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According to the textbook Alberts Molecular Biology of the Cell (5th ed., p. 902), negative feedback loops cause oscillations when they are long delayed. I just can't figure out why.
Except for that, in case of short delayed feedback, the inhibition doesn't seem to be full and to go down only mid-way, according to the attached plot. Why?
For that you would need to understand the dynamical systems theory behind the loop. The point at which the oscillation starts is called the Hopf-bifurcation. I shall explain this in simple intuitive terms. Lets assume that Protein-X activates the production of Protein-Y which in-turn causes inhibition of Protein-X production. This is a negative feedback loop.
X → Y
Y ⊣ X
Situation-1: No/little delay
You switch on the production of X. This leads to rapid production of Y which will in-turn quickly repress the formation of X. This would happen so quickly and all that would be observed is a reduction in the steady-state level of X (compared to an unregulated system). Mathematically, the eigenvalues of the system (jacobian matrix- denotes the slope of the function) evaluated at the steady state would be negative and real.
Situation-2: Some delay
In this case, it will take some time for Y to accumulate. This will cause X to shoot up above its final steady state level. This, in-turn, will cause higher production of Y. Because of higher production of Y, X will start getting repressed strongly when Y gets accumulated and starts acting. This will cause X to shoot down below the steady-state but the magnitude of this shoot-"down", will be less than the first shoot-"up". Finally, the system will settle to the steady state. This phenomenon is called damped oscillation. In this case, the eigenvalues of the jacobian are complex with negative real parts.
Situation 3: long delay
In this case the delay between production of Y and its action is so high that the system does not damp but sustains its oscillations. The eigenvalues of the jacobian are imaginary. These kind of oscillations are generally sensitive to different factors.
More robust forms of oscillations exist, called limit cycles. The network structures responsible for this are more complex than a delayed negative feedback loop. Typically it has a coupled positive and negative feedback. Understanding limit cycles is little bit more complicated and I shall not discuss them here.
Also, check out these reviews:
Tyson, John J., and Béla Novák. "Functional motifs in biochemical reaction networks." Annual Review of Physical Chemistry 61 (2010): 219.
Tyson, John J., Katherine C. Chen, and Béla Novák. "Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell." Current Opinion in Cell Biology 15.2 (2003): 221-231.
and this book:
Alon, Uri. An introduction to systems biology: design principles of biological circuits. CRC press, 2006. ISBN 9781584886426
This was our causal loop.
Fig-1 Our situation
Oscillation is simply an up and down of something, occurring in repeat with time. Now let's see why it causing an oscillation.
Fig 2. Analogy with toilet siphon. C = Cause. E = Effect. E(min)= minimum value of effect . E(max)= maximum value of effect (as much allowed by time delay).
I've compared here this process with a toilet siphon.
Here I've considered the constantly adding up water from tap; is cause. As in your case, Enzyme-E is always present at the left side.
In my case (siphon), the water-level is the effect (In your case it is phosphorylated enzyme (EP), that may present at right side, or may not, if inhibition is high.)
Step-1: starting point: cause is applied
Step-2 : Since inhibition take-place a while latter, your EP adds up (my water level goes up). The dotted inhibition sign on fig2 indicates it yet not reached to the target.
Step-3: When a certain time passed, no-more EP accumulates in your system-of interest. Because inhibition starts. In my case, siphon-effect starts, and water level decrease.
Step 4: Ongoing inhibition decrease the result (in your given-case it is phosphorylation product EP, in my case water siphon it is water level. The siphon start pump out some water.)
Step 5: At next stage, your product EP become so low-concentration, that feedback became feeble, and once stops. In my case, the water level come down below siphon input. so siphon-effect stops.
And Step 6 (Similar to step-1)the process repeats once again.
what will happen if the output from a not gate injected back to its own input
Any Feedbacks are welcome.
One of the conditions for oscillations in a negative feedback system are delays. The question is how much of a delay and what do we even mean by a delay? Imagine a sine wave signal being transmitted along the pathway:
-> x1 -> x2 -> x3 -> x4 -> etc
As the signal moves from one species to the next it will get delayed because it takes time for species to accumulate and empty. Using a sine wave is useful because the delay will be related to shifts in the sine wave itself. It is possible to show that the maximum delay a sine wave can experience at each species is 90 degrees. See figure below. The blue wave is the wave we see at x1 and the red wave is approxiamtely the one we see at x2. Notice it has been pushed to the right 90 degrees.
For the linear pathway above, the delay at x2 will be up to 90 degrees, at x3 another 90 degrees (180 relative to x1) and another 90 degrees at x4 (total of 270 relative to x1). The actual amount of delay will depend on the frequency of the wave, the higher the frequency the more likely it will experience the maximum delay.
Now consider a system where x4 negatively feedbacks to the reaction x1 -> x2.
In a real system we won't be injecting a sine wave but there will be noise and that noise will contain many frequencies including one where the phase shift at x4 will be exactly 180. Note we can't get exactly 180 at x3 because we can only approach 180. Consider now that a noisy signal has been shifted exactly 180 degrees at x4, this signal is transmitted back via the negative feedback to the reaction x1->x2. Guess what the negative feedback does, it shifts the signal by another 180 degrees (eg a 0 turns into a 1 and a 1 turns into a 0, it inverts the signal), therefore the full phase shift is now 360 degrees. This means instead of the dampening down disturbances the negative fedback will actually start to amplify them. The negative feedback has turned into a positive feedback. At this point, the system becomes unstable. The instability doesn't cause things to go to infinity however because of physical contraints and nonlinearities in the system. This explains why it is impossible to get sustained oscillations in a negative fedback with only one step or two steps, oscillations only occurs when you have three steps inside the signal loop because that is the only time you can actually get a 180 phase shift.
The other thing to note is that one other thing needs to be present for oscillations to occur, even if you get a positive feedback. That is the instability has to be amplified, if the instbility is dampend as it goes round the loop nothing will happen. Therefore the other condition, the so-called loop gain, has to be at least one. This can be achieved is there is cooperativity of the signal molecule on the inhibition site.
Biochemical switches in the cell cycle
A series of biochemical switches control transitions between and within the various phases of the cell cycle. The cell cycle is a series of complex, ordered, sequential events that control how a single cell divides into two cells, and involves several different phases. The phases include the G1 and G2 phases, DNA replication or S phase, and the actual process of cell division, mitosis or M phase.  During the M phase, the chromosomes separate and cytokinesis occurs.
The switches maintain the orderly progression of the cell cycle and act as checkpoints to ensure that each phase has been properly completed before progression to the next phase.  For example, Cdk, or cyclin dependent kinase, is a major control switch for the cell cycle and it allows the cell to move from G1 to S or G2 to M by adding phosphate to protein substrates. Such multi-component (involving multiple inter-linked proteins) switches have been shown to generate decisive, robust (and potentially irreversible) transitions and trigger stable oscillations.  As a result, they are a subject of active research that tries to understand how such complex properties are wired into biological control systems.   
