An allosteric model of circadian KaiC phosphorylation
Jeroen S. van Zon*†, David K. Lubensky†‡, Pim R. H. Altena§, and Pieter Rein ten Wolde§¶
*Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom; †Division of Physics and Astronomy, Vrije Universiteit
Amsterdam, 1081 HV, Amsterdam, The Netherlands; and §Fundamenteel Onderzoek der Materie Institute for Atomic and Molecular Physics,
Kruislaan 407, 1098 SJ, Amsterdam, The Netherlands
Edited by David R. Nelson, Harvard University, Cambridge, MA, and approved February 27, 2007 (received for review September 30, 2006)
In a recent series of ground-breaking experiments, Nakajima et al.
[Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, Iwasaki H,
Oyama T, Kondo T (2005) Science 308:414 – 415] showed that the
three cyanobacterial clock proteins KaiA, KaiB, and KaiC are sufficient in vitro to generate circadian phosphorylation of KaiC. Here,
we present a mathematical model of the Kai system. At its heart is
the assumption that KaiC can exist in two conformational states,
one favoring phosphorylation and the other dephosphorylation.
Each individual KaiC hexamer then has a propensity to be phosphorylated in a cyclic manner. To generate macroscopic oscillations, however, the phosphorylation cycles of the different hexamers must be synchronized. We propose a novel synchronization
mechanism based on differential affinity: KaiA stimulates KaiC
phosphorylation, but the limited supply of KaiA dimers binds
preferentially to those KaiC hexamers that are falling behind in the
oscillation. KaiB sequesters KaiA and stabilizes the dephosphorylating KaiC state. We show that our model can reproduce a wide
range of published data, including the observed insensitivity of the
oscillation period to variations in temperature, and that it makes
nontrivial predictions about the effects of varying the concentrations of the Kai proteins.
C
yanobacteria are the simplest organisms to use circadian
rhythms to anticipate the changes between day and night. In
the cyanobacterium Synechococcus elongatus, the three genes
kaiA, kaiB, and kaiC are the central components of the circadian
clock (1). In higher organisms, it is believed that circadian
rhythms are driven primarily by transcriptional feedback (2).
KaiC phosphorylation, however, shows a circadian rhythm even
when transcription and translation are inhibited (3). Still more
remarkably, it was recently shown that this rhythmic KaiC
phosphorylation can be reconstituted in vitro in the presence of
only KaiA, KaiB, and ATP (4). The Kai system thus represents
a very rare example of a functional biochemical circuit that can
be recreated in the test tube. It is a major open question to
explain how stable oscillations can result from the experimentally observed interactions among the different Kai proteins.
In living cells, KaiC phosphorylation increases during the subjective day and decreases during the subjective night, and this
phosphorylation in turn regulates KaiC’s activity as a global transcriptional repressor (5). KaiC forms a hexamer both in vivo and in
vitro (6); KaiA is present in the cell as a dimer (6) and KaiB as a
dimer (6, 7) or a tetramer (8). KaiC has both autodephosphorylation and weaker autophosphorylation activity, with the latter dependent on ATP binding (9–14). KaiC phosphorylation is stimulated by KaiA (10, 11, 15), whereas KaiB appears to interfere with
this effect (10–12, 16). KaiC hexamers form heteromultimeric
complexes with KaiA and KaiB dimers, but one such complex
contains no more than one KaiC hexamer (6, 7, 17). The composition of these complexes varies with an ⬇24-h period.
The striking observation of Nakajima et al. (4) of in vitro
oscillations in KaiC phosphorylation poses an obvious challenge for
modelers. Not only is there the potential for detailed comparisons
between a model’s predictions and the wealth of experimental data,
the Kai system also has several novel features. Most notably, ATP
is consumed, and the system is driven out of equilibrium, only
through the repeated phosphorylation and dephosphorylation of
KaiC. Other reactions, such as the (un)binding of KaiA and KaiB
7420 –7425 兩 PNAS 兩 May 1, 2007 兩 vol. 104 兩 no. 18
to KaiC, should thus obey detailed balance. Moreover, unlike in
most biological oscillations (18), in the Kai system the proteins are
neither created nor destroyed. This imposes significant constraints
on any model that hopes to explain the in vitro oscillations.
Several previous studies have put forward interesting ideas on
how these oscillations might occur (19–21). However, they either
require that KaiC hexamers can bind to each other to form
higher-order complexes (19, 20), a possibility ruled out by recent
experiments (6, 7), or they assume that KaiA and KaiB can each
take on multiple forms (21). In the latter case, the authors propose
that these forms may correspond to different subcellular localizations, but that suggestion cannot hold for the in vitro system.
Emberly and Wingreen (19) introduced the elegant hypothesis that
exchange of monomers among KaiC hexamers might contribute to
oscillations, an idea supported by recent observations (7). Their
own work, however, shows that such exchange by itself is insufficient
to produce sustained oscillations. Thus, there clearly is another
mechanism at work in the Kai system.
Here, we propose such a mechanism. Our model is built on two
key elements. First, we hypothesize that an isolated KaiC hexamer
already has a tendency to be cyclically phosphorylated and dephosphorylated as it flips between two allosteric states. Second, we
suggest that these noisy oscillations of individual hexamers can be
synchronized through the phenomenon of differential affinity,
whereby the laggards in a population outcompete the other hexamers for a limited number of KaiA molecules that stimulate
phosphorylation. The slowest hexamers thus speed up while the
fastest are forced to slow down, causing the entire population to
oscillate in phase.
In the rest of this article, we first show how a simple picture of
allosteric transitions in KaiC leads each hexamer to have an intrinsic
phosphorylation cycle. We then use an idealized model to introduce
the concept of differential affinity. This model shows that the
mechanism requires only a few generic ingredients, suggesting that
the same synchronization principle could be at work in other
biological systems. Finally, we turn to a more complicated model of
the Kai system. This model reproduces the phosphorylation behavior of KaiC not only in the in vitro experiments in which all three
Kai proteins are present, but also in systems where KaiA and/or
KaiB are absent. In fact, we found that the experiments on the
various subsets of the three Kai proteins strongly constrain the
model’s design. Beyond synchronizing oscillations, KaiA and KaiB
must also stabilize one or the other KaiC state by binding to it.
When this binding is strong enough, the system moreover exhibits
temperature compensation, as observed (4).
Author contributions: J.S.v.Z., D.K.L., and P.R.t.W. designed research; J.S.v.Z., D.K.L.,
P.R.H.A., and P.R.t.W. performed research; and J.S.v.Z., D.K.L., and P.R.t.W. wrote the
paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
‡Present
address: Department of Physics, University of Michigan, Ann Arbor,
MI 48109-1040.
¶To
whom correspondence should be addressed. E-mail: [email protected].
This article contains supporting information online at www.pnas.org/cgi/content/full/
0608665104/DC1.
© 2007 by The National Academy of Sciences of the USA
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0608665104
A
B
to be in the I state, where dephosphorylation occurs spontaneously.
Each monomer thus tends to go through the sequence of reactions
A 3 A-ATP 3 Ap-ADP 3 Ip-ADP 3 Ip 3 I 3 A, during which
one ATP molecule is hydrolyzed.
KaiC Hexamers. In the spirit of the MWC model (26), we assume that
Fig. 1. Model of conformational transitions in individual KaiC hexamers. (A)
Schematic free energy levels for KaiC subunits. Subunits can be in the active (A)
or the inactive (I) state. Furthermore, subunits can be phosphorylated (p) and
bind ATP or ADP. Phosphorylation favors the inactive state, and nucleotide
binding favors the active state. (B) Reaction network for a KaiC hexamer with
six phosphorylation sites. Ci and C̃i denote a hexamer with i phosphorylated
monomers in, respectively, the active and inactive state. (C) Phosphorylation
cycles for the model in B. The phosphorylation level p of a single KaiC hexamer,
as obtained by stochastic simulations (solid line), and of a population of
hexamers, obtained from the mean-field rate equations (dashed line). The
phosphorylation level p ⬅ 冱i i([Ci] ⫹ [C̃i])/冱i 6([Ci] ⫹ [C̃i]), where [Ci] is the
concentration of hexamers in state Ci. For parameter values, see SI Appendix.
Allosteric Model
In this section, we introduce a simple model of allosteric
transitions in KaiC that naturally gives rise to repeated rounds
of phosphorylation and dephosphorylation within each hexamer.
Allosteric conformational changes are widespread in biochemistry, and the conformations of members of the RecA/DnaB
superfamily, to which KaiC belongs, have been extensively
studied (22, 23).
KaiC Monomers. Although there is strong evidence that KaiC
monomers can be phosphorylated at multiple sites (10, 24, 25), most
published data do not distinguish between different phosphorylated
forms. We thus assume that KaiC monomers have only two
phosphorylation states, phosphorylated and unphosphorylated.
We postulate that an individual KaiC monomer can be in either
an active (A) or an inactive (I) conformation. Fig. 1A shows the free
energies of the different monomer states; we consider ATP binding
only to unphosphorylated and ADP binding only to phosphorylated
monomers. As the figure indicates, we assume that phosphorylation
favors the inactive over the active state. Nucleotides have a higher
affinity for monomers in the active state than for those in the
inactive state, so nucleotide binding favors the active state over the
inactive one. We also take both the transfer of a phosphate from
ATP to a KaiC monomer and the removal of the phosphate from
the monomer to be thermodynamically favorable. Taken together,
these elements allow for a phosphorylation cycle: unphosphorylated monomers prefer to be in the A state, where ATP hydrolysis
drives phosphorylation, whereas phosphorylated monomers prefer
van Zon et al.
fi ⫽ k0exp关⌬G共i兲/2兴 ⬃ c i
[1]
bi ⫽ k0exp关⫺⌬G共i兲/2兴 ⬃ c ⫺i,
[2]
where k0 sets the basic time scale and c ⫽ exp[(⌬Gp ⫺ ⌬Gu)/2].
Alternatively, one can develop an explicit transition state theory
that includes the number of bound nucleotides as one of the
order parameters for the conformational transition (see SI
Appendix). This leads to flipping rates that vary exponentially
with i just as in Eqs. 1 and 2. In either case, the rates depend
strongly on the phosphorylation level, with the consequence that
hexamers can flip from A to I only when most of their monomers
are phosphorylated, and from I to A only when most are not
phosphorylated.
In Fig. 1C, we show the time dependence of the phosphorylation
level of a single KaiC hexamer obtained by Monte Carlo simulations
of the chemical master equation (see SI Appendix) (27). Initially, the
KaiC hexamer is in the unphosphorylated active state, C0. KaiC
(de)phosphorylation clearly occurs in a cyclic fashion, with few
transitions from one conformation to the other occurring at intermediate phosphorylation. However, both the amplitude and the
period of the phosphorylation cycle are highly variable. Because of
this variability, the phosphorylation cycles of a population of
independent KaiC hexamers will quickly dephase. As a result, in
Fig. 1C the mean phosphorylation level of the KaiC population
calculated by integrating deterministic rate equations based on the
law of mass action shows no oscillatory behavior. To explain the
oscillations observed in the in vitro Kai system, the uncoupled
phosphorylation cycles of the individual KaiC hexamers need to be
synchronized.
Synchronization with Differential Affinity
The natural candidates to link the phosphorylation states of
different KaiC hexamers are the other two Kai proteins. Here,
we present a simple model in which KaiA plays this role by
catalyzing phosphorylation in the active state, while KaiB is
completely absent. This model will allow us to introduce several
important ideas without the distractions that a more faithful
description would entail. It shows synchronized limit-cycle osPNAS 兩 May 1, 2007 兩 vol. 104 兩 no. 18 兩 7421
BIOPHYSICS
C
the energetic cost of having two different monomer conformations
in the same hexamer is prohibitively large. We can then speak of a
hexamer as being in either the A or the I state. The total (free)
energy of the hexamer is simply the sum of the contributions from
its constituent monomers. Highly phosphorylated hexamers thus
prefer to be in the I state, where they will be dephosphorylated,
whereas weakly phosphorylated hexamers prefer the A state, where
they will be phosphorylated. As a result, each hexamer tends to go
through a cycle in which it is first phosphorylated, then dephosphorylated, as indicated in Fig. 1B and Fig. 8 of the supporting
information (SI) Appendix.
