Simple post-translational circadian clock models from selective sequestration
aa r X i v : . [ q - b i o . M N ] F e b Simple post-translational circadian clockmodels from selective sequestration
Journal TitleXX(X):1–9c (cid:13)
SAGE
Mark Byrne
Abstract
It is possible that there are post-translational circadian oscillators that continue functioning in the absence of negativefeedback transcriptional repression in many cell types from diverse organisms. Apart from the KaiABC system fromcyanobacteria, the molecular components and interactions required to create in-vitro (”test-tube”) circadian oscillationsin different cell types are currently unknown. Inspired by the KaiABC system, I provide ”proof-of-principle” mathematicalmodels that a protein with 2 (or more) modification sites which selectively sequesters an effector/cofactor molecule canfunction as a circadian time-keeper. The 2-site mechanism can be implemented using two relatively simple couplednon-linear ODEs in terms of site occupancy; the models do not require overly special fine-tuning of parameters forgenerating stable limit cycle oscillations.
Keywords circadian clocks, biological oscillators, sequestration, limit cycles, mathematical modeling
Introduction
There are circadian ( ≈ hr) clocks in most organismsDunlap (1999). These internal biological clocks areexperimentally characterized by sustained oscillations in oneor more measured outputs under (approximately) constantconditions (e.g., constant light or darkness). Circadianoscillators can also be entrained via external stimuli, suchthat the internal oscillator’s period and relative phase can besynchronized by the entraining stimulus Roenneberg et al.(2003). In contrast to general biochemical oscillators, theoscillatory period of a circadian clock remains almostunchanged for different constant temperatures in the relevantphysiological range (temperature compensation) Hong et al.(2007). The simplest known example of a circadian clockthat functions outside cells (in a ”test-tube”) is the 3-protein KaiABC system Tomita et al. (2005); Nakajima et al.(2005). This type of clock is termed a post-translationaloscillator (PTO) since its operation does not requiredynamical feedback from a genetic circuit. In contrast,the fundamental oscillatory design of the circadian clockin most organisms in vivo is believed to consist of atranscription-translation negative feedback loop (TTFL)which is influenced by post-translational modifications(see, for example Young and Kay (2001); Lee et al. (2001);Lowrey and Takahashi (2000)). However, negative feedback TTFLs do not generally imply sustained oscillationsKurosawa et al. (2002); Qin et al. (2010). In this context,mathematical models were originally required as proof-of-principle that the TTFL mechanism was quantitativelyviable, in the sense that simple TTFL mathematical modelscould, in principle, reproduce experimentally realisticcircadian oscillations in the relevant measured outputsGoldbeter (1995).Both simplified (2D) and more complex mathematicalmodels of eukaryotic TTFL circadian clocks have beeninvestigated Tyson et al. (1999); Forger and Peskin (2003).However, there is evidence of the existence of circadianPTOs in a variety of species absent genetic feedback loopsONeill and Reddy (2011); ONeill et al. (2011); Edgar et al.(2012) . These findings suggest that it may be usefulto investigate post-translational designs which can satisfythe defining conditions for a circadian clock Jolley et al.(2012). The elucidation of basic molecular designs whichcan create circadian clocks can presumably serve as a guideto experimental searches for possibly other protein-based”in vitro” PTOs and for investigating the apparent degreeof simplicity (or difficulty) required for intracellular post-translational molecular methods of timing. Physics Department, Spring Hill College, Mobile, AL 36608 USAEmail: [email protected]
Prepared using sagej.cls [Version: 2017/01/17 v1.20]
Journal Title XX(X)
In particular finding simplified two dimensional mathe-matical models of circadian clocks is potentially useful sincea variety of mathematical tools (phase plane analysis, Hopfbifurcation theory) can be used to analyze and visualize thesystem’s dynamics. This study demonstrates the existence oftwo simple designs whose mathematical models are perhapsthe simplest 2D circadian clock models for post-translationalmolecular oscillators. Both designs require a protein with atleast two regulatory sites, selective sequestration of an effec-tor molecule by one of the protein states, and a separationinto fast-slow kinetics for regulation of site occupancy. Thesegeneral conditions are sufficient to generate stable limit cycleoscillations in the population occupancy of the sites forreasonable rates and parameter values. Furthermore this classof selective sequestration models permits both entrainmentand temperature compensation, by assuming one of the sites(the ”fast” site) is sensitive to external perturbation while the”slow” regulatory dynamics on the other site is essentially”buffered” from external perturbations.