Modeling Biological Oscillations: Integration of Short Reaction Pauses into a Stationary Model of a Negative Feedback Loop Generates Sustained Long Oscillations
Sustained oscillations are frequently observed in biological systems consisting of a negative feedback loop, but a mathematical model with two ordinary differential equations (ODE) that has a negative feedback loop structure fails to produce sustained oscillations. Only when a time delay is introduced into the system by expanding to a three-ODE model, transforming to a two-delay differential equations (DDE) model, or introducing a bistable trigger do stable oscillations present themselves. In this study, we propose another mechanism for producing sustained oscillations based on periodic reaction pauses of chemical reactions in a negative feedback system. We model the oscillatory system behavior by allowing the coefficients in the two-ODE model to be periodic functions of time-called pulsate functions-to account for reactions with go-stop pulses. We find that replacing coefficients in the two-ODE system with pulsate functions with microscale (several seconds) pauses can produce stable system-wide oscillations that have periods of approximately 1 to several hours long. We also compare our two-ODE and three-ODE models with the two-DDE, three-ODE, and three-DDE models without the pulsate functions. Our numerical experiments suggest that sustained long oscillations in biological systems with a negative feedback loop may be an intrinsic property arising from the slow diffusion-based pulsate behavior of biochemical reactions.
Keywords: deterministic model negative feedback reaction pause slow diffusion ultradian rhythms.
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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: and the NN model as follows:
Network diagram of the p53 signalling pathway. Adapted from the KEGG signalling pathway .
It is well known that stress induces an increase in levels of p53 which in turn leads to an increase in the transcription of Mdm2 . One pathway for stabilization of p53 is via the kinase ATM, which is activated by DNA damage and phosphorylates p53 close to its Mdm2 binding site, so blocking its interaction with Mdm2 . In addition, ATM phosphorylates Mdm2 which not only interferes with its ability to bind to p53 but also enhances the degradation of Mdm2 [21, 22] providing an additional route for p53 stabilization. Another mechanism for the increase in p53 levels is the activation of ARF (known as p14 ARF in humans), a nucleolar protein that senses DNA damage . It has been shown that DNA damage disrupts the interaction of ARF with the nucleolar protein B23 (nucleophosmin) releasing ARF into the nucleoplasm so that it can bind to Mdm2 . ARF binding enhances the degradation of Mdm2, resulting in p53 stabilisation [23, 25]. ARF also responds to aberrant growth signals which are triggered by oncogenes such as Ras or Myc, although the induction of genes from the INK4a/ARF locus displays species-specific variations . Since an increase in p53 leads to an increase in Mdm2 transcription, and Mdm2 targets p53 for degradation, p53 levels are again inhibited, providing a negative feedback loop. Negative feedback loops have been found in several systems of interacting proteins (e.g. Hes1 in Notch signalling , NF-kB signalling system ) and have attracted the attention of mathematical modellers. In particular, models have been produced to analyse the oscillations of p53 and Mdm2 in previously published single-cell fluorescent reporter assays [29–34]. The single cell assays have been very informative, revealing that increasing DNA damage results in an increased number of oscillations, but not an increased magnitude in the response [30, 35]. The data also show that there is large intercellular variation with a fraction of cells showing no response or a slowly fluctuating signal. In the cells in which oscillations were detected, there was a wide fluctuation in the amplitude (about 70%) and smaller variations in the period of the peaks (about 20%) . The oscillations in these data showed a period of about 5.5 hours with a delay of about 2 hours between p53 and Mdm2 peaks (Figure 2, which is a reproduction of figure 1 from Geva-Zatorsky et al.  with permission from the publisher).
Prolonged oscillations in the nuclear levels of fluorescently tagged p53 and Mdm2 in individual MCF7, U280 cells following gamma irradiation. Reproduced with permission from Geva-Zatorsky et al. 2006, Molecular Systems Biology . A. Time-lapse fluorescence images of one cell over 29 h after 5 Gy of gamma irradiation. Nuclear p53-GFP and Mdm2-YFP are imaged in green and red, respectively. Time is indicated in hours. B. Normalised nuclear fluorescence levels of p53-CFP (green) and Mdm2-YFP (red) following gamma irradiation. Top left: the cell shown in panel A. Other panels: five cells from one field of view, after exposure to 2.5 Gy gamma irradiation.
All previous models to date have used a deterministic approach to analyse the oscillatory behaviour. These models have used differential equations and mathematical functions requiring a fairly large number of parameters with the generation of oscillations being very dependent on the range of parameter values chosen. Geva-Zatorsky et al.  constructed six different models and found that the simplest model, which contained one intermediary and one negative feedback loop with a delay, was unable to produce multiple oscillations and that it was necessary to either introduce a positive feedback loop or a time delay term (See figure 6 of Geva-Zatorsky et al. ). However, these additions were not sufficient for robustness over a wide range of parameter values. The addition of a non-linear negative feedback loop, a linear positive feedback loop or a second negative feedback loop produced models that were able to demonstrate sustained oscillations over a wide range of parameters. As the models are deterministic, the outcome only depends on the initial conditions and so they cannot be used to investigate cell-cell variability. Geva-Zatorsky et al.  incorporated some random noise in protein production in their models and found that the introduction of low-frequency noise resulted in variability in the amplitude of the oscillations as observed experimentally. Ma et al.  also incorporated a stochastic component for the DNA damage component of their model which resulted in variability in the number of oscillations. However, for a dose of 2.5 Gy, they found that the majority of cells had only one peak and that a step input of DNA damage was required to obtain sustained oscillations.
The deterministic models have been useful in showing that sustained oscillations can be produced in a system where there is at least one negative feedback loop with a delay, and a sustained signal. The signal represents damaged DNA which triggers the cellular response as long as the DNA damage persists . It has also been shown that stochasticity in protein production rates and DNA damage events can explain some of the variability in the data. However, in cellular systems, there will be random effects on all processes. Most of the previous models ignored the fact that p53 has to bind to Mdm2 for its Mdm2-dependent degradation and that it is transcriptionally inactive when bound. Instead, the models assumed that p53 degradation depended on total Mdm2 levels regardless of whether Mdm2 was bound to p53 or not. Since the regulation of p53 is dependent on its interaction with Mdm2, we would expect that the oscillatory behaviour of the system would be strongly affected by the binding affinity of Mdm2 to p53. Therefore any mechanistic model of the system should include the Mdm2-p53 complex.
Other disadvantages of the current models are that they cannot be easily modified or linked to other models and they are not very accessible to non-mathematicians. Most importantly, the majority of previous models have not clearly demonstrated how the biological mechanisms of the system contribute to the oscillatory behaviour. The deterministic models of Ma et al. 2005 , Ramalingam et. al 2007  and Ciliberto et al 2005  are based on molecular mechanisms but none of these are really suitable for a stochastic approach, since the Gillespie algorithm assumes mass action kinetics and these models contain Hill or Michaelis-Menten functions in their rate laws. We chose to build the simplest possible stochastic model using a mechanistic approach that would be particularly relevant to biologists (see Figure 2 and Methods).