The transition (or flip) rates fi for a hexamer with i phosphorylated monomers to go from the A to the I state and bi to go from
the I to the A state depend on the energy barriers to the conformational changes. If we assume that ATP and ADP exchange are
fast, so that the free energy of each state is well defined, then the
difference in free energy ⌬G between the I and A states grows
linearly with i: ⌬G(i) ⫽ i ⌬Gp ⫹ (6 ⫺ i) ⌬Gu, where the subscripts
p and u refer to the free-energy differences for phosphorylated and
unphosphorylated monomers, respectively. The natural phenomenological assumption is then that the flip rates depend exponentially on the free-energy difference:
phosphorylation; [A]/[A]T
A
1
0.8
0.4
0.2
[A]/[A]T
0
relative concentration
p
0.6
0
12
24
0.8
B
p=0.10
([ACi]/[C]T)*50
0.4
0
[Ci]/[C]T
0
1
2
3
4
5
36
time (hour)
0.8
48
C
p=0.43
0.4
60
72
0.8
D
p=0.77
0.4
0
0
6
0 1 2 3 4 5 6
0
number of subunits phosphorylated
1
2
3
4
5
6
Fig. 2. Limit cycle oscillations in KaiC phosphorylation for the simplified
model defined by Eqs. 3–5. (A) Mean phosphorylation level p and normalized
concentration of free KaiA [A]/[A]T. During the phosphorylation phase, [A]
drops almost to zero. (B–D) KaiA binding at three stages of the phosphorylation phase, marked by circles in A. KaiA favors the less phosphorylated KaiC
hexamers. [C]T, total KaiC concentration; ACi, complex of KaiA and Ci. We take
[A]T/[C]T ⫽ 0.02 and initially set [C0] ⫽ [C]T; see SI Appendix for other
parameters.
cillations in KaiC phosphorylation, provided that the concentration of KaiA is sufficiently small and that KaiA binds to KaiC
with differential affinity: KaiA should bind most strongly to
weakly phosphorylated KaiC hexamers. Although here we limit
our discussion to a particular model inspired by the Kai system,
the differential affinity mechanism is also amenable to a more
general, abstract formulation that we describe in SI Appendix,
where we also show that the oscillations arise through a supercritical Hopf bifurcation.
We assume that only a single dimer of KaiA can bind to a KaiC
hexamer, and we force every hexamer to proceed through the states
C0–C6 and C̃6–C̃0 in order (thus neglecting intermediate flips). This
yields
f6
b0
¡ C̃6, C̃0 O
¡ C0
C6 O
[3]
k̃ dps
¡ C̃i⫺1
C̃i O
k Af
[4]
k pf
Ci ⫹ A |
-0 ACi O
¡ Ci⫹1 ⫹ A 共i ⫽ 6兲.
[5]
k iAb
We use deterministic, mass-action kinetics to model the effects
of these reactions. Here, Ci and C̃i denote i-fold phosphorylated
KaiC hexamers in the active and inactive states, and A denotes
a KaiA dimer. Eqs. 3–5 describe the same processes within a
single hexamer as the diagram in Fig. 1B, with the exception that
phosphorylation of the active state now requires KaiA, which
associates with active KaiC with on and off rates kAf and kiAb
and stimulates phosphorylation with a rate kpf (Eq. 5). We
implement differential affinity by setting kiAb ⫽ kiAb␣i, with ␣ ⬎
1 (see SI Appendix).
Fig. 2A shows the mean phosphorylation level of a population of
KaiC hexamers as a function of time. In contrast to the behavior
seen in Fig. 1C, there are clear oscillations: The KaiA dimers
effectively couple the phosphorylation cycles of the individual KaiC
hexamers. During the phosphorylation phase of the oscillations,
7422 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0608665104
most hexamers are in the active form. In this state, they can bind
KaiA, which stimulates their phosphorylation. The concentration of
KaiA, however, is limited; indeed, in this part of the cycle, the
concentration of free KaiA is close to zero (Fig. 2 A). This means
that the KaiC hexamers compete with one another for KaiA. In this
competition, the complexes with a lower degree of phosphorylation
have the advantage because they have a higher affinity for KaiA.
Hence, during the phosphorylation phase, KaiA will be mostly
bound to the lagging hexamers. This is shown in Fig. 2B, where the
concentrations [Ci] and [ACi] are plotted versus i for three different
time points. The distributions do not overlap: KaiA has a clear
preference for the less phosphorylated KaiC hexamers. Because the
phosphorylation rate depends on the amount of bound KaiA,
laggards with a low degree of phosphorylation will be phosphorylated at a high rate, whereas front-runners with a high degree of
phosphorylation will be unable to increase their phosphorylation
level further. This is the essence of the differential affinity synchronization mechanism.
Full Model of the Kai System
The simple model of the previous section showed how differential affinity can synchronize the oscillations of the different
KaiC hexamers. This model, however, neglects KaiB completely
and is not consistent with the large body of experimental data on
the Kai system. Here, we present a more refined allosteric model.
The Model. The key ingredients of our model are as follows.
1. KaiA can bind to the active form of KaiC, stimulating KaiC
phosphorylation. Recent experiments suggest that, in the
absence of KaiB, KaiA binds as a single dimer to the CII
domain of the KaiC hexamer (28). Because this is the domain
containing KaiC’s phosphorylation site, it seems reasonable
that the affinity of KaiA might depend on the phosphorylation state of KaiC. We thus assume, as before, that a single
KaiA can bind to the active state of KaiC and that the affinity
of KaiA for active KaiC decreases as the phosphorylation
level increases.
2. The active state of KaiC is more stable than the inactive one. The
experiments described in refs. 3 and 7 show that in the presence
of only KaiA, KaiC becomes very highly phosphorylated. In the
absence of KaiB, KaiC should thus have no tendency to cyclically
phosphorylate and dephosphorylate. This requires that the
active state of KaiC has a lower free energy than the inactive one
(thus shifting the energy levels in Fig. 1 A from their symmetric
values).
3. KaiB can bind to the inactive form of KaiC. The resulting KaiB–
KaiC complex can then bind to and sequester KaiA. The phosphorylation behavior of KaiC in the presence of KaiB, but not
KaiA, is essentially identical to that of KaiC in the absence of
both KaiA and KaiB (11, 12). This observation strongly suggests
that KaiB does not directly affect phosphorylation and dephosphorylation rates. We propose instead the following functions
for KaiB. (i) KaiB can increase the stability of the inactive state
of KaiC by binding to it. This restores the capacity of individual
KaiC hexamers to sustain phosphorylation cycles. (ii) Strong
binding of KaiA by KaiB associated with the inactive KaiC
hexamers reduces the concentration of free KaiA dimers. This
leads to a variant of the differential affinity mechanism, which
is necessary for synchronizing the oscillations of the different
KaiC hexamers, as we clarify below. Based on the measured size
of the heteromultimeric complexes (6, 7), we assume that the
inactive form of KaiC can bind two KaiB dimers, and that B2C̃4,
B2C̃3, B2C̃2, and B2C̃1 can each bind two KaiA dimers with high
affinity. Neither assumption is critical: A model in which more
than two KaiB and two KaiA dimers can bind also generates
oscillations.
van Zon et al.
1
A
0.8
0.6
p
4. The rate of spontaneous phosphorylation is lower than that of
spontaneous dephosphorylation. The model includes spontaneous
phosphorylation and dephosphorylation of both active and
inactive KaiC. Because KaiC reaches a low phosphorylation level
in the absence of KaiA (and KaiB) (3, 7, 11), the rate of
spontaneous phosphorylation is lower than that of spontaneous
dephosphorylation.
0.4
0.2
0
This model is described by the following reactions:
[6]
bi
k iAf
k pf
-0 ACi O
¡ Ci⫹1⫹A
Ci ⫹ A |
[7]
k iAb
k̃ iBf
72
0.5
0
0
12
24
36
time(hour)
48
60
72
[9]
k̃ dps
k̃ ps
-0 B2C̃i⫹1, A2B2C̃i |
-0 A2B2C̃i⫹1.
B2C̃i |
[10]
k̃ dps
As in Synchronization with Differential Affinity above, we assume
that the reaction rates are given by deterministic, mass-action
kinetics. The most critical parameters are the (de)phosphorylation rates. They have not been directly measured but are strongly
constrained by the large number of quantitative in vitro experiments on the subsets of Kai proteins (see below). The model’s
predictions are much less sensitive to the remainder of its 39
parameters; for these, we have simply chosen plausible values
(see SI Appendix).
KaiA ⴙ KaiB ⴙ KaiC. Fig. 3A shows that our model produces sustained
oscillations in KaiC phosphorylation when all three Kai proteins are
present in the concentrations used in ref. 7. Both the period and the
amplitude of the oscillations agree well with those observed in refs.
4 and 7. Fig. 3B shows the concentrations of complexes containing
KaiA and KaiC ([AC]); KaiB and KaiC ([BC]); and KaiA, KaiB,
and KaiC ([ABC]), as a function of time. In the phosphorylation
phase of the oscillations, KaiA binds to KaiC and stimulates its
phosphorylation. At the top of the phosphorylation cycle, where
KaiC hexamers flip from the active to the inactive state, KaiA is
released and KaiB binds to the inactive KaiC hexamers. The binding
of KaiB stabilizes the inactive form of KaiC, preventing phosphorylation by KaiA. One critical role of KaiB is thus to allow the KaiC
hexamers to enter the dephosphorylation phase of the cycle.
Fig. 3B also shows that after [BC] has increased, [ABC] increases.
This is because B2C̃4–B2C̃1 can bind KaiA. This illustrates the
second function of KaiB: KaiB that is bound to KaiC also sequesters
KaiA. This leads to a form of the differential affinity mechanism at
the end of the dephosphorylation phase of the cycle: The KaiC
hexamers that are still in the inactive form (the laggards) will take
away KaiA from those hexamers that have already flipped from the
inactive to the active state (the front-runners). This reduces the
phosphorylation rate of the front-runners, allowing the laggards to
catch up.
In our model, differential affinity acts at the bottom of the
dephosphorylation phase of the cycle and throughout the phosphorylation phase. From the perspective of synchronizing the
oscillations of the different hexamers, the ideal would be an
ever-decreasing affinity between KaiA and KaiC, even as a given
hexamer passes through the same sequence of states again and
again. Thermodynamics, however, dictates that the affinity of KaiA
Fig. 3. Sustained oscillations in the full model defined by Eqs. 6–10. (A) The
mean phosphorylation level p of KaiC shows a stable 24-h rhythm. (B) Kai
complexes. At t ⫽ 0, KaiC is fully unphosphorylated: [C0] ⫽ [C]T; [A] ⫽ [A]T; [B] ⫽
[B]T. The average phosphorylation then increases as KaiA binds KaiC and stimulates phosphorylation. Next, the amount of KaiB–KaiC complex ([BC]) increases at
high phosphorylation as KaiB binds to the inactive state of KaiC. Subsequently,
KaiA is sequestered into a KaiA–KaiB–KaiC complex (ABC). The total concentrations equal those used in the in vitro experiments described in ref. 3: [C]T ⫽ 0.58
␮M; [A]T ⫽ 1.75 ␮M; [B]T ⫽ 0.58 ␮M, corresponding to [A]T ⫽ [C]T and [B]T ⫽ 3[C]T.
For other parameter values, see Table 2 of SI Appendix.
for KaiC must increase somewhere in the cycle. In our model, this
happens at the top of the inactive branch, where B2C̃6 and B2C̃5 do
not bind KaiA, but B2C̃4 does have a high affinity for KaiA. To
obtain agreement with experiment, it is both necessary and sufficient for differential affinity to act on the inactive branch, although
differential affinity on the active branch does enhance the oscillations’ amplitude.
KaiA ⴙ KaiC. Fig. 4 shows that, in the presence of only KaiA, initially
unphosphorylated KaiC reaches a phosphorylation level of ⬇90–
95% after 6–8 h, in good quantitative agreement with experiment
(7). In our model, KaiC is biased toward the active state, and KaiA
binding increases the stability of the active state even further. This
explains the high steady-state phosphorylation level when only
KaiA is present.