Results
Post-translational oscillatory mechanisms fromtwo protein sites and selective sequestration
For simplicity assume a ”core” clock protein ( X ) withtwo modification sites ( a = 1 and b = 2 ) in which the rateof modification of the residues can be altered by another”effector” molecule ( Y ). For the sake of generality wesuppose the molecule that is transferred to each site of theprotein remains unspecified (e.g., a phosphoryl group, ATP,or an oxygen atom) - for the KaiC protein there are twophosphorylation sites per KaiC monomer and the rate of(auto) phosphorylation of the sites is affected by anotherprotein (KaiA) in a hyperbolic manner Iwasaki et al. (2002);Rust et al. (2007). We propose two general simple designsfor an autonomous clock using sequestration of Y based onthe occupancy of the two sites (Fig 1).In one oscillatory scheme each site is independentlymodified and the kinetics proceed at different rates on the twosites; in a second class of models the occupancy of one site istransferred to the other site via some unspecified mechanism(e.g., an intra-protein transfer). Assume the populationkinetics of the two sites follows 1st order kinetics so that thefractional occupancy of each site (0 < x i < , e.g., degree ofphosphorylation) in the population is described as follows: dx i /dt = k i (1 − x i ) − k − i x i ( i = 1 , (1) Figure 1.
Two possible simple designs for a protein-basedclock. In design I, two modification sites on the protein (notnecessarily adjacent) are both regulated by [Y]. Filled circlesindicate occupancy of a protein site, by addition of a smallmolecule to that residue (for example). In design II theoccupancy of one site (regulated by [Y]) is transferred to asecond site. Only one of the four protein states sequesters theeffector molecule, Y.) where i labels each modification site and a constant time-independent decay (e.g., dephosphorylation) term is allowedin the kinetics. In general the modification rate(s) on thesite(s) are assumed to vary as smooth, monatonic functions, f , of the effector concentration, which we parameterize as k i = k i,max f ([ Y ]) where < f ([ Y ]) < ; hyperbolic andlinear regulatory functions of the rates are examined below.An individual protein ( X ) with two modification sites canbe in one of four states: ( a, b ) = (0 , , (1 , , (0 , , (1 , with zero indicating an un-occupied site and one occupied.Population statistical arguments imply the average fractionof proteins in each of the four states is x (0 , = (1 − x )(1 − x ) x (1 , = x (1 − x ) x (0 , = x (1 − x ) x (1 , = x x (2)Now suppose the modification rates ( k i ) of both sitesfollow Michealis-Menten regulatory kinetics, dependinghyperbolically on the concentration of the effector molecule ( Y ) : k i = k i,max [ Y ][ Y ] + K . (3)
Prepared using sagej.cls
Byrne O cc u p a n c y O cc u p a n c y Time (days)
Site 1 (“fast” site)Site 2 (“slow” site)(0,0) state(1,0) state (0,1) state(1,1) state
Time (days) (A)(B)
Figure 2.
Design I with independent (hyperbolic) modificationof the sites by [Y]. (A) Example population occupancy kineticsfor the two protein sites (rates and parameters are in the maintext). (B) Dynamics for the corresponding 4 protein states.