The aims of building a stochastic mechanistic model are three-fold. First, we wanted to see if a simple model with stochastic effects would produce sustained oscillations without the need to introduce additional feedback loops or non-linear functions. Second, we wanted the model to be based on the mechanisms that have been proposed by biologists and could be easily understood by the non-mathematically inclined. Third, we wanted to build a model that can be easily incorporated into a larger model such as our earlier model of the ubiquitin-proteasome system . In order to achieve these objectives we used the Systems Biology Markup Language (SBML) . SBML is a well-known modelling standard, allowing models to be shared in a form that other researchers can use even in a different software environment. Since both ATM and ARF activation have been proposed as mechanisms for stabilising p53 after DNA damage, we developed two independent models to see whether oscillations would result from either of these mechanisms. The ARF model is simpler, and so we introduce this model first and show how it can be modified to produce the ATM model.
There are many tools available for creating and running SBML models (see http://www.sbml.org). We chose to use the Biology of Ageing e-Science Integration and Simulation system (BASIS) [40, 41] to store the models, run simulations and store results. The advantage of this system is that it is user friendly, it can be freely accessed by a web browser, and allows easy sharing of models.
Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops
Feedback controls are central to cellular regulation. Negative-feedback mechanisms are well known to underline oscillatory dynamics. However, the presence of multiple negative-feedback mechanisms is common in oscillatory cellular systems, raising intriguing questions of how they cooperate to regulate oscillations. In this work, we studied the dynamical properties of a set of general biochemical motifs with dual, nested negative-feedback structures. We showed analytically and then confirmed numerically that, in these motifs, each negative-feedback loop exhibits distinctly different oscillation-controlling functions. The longer, outer feedback loop was found to promote oscillations, whereas the short, inner loop suppresses and can even eliminate oscillations. We found that the position of the inner loop within the coupled motifs affects its repression strength towards oscillatory dynamics. Bifurcation analysis indicated that emergence of oscillations may be a strict parametric requirement and thus evolutionarily tricky. Investigation of the quantitative features of oscillations (i.e. frequency, amplitude and mean value) revealed that coupling negative feedback provides robust tuning of the oscillation dynamics. Finally, we demonstrated that the mitogen-activated protein kinase (MAPK) cascades also display properties seen in the general nested feedback motifs. The findings and implications in this study provide novel understanding of biochemical negative-feedback regulation in a mixed wiring context.
Feedback regulations are arguably the most common control mechanisms employed by cellular systems, ranging from metabolic to transcriptional to signalling levels [1,2]. Feedback provides cells with self-adjusting capability essential for their survival in fluctuating environments. Failure of proper feedback regulation can lead to diseased states of the cells . A large body of research has thus been dedicated to studying feedback regulation and the implications at cellular levels. While positive feedback is critical in giving rise to multiple stable steady states, underlining switch-like behaviours and prolonging signals , negative feedback is essential in maintaining homeostasis and generating oscillatory dynamics [1,4].
Despite concrete progress, much is still to be understood about the regulation and function of feedback mechanisms in many cellular systems. Advances in molecular techniques are providing better understanding of the mechanistic picture of the cells, which show that feedback mechanisms often appear tangled up and act cooperatively rather than in isolation. Examples of cellular systems in which several negative-feedback loops intertwine to coordinate complex dynamics are ubiquitous. At least three different negative-feedback mechanisms are found to regulate the oscillatory nucleo-cytoplasmic shuttling of the transcriptional nuclear factor kappaB . More than two negative-feedback loops, acting at different layers of the cascade, were also described in the extracellular signal-regulated kinase (ERK) pathway , which may underline the oscillations recently observed in certain cell types [7,8]. Furthermore, multiple negative-feedback structures are prominent in the synthesis pathways of many common amino acids, where the amino acids regulate their own production by concurrently inhibiting multiple upstream reaction steps [9,10]. For instance, the tryptophan operon has evolved seemingly redundant negative-feedback loops, but which appear to be providing distinct dynamical and noise-related functions [11–13]. The cooperative feedback structures exemplified above often underline intricate connections between molecular components within cellular pathways as well as crosstalks between them. Breakdown of such concerted operation could lead to malfunctions in the harbouring system and results in diseased state of the cell. Although studies looking at linked-feedback structures, e.g. mixed positive and negative feedbacks, have been carried out to some extent, their number remains limited [14–16]. Structures of mixed negative feedbacks have received far less attention, creating gaps in our understanding concerning their dynamical behaviour.
In the current work, we set out to fill some gaps by studying in detail the dynamical properties of various general dual-negative-feedback motifs with nested structures, using combined analytical and computational approaches. Schematic of the studied motifs and the single-feedback loop counterpart are given in figure 1. Each motif consists of two negative-feedback loops acting simultaneously at different layers of control. All motifs share the outer feedback loop but differ in the second inner loops, three of them are auto-repression feedback loops. We aimed to understand in these coupled-feedback contexts, how the individual loops contribute to the regulation of oscillations’ existence and of the basic quantitative characteristics including the period (frequency), amplitude and average (mean) value (electronic supplementary material, figure S1a). Of particular interest, we asked if the coupled-feedback structures provide any enhanced robustness to the control of oscillation. To this end, we derived analytical conditions based on which parameter regimes leading to distinct dynamical characteristics can be obtained. Crossing between these regimes corresponds to qualitative change of the system's behaviours including fixed-point steady state and sustained oscillations, which are the prominent dynamics in the considered motifs. As a specific example, similar questions were investigated in the well-conserved mitogen-activated protein kinase (MAPK) signalling cascades where a nested negative-feedback structure is at work [6,17].
Figure 1. Kinetic schemes of the studied single- and dual-negative feedback motifs. Activation, inhibition and degradation are indicated by normal, barred and dashed arrows, respectively. The precursor S0 is assumed to be consantly buffered. A single-loop motif where species Si inhibits Sj is denoted Fij (i, j = 1 … 3 i = j indicates auto-regulatory loop where Si represses itself). Fij,kh denotes a dual-feedback motif where Si inhibits Sj and Sk inhibits Sh (i, j, k, h = 1 … 3). Following this notation, the feedback systems in this figure are referred to as F31, F31,32, F31,21, F31,33, F31,22 and F31,11. (Online version in colour.)