(KaiB ⴙ) KaiC. Fig. 4 also shows that the phosphorylation behavior
of KaiC in the presence of KaiB is very similar to that of KaiC alone,
as observed (11, 12). Our model can explain this observation by
assuming that the spontaneous dephosphorylation rate of the two
1
C alone
C+A
C+B
C+A+B
0.8
0.6
BIOPHYSICS
k dps
van Zon et al.
60
B
[8]
k̃ ps
-0 Ci⫹1, C̃i |
-0 C̃i⫹1
Ci |
k̃ dps
48
k̃ iAb
k ps
k̃ ps
36
time(hour)
p
k̃ iBb
24
AC
BC
ABC
1
k̃ iAf
-0 B2C̃i, B2C̃i ⫹ 2A |
-0 A2B2C̃i
C̃i ⫹ 2B |
12
B
fraction KaiC bound
in complexes
fi
Ci |
0 C̃i
0
0.4
0.2
0
0
4
8
12
time(hour)
16
20
24
Fig. 4. KaiC phosphorylation in the absence of KaiA and KaiB (C alone), in the
presence of KaiA (C⫹A), in the presence of KaiB (C⫹B), and in the presence of
both KaiA and KaiB (C⫹A⫹B). For C⫹A, KaiC is initially fully unphosphorylated; for C alone and C⫹B, KaiC is initially fully phosphorylated (see also ref.
7). Parameters are as in Fig. 3.
PNAS 兩 May 1, 2007 兩 vol. 104 兩 no. 18 兩 7423
1
Varying Flip Rates
fi x 5
0.8
i
i
i
12
24
36
time(hour)
48
60
72
1
1
0
Varying Dissociation Constants
0.8
x5
x1
x 1/5
p
0.6
0.4
0.2
0
12
24
36
time(hour)
48
2
60
72
Fig. 5. Temperature-compensated oscillation period. The period of KaiC
phosphorylation changes by 10% when the forward ( fi) or backward (bi) flip
rates are changed by a factor 25 (A) and by ⬍5% when the dissociation
constants for all KaiA and KaiB binding reactions are simultaneously changed
by a factor 25 (B). Parameters are as in Fig. 3.
KaiC conformations is the same and is unaffected by KaiB binding,
which only stabilizes the inactive state with respect to the active one.
Temperature Compensation. A striking feature of the in vitro oscil-
lations of the Kai system is that they are temperature-compensated
(4). Specifically, as the temperature is increased from 25°C to 35°C,
the period of the oscillations decreases by only 10%. In general, the
oscillation period of a network depends on the rates of all of the
reactions in the system. In principle, one could try to achieve
temperature compensation by balancing the temperature dependencies of all of these rates (29). We have adopted a different
approach that is motivated by the fact that the (de)phosphorylation
reactions are each individually temperature-compensated (3): The
phosphorylation time courses of KaiC alone and of KaiC with KaiA
change little between 25°C and 35°C. Indeed, the key idea of our
approach is to construct the model so that the oscillation period is
determined by those rates that are known from experiment to be
robust against temperature variations while leaving it insensitive to
the other rates, which might vary with temperature.
A natural idea is to demand that the rates that can vary with
temperature be much faster than the (de)phosphorylation rates, so
that the period is dominated by the latter, which are temperaturecompensated. This leads to the following ingredient.
5. All (un)binding rates and the flip rates f6 and b0 are much faster
than the (de)phosphorylation rates. Most conformational transitions are made at the top and bottom of the cycle; the period
is thus less sensitive to flip rates other than f6 and b0.
Even when the (un)binding reactions between the Kai proteins
are fast, however, the period can still depend on the ratios of
their rates (the dissociation constants), which will vary with
temperature. The period becomes independent of the dissociation constants if all binding reactions go to completion. This
occurs when the dissociation constants are much smaller than
typical protein concentrations; in this limit, a change in the
dissociation constants will have no appreciable effect on the
fraction of bound proteins. We thus require the following.
6. The affinities among the Kai proteins are high. KaiA, the least
abundant of the three proteins, will then be almost entirely
bound up in complexes with KaiB and KaiC, in agreement
with ref. 7. As long as the relative magnitudes of the dissociation constants do not change with temperature, the com7424 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0608665104
2
1
0
0.5
1
[A]T/[C]T
0
1.5
0
0.5
[A] /[C]
T
1
1.5
T
D
C
20
1
25
30
0.5
0
0
24
48
time (hour)
72
average phosphorylation
0
T
b x 1/5
3
T
0.2
0
Amplitude
4
3
[B] /[C]
original
b x5
[B]T/[C]T
0.4
p
B
B
Period
4
f x 1/5
0.6
0
A
average phosphorylation
A
0.2
1
0.4
0.6
24
48
time (hour)
72
0.5
0
0
Fig. 6. KaiC oscillations as a function of KaiA and KaiB concentration. (A and
B) Period (A) and amplitude (B) of oscillations in KaiC phosphorylation as a
function of the concentration of KaiA and KaiB. The dashed curve shows the
location of the supercritical Hopf bifurcation that gives birth to the oscillations, and the color scales give period in hours and amplitude of p oscillation.
Note the appearance of a small region of bistability (solid line; see also SI
Appendix) at low [A]T and [B]T. The remaining parameters are as in Fig. 3. (C)
KaiC oscillations as a function of KaiA concentration. Results are shown for
[B]T ⫽ 3[C]T and [A]T ⫽ 0.2[C]T (green), 0.6[C]T (blue), [C]T (black), and 1.4[C]T
(red). (D) KaiC oscillations as a function of KaiB concentration. Results are
shown for [A]T ⫽ [C]T and [B]T ⫽ 1.2[C]T (purple), 2.1 [C]T (yellow), and 3[C]T
(black).
position of these complexes will moreover be unaffected. The
phosphorylation rates, which depend on [ACi], are then
robust to changes in temperature. Another important consequence of this condition is that a proportional increase in all
of the protein concentrations will have no effect on the
oscillations, as has been observed (7).
Because no data on the temperature dependence of the
dissociation constants and flip rates exists, we made the following estimate. We assumed that both the binding energies and the
energy barriers for the conformational transitions are at most 50
kBT. If the temperature is changed from 25°C to 35°C, the
dissociation constants and flip rates can then change by about an
order of magnitude. To test whether our model is robust against
such perturbations, we have varied both dissociation constants
and flip rates by a factor of 5 in each direction. Fig. 5 shows that
our model withstands these trials: The period varies by ⬇5–10%,
in very good agreement with the experiment described in ref. 4.
This is strong evidence that conditions 5 and 6, together with
temperature-compensated (de)phosphorylation rates, are sufficient for temperature-compensated oscillations.
KaiC Dynamics as a Function of KaiA and KaiB Concentration. Fig. 6
shows the behavior of our model as a function of the total KaiA and
KaiB concentrations [A]T and [B]T. For [A]T ⱗ 0.5[C]T, the system
exhibits no oscillations. At around [A]T ⫽ 0.5[C]T, the system starts
to oscillate via a supercritical Hopf bifurcation with a period of ⬇35
h (see SI Appendix for details on the bifurcation analysis). As the
KaiA concentration is increased, the period monotonically decreases. In contrast, the amplitude first increases to reach a maximum at around [A]T ⫽ 0.85[C]T, then decreases until oscillations
disappear at around [A]T ⫽ 1.25[C]T. The dynamics as a function
of the KaiB concentration are markedly different. Fig. 6 shows that
van Zon et al.
Discussion
We have presented an allosteric model of KaiC phosphorylation
that can describe a wealth of experimental data on the Kai
system. Its foundation is the assumption that each KaiC hexamer
can exist in two distinct conformational states, an active one in
which it tends to be phosphorylated and an inactive one in which
it tends to be dephosphorylated. Because of the interplay
between nucleotide binding, which favors the active state, and
phosphorylation, which favors the inactive state, each individual
hexamer will repetitively gain and lose phosphate groups. However, if macroscopic oscillations are to be observed, the phosphorylation cycles of the individual hexamers must be synchronized. We introduced a mechanism, called differential affinity,
which, in contrast to some previous models (19, 20), allows for
synchronization even in the absence of direct interactions between hexamers. The key idea is that although all KaiC hexamers
1. Ishiura M, Kutsuna S, Aoki S, Iwasaki H, Andersson CR, Johnson CH, Golden
SS, Kondo T (1998) Science 281:1519–1523.
2. Dunlap JC, Loros JJ, DeCoursey PJ (2004) Chronobiology: Biological Timekeeping
(Sinauer, Sunderland, MA).
3. Tomita J, Nakajima M, Kondo T, Iwasaki H (2005) Science 307:251–254.
4. Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, Iwasaki H, Oyama T,
Kondo T (2005) Science 308:414–415.
5. Nakahira Y, Katayama M, Miyashita H, Kutsuna S, Iwasaki H, Oyama T, Kondo
T (2002) Proc Natl Acad Sci USA 101:881–885.
6. Kageyama H, Kondo T, Iwasaki H (2003) J Biol Chem 278:2388–2395.
7. Kageyama H, Nishiwaki T, Nakajima M, Iwasaki H, Oyama T, Kondo T, (2006)
Mol Cell 23:161–171.
8. Hitomi K, Oyama T, Han S, Arvai AS, Getzoff ED (2005) J Biol Chem
280:19127–19135.
9. Nishiwaki T, Iwasaki H, Ishiura M, Kondo T (2000) Proc Natl Acad Sci USA
97:495–499.
10. Iwasaki H, Nishiwaki T, Kitayama Y, Nakajima M, Kondo T (2002) Proc Natl Acad
Sci USA 99:15788–15793.
11. Xu Y, Mori T, Johnson CH (2003) EMBO J 22:2117–2126.
12. Kitayama Y, Iwasaki H, Nishiwaki T, Kondo T (2003) EMBO J 22:2127–2134.
13. Nishiwaki T, Satomi Y, Nakajima M, Lee C, Kiyohara R, Kageyama H, Kitayama
Y, Temamoto M, Yamaguchi A, Hijikata A, et al. (2004) Proc Natl Acad Sci USA
101:13927–13932.
14. Hayashi F, Itoh N, Uzumaki T, Iwase R, Tsuchiya Y, Yamakawa H, Morishita M,
Onai K, Itoh S, Ishiura M (2004) J Biol Chem 279:52331–52337.
van Zon et al.
compete to bind KaiA, which stimulates phosphorylation, the
laggards in the cycle are continuously being favored in the
competition. This mechanism is most effective when KaiB and
KaiC bind KaiA very strongly. It is also precisely in this limit that
the oscillation period becomes insensitive to changes in the Kai
proteins’ affinities for each other. Differential affinity and
temperature compensation are thus intimately connected. The
mechanism of driving two-body reactions to saturation is, however, more general; it could, for instance, be used to make
temporal programs of gene expression robust against temperature variation (30).
In S. elongatus, the concentration of KaiA dimers is ⬍10% of that
of KaiC hexamers (12). Our model predicts that in this regime, the
in vitro oscillations of ref. 4 disappear. The very recent experiments
described in ref. 7 support this prediction: They unambiguously
demonstrate that in vitro, the oscillations cease to exist if the
concentration of the KaiA dimers is ⬍25% of that of the KaiC
hexamers. Clearly, in vivo, other processes are at work. It is known,
for instance, that both the subcellular localization of the Kai
proteins (12) and KaiC’s role as a transcriptional repressor (5) affect
circadian rhythms, as do other clock proteins such as SasA (6). It is
tempting to speculate, however, that these additional effects merely
shift the phase boundaries of the model presented here without
changing its basic mechanism. One could imagine, for example, that
a combination of KaiB localization to the cell membrane and
competitive binding by molecules like SasA could reduce the
number of sites available to sequester KaiA, thus allowing the
oscillator to function at lower KaiA concentrations.
