range of coupling strength for synchronized oscillation changes along with the time delay. In addition, the period of oscillations produced by the coupled oscillator does not change much with either the coupling strength or the time delay; the period of synchronized oscillations changes only a little as the coupling strength increases (the right inset in Fig. 3A). In this case, the range of the coupling strength for synchronized oscillation is also wider compared to that of PP coupling (compare two insets in Fig. 3A). The characteristics of amplitude are similar to those of period (Fig. 3B). For PP (Fig. 3B, left), the variance of the amplitude of the resulting synchronized oscillations is very large as the coupling strength changes (the left inset in Fig. 3B). In addition, the amplitude of synchronized oscillation increases rapidly and then oscillation switches off as the coupling strength increases. Compared to PP, NN also shows different features in terms of amplitude (Fig. 3B, right). As the coupling strength increases, the amplitude decreases very slowly. Hence, coupled oscillators of type NN produce synchronized oscillations with almost constant period and amplitude, even if the coupling strength changes. In other words, NN can induce more stable oscillation patterns with respect to parameter changes, especially for the coupling strength.

Fig. 2.

The parameter regions of coupling strength and communication time delay between two oscillators for synchronized oscillations with respect to each coupling type. Insets show the average of the phase differences of two oscillators obtained from 100 perturbations of the parameters. (A) PP coupling type. (B,C) Temporal oscillation patterns of the two oscillators with coupling strength f=3 with communication time delay of τ=0 cycles (B) and τ=0.5 cycles (C). (D) NN coupling type. (E,F) Temporal oscillation patterns with communication time delay of τ=0 cycles (E) and τ=0.5 cycles (F). (G) Two oscillators with a short communication time delay can be synchronized by PP coupling, whereas those with a long communication time delay can be synchronized by NN coupling. Violet lines denote X1; green lines denote X2 before (dotted lines) and after (solid lines) coupling.

In addition to the period and amplitude, we also investigated the noise robustness (Fig. 3C). Following application of noise to the amplitude and period of one of the pair of coupled oscillators, we found that the amplitude of type PP remained close to the undisturbed range, whereas the amplitude of type NN showed excursions well outside the range. Type PP lost cycle period and synchronization more than type NN, although type PP regained synchronization more rapidly than NN coupling after the noisy stimulus disappeared.

We can conclude that the consequences of coupling oscillators with PP or NN show contrasting features with respect to stability of period and amplitude (favored by type NN), toggling of the oscillation phenomenon with changing coupling strength (favored by PP), robustness to noise (favored by PP for amplitude but not period), and effects of delay in feedback. To investigate a design principle in synchronized oscillators, we tested the effect of selection on the evolution of a number of digital oscillators that result in synchronized oscillations by coupling, and compared the resulting coupled structure obtained from such artificial evolution with those identified in Table 1 involving a range of situations.

Fig. 3.

Synchronization properties of coupled oscillators with various coupling parameters or artificial evolution. (A) The period of the resulting synchronized oscillations produced by the coupled oscillators for each time delay and coupling strength in coupling types PP (left) and NN (right). Insets show the change of period as the coupling strength varies when the time delay is fixed to 0.13 (left) or 0.5 (right) cycles. (B) The amplitude of the resulting synchronized oscillations produced by the coupled oscillators for each time delay and coupling strength in coupling types PP (left) and NN (right). Insets show the change of amplitude as the coupling strength varies when the time delay is fixed to 0.13 (left) or 0.5 (right) cycles. (C) Response patterns when we apply random perturbation (upper panel) to one node (Y1) of the synchronized oscillators for coupling types PP (left) and NN (right) with f=3.0. (D) The numbers of different coupling types (PP, NN and PN) observed out of 1000 artificial evolutions of digital oscillation circuits for synchronized oscillations. (E) The average time delay of each coupling type (PP and NN) calculated from the evolved digital oscillators. Error bar denotes s.d. Violet lines denote X1; green lines denote X2 before (dotted lines) and after (solid lines) coupling.

Artificial evolution of digital oscillation circuits follows the same design principle for synchronized oscillations

Biology and engineering share many interesting design principles (Gardner et al., 2000; Tyson et al., 2003; Guantes and Poyatos, 2006; Kim, T.-H. et al., 2007). We simulated evolution using an evolvable digital system. We first generated a number of digital oscillation circuits (corresponding to biological oscillators), evolved the circuits using a genetic algorithm (Kim, J. et al., 2008) (see supplementary material Fig. S5) and examined the resulting Boolean logic between two digital oscillation circuits. Experimental results from 1000 simulation runs showed that about 60% of the evolved synchronized digital oscillators have coupled oscillator circuits of type PP (Fig. 3D) and that the average time delay between oscillators is short (Fig. 3E), whereas about 5% have the coupling type NN (Fig. 3D) and the time delay is about ten times longer than that of PP (Fig. 3E). These results are consistent with the analysis results of the biological systems above. Here, the prevalence of PP is due to the evolution process that selects in its early stage a simpler structure (combination logic gradually evolves from simple to complex), which means a relatively short communication time delay between oscillators, and this supports our previous analysis. Hence, we can conclude that artificial evolution of digital oscillation circuits also follows the same design principle as biological evolution for synchronization of biological oscillators.