Letting site a represent the ’fast’ site ( k ,max >> k ,max )breaks the symmetry in the model and sequestration of Y bythe (0 , protein states can generate sustained oscillationsdepending on the rate constants and model parameters. Asimple ”rigid” model of sequestration is to instantaneouslyalter the concentration of Y according to the concentration of X proteins in one of the four protein states Rust et al. (2007).If an average of N molecules of Y are sequestered per X protein in the state x (0 , , the non-sequestered concentrationof [ Y ] a time t is given by: [ Y ] = max { [ Y ] − N [ X ] x (0 , , } (4)where [ Y ] indicates the initial concentration of molecule [ Y ] . For this mechanism, the dependence of the systemdynamics in terms of the model parameters and rates is bestseen by rescaling equations (2) and (4) by the fixed proteinconcentration [ X ] and the parameter N : k i = k i,max ˜ Y ˜ Y + ˜ K (5)and ˜ Y = max { ˜ Y − x (1 − x ) , } (6)where ˜ Y ≡ [ Y ] N [ X ] is the dimensionless scaled fraction ofthe effector, Y ; ˜ Y ≡ [ Y ] N [ X ] is the equivalent dimensionlessscaled fraction of initial effector [ Y ] , and ˜ K ≡ KN [ X ] is adimensionless Michaelis constant. The system dynamics is encoded by the two fixed parameters ( ˜ Y and ˜ K ) and three(ratios) of rate constants (since one rate constant can beabsorbed into a dimensionless time parameter by re-scalingboth differential equations in Eqn.(1) by, e.g., /k ,max ).For example, sustained ≈ hr oscillations are reproducedusing the parameters ˜ Y = 0 . , ˜ K = 0 . and the rates k ,max = 0 . hr − = k − , k ,max = 0 . hr − = k − (Fig2). Numerical integration suggests a limit cycle upon varyingthe initial conditions (Fig 3A). Sustained oscillations arepossible as a result of an autocatalytic negative feedbackloop (from sequestration), previously suggested in severalmathematical models of the KaiABC clock Clodong et al.(2007), van Zon et al. (2007), Rust et al. (2007). In this 2Dmodel the qualitative description for sustained oscillationsis that the slow kinetics on one site allows the site withrapid kinetics to reach near maximal occupancy beforesequestration takes effect. Once sequestration starts to occurfrom the slow formation of (0 , states, loss of occupancy onthe fast site drives the transition (1 , → (0 , . This causesfurther sequestration of Y and increases the rate of furtherloss of occupancy on the fast site creating additional (0 , states (auto-catalysis). Slow loss of occupancy from the 2ndsite assures that the 1st site becomes largely unoccupieduntil de-sequestration of Y occurs simultaneous with thetransition (0 , → (0 , . Numerical investigations of thesolution space of this model (Eqs. 1,4 and 5) indicatethat the existence of oscillations is quite sensitive to theparameters ˜ Y and ˜ K (Fig 3D). Qualitatively as the relativeconcentration of initial effector is lowered (for constant ˜ K ) the oscillatory period decreases because sequestrationof Y takes less time; for fixed initial effector, increasingthe Michaelis constant delays the onset of sequestrationresulting in a longer oscillatory period. Numerical solutionsindicate that sustained oscillations require a ”fast-slow”separation of timescales for the maximum modification rateson the two sites (Fig 3B). Varying both k ,max and k ,max over the range [0 , hr − in steps of . hr − (andsetting k − = k ,max , k − = k ,max ) suggests sustainedoscillations occur when k ,max < . k ,max (Fig 3B). Asexpected the oscillatory period is only weakly dependent on k ,max but strongly dependent (power-law dependent with apower ≈ -0.75) on k ,max (Fig 3C), which is important ininterpreting both temperature compensation and entrainmentin these clock models.Interestingly there is an even simpler class of sequestrationmodels that can generate stable limit cycles (and might beemployed biochemically). Consider the ”transfer” design inwhich the molecule occupying site 1 is transferred to site 2 Prepared using sagej.cls
Journal Title XX(X) x x (A) k (hr -1 ) k , m a x ( h r - ) (B) k (hr -1 ) k = 0.4 hr -1 k = 1.0 hr -1 P e r i o d ( h r ) (C) Y = [Y] /(N[ X ]) ~K = (cid:0) / ( N [ (cid:1) ]) (cid:2) (D) .1 (cid:3)2 (cid:4)3 (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10)(cid:11) (cid:12) (cid:13) (cid:14) (cid:15) Figure 3.