2. Models and methods
2.1. Feedback motifs and mathematical description
Early work on modelling feedbacks in gene-related systems started with that of Goodwin . The first repressive model of transcriptional regulation where a gene's expression is inhibited by its own protein product was proposed. This simple model has provided a practical framework for a plethora of subsequent studies exploring the dynamics of negative-feedback genetic systems [19–24]. Here, we extended the Goodwin framework to describe feedback systems with multiple negative mechanisms. A multi-loop system is described by the following differential equations:
In equation (2.1), there is a subindex set Ik (k = 1 … m), which describes the inhibition of Sk via different feedback loops by the species denoted by the subindex k. For instance, I1 = <2, 3>if S1 is repressed by S2 and S3. Figure 1 displays the scheme of the dual-negative-feedback motifs of length three considered in this study. The ordinary differential equations (ODE) describing their dynamics following the same notations as in equation (2.1) are given in §1 of the electronic supplementary material. The input S0, which can be a gene, ligand or signalling protein that triggers sequential activation of a transcriptional or signalling cascade, was included in parameter k1 in (2.1) for convenience. Degradation and synthesis are assumed to follow first-order kinetics. Since the half-saturating constant K is inversely correlated with the inhibiting Hill function, we refer to K as the reciprocal feedback strength of the respective loop [25,26]. Biologically, K represents the binding constant between the repressor and the repressed target . A small value of K indicates that the feedback is strong while large K implies weak inhibition. On the other hand, the Hill coefficient n modulates the degree of function sigmoidity and signifies the level of nonlinearity exhibited by the feedback loop.
2.2 . Analytical analysis of dynamics based on the Routh–Hurwitz theorem
Biochemical systems can exhibit diverse types of dynamical behaviour including fixed point, mono- or bi-stable steady states, oscillations (limit cycles) and even irregular fluctuation (chaos) [27,28]. Sustained oscillations and fixed-point steady states are often the prominent dynamics displayed by negative-feedback systems which are separated by Hopf bifurcation [21,24]. Four motifs in figure 1a,d–f can be categorized as monotone cyclic negative-feedback systems and thus have been proved to only exhibit fixed point or sustained oscillations . Linear stability analysis hence provides a convenient way to detect oscillations in these systems. For the remaining dual-feedback motifs that are non-monotone cyclic (figure 1b,c), further methods are required to validate oscillations, which we will discuss below. We focus here on how the transition between these two dynamics is governed for the considered motifs.
Dynamical insights of a system can be obtained by linearizing the governing differential equations around the equilibrium state and probing the eigenvalues of the resulting Jacobian matrix, J, given by :
If the real parts of all J's eigenvalues are negative, the system's equilibrium is deemed stable and reaches a fixed-point steady state, whereas if any of the real parts are positive, system stability is lost and in the motifs considered here, the system moves into a limit cycle regime where it oscillates. The eigenvalues are usually obtained numerically and their real part values are assessed. To facilitate analytical treatment, we instead employ the Routh–Hurwitz theorem [29,30] which states that the eigenvalues, essentially the roots of the characteristic polynomial |J − λI| = λ n + α1λ n −1 + … +αn−1λ + αn = 0, all possess negative real parts if and only if the following condition holds true
where Δk(k = 1 … n) are the determinants of the properly constructed matrices
Equation (2.4), therefore, represents the sufficient condition for oscillations in the dual-feedback motifs while it is only a necessary condition in motifs F31,32 and F31,21. To verify that (2.4) also constitute the sufficient condition for oscillations in these systems, we employed analysis based on the Floquet theory that globally assesses the stability of the obtained periodic solutions [31–33]. The mathematical and computational background of the Floquet analysis is given in the electronic supplementary material. This method was numerically implemented in M athematica v. 8.0 (code can be provided on request). In the following sections, we will present first the analytical analysis based on (2.4) for the single-loop system F31 followed by the exemplified dual-loop motif F31,32. Detailed derivations for the remaining motifs are provided in the electronic supplementary material.
3.1. Dynamical analysis of the single-loop system F31
We begin by examining the single negative-feedback motif F31, where the first species in the cascade S1 is inhibited by the last species S3. The ODE model of this motif is given in §1, electronic supplementary material. Systems of this type have been previously studied assuming equal degradation rates [24,34], yielding informative results but are limited in their biological implication owing to the oversimplified assumption. Our following analysis assumes no such restricting assumption and was aimed to facilitate an extension to motifs of multiple-feedback loops. According to the Routh–Hurwitz theorem, F31 reaches the steady state if and only if
Using equation (3.2), the coefficients α1, α2, α3 are given by
This introduction of a composite variable M1 (0 < M1 < 1), as will soon become apparent, allows us to ignore possible complications involving x3 and the related rational functions such as those in equations (3.2) and (3.3), which greatly facilitates our treatment. Since α1, α2 > 0, equation (3.1) means that oscillatory dynamics arise if α1α2 − α3 < 0. Substituting α1, α2, α3 into this condition gives
Rearranging x3 in terms of M1, K1 and n1 from equation (3.4) and substitute x3 into equation (3.3), we have:
Because dF(M1)/dM1 < 0 for M1∈[0,1], F(M1) strictly decreases with M1 the oscillations condition, equation (3.5), is transformed into the following equivalent form:
The symbolic, parametric condition equation (3.6) governs the bifurcation points separating the dynamics of fixed-point steady state and limit cycles of the single-loop system. It is immediately clear that the system can only oscillate if the feedback loop is sufficiently strong (small enough K1). Moreover, the Hill coefficient n1 must be greater than g for any chance of oscillations to realize. The latter observation is consistent with the previous derivation by Griffith .
3.2. Dynamical analysis of the dual-feedback systems
Demonstrating this, we describe here treatment of the dual-feedback system F31,32, while only the main results are reported for other motifs whose detailed derivations are given in the electronic supplementary material. Motif F31,32 contains a second, inner-feedback loop, where the cascade product S3 also inhibits species S2. The governing ODEs are presented in §1, electronic supplementary material. This newly incorporated loop is also modelled by Hill kinetic function in which the inverse of the parameter K2 describes the inhibition strength, and n2 measures the nonlinearity level of the feedback regulation. The steady-state solution of the system variables can be obtained by setting the RHS of the differential equations to zero which leads to
Introducing and from equation (3.7) we obtain
Since x3 can be written as , substitute this into equation (3.8) and rearrangement yields
It is convenient here to report similar conditions obtained for the remaining dual-feedback motifs (see electronic supplementary material for derivation).
3.3. Addition of the inner-feedback loops reduces and potentially eliminates oscillatory dynamics
It can be observed that equations (3.11–3.15) share a common form despite the difference in the coefficients of the M2-related terms inside function F. Moreover, an extra term (1−M2) appears on the RHS of these equations that does not appear in (3.10) derived earlier for F31. It turned out that this additional term and those inside the function F brings about important implications in how the inner negative-feedback loops affect system dynamics in the dual-feedback motifs, as shown below.
First, since the composite variable M2 belongs to the [0,1] interval, the following holds true for all dual-feedback motifs:
Note that the form of β are motif-specific and γ = 0 for F31,21 and F31,32. Equation (3.16) indicates that K1 < F(g/n1) is a necessary condition for oscillations in the dual-feedback motifs which, in comparison with condition equation (3.6), implies that addition of any of the negative-feedback loops: S3 repressing S2, S2 repressing S1, or auto-repression of S1, S2 or S3 would not produce oscillations if the F31 does not oscillate in the first place. Equation (3.16) also suggests that the dual-feedback structures reduce the effective upper-bound threshold of K1 possible for oscillations, thereby shrinking the oscillations domain compared with the single-loop motif.