Finally, our model makes a number of predictions that could be
verified experimentally. One clear prediction is that KaiC can exist
in two distinct conformational states. Moreover, our model suggests
that KaiC binds KaiA and KaiB very strongly, with dissociation
constants that depend on the conformational state and phosphorylation level of the KaiC hexamer. But perhaps the strongest test of
our model concerns the KaiC oscillation dynamics as a function of
the KaiA and KaiB concentrations (see Fig. 6): We predict that the
oscillations will disappear when the KaiA concentration is increased
but not when the KaiB concentration is increased.
We thank Daan Frenkel and Martin Howard for a critical reading of the
manuscript and the Aspen Center for Physics for its hospitality to D.K.L.
early in this project. The work was supported by Fundamenteel Onderzoek der Materie/Nederlandse Organisatie voor Wetenschappelijk
Onderzoek.
15. Hayashi F, Ito H, Fujita M, Iwase R, Uzumaki T, Ishiura M (2004) Biochem
Biophys Res Commun 316:195–202.
16. Williams SB, Vakonakis I, Golden SS, LiWang AC (2002) Proc Natl Acad Sci USA
99:15357–15362.
17. Iwasaki H, Taniguchi Y, Ishiura M, Kondo T (1999) EMBO J 18:1137–1145.
18. Goldbeter A (1996) Biochemical Oscillations and Cellular Rhythms: The Molecular
Bases of Periodic and Chaotic Behaviour (Cambridge Univ Press, New York).
19. Emberly E, Wingreen NS (2006) Phys Rev Lett 96:038303.
20. Mehra A, Hong CI, Shi M, Loros JJ, Dunlap JC, Ruoff P (2006) PLoS Comput
Biol 2:e96.
21. Kurosawa G, Aihara K, Iwasa Y (2006) Biophys J 91:2015–2023.
22. Wang J (2004) J Struct Biol 148:259–267.
23. Bell CE (2005) Mol Microbiol 58:358–366.
24. Nishiwaki T, Satomi Y, Nakajima M, Lee C, Kiyohara R, Kageyama H, Kitayama
Y, Temamoto M, Yamaguchi A, Hijikata A, et al. (2004) Proc Natl Acad Sci USA
101:13927–13932.
25. Xu Y, Mori T, Pattanayek R, Pattanayek S, Egli M, Johnson CH (2004) Proc Natl
Acad Sci USA 101:13933–13938.
26. Monod J, Wyman J, Changeux J-P (1965) J Mol Biol 12:88–118.
27. Gillespie DT (1977) J Phys Chem 81:2340–2361.
28. Pattanaeyk R, Williams DR, Pattanayek S, Xu Y, Mori T, Johnson CH, Stewart
PL, Egli M (2006) EMBO J 25:2017–2028.
29. Ruoff P, Rensing L, Kommedal R, Mohsenzadeh S (1997) Chronobiol Int
14:499–510.
30. Shen-Orr S, Milo R, Mangan S, Alon U (2002) Nat Genet 31:64–68.
PNAS 兩 May 1, 2007 兩 vol. 104 兩 no. 18 兩 7425
BIOPHYSICS
a minimum KaiB concentration of about [B]T ⫽ [C]T is needed to
sustain oscillations. Above that threshold, neither the period nor the
amplitude depend strongly on [B]T.
The different effects of varying [A]T and [B]T can be understood
from the different roles the two dimers play in our model. KaiA
stimulates the phosphorylation of KaiC. If the total KaiA concentration is very low, the phosphorylation rate will thus be so slow that
it is counterbalanced by the spontaneous dephosphorylation rate.
If, on the other hand, the total KaiA concentration is very high, the
mechanism of differential affinity no longer functions, because it
relies on competition for a limited amount of KaiA. The function
of KaiB is to stabilize inactive KaiC and to sequester KaiA. As long
as enough KaiB is available to perform these functions, the period
and amplitude will not depend on the KaiB concentration.
Interestingly, the very recent experiments described in ref. 7 give
strong support for our model. In particular, these experiments show
that when the KaiA and KaiB concentrations are reduced from
their standard values by a factor of 4 and 3, respectively, all
oscillations cease in very good agreement with our results. We
further predict that there is an upper bound on the KaiA concentration, but not on the KaiB concentration, for oscillations to exist.
Moreover, although the amplitude and the period of the oscillations
do not depend in our model on [B]T, they do depend in a very
specific manner on [A]T. These dependencies on the KaiA and KaiB
concentrations are direct consequences of the basic roles of these
proteins in our model. They thus represent some of our most robust
and important predictions.
Supporting Information Appendix
An allosteric model of KaiC phosphorylation
Jeroen S. van Zon, David K. Lubensky, Pim R. H. Altena, and Pieter Rein ten
Wolde
In this appendix, we provide background information on our model of the in vitro
Kai system and the calculations that we have performed. We will closely follow
the outline of the main text: section I corresponds to the section Allosteric Model
of the main text; section II corresponds to Synchronization with Differential
Affinity; section III corresponds to Full Model of the Kai system.
1
Allosteric Model
In this section, we discuss in more detail our model of conformational transitaions of individual KaiC hexamers presented in Allosteric Model. We first
present a statistical-mechanical description of the allosteric model. This allows us to describe the thermodynamics of the phosphorylation cycle. We then
present a model based on concepts of transition state theory that allows us to
describe the dynamics of the phosphorylation cycle, in particular the dynamics
of the conformational transitions. Lastly, we briefly discuss how we have performed the simulations on the model in Fig.1B, the results of which are shown
in Fig.1C.
1.1
Thermodynamics: A statistical-mechanical model
The allosteric model relies on the following key assumptions:
1. Each of the N = 6 monomers of a hexamer is either in an active or in an
inactive conformational state.
2. The conformations of the monomers are strongly coupled, such that all
monomers of a hexamer are in the same conformational state at all times.
3. Both in the active and inactive state, monomers can be (de)phosphorylated
and (un)bind nucleotides. We assume that nucleotides have a higher affinity for monomers in the active state than for those in the inactive state.
Consequently, nucleotide binding enhances the stability of the active state
with respect to the inactive one. In contrast, we assume that phosphorylation favors the inactive state.
4. Nucleotide exchange is faster than phosphorylation and therefore in thermodynamic equilibrium on the time scale of phosphorylation.
The model makes the following further assumptions that are of secondary importance:
1. Each monomer has two phosphorylation states, phosphorylated and unphosphorylated. Each hexamer thus has N = 6 phosphorylation sites.
1
2. Phosphorylation of the different monomers of a hexamer occurs sequentially around the hexamer.
3. The unphosphorylated monomers can bind ATP, while the phosphorylated
monomers can bind ADP.
This leads to the following partition function for a hexamer that it is in
a conformational state α, at phosphorylation level p, with q ATP and r ADP
molecules bound:
q
α
N −p α,T
Z α (p, q, r) = e−βN Em
[ATP]/KD
×
q
p p r
α,P
α,D
[Pi]/KD
[ADP]/KD
.
(11)
r
α
is the energy of an unphosphorylated
Here, β is the inverse temperature, Em
α,s
is the dissociation
monomer in state α (with no nucleotide bound), and KD
constant for the binding of species s to the hexamer in state α; T denotes ATP,
D ADP, and P a phosphate group Pi. Since nucleotide exchange is assumed to
be fast, it is meaningful to integrate over the number of nucleotides. This yields
the following partition function for a hexamer in state α with phosphorylation
level p:
Z α (p)
N −p
α
α,T
= e−βN Em 1 + [ATP]/KD
×
h
ip
α,P
α,D
.
[Pi]/KD
1 + [ADP]/KD
(12)
The (excess) chemical potential of a hexamer in conformational state α at
phosphorylation level p is given by
µα (p) = −kB T ln[Z α (p)].
(13)
We stress that the chemical potential of a KaiC hexamer depends upon the
chemical potentials (the concentrations) of the nucleotides: µα (p) = µα (p; µT , µD , µP ).
We will now consider the symmetric model of Fig.1A. In this model, the
energy levels of the active and inactive conformational state are mirror images
of each other. This is for reasons of clarity, and not because it is essential. In
fact, the model of Full Model of the Kai System is asymmetric, with the active
state being more stable than the inactive one.
Fig. 7 shows the chemical potentials of a KaiC hexamer in the active and
inactive state, respectively, as a function of the phosphorylation level, for the
energy diagram shown in Fig.1A. Here, µα (p = 6) − µα (p = 0) corresponds
to the free-energy change upon fully phosphorylating a KaiC hexamer in conformational state α at constant chemical potentials of the ATP, ADP and Pi
molecules, but without an ATP hydrolysis reaction (we thus consider the reaction
KaiC + Pi → KaiCPi) . It is seen that the free energy increases markedly for
both conformational states, meaning that the probability that a hexamer would
2
30
Active
Inactive
20
α
µ (p)
10
0
-10
-20
0
1
2
3
p
4
5
6
Figure 7: The chemical potential of a KaiC hexamer as a function of the phosphorylation level p, for both the active and inactive conformational state, and
for the symmetric model of Fig.1A. The chemical potential µα (p) is given by
Eq. 13.
fully phosphorylate spontaneously, is essentially zero. Indeed, the essence of our
allosteric model is that in the active conformational state, the energy from ATP
hydrolysis is used to phosphorylate the KaiC hexamer, while in the inactive
state dephosphorylation occurs spontaneously.
When ATP is hydrolyzed to p-fold phosphorylate a single KaiC hexamer in
the active state, the total change in free energy of the system is
∆GA (p) = µA (p) − µA (0) + p(µD + µP − µT ),
(14)
where µA (p) is given by Eq. 13. If in the active state binding of both ADP
and ATP is strong (i.e. [ADP]/KA,ADP
, [ATP]/KA,ATP
1), then the above
D
D
expression reduces to
∆GA (p)
= p(−∆GA,T + ∆GA,P + ∆GA,D + ∆Ghydro )
= p∆GA
m;AATP→Ap ADP .
(15)
α,s
Here, ∆Gα,s = +kB T ln KD
is the binding free energy of species s and ∆Ghydro
is the standard reaction free energy of an ATP hydrolysis reaction. The overall
free energy change ∆GA (p) corresponds to that of p phosphotransfer reactions
AATP → Ap ADP on the active KaiC hexamer (A denotes a subunit in the
active state). The free-energy change be understood by noting that in the limit
of strong nucleotide binding considered here, the unphosphorylated monomers
are essentially always occupied by ATP, while the phosphorylated monomers
are essentially always occupied by ADP; see also Fig.1A.
Fig. 8 shows the free energy of the system in the presence of ATP hydrolysis. In the active conformational state, KaiC binds ATP. ATP hydrolysis then
3
0
Active
Inactive
phosphorylation with ATP hydrolysis
-20
∆G(p)
∆G of 6 ATP -> 6 ADP + 6 Pi
ADP release
-40
spontaneous dephosphorylation
ATP binding
-60
0
1
2
3
p
4
5
6
Figure 8: The free energy of the system as a function of the phosphorylation level
in the presence of ATP hydrolysis for the symmetric model of Fig.1A. The solid
lines denote the path of the system. Driven by ATP hydrolysis, a KaiC hexamer
is phosphorylated in the active state. When the KaiC hexamer is (nearly) fully
phosphorylated, it flips from the active to the inactive conformational state. In
the inactive state, ADP is released and the hexamer dephosphorylates spontaneously. At low phosphorylation levels, the hexamer flips back to the active
state. The active hexamer rebinds ATP and the phosphorylation cycle starts
over again. The dotted lines correspond to the energetically unfavorable path
of driven phosphorylation of inactive KaiC and spontaneous dephosphorylation
of active KaiC.
drives the phosphorylation of the hexamer, and the reduction in free energy
of the whole system is given by Eq. 15. When the hexamer is (nearly) fully
phosphorylated, it flips to the inactive conformational state. In the inactive
state, nucleotide binding is weak and, as a result, ADP is released. The hexamer now dephosphorylates spontaneously;
binding is weak,
h since nucleotide
i
I,P
the free-energy change is given by 6kB T ln [Pi]/KD
(see Eqs. 12 and 13). At
low phosphorylation levels, the inactive hexamer flips back towards the active
conformational state. KaiC rebinds ATP and the phosphorylation cycle starts
over again. After one full phosphorylation cycle, the free energy of the system
has been reduced by the free-energy change corresponding to 6 ATP hydrolysis
reactions: ∆G = 6 (∆Ghydro + kB T ln ([ADP][Pi]/[ATP])).