Model I: (A) Stable sample limit cycle trajectories for different initial conditions using the rates and parameters from fig 2.(B) Parameter region allowing oscillations and the oscillatory period in hours assuming ˜ Y = 0 . , ˜ K = 0 . (C) An elaboration ofPanel (B), indicating sensitive dependence of period (power-law) to the ”slow” rate constant for site 2 modification (D) Parameterregion allowing oscillations and the oscillatory period in hours for a given dimensionless Michaelis constant and scaled initialconcentration of effector, assuming k ,max = 0 . hr − = k − , k +2 ,max = 0 . hr − = k − . at rate k T so that the model is now described by a class ofODEs: dx /dt = k ([ Y ])(1 − x ) − k T x − k − x dx /dt = k T x − k − x (7)where terms for both transfer and loss of occupancyfor site 1 have been included. In these designs the rate k can depend linearly on the substrate ( k = k +1 ˜ Y ) andgenerate limit cycle oscillations (hyperbolic variation canalso generate stable limit cycles). Non-dimensionalizing( τ ≡ k +1 t ) and neglecting loss of occupancy withouttransfer (for the moment), k − = 0 , gives the followingsimple system ( α ≡ k T k +1 , β ≡ k − k +1 ), where : f ( x , x ) ≡ dx /dτ = ˜ Y (1 − x ) − αx g ( x , x ) ≡ dx /dτ = αx − βx (8) with ˜ Y given by Eqn.(5). For example there are stablelimit cycle oscillations for α = 0 . , β = 0 . , ˜ Y = 0 . ;setting k +1 = 0 . hr − generates a circadian timescale forthe oscillations (Fig 4, 5A). Allowing loss of occupancyfrom site 1 lowers the oscillation amplitude and can resultin damped oscillations; examples for k − = 0 . hr − and k − = 0 . hr − are shown in Fig 5B.The simple dynamical system can be studied analytically- see Appendix for a linear perturbative analysis about thesteady-states. Direct integration of the ODEs confirms theregions of instability and oscillatory period estimates froma linear stability analysis of the dynamical system (seeAppendix Fig 9). As an example, consider ˜ Y = 0 . . Linear(in)stability constraints suggest oscillations generally require β < . , α < . and β < α (Fig 6). The dimensionlessoscillatory period ( ˜ P ) is fit well ( R > . ) by differentpower-law functions for a given β : ˜ P ≈ Aα B with ( A, B ) =(2 . , − . for β = 0 . ; ( A, B ) = (3 . , − . for β = 0 . ; and ( A, B ) = (2 . , − . for β = 0 . . Thephysical period is P = ˜ Pk +1 which, in this regime of Prepared using sagej.cls
Byrne Site 1 (“fast” site)Site 2 (“slow” site)(0,0) state(1,0) state (0,1) state(1,1) state O cc u p a n c y O cc u p a n c y Time (days)Time (days) (A)(B)
Figure 4.
Design II with transfer of occupancy from site 1 tosite 2. (A) Example population occupancy dynamics for the twoprotein sites (rates and parameters are in the main text). (B)dynamics for the corresponding four protein states.
Figure 5.
Design II: (A) Stable sample limit cycle trajectories fordifferent initial conditions using the rates and parameters fromfig 4. (B) Allowing both loss of occupancy from site 1 andtransfer to site 2; k − = 0 . hr − (thin trace, dampedoscillations) and k − = 0 . hr − with slightly reducedamplitudes compared to Fig 4A traces (thicker red and black) parameters for oscillations gives an approximate physicaloscillatory period: P ≈ Ak T B k +1 − B − (9)For example, for β = 0 . : P ≈ . k − . T k . (10)Thus the oscillatory period in this class of models isprimarily set by the ”transfer” rate ( k T ). Figure 6.