Secondly, consider the case when the second-feedback loop is weak. In this scenario, K2 is large that makes M2 ≈ 0. The RHS of equations (3.11–3.15) then become F(α). The condition for oscillations in the coupled-loop motif F31,32 thus approaches that of the single-loop motif F31 in the limiting case of weak feedback inhibition by the inner loop. On the other hand, when the inner loop is strong, K2 is large resulting in M2 ≈ 1. The RHS of equations (3.11–3.15) then become null which would lead to unsatisfactory conditions for oscillations if K1 > 0. An important and rather unexpected consequence is that the inner-feedback loop, when sufficiently strong could completely eliminate oscillations by overriding the oscillatory dynamics triggered by the outer feedback loop. To verify these predictions, we performed numerical simulations of all the motifs with arbitrary parameter sets, which indeed showed that pre-existing oscillations in the single-loop system F31 can be abolished when incorporating a sufficiently strong second-feedback loops whereas oscillations are maintained for weak inhibition levels (electronic supplementary material, figure S2).
3.4. Dynamics partitioning revealed differential oscillations-reducing efficiency by the dual-feedback motifs
Although equation (3.11–3.15) allowed us to draw new insights into the effects of the nested-feedback structures on the appearance likelihood of oscillations, the presence of M2 in these equations complicates the derivation of conditions that involve only the model parameters. Below, we further refine these equations to achieve this aim.
Again, we use the system F31,32 as an illustrative case. Since x3 can also be written as , equation (3.8) enables us to express M1 in terms of M2 as follows
Since F2(M2) strictly increases with M2 owing to dF2(M2)/dM2 > 0 for M2∈[0,1] (see the electronic supplementary material,), derivation of the parameters-only condition that governs oscillatory dynamics for the considered motif is simplified to determination of values of the variable M2 satisfying equation (3.10) or the equivalent form
Such ranges of M2 cannot be obtained analytically but can be quickly obtained by numerically solving equation (3.19), and can be conveniently visualized on the M1 versus M2 coordinate where intersection of the functions G(M2) and α + βM2 are shown (figure 2a). Plugging the obtained numerical value of M2 into equation (3.18) allows us to readily compute the value of K1 that result in oscillations given any parameterization of the remaining model parameters. Bifurcation plots that partition the parameter space (i.e. K1 versus other parameters) can therefore be constructed and based on these, we will be able to perceive how changes in certain parameters affect system dynamics and bring about oscillations. It should be noted that the form of the function F2 is slightly different across the motifs, as detailed in the electronic supplementary material.
Figure 2. (a) Visualization of the functions on both sides of equation (3.19), alpha (in Greek symbol) +βM2 and G(M2), on the M1 versus M2 coordinate which are treated here as dependent and independent variables. Range of M2 indicated as ‘oscillating’ is obtained by determination of the functions’ intersection point within the 1 × 1 square. This range of M2 gives rise to oscillations in system F31,32 as explained in the text. Similar plots for the remaining dual-feedback motifs are presented in electronic supplementary material, figure S3. Parameter values used for plotting are K1 = 1(nM), K2 = 2 (nM), k1 = 1 (nM min −1 ), k2 = k3 = 1 (min −1 ), kd1 = kd2 = kd3 = 0.2 (min −1 ), n1 = 15, n2 = 4. (b) Bifurcation diagrams of fixed-point and oscillatory dynamics projected onto the K1 versus K2 plane superimposed for all dual-feedback motifs. Parameter values used for plotting are k1 = 1(nM min −1 ), k2 = k3 = 1(min −1 ), kd1 = 0.15(min −1 ), kd2 = 0.22(min −1 ), kd3 = 0.18(min −1 ), n1 = 9, n2 = 2. Note that the same parameter units are used for other figures hereafter. (Online version in colour.)
Figure 2b compares the partition of systems dynamics (fixed-point steady state and oscillations) on the K1 versus K2 coordinate for all five dual-negative-feedback motifs. Consistently in all these motifs, a second loop with strong inhibition, signified by small K2, tends to bring the system out of the oscillation domain. In fact, we observed an increasing effect where stronger second loops result in smaller size of the oscillation domain. Since the same value of n2 was used for all motifs in figure 2b, it reveals that the ability to diminish oscillations is weakened as the second loop moves upstream of the cascade. Indeed, the addition of S3's auto-inhibition loop gives the smallest oscillations domain, whereas it is largest with auto-inhibition of S1. Furthermore, the figure shows, as predicted, that K1 approaches a threshold value (F(g/n1)) at large K2, which is independent of n2.
3.5. Dependence of oscillations on the degradation rates and optimization of oscillation's appearance
The rates of degradation have been suggested to play important roles in influencing the existence of oscillations in feedback systems [35–37]. Symmetry among elements within a feedback loop was predicted to increase the occurrence likelihood of oscillations . It is, therefore, interesting to study how the degradation rates affect systems dynamics in our dual-feedback context. Figure 3 shows on the K1 versus kdi bifurcation diagrams (i = 1 … 3) that the threshold K1 depends biphasically on the degradation rates in all motifs. The bell-shaped oscillation domains suggest that oscillatory behaviour arises only over a bounded range of the degradation rates. The three-dimensional bifurcation plot (figure 3d) of K1 in response to simultaneous changes in pairs of the degradation rates further shows a cone-shaped oscillation domain. The oscillation domain of all the motifs projected in the space of the degradation rates would therefore form a restricted volume, whose size is expectedly diminished as the second-feedback loop's inhibition intensity increases. The ability to oscillate is only realized when the rates of degradation are tuned within this enclosed domain, conferring oscillatory dynamics much less probable in the considered motifs compared with steady-state equilibrium dynamics. The biphasic dependence of the threshold K1 on the rates of degradation also implies an optimal strength of the S3-repressing-S1 feedback inhibition at which the appearance of systems oscillations is maximized. Importantly, this optimal state is achieved at intermediate levels of the turnover rates of the components within the loop.
Figure 3. (a–c) Two-dimensional bifurcation diagrams of systems dynamics projected onto the plane of K1 versus rates of degradation superimposed for all dual-feedback motifs and compared with the single-feedback one. Oscillations domains are shaded. The results are consistent with observations on the K1 versus K2 plane shown in figure 2b. (d) Three-dimensional bifurcation plot that shows cone-shaped dependence of the threshold value of K1 dependence on a pair of degradation rates, kd1 and kd2, further illustrating the enclosing property of the oscillation domain in the degradation parameters’ space. Parameter values used for plotting are K1 = 1, K2 = 5, k1 = k2 = k3 = 1, kd1 = kd2 = kd3 = 0.2, n1 = 9, n2 = 2. Note that the inner loops in all the dual-feedback motifs are specified with the same value of K2 and n2 for ease of comparison. (Online version in colour.)