1.2
Dynamics: a transition-state theory of the conformational transitions
So far, we have discussed the thermodynamics of the phosphorylation cycle. We
will now discuss the dynamics of the cycle, in particular the dynamics of the
conformational transitions. This is important, because while the large ampli-
4
tude oscillations as observed experimentally require that the hexamers should
not flip at intermediate phosphorylation levels, the stability of one conformational state with respect to that of the other, does change in sign at intermediate
phosphorylation levels: in the symmetric model considered here and in Fig.1A,
the active state is more stable for p < 3, while the inactive state is more stable
for p > 3. How can we explain that the conformational transitions predominantly occur when the hexamers are either nearly fully phosphorylated or fully
unphosphorylated?
This is a difficult question to answer, because it requires knowledge of the
microscopic dynamics of the transition paths between the conformational states.
However, if we assume that nucleotide binding is an important component of
the reaction coordinate that describes the conformational transitions, then we
can make an estimate of the flipping rates using a mesoscopic model based on
concepts from transition-state theory [1].
If nucleotide binding contributes to the reaction coordinate of the conformational transitions, then we cannot integrate it out as we have done so far.
To derive the flipping rates, we start by considering the free-energy difference
between two conformational states with the same number of nucleotides bound:
∆G(p, q, r) = N ∆Em + p∆Ep + q∆ET + r∆ED .
(16)
A,s
I,s
I
A
and ∆Es = kB T ln KD
/KD
. If we assume, for simplic−Em
Here, ∆Em = Em
ity, that the model is symmetric, ∆Em = 0, and that the difference in binding
energy between the active and inactive state is the same for ATP and ADP,
∆ET = ∆ED = ∆ET/D , then the above expression reduces to:
∆G(p, n) = p∆Ep + n∆ET/D ,
(17)
where n is the number of nucleotides that are bound. We iterate that phosphorylation favors the inactive state, and hence ∆Ep > 0, while nucleotide binding
favors the active state, ∆ET/D < 0. We can use the above expression to estimate the flipping rate if we assume that nucleotide binding is the dominant
reaction coordinate for the flipping process.
This is illustrated in Figs. 9 and 10, for the case |∆Ep | = −|∆En |. Fig. 9
shows a sketch of the free-energy surface ∆G(p = 3, n, c) of a three-fold phosphorylated hexamer as a function of the number of bound nucleotides, n, and as
a function of an order parameter that describes the conformational state of the
hexamer, c; the parameter c is zero if the hexamer is in the inactive state and
one if it is in the active state. Clearly, we do not know what would be the best
order parameter to describe the conformational transition, let alone what the
free energy would be as a function of this order parameter for different values
of n. Nevertheless, we do have some knowledge of the free-energy surface: we
know how the free energy ∆G(p, n, c) changes as a function of n for c = 0 and
for c = 1 – this is given by the free energy of nucleotide binding to the inactive and active state, respectively; this free energy is related to the log of the
partition function in Eq. 11. We therefore make the minimal assumption that
the free-energy surface ∆G(p, n, c) is given by a linear interpolation between the
5
6
E
Active state
1
4
0.8
2
0.6
c
0.4
0
0
Inactive state
2
0.2
4
n
60
Figure 9: The free energy of a KaiC hexamer for a phosphorylation level of
p = 3, as a function of the number of bound nucleotides n and as a function
of an order parameter c that denotes the conformational state of the KaiC
hexamer: it is zero if the hexamer is in the inactive state and one if it is in
the active state. In the inactive state, essentially no nucleotides are bound, and
n ≈ 0, while in the actives state, because of strong nucleotide binding, n ≈ 6. It
is seen that in order to flip from the inactive to active state, the system has to
cross a free-energy barrier; the transition state is denoted by the red cross. We
imagine that the height of the barrier to go from the inactive to active state is
given by the free-energy to add three nucleotides, while the barrier to flip from
active to inactive is given by the free energy to remove 3 nucleotides.
two functions ∆G(p, n, 0) and ∆G(p, n, 1). This leads to the surface shown in
Fig. 9.
In the active state, nucleotides bind the hexamer very strongly and, consequently, n ≈ 6, while in the inactive state nucleotides bind the hexamer rather
weakly and n ≈ 0. The two (meta)stable states of the hexamer are thus the active state with six nucleotides bound and the inactive state with no nucleotides
bound. These two states are separated by a “transition-state” surface: in order to go from one (meta) stable state to the other, the system has to cross a
free-energy barrier. We assume that the transition state is given by the saddlepoint in the free-energy surface ∆G(p, n, c), as shown in Fig. 9. This means
that both the location and the height of the free-energy barrier for flipping are
determined by that number n∗ for which the two states become equally stable,
∆G(p, n∗ ) = 0 (see Fig. 9 and Eq. 17). Clearly, the location of the transition
state depends upon the phosphorylation level p of the hexamer: in the symmetric model considered here, the two conformational states are equally stable if
the number of bound nucleotides is n∗ = p (see Fig. 10 and Eq. 17). The height
of flipping from the active to inactive state is thus given by
"
#
X
∗
A
A
β∆GA→I (p) = − ln
Z (p, q, r)δ(q + r − p)/Z (p, 0, 6) ,
(18)
q,r
6
20
Inactive
p=0
Active
10
∆G(p,n)
p=3
0
Inactive
Active
Active
p=6
-10
-20
Inactive
0
1
2
3
n
4
5
6
Figure 10: The free energy of the active and inactive state as a function of
the number of bound nucleotides, n, for three different phosphorylation levels,
p = 0, 3, 6. In the active state, n ≈ 6, while in the inactive state, n ≈ 0. The free
energy of p-fold phosphorylatedP
hexamer in state α with n nucleotides bound is
given by ∆Gα (p, n) = −kB T ln q,r Z α (p, q, r)δ(q + r − n), where Z α (p, q, r) is
given by Eq. 11.
while the barrier height for the reverse transition is given by
"
#
X
∗
I
I
β∆GI→A (p) = − ln
Z (p, q, r)δ(q + r − p)/Z (p, 0, 0) .
(19)
q,r
Here, Z α (p, q, r) is given by Eq. 11. In words, if a p-fold phosphorylated hexamer
is in the active state, with 6 nucleotides bound, then in order to flip to the
inactive state with no nucleotides bound, it has to cross a barrier with a height
that corresponds to the energetic cost of removing 6−p nucleotides. Conversely,
the height of the barrier for an inactive hexamer, with no nucleotides bound,
to flip to the active state, is given by the free energy to add p nucleotides.
Neglecting entropic factors, the height of the free-energy barrier thus scales
linearly with the phosphorylation level, leading to the exponential flipping rates
of Eqs.1 and 2.
1.3
Numerical calculations on the allosteric model
The chemical reactions of the model in Fig.1B are:
fi
ps
dps
e i , Ci k→
e i k̃→
e i−1
Ci C
Ci+1 , C
C
(20)
bi
Here, Ci corresponds to an active KaiC hexamer with phosphorylation level
e i corresponds to an inactive KaiC hexamer with phosphorylation level
i, while C
7
i. The first, reversible reaction corresponds to the conformational transitions of
the KaiC hexamers with forward and backward flipping rates fi and bi , respectively, the second corresponds to phosphorylation of active KaiC at rate kps ,
while the third reaction corresponds to dephosphorylation of inactive KaiC at
rate k̃dps .
To study the phosphorylation behavior of a single KaiC hexamer, we cannot
use macroscopic rate equations based on the law of mass action: these equations
would correspond to the average of a population of KaiC hexamers. To simulate
the behavior of a hexamer, we have performed kinetic Monte Carlo simulations of
the zero-dimensional chemical master equation corresponding to the reactions in
Eq. 20 [2]. The solid line in Fig.1C corresponds to the results of those stochastic
simulations.
To study the time evolution of the average phosphorylation level of an ensemble of KaiC hexamers, we have used macroscopic rate equations based on
the law of mass action. The chemical rate equations that correspond to Eq. 20
are:
d[Ci ]
dt
ei ]
d[C
dt
ei ] − fi [Ci ]
= kps [Ci−1 ] − kps [Ci ] + bi [C
(21)
ei−1 ] − k̃dps [C
ei ] + fi [Ci ] − bi [C
ei ]
= k̃dps [C
(22)
The dashed line in Fig.1C corresponds to the numerical results of propagating
these ordinary differential equations.
The results in Fig.1C were obtained with the following values for the parameters: kps = 0.01hr−1 , k̃dps = 0.05hr−1 , fi = 0.1N −i hr−1 and bi = 0.1i hr−1 .
2
Simple Models with Differential Affinity
In this section, we first provide background information on the simplified model
of the Kai system discussed in section Synchronization with Differential Affinity
of the main text. We then briefly discuss a more generic class of differential
affinity models.
2.1
A minimal differential affinity model of the Kai system
In Synchronization with Differential Affinity, we assume that only a single KaiA
dimer can bind to an active KaiC hexamer. The chemical reactions of this model
are given in Eqs.3-5. They correspond to the following mass-action kinetic
e i ] of KaiC in the active and inactive
equations for the concentrations [Ci ] and [C
8
states, [ACi ] of the KaiA-KaiC complex, and [A] of free KaiA:
d[Ci ]
dt
= kpf [ACi−1 ] − k Af [A][Ci ] + kiAb [ACi ]
e 0 ] − δi,6 f6 [C6 ]
+δi,0 b0 [C
d[ACi ]
dt
e i]
d[C
dt
d[A]
dt
(23)
= −kpf [ACi ] + k Af [A][Ci ] − kiAb [ACi ] (i 6= 6)
(24)
e i−1 ] − [C
e i ]) − δi,0 b0 [C
e 0 ] + δi,6 f6 [C6 ]
= k̃dps ([C
(25)
= −[A]
5
X
Af
k [Ci ] +
i=0
5
X
(kiAb + kpf )[ACi ] ,
(26)
i=0
where δi,j is the Kronecker delta, k̃dps is the spontaneous dephosphorylation
rate, kpf is the rate of phosphorylation catalyzed by KaiA, and fi and bi are
the flipping rates as defined in Eqs.1 and 2. The rates of KaiA binding to and
unbinding from active KaiC are respectively k Af and kiAb , with the latter dependent on the number of i of phosphorylated monomers in the KaiC hexamer.
Because we generally choose parameters such that (un)binding of KaiA to KaiC
is much faster than (de)phosphorylation, it is an excellent approximation to
assume that these binding reactions are equilibrated. In this case, we do not
have to keep track of [Ci ] and [ACi ] separately; instead, we obtain dynamical
equations for the total concentration [Ci ]T = [Ci ] + [ACi ] of KaiC in the active
e i ]:
state and for [C
d[Ci ]T
dt
=
kpf [A]
kpf [A]
[Ci−1 ]T −
[Ci ]T
Ki−1 +[A]
Ki +[A]
e i ] − δi,6 fi [Ci ]T
+δi,0 bi [C
(27)
e
d[Ci ]
e i−1 ] − [C
e i ]) − δi,0 bi [C
e i ] + δi,6 fi [Ci ]T
= k̃dps ([C
(28)
dt
along with a constraint equation giving the free KaiA concentration [A] implicitly:
[A] +
5
X
[A][Ci ]T
K
i + [A]
i=0
=
[A]T .
(29)
Here [A]T is the total concentration of KaiA and Ki = kiAb /k Af is the dissociation constant for KaiA binding to i-fold phosphorylated KaiC in the active
state. The i dependence Ki = K0 αi with α > 1 gives differential affinity. Here,
and in the next section, the differential equations were solved using Matlab.
We note that the assumption that KaiA binding is fast is convenient for some
purposes but not essential; we have verified that the model’s predictions are the
same whether we use Eqs. 23–26 or the reduced set Eqs. 27–29.
The results in Synchronization with Differential Affinity were obtained using
the parameter values in Table 1. Note that the factors k0Ab and α in the definition
kiAb = k0Ab αi are equal to 10hr−1 and 10, respectively.