Approximate region of stable limit cycle oscillationsfor design II from numerical integration; colormap showsdimensionless period as the ratio of rates are varied (fixed ˜ Y = 0 . ) and the inset shows the approximate power-lawdependence of the dimensionless period on α . Entrainment and Temperature Compensation
Circadian clocks are both sensitive to external perturbations(can be entrained) yet retain near period invariance undervaried environmental conditions, such as temperature fluc-tuations. In these simple models we can suggest a possiblemechanism; the oscillator period is ”mildly” sensitive toexternal perturbations so that the oscillation period is approx-imately constant and set by the ”slow” internal dynamics,while the ”fast” dynamics (e.g., k +1 ) incorporates exter-nal perturbations. In this model oscillator, all perturbations(metabolic, light or dark pulses, temperature, etc.) interactonly through the modification rates, assuming the pertur-bation does not alter the relative protein abundances ( [ Y ][ X ] ).In these models the change in period ( δP ) in terms of anyperturbation in rates ( δk j ) is, to 1st order, δP = X j ∂P∂k j δk j (11)This perturbation should be approximately zero for”compensation” to be effective (temperature or otherperturbations). There are two generic possibilities; one is thatthe effective rates in the model are ”trivially” insensitive tothe perturbation ( δk j ≈ ) due to structural properties of theprotein(s). However if the system were completely ”trivially”compensated then entrainment would not be possible (exceptby protein and/or effector abundance variation). The otheris that the system is ”tuned” to some degree so that oneperturbation which tends to increase the period is counteredby another that decreases the period Ruoff (1992)). Moregenerally, in the parameter space of P ( k j ) approximately Prepared using sagej.cls
Journal Title XX(X)
Figure 7. (A). Approximate period invariance of the oscillationsas the fast rate ( k +1 ) is varied while the transfer rate ( k T ) isslightly adjusted according to Eqn (11) in the text with B ≈ − . . (B). Sample site 2 occupancy oscillationscorresponding to the variation of k +1 shown in Panel A (largeroscillation amplitudes correspond to larger k +1 ). flat regions correspond to approximate period-invariant sub-spaces of the parameter space. The approximate power-law dependence in these models (Eqn 9) displays thiscompensatory mechanism with the constraint that thedynamical system’s period remains invariant when settingthe sum of partials to zero in the power-law approximation: δk T k T = (1 + 1 /B ) δk +1 k +1 (12)For example, consider initial unperturbed rates of k +1 =0 . hr − and k T = 0 . hr − for β = 0 . . Numerically theperiod is P ≈ . hrs . Simulations suggest a doubling ofthe site 1 rate (typical for a 10 ◦ C increase), k +1 = 1 . hr − is almost compensated ( P new ≈ . hrs ) by about a percent increase in the transfer rate (correspondingto B ≈ − . , k T = 0 . hr − ), Fig 7A. It is clear fromdirect integration of the model that period compensationis possible as predicted by Eqn 11 (Fig 7A,B); as thefast-rate is further lowered below k +1 ≈ . hr − , theoscillations start damping with increasing period. If weassume the rate doubling on the 1st site corresponds toa 10 ◦ C increase, these parameters give a Q10 (for theperiod) of about . . Since these are tuned parameters,a desired Q10 can be selected by a judicious choice ofrate compensation, corresponding to a choice of activationenergy threshold(s) for regulation of site occupancy using the Arrhenius-Boltzmann temperature-dependence of therates ( k i ∝ A i exp( − E i / ( kT ) ), as previously described forgeneral biochemical kinetics (Ruoff (1992)).Entrainment in the model is examined using bothcontinuous and discrete (pulse) perturbations. For smallamplitude perturbations, ”stable” entrainment is possiblewithin an approximate range of 19 to 26hrs for theseparameters. The following external driving function wasassumed: δk +1 k +1 = Asin (2 πt/τ ) with amplitude, A = 0 . chosen. The transfer rate was modified to retain nearperiod invariance in the absence of the external continuoussinusoidal perturbation ( B ≈ − . in eqn 11). For example,the period shifts from . hrs (unperturbed) to . hrs for an external τ = 24 hr rhythm and from . hrs(unperturbed) to . hrs for an external τ = 19 hr rhythm(Fig 8A). Entrainment is also examined using 2-hr pulsereductions in the rates. Phase response curves (PRCs) areconstructed using 2-hr pulses by transiently reducing thethe on-rate starting on day 4.5 of the unperturbed oscillator(Fig 8 B,C). The sample PRCs in Fig 8 show reductionsof k +1 = 1 . hr − to { . , . , . } hr − starting on day4.5 of the unperturbed oscillator over one oscillations cycleand computing the phase shift relative to the unperturbedcontrol on day 10 of the oscillation. The transfer rate wasalso adjusted as previously described according to eqn 11using B = − . . The long ”dead phase” of the PRC in thesemodels is because sequestration already abrogates the fastrate ( k +1 ≈ ) so that further reductions in the rate by anexternally applied down-pulse in this rate has little effectduring the time interval of significant sequestration. Discussion