Next, we ask the question, what distribution of the degradation rates would lead to maximal oscillations given a fixed steady-state level of the output S3? This comparative criterion is realistic since cellular pathways typically maintain a functional range of key proteins. For the single-loop motif F31, constant x3 can be brought about by fixing the product of the rates of degradation (see equation (3.3)). We found that in this case, oscillations are most likely under conditions of equal degradation rates, which agrees with previous prediction  (see proof in §2.1, electronic supplementary material). For the dual-feedback systems, F31,32 and F31,33, constant steady-state S3 levels can be similarly guaranteed by fixing the product kd1kd2kd3. We proved analytically that oscillations are most likely for F31,32 when kd2 = kd3 at any given kd1. On the other hand, F31,33 is most prone to oscillations when kd1 = kd2 at a given kd3 (§2.2, 2.3, electronic supplementary material). To verify these conclusions, e.g. for S31,32, we generated numerous parameter sets that differ only in kd2, kd3 but keep their product fixed. The corresponding value of the threshold K1 was then computed for each set (electronic supplementary material, figure S2c) which indeed shows that the threshold K1 is maximized at comparable levels of kd2 and kd3. Interestingly, similar analysis shows that K1 is also maximized at comparable <kd1, kd2> (electronic supplementary material, figure S2a) and <kd1, kd3> (electronic supplementary material, figure S2b), suggesting that oscillations in F31,32 are most probable under equal degradation rates of all species (kd1 = kd2 = kd3). This is also the case for the system F31,33, as shown in electronic supplementary material, figure S2d–f. For the dual-feedback motifs F31,21, F31,22 and F31,11, constant steady-state S3 is brought about either by fixing <kd3, kd1kd2>, <kd3, kd1kd2> or <kd1, kd2kd3>. We found that oscillations are most likely for F31,21, F31,22 and F31,11, when kd1 = kd2, kd1 = kd2 and kd2 = kd3. Analytical proofs of these predictions are shown in §2.4 and 2.5 of the electronic supplementary material, and also confirmed by computational simulations (figure 4 and electronic supplementary material, S3). These results suggest that in systems with nested-feedback structure, symmetry among the elements of the loops also promotes systems oscillations.
Figure 4. Dependence of the appearance of systems oscillations on the distribution of degradation rates. The dual-feedback F31,21 is used here as a showcase, see electronic supplementary material, figure S4, for the remaining motifs. (a) Time-course simulations showing that the constant steady-state level of S3 is maintained by the condition of fixed kd1·kd2, as discussed in the text. Each curve represents S3′s temporal evolution simulated for one parameter set, with different <kd1, kd2> but fixed kd1·kd2 = 10, and the remaining parameter values are K1 = 1, K2 = 0.5, k1 = 10, k2 = 2, k3 = 1, kd1 = 2.5, kd2 = 2, kd3 = 3, n1 = 10 and n2 = 5 (note that these values were chosen arbitrarily to illustrate our analytical results a total of 200 parameter sets were generated). (b) For each parameter set discussed, we calculated the threshold value of K1 which is shown as a dot whose size indicates its corresponding magnitude. Since kd1·kd2 is fixed, all the dots line up on the log coordinate of kd1 and kd2. The dots appear symmetrical and the largest dots are at the centre of the sets, showing that oscillations are most likely for sets with comparable kd1 and kd2, and maximized with equal kd1 and kd2.
3.6. Higher nonlinearity of both feedback loops enhances oscillations
Negative-feedback regulation requires a sufficient degree of nonlinearity for oscillations to arise . This is evident from the condition n1 ≥ g needed for oscillations in the single-loop and in the tested dual-feedback systems. Since in the dual-feedback motifs, the feedback strength of the individual loops brought about opposing effects on systems oscillations, we asked if the loops’ nonlinearity levels behave in the same way. Figure 5 shows that this is not the case. In all the tested dual-feedback motifs, increasing the Hill coefficients n1 and n2 both enlarged the oscillations domain projected here on the K1 versus K2 coordinate, leading to enhanced oscillations. However, the underlying mechanisms of this enhancement are different. While higher n1 supports oscillations by raising the threshold K1, allowing oscillations possible at an even weaker S3-repressing-S1 loop strength, higher n2 enables oscillations at even stronger intensity of the inner loops. Taken together, the result implies that although addition of stronger inner loops diminished oscillations, raising its nonlinearity level tolerated this effect and thus promoted oscillations instead.
Figure 5. Effects of the feedback loops’ nonlinearity levels on oscillations observed on the K1 versus K2 bifurcation plot. For all the dual-feedback motifs, shaded oscillations domains are compared between a reference case (black) and when the first (red) or second loop (blue) is increased in Hill coefficient. The remaining parameter values for plotting are K1 = K2 = 1, k1 = k2 = k3 = 1, kd1 = kd2 = kd3 = 0.2. (Online version in colour.)
3.7. Regulation of quantitative properties of oscillation dynamics in the dual-feedback structures
An oscillatory dynamic bears a number of basic quantitative characteristics such as its frequency (period), amplitude (difference between peak and trough) and the average value (mean electronic supplementary material, figure S1a). Recent studies report that these metrics may play important roles in directing cellular responses. In yeast, the level of extracellular calcium was found to modulate the nuclear translocation frequency of transcriptional factor Crz1 and this frequency directly controls expression of multiple target genes . In calcium and endocrine oscillations, the frequency was also thought to contain key functional signal [4,40,41]. In other cases, the amplitude plays key physiological roles. Allada et al.  showed that a clock gene mutant in Drosophila melanogaster that produces persistent oscillations but with reduced amplitude led to behavioural malfunction. Previous computational work on mixed positive-negative feedback oscillators revealed such coupling enabled robust tuning of the frequency at stable amplitude . It is therefore of our interest to investigate how the three metrics (period, amplitude and mean) of the oscillations dynamics are controlled in the mixed negative-feedback motifs.
First, to understand how the individual loops affect the oscillation pattern in the dual-feedback motifs, the feedback strength of either loop was varied and the period, amplitude and mean value of the oscillatory level of the species S3 were computed (see §4.1, electronic supplementary material, for computational procedure). The parameter values used in our calculation was based on the physiological values obtained for the endogenous circadian clock system in Neurospora crassa . The molecular basis of this system is described in electronic supplementary material, figure S1b, which follows the general motif F31. Our computational result showed that the period increases with stronger feedback of the outer loop, while decreases towards zero (i.e. loss of oscillations) at higher feedback of the nested loop (figure 6a). This was observed for all the dual-feedback motifs which is consistent with our previous finding that the second-feedback loops can eliminate oscillations when sufficiently strong. We observed a biphasic dependence of the amplitude on the outer loop, indicating existence of an optimal strength a maximal amplitude (figure 6b). On the other hand, the inner loops decrease the amplitude across all motifs as expected. Also as expected, both loops decrease the average value of the oscillatory S3 concentration (figure 6c).