9
Table 1: List of parameter values for the simple model discussed in Synchronization
with Differential Affinity.
kpf
k̃dps , f6 , b0
k Af
kiAb
[A]T
[C]T
2.2
13.6 hr−1
0.908 hr−1
1
3.45·10 3 M−1 hr−1
10i+1 hr−1
0.012 µM
0.58 µM
Parameter dependence and bifurcation behavior
It is natural to ask how the model’s behavior changes as the parameters are
varied from the specific values listed in Table 1. This task is facilitated by
its simple cyclic structure, which allows one to prove that the system of equations 23–26 always admits exactly one fixed point. As the total concentration
[A]T of KaiA is increased from zero, this fixed point becomes unstable through a
supercritical Hopf bifurcation. The resulting stable limit cycle persists through
a fairly broad range of [A]T values before finally disappearing at a second supercritical Hopf bifurcation where the unique fixed point regains its stability.
This behavior can be understood as follows: If [A]T is too large, then the concentration of KaiA is no longer limiting, and differential affinity cannot act to
synchronize oscillations. On the other hand, if [A]T becomes too small (while
the other parameters are held fixed), then the phosphorylation reactions on the
active branch will proceed too slowly compared to the dephosphorylation reactions on the inactive branch. The first fully phosphorylated hexamers will then
be dephosphorylated and return to state C0 before the lagging KaiC complexes
reach state C6 . At this point, the front runners—the hexamers in state C0 —will
win the competition for the limited amount of KaiA over the laggards that still
haven’t finished the active branch. Whereas normally with differential affinity
the slowest hexamers speed up and the fastest are forced to slow down, now the
opposite scenario arises: the front runners will progress faster around the cycle
than the laggards. This will desynchronize the cycles of the different hexamers.
In KaiC Dynamics as a Function of the KaiA and KaiB Concentration we
discuss a different scenario for the breakdown of oscillations at small [A]T that
depends on the possibility of spontaneous phosphorylation and dephosphorylation reactions on the active branch. Here, those reactions are absent, and
as a result, oscillations persist down to lower [A]T before eventually vanishing
because some KaiC hexamers begin to return to state C0 too quickly.
Varying other parameters has similar effects. For example, as the parameter
K0 in Ki = K0 αi is increased, the limit cycle will eventually collapse into the
fixed point in a Hopf bifurcation. Indeed, if the Ki become too large, then
KaiC hexamers cannot efficiently bind all of the available KaiA, and differential
affinity is no longer possible. Fig. 11 shows a 2-parameter bifurcation diagram as
10
−4
x 10
8
7
Stable Fixed Point
6
K0/[A]T
5
4
3
Stable Limit Cycle
2
1
0
0
0.01
0.02
0.03
0.04
0.05
[A] /[C]
T
T
Figure 11: Two-parameter bifurcation diagram as a function of the ratios
[A]T /[C]T and K0 /[A]T for the model of Eqs. 23–26. The solid line gives the
locus of a supercritical Hopf bifurcation; to one side of the line, the systems’s
unique fixed point is stable, and as the line is crossed, that fixed point loses its
stability to a newly-born limit cycle. Note that, even as K0 → 0, oscillations
do not persist for [A]T /[C]T too large or too small.
a function of the dimensionless ratios [A]T /[C]T and K0 /[A]T that demonstrates
these effects. We will see that similar bifurcation behavior reappears in more
realistic models of the Kai system.
2.3
Generic differential affinity model
The simple model discussed above and in Synchronization with Differential
Affinity can be seen as one example of a more generic class of models that
use the same mechanism to synchronize oscillations. Here, we briefly discuss
this broader perspective on differential affinity. A fuller mathematical analysis
will appear in a forthcoming publication.
We begin by considering the following cycle:
C0 → C1 → . . . → CN −1 → CN → C0 ,
(30)
where Ci is a protein that has been i-fold covalently modified. At least two of
the reactions require a catalyst A acting with Michaelis-Menten kinetics, and
those steps that are not catalyzed by A are simple first-order reactions.
Suppose that the reactions C0 → C1 → . . . → Cj require the catalyst A
(with j < N ). Then, one might imagine that this system will oscillate if a) the
concentration of A is sufficiently low, and b) the dissociation constants Ki for the
binding of A to Ci satisfy K0 < K1 < . . . < Kj . These two conditions together
ensure that A first binds to C0 and catalyzes the reaction C0 → C1 ; only when
11
1
[C0]
0.9
[C1]
[A]
0.8
[Ci]/[C]T; [A]/[A]T
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Time (a.u.)
Figure 12: Limit-cycle oscillations in the generic model of Eqs. 31–35. As in
Fig.2, the concentration [A] of free A is almost zero except during a brief period
when the concentrations [C0 ] and [C1 ] of the forms of C that can bind A are
small. Initially, [C0 ] = [C]T and [A] = [A]T . Parameters (in arbitrary units):
[C]T = 1, [A]T = 0.02, K0 = 0.001, K1 = 1 k0 = 10, k1 = 100, and kps = 1.
the concentration of C0 has dropped almost to zero does A begin to bind to C1
and catalyze the next reaction, and so on until state Cj is reached. It turns out,
however, that these requirements alone are not sufficient. In addition, we must
at a minimum demand that c) the distribution of arrival times of different C
molecules at Cj is not too wide compared to the average time to travel from Cj
back to C0 and d) that the distribution of arrival times back at C0 is not too
broad. If the former condition, c), does not hold, then the fastest C molecules
will arrive at C0 before some of the slower C molecules reached Cj . As we noted
in the previous subsection, oscillations cannot survive such a situation: Because
A binds most strongly to C0 , the fastest C molecules will siphon A away from
the slow molecules that still need A to progress around the cycle; these will then
slow down further until all synchronizing effect of differential affinity has been
lost. Similarly, differential affinity fails when the arrival of C molecules at state
C0 is too spread out (condition d). Then, there are relatively few C’s competing
for A molecules at any given instant, and even those with the highest number
i of modifications can continue upwards towards Cj . Thus, for oscillations to
occur, N − j must be neither too small nor too large.
We can find limit cycles in this model with j as small as 1 (corresponding
to A binding to the two states C0 and C1 ) and N as small as 4. Fig. 12 shows
12
oscillations in such an N = 4, j = 1 model governed by the dynamical equations
d[C0 ]
dt
d[C1 ]
dt
d[C2 ]
dt
d[Ci ]
dt
[A]T
[A][C0 ]
K0 + [A]
[A][C0 ]
[A][C1 ]
= k0
− k1
K0 + [A]
K1 + [A]
[A][C1 ]
= k1
− kps [C2 ]
K1 + [A]
= kps [C4 ] − k0
= kps ([Ci−1 ] − [Ci ]) (i = 3, 4)
=
[A] +
1
X
[A][Ci ]
,
Ki + [A]
i=0
(31)
(32)
(33)
(34)
(35)
where we have immediately made the assumption that A binding and unbinding
to C is fast and slaved A to the concentrations {[Ci ]}. Although the oscillations
persist only up to N = 7 if the other parameter values are kept constant, there
appears to be no maximum allowed value of N if the rate kps of the transitions
Cj → Cj+1 → . . . → CN → C0 is allowed to decrease as 1/(N − j). Indeed, in
this case, the mean time to travel from Cj to C0 remains constant, while the
distribution of travel times becomes narrower, which only enhances oscillations.
One could imagine many variations on the model just described. For example, we anticipate that for appropriate parameter values, oscillations will also
occur when A binds to and is sequestered by C0 but is not required for the
transition C0 → C1 , while still catalyzing the reaction C1 → C2 . One might
also consider cases in which differential affinity acts on two separated blocks of
reactions C0 → C1 → . . . → Cj and Cn → Cn+1 → . . . → Cn+k . In each case,
the same basic principles of differential affinity are likely to be at work.
The models of the Kai system in the main text are clearly related to this
generic class of models. The phosphorylation cycles of both the minimal differential affinity model in Synchronization with Differential Affinity and of the
complete Kai model in Full Model of the Kai System have an active branch
where KaiC is phosphorylated, and an inactive branch where KaiC is dephosphorylated. Also in both models, KaiA catalyzes the phosphorylation reactions
on the active branch. Both ingredients are inspired by experimental observations; together they give a concrete example of how the abstract cycle of Eq. 30
might be implemented. However, the discussion of the generic model above
shows that from the perspective of synchronizing the oscillations, there is no
need to make a distinction between an active and an inactive branch. Indeed,
the same formalism could be applied to cycles made of more than two allosteric
conformations, or of multiple different sorts of covalent modifications, or created
in any number of other ways. The differential affinity mechanism thus has the
potential to be generalized far beyond the Kai system.
13
3
Full Model of the Kai System
3.1
Model: the chemical rate equations
The chemical reactions that describe the full model of the Kai system are given
by Eqs.6-10. Using the law of mass action, this leads to the following set of
e i ],
macroscopic chemical rate equations for the concentration of [Ci ], [ACi ], [C
e
e
[Bm Ci ], and [Am Bm Ci ]:
d[Ci ]
dt
= kps[Ci−1 ]+kdps[Ci+1 ]−(kps +kdps )[Ci ] + kpf [ACi−1 ]
e i ] − k Af[A][Ci ] + k Ab[ACi ]
−fi [Ci ]+bi [C
i
i
d[ACi ]
dt
e i]
d[C
dt
= kiAf[A][Ci ] − kiAb[ACi ]−kpf [ACi−1 ]
(38)
e i−1 ]+ k̃dps [Bm C
e i+1 ]−(k̃ps + k̃dps )[Bm C
e i]
= k̃ps [Bm C
e i ] − k̃iBb [Bm C
e i]
+k̃iBf [B]m [C
Af
m
Ab
e i ] + k̃i [Am Bm C
e i]
−k̃i [A] [Bm C
e i]
d[Am Bm C
dt
(37)
e i−1 ]+ k̃dps [C
e i+1 ]−(k̃ps + k̃dps )[C
e i ]+fi [Ci ]−bi [C
e i]
= k̃ps [C
e i ] + k̃iBb [Bm C
e i]
−k̃iBf [B]m [C
e i]
d[Bm C
dt
(36)
(39)
e i−1 ]+ k̃dps [Am Bm C
e i+1 ]−(k̃ps + k̃dps )[Am Bm C
e i]
= k̃ps [Am Bm C
e i ] − k̃ Ab [Am Bm C
e i]
+k̃iAf [A]m [Bm C
i
(40)
Here, the concentrations of free KaiA and KaiB, [A] and [B], are given by:
[A] =
[A]T −
6
X
e i ])
([ACi ] + m[Am Bm C
(41)
e i ] + m[Am Bm C
e i ])
(m[Bm C
(42)
i=0
[B]
=
[B]T −
6
X
i=0
The phosphorylation and dephosphorylation rates on the active branch are kps
and kdps , respectively, and the flipping rates are fi and bi . The active state
can bind KaiA with forward and backward rates kiAf and kiAb , and KaiA can
catalyze phosphorylation with the rate kpf . We assume that in the inactive
state KaiC can bind m = 2 KaiB molecules with forward and backward rates
k̃iBf and k̃iBb , respectively. This KaiB-KaiC complex can then sequester m = 2
KaiA molecules with forward and backward rates k̃iAf and k̃iAb .
14
3.2
Model: The free-energy difference between the active
and inactive state of KaiC
In the presence of only KaiA, KaiC is phosphorylated to a very high level of
90 − 95%. This requires that the active state of KaiC be more stable than the
inactive one. However, in order for KaiC to sustain oscillations in the presence
of both KaiA and KaiB, KaiC should be able to flip from the active state to
the inactive one at higher phosphorylation levels. This means that the freeenergy difference ∆G(p) between KaiC in its active and inactive conformational
states should be strongly negative, but also that it should rapidly decrease
in magnitude as KaiC becomes fully phosphorylated. A simple model that
captures this is one in which the addition of each phosphate group decreases
the free-energy difference by an amount ∆Ep , and moreover where the creation
of an interface between a phosphorylated and an unphosphorylated unit also
destabilizes the inactive state by an additional amount : upon the addition of
the last phosphate group, two such interfaces are removed, leading to an extra
change in the free-energy difference of 2 in favor of the inactive state. This leads
to the following expression for the free-energy difference between the active and
inactive state:
X
∆G(p, q, r) = N ∆Em + p∆Ep − ni nj + q∆ET + r∆ED .