The model designs and simulations in this paper suggest thatthe molecular interactions required for a post-translationalcircadian clock could be surprisingly simple. In thesedesigns, selective sequestration of an effector (regulator) ofprotein site occupancy and a separation of time-scales inthe dynamics of regulation of the protein sites appear to besufficient for generating sustained oscillations. The modelcan be easily generalized to a protein with N regulatorysites of which a subset of the N states selectively sequestersan effector (or multiple effectors) for finer clock regulation,coupling to other proteins, improved ”buffering” of the slowdynamics, etc. Already, with just 2 protein sites and 4 statesit is possible to both entrain the oscillations to externalcues and have the limit cycle oscillations compensatefor fluctuations in the fast dynamical variable (e.g., fromtemperature or metabolic perturbations). A limitation of the Prepared using sagej.cls
Byrne Figure 8. (A). Entrainment to externally driven oscillations of the rate k +1 starting on day 4; black = control, orange = 24hr, green =19hr. (B). Sample traces including pulse reductions in the fast rate (setting k +1 = 0 for 2-hr intervals starting on day 4.5). (C).Sample phase response curves (PRCs) for varying amplitude pulse reductions (measured on day 10) with k T varied according toeqn 11. model is that much complexity is encoded in the effectiverate constants, including energy regulation and ATP-ADPdynamics which were not included in these models to keepthe parameter space as small as possible. In particular forcircadian clocks, the effective rates are much slower thantypical rates from enzyme or binding kinetics ( s − or less),whereas TTFL models incorporate a natural several hr timescale that is part of the transcription-translation process.In the KaiABC clock, the slow ATPase dynamics of KaiCare implicated in the slow characteristic oscillatory timescale(Terauchi et al. (2007)) and a similar slow ATP hydrolysiswould likely be involved in setting the effective rates inthese models. Another limitation in these models, also usedto reduce the potential state-space of dynamical variables,is that sequestration was not explicitly modeled using massaction (for example); the linear sequestration model by oneof the protein states (eqn 5) is an effective and direct methodto simulate selective sequestration (Rust et al. (2007)) butintroduces a hard cutoff in these models reflected in the fastrate abruptly shifting to zero.A prediction of the simplest 2-site/4-state version of themodel is that evolutionary mechanisms should have selectedrates such that the existence of oscillatory behavior is robustunder typical fluctuations of the effective rates; the ”transfer”model predicts the most likely values of α in the range . to . and small β ( < . ). Design I with hyperbolicregulation suggests ”fast” effective modification rates of . to . hr − and a ”slow” modification rate less than / of these values. Presumably, similar to the KaiABC clock,slow conformational dynamics of the protein(s) structure isimplicated in the slow regulatory (and period-determining)dynamics. A further prediction of these models is therather general approximate power-law dependence of theoscillation period on the slow rate, which might be testedusing various clock mutants with ”cloistered” regulatorysite(s). Perhaps these and similar models will assist andencourage the experimental search for in-vitro protein-basedoscillators beyond the remarkable KaiABC cyanobacterialclock. Methods
Numerical integration of the ODEs was implemented inFortran (GNU Fortran G77, Free Software Foundation) using4th-order Runge-Kutta. The sequestration constraint (eqn5) was implemented using a threshold of . for ˜ Y .In-house fortran code was written for scanning parameterspace, peak finding, estimating oscillatory periods (typicaluncertainty < . hr ) , and applying perturbations (fig 8).For figures 7 and 8, the initial conditions, x ( t = 0) = 0 . x ( t = 0) were used. Other parameters are listed in the maintext. Colormap figures (Fig 3B,D, Figs 6,9) were generatedusing Matlab (The MathWorks, Inc., Natick, Massachusetts,United States). Prepared using sagej.cls
Journal Title XX(X)
Figure 9.