Figure 6. Regulation of the oscilations’ period, amplitude and mean value in the dual-feedback motifs. (a–c) Dependence of the period (min), amplitude (nM) and mean value (nM) on simultaneous changes in the (reciprocal) strength of the outer (K1) and inner (K2) feedback loop. Parameters used are k1 = k2 = k3 = 1, kd1 = kd2 = kd3 = 0.2, n1 = 15, n2 = 2. (d–f) Same data obtained in (a–c) plotted in parametric plots for the period, amplitude and mean. (d) Each curve has different outer loop strength but identical inner loop strength lower curves have stronger inner loop. (e,f) Each line has different inner loop strength, but identical outer loop strength the lower curves have stronger outer loop. (Online version in colour.)
Next, we simultaneously varied the strength levels of both the outer and inner loops and recorded the period, amplitude and mean value. Plotting one metric against another in parametric planes revealed interesting observations (figure 6d–f). We found that in all the dual-feedback motifs, altering the short inner loop allowed the system to widely modulate the period or amplitude while keeping the mean value at near-constant levels, indicated by the almost perfect horizontal lines in figure 6e,f. On the other hand, varying the long outer loop allowed the systems to modulate the period while keeping the amplitude relatively stable. This effect is increased when the systems are well within the oscillating regime, indicated by the lines inside the box in figure 6d. Furthermore, higher inner loop seemed to make the tunability more pronounced (lower lines, figure 6d). These observations suggest that having coupled negative feedbacks provides the systems with robust ways to adjust different features of the oscillation dynamics, an ability similar to that of the mixed positive–negative motifs . Such robustness may be the key in enabling the systems to decouple functional controls regulated by different features of the oscillation dynamics. To see to what extent the above findings are dependent on parameterization, we carried out similar analysis for other random parameter sets generated from the basal set (electronic supplementary material, figure S5a and §3). We found that the findings still hold true for these sets, suggesting that they are likely consequences of the motifs’ topology.
3.8. The mitogen-activated protein kinase signalling cascade: an example
The MAPK signalling cascades are among the best-studied signalling systems in cells, which play crucial roles in many cellular processes including proliferation and survival [6,17]. Kholodenko  predicted that sustained oscillations can occur in these cascades under physiological parametric conditions. This prediction has been backed up by recent experimental reports [8,44]. The MAPK cascade model analysed by Kholodenko  however contained only a single negative-feedback loop caused by inhibition of the active form of the upstream GTPase (e.g. RasGTP) by the activated MAPK protein, while infact multiple negative-feedback loops are at work . To further understand oscillation dynamics of the MAPK cascades in multi-loop context, we analysed an extended MAPK model. We considered an additional negative feedback brought about by inhibition of the MAPK kinase kinase (MKK) by the active MAPK protein (ppMAPK, figure 7a). For instance, in the extracellular signal-regulated kinases (ERK) pathway, an exemplary MAPK cascade, the second negative feedback is caused by phosphorylation of Raf-1 on inhibitory sites by active ERK [6,17].
Figure 7. Analysis of oscilations in the dual-feedback MAPK cascades. (a) Kinetic scheme of the MAPK cascades extended from Kholodenko  p- and pp- denote single and double phosphorylation of the MAPK protein, its kinase (MKK) and its kinase kinase (MKKK). Parameters KI1, KI2 describe the reciprocal strength of the outer- and inner-feedback loop. (b) Bifurcation diagrams of fixed-point steady state (blue) and oscillatory dynamics (white) projected onto the KI1 versus KI2 plane. Parameter values used are given in electronic supplementary material, table S1. (Online version in colour.)
The ODE and parameter values of the extended Kholodenko model are given in electronic supplementary material, tables S1 and S2. The feedback strength of the two loops can be adjusted by varying parameters KI1 and KI2. Our numerical stability analysis showed that these feedback loops affect oscillations differently (figure 7b). Higher strength of the outer loop tends to move the system into the oscillatory domain, thereby enhancing oscillations, while a stronger inner loop tends to move the system out of the oscillatory domain, thereby weakening oscillations (figure 7b). Similar to the general dual-feedback motifs, a sufficiently strong inner loop can destroy oscillations in the MAPK cascades. However, unlike in the general motifs, where the outer loop can always rescue oscillations, in the MAPK cascades, there exists a barrier strength of the inner loop over which the system fails to oscillate regardless of how strong the outer loop is. This difference may be due to the different wiring and level of complexity in the MAPK model. We further computed the period, amplitude and mean value of the oscillatory ppMAPK levels in response to increasing strength of the outer and inner loop (electronic supplementary material, figure S6). Interestingly, both loops reduced the oscillations period and mean value. The outer loop also exhibited a bi-phasic modulation of the amplitude as expected, while the inner loop consistently decreased the amplitude.
Understanding the emergence of functional behaviour from the dynamical controls of cellular networks is of fundamental importance in biology. It is, therefore, critical to dissect how certain system dynamics are brought about and how they are regulated. Oscillations are ubiquitously observed in cells and have been shown to play important roles in a broad array of cellular processes [45,46]. The oscillatory behaviour produced can be time-dependent, as in glycolytic oscillations , the circadian rhythms , the cell cycles , or space-dependent, as in calcium waves [50,51] and neuronal oscillations . In all of these examples, oscillations are driven by some sort of negative-feedback mechanism. However, in many oscillating systems, their circuits contain more than one negative-feedback loop which begs the question how they might cooperate to control oscillations. In the current work, we studied the emergence and control of oscillation dynamics in the context of coupled, nested negative feedback biochemical motifs. By comparative examination of the dual-loop and single-loop structures, we aimed to dissect the relevant roles of the individual feedback loops. To this end, we employed combined analytical and numerical methods.
Most interestingly, we found that the individual-feedback loops regulate oscillations in opposing manners in all of the general dual-feedback motifs. The long, outer loop promotes oscillations whereas the short, inner loops strictly inhibit oscillations. Such oscillations-inhibiting effects, when sufficiently strong, can dominate the oscillation-generating effect of the outer loop and completely destroy oscillations. Coupling multiple negative feedbacks together thus can suppress rather than enhance oscillations as found previously . Note that other ways of negative-feedback coupling such as mutual inhibition can also inhibit oscillations and promote bistability instead , as they behave like a positive feedback. Such wiring is, however, structurally different to the nested-feedback motifs considered here, highlighting that different ways of coupling may yield similar system responses yet through distinct mechanisms. Our analytical proof further showed that the above observation is parameter independent and rather a feature of the network topology. Comparison of the oscillations domain among the dual-feedback motifs under the same parametric specification of the inner-feedback loop revealed that as it moves further downstream of the cascade, it represses oscillations more strongly. Auto-inhibition of the final component of the cascade, therefore, reduces oscillations most efficiently.