Conclusion

Synchronized oscillations are important for biological functioning (Glass, 2001; Buzsaki and Draguhn, 2004; Schoffelen et al., 2005; Mara et al., 2007), and although the biochemical and genetic components involved in many types of interactions have been characterized, the key factors and parameters involved in synchronization have not been explored in detail. Here, we show a design principle of a synchronization mechanism, the advantage of a particular coupling structure for synchronization, and the consequences of changes in synchronization time (Fig. 1), time delay (Fig. 2), the switching and loss of synchronized oscillations (Fig. 3), and a group of parameters that are relevant to diverse biological systems in which synchronized oscillation of independent oscillators has been observed (Table 1).

In summary, PP coupling is advantageous when local oscillators are connected with a short time delay, whereas NN coupling is required for a longer time delay. In addition, PP coupling results in more robust synchronized oscillations with respect to noise disturbances compared to NN coupling. However, PP coupling can induce a large fluctuation in the amplitude and period of the resulting synchronized oscillation depending on the coupling strength, whereas NN coupling ensures almost constant amplitude and period irrespective of the coupling strength. The biological significance of our findings can be found from the examples shown in Table 1. For instance, oscillators coupled through PP (such as segmentation clocks and circadian clocks) might have been evolved to robustly oscillate with respect to noisy cue signals, whereas oscillators coupled through NN (such as ovulation oscillators and insulin secretion oscillators) might have been evolved to be robustly synchronized for a wide range of variation in the communication signal between the oscillators. Moreover, our results imply that independent oscillators can be synchronized with only weak interactions, which might include small intercellular fluxes between identical pathways. Furthermore, with estimates showing that a high proportion of genes show some sort of diurnal rhythm (McDonald and Rosbach, 2001; Mockler et al., 2007), our results have significant implications by providing a mechanism whereby a large number of oscillating cellular events can become synchronized when only weakly coupled to the environment-driven master oscillator. The abrupt transitions, which biologically correspond to robust switching behavior, were noted with respect to the transition between unsynchronized (UO), synchronized (SO) and no (NO) oscillations as coupling strength increases (Fig. 1C, Fig. 3). There is an interesting study on brain rhythms (Kopell et al., 2000), supporting our design principle, in which it was shown that a synchronized β-rhythm is generated by PP coupling of adjacent (thereby having a short time delay) neurons, whereas it is generated by NN coupling of distant (thereby having a long time delay) neurons. The contrasting synchronization of oscillators between different organisms is shown with the short time delay but robust synchronization seen in Dictyostelium synchronization via cAMP diffusion [see Kim, J. et al. (Kim, J. et al., 2007), with PP coupling] whereas synchronization of much longer ovulation cycles in mammals involving pheromone signaling is supported by NN feedback (Stern and McClintock, 1998; Brennan and Zufall, 2006). We have further found that artificial evolution of random coupling among digital oscillation circuits also follows the same design principle we unraveled. From these results, we conclude that a particular type of coupling might have been selected according to evolutionary requirements such as time delay of the communication between oscillators and the dynamic characteristics of resulting synchronized oscillation patterns. Recently, engineered control of cellular function through the design and manipulation of genetic oscillators is within the reach of current technology (Hasty et al., 2000; Hasty et al., 2001). Since most biological oscillators are originated from genetic oscillators, our results can provide a useful experimental guide for synchronization study of genetic oscillators, particularly for engineering synchronized genetic oscillators.

Materials and Methods

As most of the biochemical reactions such as gene transcriptional regulations can be approximated by Hill-type stimulus-response curves (Yagil and Yagil, 1971; Lemmer et al., 1991; Gardner et al., 2000), biological oscillators can also be described by such Hill-type equations. There have been a number of experimental case studies showing the validity of the Hill-type modeling of various biological oscillators such as circadian oscillators (Goldbeter, 1995; Scheper et al., 1999; Ruoff et al., 2001; Smolen et al., 2001; Forger and Peskin, 2003; Smolen et al., 2004; Gonze et al., 2005; Locke et al., 2005; Locke et al., 2006; Bernard et al., 2007; Kuczenski et al., 2007; Leise and Moin, 2007; Bagheri et al., 2008), calcium oscillators (Tang and Othmer, 1994; Friel, 1995; Li and Wang, 2007), segmentation clocks (Meinhardt and Gierer, 2000; Rida et al., 2004; Yoshiura et al., 2007; Zeiser et al., 2007; Momiji and Monk, 2008) and NF-κB oscillators (Krishna et al., 2006; Ashall et al., 2009) (see supplementary material Table S1 for details). So, we have adopted such a well-established Hill-type mathematical model in this paper to explore the general design principles of synchronized biological oscillations. Since the time delays between oscillators are important factors for synchronization, we used delayed differential equations for our mathematical models.

In particular, we have constructed the PP model as follows: Embedded Image and the NN model as follows: Embedded Image where we set H=3, V1=V2=V3=V4=1, time delay τ12=τ (0<τ<15), τ3456=2, coupling strength F12=F21=f (0<f<8), K31=K42=K13=K24=0.5, Kd1=Kd2=Kd3=Kd4=0.5, and Kb1=Kb2=Kb3=Kb4=0.1 for simplification. The delayed differential equations were solved numerically by dde23 in MATLAB 7.0 (R14). Further details are available on request.

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Ministry of Education, Science and Technology (MEST) through the BRL (Basic Research Laboratory) grant (2009-0086964), the Systems Biology grant (20090065567) and the 21C Frontier Microbial Genomics and Application Center Program (Grant MG08-0205-4-0).

Footnotes

  • Accepted November 19, 2009.

References

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