Predicted oscillatory parameter space anddimensionless oscillatory period (colormap) for design II usinganalytical solutions for the fixed point(s) of the dynamicalsystem (eqn 7) and applying the conditions for (linear) instabilityof the steady-state(s) from the matrix, M . Appendix: Linear Perturbative Analysis ofSimple Transferase Model
In this section a linear perturbative analysis about the steady-state solutions of the design II models (Eqn 7) is used tosuggest constraints on the parameters that permit (or donot permit) limit cycle oscillations. The fixed points ( x ∗ i ∋ dx ∗ i /dτ = 0 ) are solvable analytically (from a cubic) and arefunctions of two effective parameters only, β and ˜ Y βα . Alinear perturbation about the steady-states yields a stabilitymatrix, evaluated at the steady state(s): M = x ∗ (1 − x ∗ ) − ( ˜ Y + α ) − x ∗ (2 − x ∗ ) α − β ! Potential oscillatory solutions with instability of the steadystate (unstable spiral) requires
T rM > and DetM > ( T rM ) . Since x ∗ = αβ x ∗ and max { (1 − x ∗ ) x ∗ } = 1 / this implies < β ( ˜ Y α + βα + 1) < / (13)Since each term is positive this implies β < / . Furtherconstraints are given using Bendixson’s theorem; the sum ofpartial derivatives is ∂f∂x + ∂g∂x = − ( α + β + ˜ Y ) + 2 x (1 − x ) . (14)As the latter quantity is confined on the interval [0 , assuming the domain < x i < this sum of partials will be strictly negative (and thus not permit limit cycles) unless < α + ˜ Y < , (15)Since each parameter is positive, α < , ˜ Y < . Directnumerical evaluation of the steady states and applying theconditions for instability as functions of α and β givetighter ranges of instability consistent with these analyticalconstraints (Fig 9). The imaginary component( ω ) of theeigenvalues of M (for each unstable steady-state fromsolving the cubic and applying the instability criteria) givesa ”local” perturbative approximation to the dimensionlessperiod ( ˜ P = πω ) as functions of α and β (see Fig 9colormap). The estimate of the linear perturbative analysisis in good agreement with the numerical estimate of theoscillatory parameter space and corresponding oscillationperiods from direct numerical integration of the ODEs. References
Clodong S, D¨uhring U, Kronk L, Wilde A, Axmann I, Herzel H andKollmann M (2007) Functioning and robustness of a bacterialcircadian clock.
Molecular systems biology
Cell
Nature
Proceedings of the NationalAcademy of Sciences
Proceedings of the RoyalSociety of London. Series B: Biological Sciences
Proceedingsof the National Academy of Sciences
Proceedings of the NationalAcademy of Sciences
Cell reports
Journal of theoretical biology
Prepared using sagej.cls
Byrne Lee C, Etchegaray JP, Cagampang FR, Loudon AS and Reppert SM(2001) Posttranslational mechanisms regulate the mammaliancircadian clock.
Cell
Annual reviewof genetics
Science
Nature
Nature
PLoS biology
Journal of Biological Rhythms
Journal of Interdisciplinary CycleResearch
Science
Proceedings of the National Academyof Sciences
Science
Biophysical journal
Proceedings of the National Academy of Sciences
Nature Reviews Genetics
Acknowledgements
I appreciate comments and discussion on early work on thisproject from Carl Johnson, Tetsuya Mori, Ximing Qin and YaoXu (Vanderbilt U.). I also express my appreciation to Spring HillCollege (and especially Dr Lesli Bordas) for enabling a ”sabbatical”leave during the Spring 2020 semester.