(43)
hi,ji
Here, ni denotes the phosphorylation state of unit i – ni = 1 if unit i contains a phosphate group and zero otherwise – and the sum hi, ji includes all
nearest neighbor units of KaiC. For the full model, ∆Em , ∆ET , ∆ED < 0, and
∆Ep , > 0. For simplicity, we assume that ∆Ep = 2. This yields the following expressions for the transitions between the active and inactive states: for
the forward rate we have fi = δγ N −i for 1 ≤ i < N and fi = 10 · δγ N −i for
i = 0, N (this is due to, respectively, the creation or removal of the interfaces
between phosphorylated and unphosphorylated units). The backward rate bi is
independent of i and much larger than the forward rate fi , so that the stability
of the inactive state is only due to binding of KaiB.
3.3
Setting the parameters
Using the expressions for fi and bi just derived, the full model contains 39
parameters. However, their values are very much constrained by the large body
of experimental data on this system. We now describe how we have determined
the parameters, and how critical their precise values are for the behavior of the
model. Unless indicated otherwise, the exact values of the parameters for the
full model are summarized in Table 2.
Concentrations The concentrations of KaiA and KaiB dimers are [A]T = 0.58µM
and [B]T = 1.75µM, respectively, and the concentration of KaiC hexamers is
[C]T = 0.58µM. This corresponds to a concentration ratio of (KaiA dimers):(kaiB
dimers):(KaiC hexamers) = 1:3:1. The corresponding monomer concentrations
15
Table 2: List of parameter values for the model discussed in Full Model of the Kai
System.
kps , k̃ps
kdps , k̃dps
kpf
fi
bi
kiAf
kiAb
k̃iBf
k̃iBb
k̃ Af
k̃ Ab
[A]T
[B]T
[C]T
0.025 hr−1
0.4 hr−1
1.0 hr−1
−5
−5
−4
−3
−2
−1
{10 , 10 , 10 , 10 , 10 , 10 , 10} hr−1
100 hr−1
8
1.72·10 M−1 hr−1
{10, 30, 90, 270, 810, 2430, 7290} hr−1
2.97·1012 × {0.01, 1, 1, 1, 1, 1, 1} M−2 hr−1
1·102 × {10, 1, 1, 1, 1, 1, 1} hr−1
18
2.97·10 × {0, 1, 100, 100, 1, 0, 0} M−2 hr−1
100 hr−1
0.58 µM
1.75 µM
0.58 µM
are 1.17µM KaiA, 3.5µM KaiB and 3.5µM KaiC. Oscillations in phosphorylation have been observed for these concentrations in the in vitro experiments of
Tomita et al. [3]. However, it should be noted that for the results in this article
the ratios of KaiA and KaiB to KaiC, [A]T /[C]T and [B]T /[C]T respectively, are
more important than absolute concentrations. Hence, we often express concentrations in units relative to [C]T .
Flipping rates Following the discussion in the last paragraph of the previous
section on the model for the free-energy difference between the active and inactive state of the KaiC hexamers, we next discuss the values of the flipping rates.
As discussed in that paragraph, it is important that: 1) in the absence of KaiB,
the active state has a lower free energy than the inactive one; 2) the hexamers
should not flip at intermediate phosphorylation levels. In the previous section,
we presented a model for the free-energy difference between the active and inactive state. Here we give the values of the flipping rates that are consistent
with this free-energy difference, which is requirement 1), and with requirement
2). The forward rate is fi = δγ N −i for 1 ≤ i < N and fi = 10 · δγ N −i for
i = 0, N , with γ = 0.1 and δ = 2hr−1 . The backward rate bi is independent of
i and given by b = 100hr−1 . As long as the two requirements of this paragraph
are satisfied, however, the precise values of the flipping rates are not important
for the behavior of the model.
KaiC alone and KaiC + KaiB: Spontaneous (de)phosphorylation rates
The spontaneous phosphorylation and dephosphorylation rates are chosen such
that the phosphorylation behavior of KaiC in the absence of KaiA and KaiB and
that in the presence of only KaiB agrees well with experiment (see Fig. 4). This
yields the following rates for, respectively, the spontaneous phosphorylation and
16
spontaneous dephosphorylation reactions, both for active and inactive KaiC:
kps = k̃ps = 0.025hr−1 and kdps = k̃dps = 0.4hr−1 . The identical rates for
active and inactive KaiC ensure that the phosphorylation behavior of KaiC in
the absence of KaiA and KaiB is the same as that of KaiC in the presence of
KaiB (which stabilizes the inactive branch). The values of these rate constants
are not free and have to be carefully chosen, because their sum determines the
relaxation rate of the phosphorylation level of KaiC (in the absence of KaiA),
while their ratio determines the steady-state phosphorylation level of KaiC (in
the absence of KaiA).
KaiC + KaiA: rates of KaiA-catalyzed phosphorylation reactions The
phosphorylation rate of KaiC in the presence of KaiA is determined by two
factors: 1) the binding affinity of KaiA for KaiC; 2) the rate kpf for the KaiAcatalyzed phosphorylation reaction. As discussed in the main text, both the
mechanism of differential affinity and temperature compensation require that
the binding affinities of KaiA for KaiC be high. Given these high binding affinities, the rate at which KaiC is phosphorylated in the presence of KaiA is mostly
determined by kpf . This rate is thus chosen such that the phosphorylation rate
of KaiC in the presence of KaiA agrees with experiment. This gives kpf = 1hr−1 .
This rate constant cannot be freely chosen, because it directly affects the phosphorylation rate of KaiC in the presence of KaiA.
The mechanism of differential affinity requires that the affinity of KaiA for
KaiC be high, but decrease substantially as the phosphorylation level of KaiC
increases. Provided that these constraints are satisfied, the precise values are of
less importance. We have chosen the following expressions for the forward rate
kiAf and backward rate kiAb : kiAf = 1.72 · 108 M−1 hr−1 and kiAb = k0Ab αi , with
α = 3 and kiAb = 10hr−1 .
The binding of KaiA and KaiB to inactive KaiC The stoichiometries of
the complexes of KaiC bound to KaiB and KaiA are not known. We assume
e i binds m = 2 KaiB dimers, while the complex B2 C
e i can
that inactive KaiC, C
sequester m = 2 KaiA dimers. The value of m = 2 is not very critical as long as
enough KaiA can be sequestered by inactive KaiC to limit the amount of KaiA
available for KaiC phosphorylation. We find that for m > 2 the system also
oscillates, although the phase boundaries, as shown in Fig.6, do shift. Clearly,
more experiments are needed to resolve the compositions of these complexes.
Temperature compensation requires that the binding affinities of KaiB for
KaiC be high. But as long as this requirement is fulfilled, the precise values
are of less importance. We assume that KaiB binds inactive KaiC with forward
rate kiBf = 2.97·1012 × {0.01, 1, 1, 1, 1, 1, 1}M−2 hr−1 and backward rate kiBb =
e 0 is to ensure
1 · 102 × {10, 1, 1, 1, 1, 1, 1}hr−1 . The low affinity of KaiB for C
that unphosphorylated hexamers that are in the inactive state can rapidly flip
towards the active state.
As explained in the main text, it is important that the affinity of KaiA
for KaiB is low for KaiB associated with the highly phosphorylated inactive
KaiC hexamers, but increases strongly for KaiB that is bound to the lessphosphorylated inactive KaiC hexamers; the precise values of the dissocia-
17
tion constants are not critical. We have chosen the following forward rate k̃iAf
and backward rate k̃iAb : k̃iAf = 2.97 · 1018 × {0, 1, 100, 100, 1, 0, 0}M−2 hr−1 and
e 0 is to ensure that unphosk̃iAb = 100hr−1 . The low affinity of KaiA for B2 C
phorylated hexamers that are in the inactive state can rapidly flip towards the
active state.
Varying the parameters with temperature In order to calculate how the
oscillation period in our model varies with temperature, we would have to know
how the rate constants would vary with temperature. This would require knowledge of the activation barrier for each of the reactions, which we do not have.
We can, however, test how sensitive the oscillation period is to changes in the
rate constants. Here, one could choose various strategies: one could vary each
of the rate constants individually, or one could vary all the rate constants simultaneously, either in a correlated or an uncorrelated manner. We have performed
many such tests, and they all reveal that the model is very robust to changes in
temperature (given the experimentally observed insensitivity of the phosphorylation rates to changes in temperature (see main text)). Since the dissociation
constants and the flip rates enter the model in very different ways, we show in
the main text how the oscillations change when we vary these two groups of
parameters.
When we change the dissociation constants of KaiA and KaiB binding to
examine the effects of temperature changes, we simultaneously change the dise i , by a factor C, and the association constants for KaiA binding, Ki and K
√
sociation
and
dissociation
rates
for
KaiB
binding,
kiBf and kiBb , by 1/ C and
√
C, respectively. Because KaiB (un) binding is fast, only the ratios of their
rates—the dissociation constants—matter, and this is a choice that is consistent
with changing the dissociation constants by a factor C.
3.4
Reduced model
When KaiA binding and unbinding is sufficiently rapid, it is possible to further
simplify Eqs. 36-40. In this case, we only explicitly take into account the binding
of KaiB to the inactive KaiC hexamers and we assume that both KaiA association to the active KaiC hexamers and KaiA sequestration by KaiB bound to the
inactive hexamers is very rapid and can be treated as in chemical equilibrium.
This leads to the following reduced set of macroscopic chemical rate equations
for the total concentration [Ci ]T of KaiC in the active state, both with and
e i ], and
without bound KaiA, the concentration of KaiC in the inactive state, [C
e
the total concentration [Bm Ci ]T of KaiC in the inactive state with KaiB bound,
both with and without bound KaiA:
18
d[Ci ]T
dt
dps
ps
[Ci+1 ]T −(σips +σidps )[Ci ]T
[Ci−1 ]T +σi+1
= σi−1
e i]
−σiFf [Ci ]T +σiFb [C
e i]
d[C
dt
(44)
e i−1 ]+ k̃dps [C
e i+1 ]−(k̃ps + k̃dps )[C
e i ]+σiFf [Ci ]T −σiFb [C
e i]
= k̃ps [C
−kiBf ([B]T − m
m e
e
i [Bm Ci ]T ) [Ci ]
P
+
e m [Bm C
e i ]T
kiBb K
i
m
e + [A]m
K
(45)
i
e i ]T
d[Bm C
dt
e i−1 ]T + k̃dps [Bm C
e+1 ]T −(k̃ps + k̃dps )[Bm C
e i ]T
= k̃ps [Bm C
+kiBf ([B]T − m
Bb e m
e
e i ]T )m [C
e i ] − ki Ki [Bm Ci ]T
[B
C
m
i
m
e
Ki + [A]m
P
(46)
where the concentration of free KaiA, [A], is given by:
[A] +
6
6
X
X
e i ]T
[A][Ci ]T
[A]m [Bm C
+m
− [A]T = 0
m
m
e
Ki + [A]
i=0 Ki + [A]
i=0
(47)
The effective (de)phosphorylation rates on the active branch are given by σips =
(kps Ki + kpf [A])/(Ki + [A]) and σidps = Ki kdps /(Ki + [A]). The dissociation
e i are given by Ki = k Ab /k Af and K
e i = k̃ Ab /k̃ Af . The
constants Ki and K
i
i
i
i
Ff
effective flipping rates are given by σi = fi Ki /(Ki + [A]) and σiF b = bi , where
fi and bi are the forward and backward flipping rates. We have confirmed that
for sufficiently large k Af , k Ab and k̃ Af , k̃ Ab this set of rate equations gives results
that are identical to those in Eqs. 36-40. Unless indicated otherwise, the results
in the main text are obtained by numerically solving Eqs. 44-46.