Having showed that the shorter loop of the dual-feedback structures can abolish oscillations driven by the longer loop, it is useful to understand this in an intuitive way. Negative feedback requires sufficient delay and nonlinearity to give rise to oscillations . Our simulations show that removing the outer loop from the dual-feedback motifs rendered them incapable of exhibiting oscillations regardless of parameter choice (data not shown). The failure of these short loops to maintain oscillations on their own may be due to the lack of delay, a cause that is likely to also underline their ability to suppress oscillations in the coupled structures. To test this idea, we constructed variant motifs of the system F31,11 which extend the auto-inhibition loop of S1 by one or two intermediate components (electronic supplementary material, figure S1c–d). Addition of one intermediate component still reduced oscillations while having two intermediates introduced enough delay to prevent the extended loop from having an oscillations-reducing property. One important consequence then follows regarding the modelling of feedback systems in general. Models of biochemical systems are in many cases greatly simplified. Quite often, multiple activation steps occurring via intermediate components are described as a single step for convenience without explicitly taking into account the time delay. We, however, showed that description of a negative feedback with one or more intermediate components could result in markedly different effects of the loop towards system dynamics. Time delay, if significant, thus should be considered in the model for faithful representation of the system in reality.
It also follows from our results that motifs formed by combining the outer feedback loop and any combination of the secondary inner loops would behave similarly to the dual-feedback motifs in their ability to suppress oscillations. The opposing roles of the coupled feedbacks suggest intriguing implications regarding the evolution of negative-feedback structures. In principle, changes in system dynamics such as loss of oscillatory behaviour can be brought about by mutations that break the oscillation-generating feedback loop, or alternatively by the emergence of new feedback loops with oscillation-abolishing property. Under certain scenarios, the latter strategy may be more advantageous than the former, for example, when the newly evolved loops provide additional benefit to the cells. Since measurements like period, amplitude and mean value are intrinsic features of an oscillatory dynamics, we went on to examine how these metrics are regulated by the feedbacks interplay in the linked-feedback motifs. We found that the inner loop enabled flexible modulation of the period or amplitude, while keeping the mean value at near-constant levels. More interestingly, in presence of strong inner loop, varying the outer loop allowed the systems to significantly change the period, while keeping the amplitude relatively stable. Such features were not possible in the single-loop motif. Our findings suggest that having nested-feedback structures provides robust controls of oscillation dynamics and its basic quantitative characteristics. Such robustness may have underlined the evolution of some biochemical oscillators possessing these topological structures. To further our investigation in a more biologically realistic system, we asked similar questions in the MAPK signalling cascades that were known to regulate important cellular processes and contain multiple negative-feedback loops in a nested structure. To this end, we extended an existing mathematical model for the dual-feedback MAPK cascade. Our analysis showed that despite the fact that the MAPK model is much more intricate, its negative feedbacks also exhibit opposing regulation of oscillations as in the general motifs. It would be interesting to test this prediction experimentally in the future.
Parameter-sensitivity analysis is of great importance in understanding biological networks. The derivation of the analytical conditions that govern the occurrence of oscillations have allowed us to construct useful bifurcation plots based on accurate knowledge of when switching between different dynamical regimes occurs. Such knowledge is potentially useful in experimental design. It can also aid in the engineering of synthetic circuits with desirable targeted dynamical behaviour, since one could effectively choose appropriate points for perturbation to attain desired dynamics. In particular, we observed that oscillations only occur over limited ranges of degradation rates. The fact that each of these rates must be tuned within its viable range for oscillations to occur suggests that oscillatory behaviour is a strict parametric requirement. Its emergence cannot be guaranteed by the presence of negative feedback alone but depends greatly on the evolutionary tuning of the stability of the systems components, which may imply why oscillations are not observed in all systems with negative-feedback regulations. We went on to examine which distribution of the degradation rates would maximize oscillation occurrence, given they are required to maintain a constant steady-state level of the cascade output. We showed that for the single- as well as the dual-feedback motifs, symmetry among the elements within the feedback loops tends to promote oscillations, consistent with previous reports .
In this paper, we employed deterministic modelling as the approach of choice and thus molecular fluctuations (noise) were neglected. However, these fluctuations are inherent features of biochemical networks and may be important in functioning of the cells . Negative feedbacks have been known to suppress noise [55,56], understanding the behaviour of noise in nested feedback structures is an interesting topic. In our recent work , we explored how noise is propagated and managed in the prokaryotic tryptophan operon system of E. coli, a model system governed by three nested negative-feedback loops. Interestingly, it was found that the individual loops in this system do not all reduce noise, but some actually enhance noise levels of the system output. This work was, however, only concerned with noise when the studied system is outside the oscillating domain. Therefore, we further examined how noise might affect oscillations in the general dual-feedback motifs considered in the current work. Stochastic simulations showed that noise can extend the Hopf bifurcation point in both single- (F31) and dual-feedback systems (F31,32), producing oscillations at parameter values that does not lead to oscillations in a deterministic setting. Comparing the bifurcation points in F31 and F31,32 at different levels of the inner loop showed that the points were not altered, suggesting that addition of the second loop in the nested arrangement had no effect on the stochastic bifurcation points. Further understanding concerning the interplay between noise and oscillations in coupled-feedback settings should be among the focuses of future systems biology studies.
Recent progress in systems biology relies a great deal on numerical analysis and simulations. However, two important bottlenecks are identified with this approach: the lack of reliable parameter values and the limited exploring capability of numerical simulations. The former hurdle can be overcome by development of increasingly advanced experimental techniques while one way to help reduce the second limitation is to develop novel analytical methods for quantitative analysis of cellular systems. Analytical tools can surmount brute-force simulations and boost the capacity to explore systems dynamics over much wider parameter space. The findings and implications in this study, as a result of such approach, have provided novel understanding of negative-feedback regulation in coupling context.
The success of systems analysis of hematopoiesis will depend upon technologic breakthroughs and collaborations between the biologic and physical sciences that yield accurate predictions and emergent properties. With each discipline using a different language, this is easier said than done. Changes in undergraduate, graduate, and medical curricula must be implemented to train a new generation of biomedical researchers fluent in quantitative or engineering disciplines. 93, –95 Systems biology requires a balance between models sufficiently complex to describe a system and yet simple enough to be clinically useful. Understanding large quantities of data well enough to validate a model is especially challenging. The development of Systems Biology Markup Language (SBML) has made it easier to develop biology-oriented software packages, such as COPASI, Simmune, MetaCore, and Cytoscape, which aid model building and data analysis. 18,96, , –99 Since 2001 the number of such packages developed for systems biology has grown from 5 to more than 170. With computational power becoming ever greater and cheaper, the number and diversity of such software packages will only increase, bringing within their scope models that may not be impossible to validate with current technology. At present, most models of hematopoiesis are built at a single scale, for example, cellular or molecular. The future lies in building models that span multiple scales, incorporating more of the connections that exist between them and thereby being able to account for some of the complexity that arises from the connections. Among the fundamental questions in normal and leukemic hematopoiesis that systems biology will address are: integration of signaling pathways, circuits, and networks that determine cell fate, multiscale modeling of stem cell plasticity, synthesis of genetic and epigenetic data, global analysis of phosphoproteins, dynamics of hematopoiesis in the bone marrow microenvironment presented in 3-dimensional imaging, and cellular engineering to expand selective blood cell compartments for therapy. The complexity or density of experimental data will demand a systems approach.