3.5
Bifurcation analysis
We have performed a bifurcation analysis of the full model. To this end, we
use Eqs. 36-40. However, the
equations in Eqs. 36-40
P system of differential
e i ]+[Bm C
e i ]+[Am Bm C
e i ]) = [C]T .
obeys the conservation law ([Ci ]+[ACi ]+[C
As a consequence, a linear stability analysis would always yield at least one
eigenvector with eigenvalue zero, which complicates detection of bifurcation
points. To eliminate the zero eigenvalue associated with the conservation of
KaiC, we express the concentration of one of the KaiC complexes in terms of
the concentrations of the other KaiC complexes. We have chosen [ACN ] to
take this role. Thus, [ACN ] is not a separate dynamical variable, but is instead
defined by:
[ACN ] = [C]T −
5
X
i=0
[ACi ]−
6
X
e i ] + [Bm C
e i ] + [Am Bm C
e i ])
([Ci ] + [C
i=0
19
(48)
Figure 13: Bifurcation diagram of the full model of Eqs. 36–40 as a function of
[A]T , for different values of [B]T . Stable fixed points and unstable fixed points
are indicated by solid lines and dashed lines, respectively. When the fixed points
change stability, either saddle-node bifurcations (squares) or Hopf bifurcations
(circles) occur. Apart from [A]T and [B]T , all other parameters are as shown in
Table 2.
Numerical continuation of the fixed points and limit cycles was performed with
the software package XPPAUT [4], which incorporates the numerical continuation routines from AUTO [5]. Throughout this section, unless otherwise noted,
the parameters that are not being varied take the values given in Table 2.
In Fig. 13 we show the bifurcation diagram of the full model, defined by
Eqs. 36-40, as a function of [A]T , for different values of [B]T ; the phase diagram
is shown in Fig. 14. For very small KaiB concentration, [B]T < 0.038[C]T ,
the system has a single, stable fixed point for all [A]T (Fig. 13). For higher
KaiB concentration, 0.038 < [B]T /[C]T < 0.61, the system is bistable for a
range of [A]T (see Figs. 13 and 14): it has one unstable steady state and two
stable steady states, corresponding to different degrees of KaiC phosphorylation. At the boundaries of this bistable region, a stable and unstable fixed
point merge via a saddle-node bifurcation (Fig. 13). We discuss the origin of
this bistable regime in more detail below. For even higher KaiB concentration,
0.61 < [B]T /[C]T < 1.44, one of the two stable fixed points, namely that with
the lower phosphorylation level, becomes unstable for a range of KaiA concentrations. This stable fixed point becomes unstable via a supercritical Hopf bifurcation and gives rise to a limit cycle. Thus, in this window of KaiA and KaiB
concentrations, the system has one stable fixed point at high phosphorylation
level and one limit cycle. For yet larger KaiB concentrations, [B]T > 1.2[C]T ,
the system has only one unstable fixed point surrounded by a limit cycle for a
20
4
[B]T/[C]T
3
OSCILLATIONS
SINGLE
STEADY
STATE
SINGLE
STEADY
STATE
2
1
STABLE FIXED
POINT+LIMIT CYCLE
BISTABILITY
0
0
0.5
1
1.5
[A]T/[C]T
Figure 14: Phase diagram of the full model. In the region enclosed by the dashed
grey lines, the system possesses a stable limit cycle. In the region enclosed by
the solid black lines, the system has three fixed points, of which two are stable
in the absence of a limit cycle. Where the two regions overlap, a single stable
fixed point coexists with a limit cycle.
range of [A]T ; again, the limit cycle appears and disappears at low and high
[A]T , respectively, via a supercritical Hopf bifurcation. This limit cycle corresponds to the circadian oscillations discussed in the main text. Fig. 14 shows
that this oscillatory regime with only one limit cycle has a lower and an upper
bound on the KaiA concentration, but no apparent upper limit on the KaiB
concentration. In contrast, both the bistable regime and the regime in which
a limit cycle coexists with a stable fixed point, occur only over a fairly narrow
range of KaiA and KaiB concentrations.
In Fig. 15, we examine the properties of the limit cycle that is created at the
Hopf bifurcation. It is possible to do numerical continuation of the limit cycle
in the vicinity of the Hopf bifurcation as is shown in Fig. 15A. This analysis
shows that the limit cycle is stable and that the bifurcation is thus supercritical.
Further away from the Hopf bifurcation the numerical continuation algorithm
fails to converge. However, as shown in Fig. 15B and C, by directly solving the
differential equations 36–40 we can nonetheless show that the system continues
to converge to a stable limit cycle. Because of the fact that the algorithm cannot
continue the limit cycle all the way from one Hopf bifurcation to the other, we
cannot strictly rule out the possibility that it undergoes further bifurcations.
Nevertheless, we find by direct integration of the differential equations that both
the period and amplitude of the limit cycle vary smoothly with [A]T between
the Hopf bifurcations. Fig. 16 shows the results of this analysis.
21
1
0.3
B
A
p
0.25
0
0.2
[C4]/[C]T
0.5
0
5
1
10
15
time (days)
0.15
20
C
0.8
0.1
c
0.6
0.4
0.05
0.2
0
0.42
0.43
0.44
[A]T/[C]T
0
0.45
0
0.2
0.4
0.6
0.8
1
p
Figure 15: Limit cycle in the full model for [B]T /[C]T = 3. (A) Bifurcation
diagram of [C4 ] in the vicinity of the Hopf bifurcation as obtained by numerical
continuation of the limit cycle. The stable and unstable fixed points are indicated by a solid black line and a dashed grey line, respectively. The minimum
and maximum values of [C4 ] along the limit cycle are shown as thick black lines.
The limit cycle is stable, indicating a supercritical Hopf bifurcation. Here, we
choose to plot [C4 ] for convenience, and the concentrations of other components
of the system show similar behavior close to the Hopf bifurcation. (B) and (C)
Limit cycle for [A]T /[C]T = 1 and [B]T /[C]T = 3, obtained by numerical integration of Eqs. 36-40. (B) Phosphorylation p in time. (C) Phosphorylation p
versus c, the fraction of KaiC in the active state.
We now discuss the origin of bistability for intermediate [A]T and [B]T (see
Fig. 14). Fig. 17 shows four typical time traces of the phosphorylation level
of KaiC when the system is in the bistable regime. The different time traces
correspond to different initial conditions. These initial conditions differ in the
phosphorylation level of KaiC. Indeed, the initial degree of phosphorylation
largely determines which one of the stable fixed points the system converges
to. For low initial phosphorylation, the system converges to a steady-state
phosphorylation level of ps = 0.5, while for high initial phosphorylation, it
converges to a phosphorylation level of ps = 0.9. These two steady states differ
not only in the average phosphorylation level of KaiC, but, importantly, also in
the concentration of free KaiA: for ps = 0.5, [A] is small, while for ps = 0.9, [A]
is large.
To understand the origin of the difference between the two steady states, it
22
0.8
0.7
0.6
p
0.5
0.4
0.3
0.2
0.1
0.4
0.5
0.6
0.7
0.8
[A]T/[C]T
0.9
1
1.1
1.2
Figure 16: Amplitude of the stable limit cycle obtained by direct integration
of the differential equations as a function of [A]T , for [B]T = 3[C]T . The blue
squares give the locations of the two Hopf bifurcations; each pair of black dots
represents the minimum and maximum phosphorylation reached in one cycle of
the oscillation. These minima and maxima vary smoothly with [A]T , suggesting
that the limit cycle that is born at one supercritical Hopf bifurcation does not
undergo any further bifurcations before dying out at the other Hopf bifurcation.
should be realized that: a) KaiB is needed to stabilize inactive KaiC, but its
concentration is fairly low in the region where there is bistability; b) KaiA is
needed to phosphorylate active KaiC, but its concentration is also fairly low.
In the low ps state, most KaiC hexamers initially have a low degree of
phosphorylation. These hexamers will bind KaiA, which will stimulate their
phosphorylation. However, because [A]T is low, most of the available KaiA is
sequestered by the weakly phosphorylated KaiC hexamers. Those hexamers
that try to move up the phosphorylation ladder have a lower affinity for KaiA,
and can therefore not compete for KaiA with the weakly phosphorylated KaiC
hexamers. Their phosphorylation rates will thus be low, and counterbalanced
by the spontaneous dephosphorylation rate. As a consequence, the overall phosphorylation level will be low.
In the high ps state, most KaiC hexamers go through the approximate cycle
e 6 → B2 C
e 5 → AC5 . In the high ps state, most KaiC hexamers
AC5 → C6 → B2 C
initially have a high degree of phosphorylation. The available KaiA dimers
23
1
0.8
p
0.6
1 subunit phosphorylated
2 subunits phosphorylated
3 subunits phosphorylated
4 subunits phosphorylated
0.4
0.2
0
0
12
24
36
48
60
time (hour)
72
84
96
Figure 17: Bistability in the full model for [A]T /[C]T = 0.48 and [B]T /[C]T =
0.5. The degree of phosphorylation in time is shown for different initial conditions, [Ci (0)] = [C]T for i = 1, 2, 3, 4. For low initial phosphorylation, the
system converges to a steady state at ps ≈ 0.5. For high initial phosphorylation
the system converges to another steady state at ps ≈ 0.9.
will be able to fully phosphorylate these hexamers before they flip towards
the inactive state. On the inactive branch, these hexamers need KaiB to be
stabilized. However, because [B]T is low, the inactive hexamers will not be
stabilized very strongly, and will therefore flip back towards the active state.
At this point, the concentration of free KaiA is close to [A]T , because there
are no hexamers with a low phosphorylation level, which could bind KaiA.
Because the concentration of free KaiA is relatively high, the hexamers that
have just flipped back towards the active state can be rephosphorylated, and
the cycle starts again. This situation clearly illustrates the important role of
KaiB. Without KaiB the inactive branch is not stable, and the full allosteric
cycle will be cut short. This will eliminate the capacity of the system to generate
macroscopic oscillations.
The above explanation of bistability in the Kai system is similar to the
mechanisms that have been proposed for generating bistability in the MAPK
[6] and the CAMKII [7] systems. Both in these systems and in the Kai system, a
protein can be phosphorylated at multiple sites and the concentration of either
the kinase, as in the Kai or the MAPK system, or the phosphatase, as in the
CAMKII system, is limiting. In one steady state, these enzymes are completely
saturated, while in the other enough remains free to act on the few substrates
in need of covalent modification.
Fig. 14 summarizes the behavior of the full system. The full model has a limit
cycle for a broad range of concentrations. Although the range of [A]T for which
oscillations are observed decreases slightly with increasing [B]T , we found no
indication that oscillations cease for higher [B]T . This is in agreement with the
24
passive role played by KaiB in stabilizing the inactive branch and sequestering
KaiA. The region for which bistability occurs is much smaller. Furthermore,
as we have discussed above, the occurrence of the bistable regime does depend
upon details of the model, such as the extent to which KaiB sequesters KaiA
and stabilizes the inactive state when bound to KaiC. Further experiments will
be needed to determine whether bistability really occurs in the Kai system.
References
[1] Chandler D (1987) Introduction to modern statistical mechanics (Oxford
University Press, New York).
[2] Gillespie DT (1977) J Phys Chem 81, 2340-2361.
[3] Kageyama H, Nishiwaki T, Nakajima M, Iwasaki H, Oyama T, Kondo T
(2006) Moll Cell 23, 161-171.
[4] Ermentrout B (2002) Simulating, Analyzing, and Animating Dynamical
Systems (Society for Industrial and Applied Mathematics, Philadelphia).
[5] Doedel E, Paffenroth R, Champneys A, Fairgrieve T, Kuznetsov Y, Sandstede B, Wang X (2001) AUTO 2000: Continuation and Bifurcation Software
for Ordinary Differential Equations (with HOMCONT). Technical Report.
(Caltech, Pasadena, CA).
[6] Markevich NI, Hoek JB, Kholodenko BN (2004) J Cell Biol 164, 353 - 359.
[7] Miller P, Zhabotinsky AM, Lisman JE, Wang XJ (2005) PLoS Biology 3,
705-717.
25