Papers in Physics, vol. 6, art. 060012 (2014)

Received: 3 August 2014, Accepted: 28 October 2014
Edited by: G. Mindlin
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.060012

www.papersinphysics.org

ISSN 1852-4249

Nonlinearity arising from noncooperative transcription factor binding
enhances negative feedback and promotes genetic oscillations

Iván M. Lengyel,1 Daniele Soroldoni,2, 3 Andrew C. Oates,2, 3 Luis G. Morelli1∗

We study the effects of multiple binding sites in the promoter of a genetic oscillator. We
evaluate the regulatory function of a promoter with multiple binding sites in the absence
of cooperative binding, and consider different hypotheses for how the number of bound
repressors affects transcription rate. Effective Hill exponents of the resulting regulatory
functions reveal an increase in the nonlinearity of the feedback with the number of binding
sites. We identify optimal configurations that maximize the nonlinearity of the feedback.
We use a generic model of a biochemical oscillator to show that this increased nonlinearity
is reflected in enhanced oscillations, with larger amplitudes over wider oscillatory ranges.
Although the study is motivated by genetic oscillations in the zebrafish segmentation clock,
our findings may reveal a general principle for gene regulation.

I. Introduction

Cells can generate temporal patterns of activity
by means of genetic oscillations [1, 2]. Genetic os-
cillations are biochemical oscillations in the levels
of gene products [3–14]. They can be produced
by negative feedback regulation of gene expression,
in which a gene product inhibits its own produc-
tion directly or indirectly [15]. Such autoinhibi-
tion is often performed by transcriptional repres-
sors, proteins or protein complexes that bind the
promoter of a gene and inhibit the transcription of
new mRNA molecules [16,17], see Fig. 1. A theoret-

∗Email: morelli@df.uba.ar

1 Departamento de F́ısica, FCEyN UBA and IFIBA, CON-
ICET, Pabellón 1, Ciudad Universitaria, 1428 Buenos
Aires, Argentina.

2 MRC-National Institute for Medical Research, The
Ridgeway, Mill Hill, NW7 1AA London, UK.

3 Department of Cell and Developmental Biology, Univer-
sity College London, Gower Street, WC1E 6BT London,
UK.

ical description of biochemical oscillations requires
a delayed negative feedback, together with suffi-
cient nonlinearity and a balance of the timescales
of the different processes involved [18]. Delays may
occur naturally in transcriptional regulation, since
the assembly of gene products involves intermedi-
ate steps [16,19,20]. Nonlinearity refers to the pres-
ence of nonlinear terms in the equations describing
the dynamics. Such nonlinear terms may occur in
the equations due to the presence of cooperative
biochemical processes, where cooperativity is un-
derstood as a phenomenon in which several com-
ponents act together to orchestrate some collective
behavior [21]. Although some processes giving rise
to nonlinear terms are known, in general it is still
an open question how nonlinearity is built into ge-
netic oscillators.

A compelling model system for genetic oscilla-
tions is the vertebrate segmentation clock [22–25].
This is a tissue-level pattern generator that controls
the formation of vertebrate segments during em-
bryonic development [25, 26]. The spatiotemporal
patterns generated by the segmentation clock are

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

thought to be initiated by a genetic oscillator at the
single cell level [7,9,10,27–29]. In this genetic oscil-
lator, negative feedback is provided by genes of the
Her family, which code proteins that form dimers
and can act as transcriptional repressors [30–34].
The time taken from transcription, translation and
splicing may introduce the necessary feedback de-
lays in zebrafish and mouse [19, 34–37]. One source
of nonlinearity in the segmentation clock oscilla-
tor is the dimerization of gene products that bind
the promoter of cyclic genes [32]. However, this
may be insufficient to generate the observed oscil-
lations in the levels of gene products. One way to
increase nonlinearity would be cooperative binding
of repressors to regulatory binding sites at the pro-
moter [21, 38, 39]. Cooperative binding to multi-
ple binding sites can make the negative feedback
steeper [40]. A similar effect occurs with ultra-
sensitivity in phosphorylation cascades [41]. The
presence of clusters of binding sites for transcrip-
tion factors may be a common motif in gene regula-
tion [42], and cooperativity in transcription factor
binding has been reported for some systems [43].

In zebrafish, multiple binding sites for Her dimers
have been identified in the promoter region of
Her1, Her7, and other genes of the Notch path-
way [32, 44, 45]. However, there is no evidence for

❜�

c

A B C

0 50

1

2

d
im

e
r 

c
o
n
c
e
n
tr

a
ti
o
n

time
1 2

0.6

1.0

1.4

x
(t

)

x(t - ✁)

✁

x
x

gene

Figure 1: Delayed autoinhibition can produce ge-
netic oscillations. (A) The gene (light blue box)
is transcribed and translated into gene products x,
with a delay τ. In this example, gene products form
dimers that act as transcriptional repressors, in-
hibiting transcription of the gene (blunted arrow),
and decay at a rate c. (B) and (C) Numerical so-
lutions of Eq. (9) describing the oscillator in A.
(B) Phase space: monomer concentration x(t) vs.
the delayed concentration x(t−τ) settled to a limit
cycle. (C) Dimer concentration oscillates as a func-
tion of time. Parameters in B, C: bP = 2, τ = 1,
c = 1, x0 = 1, N = 12, M = 6.

cooperative binding of transcriptional regulators in
the case of the zebrafish segmentation clock. There-
fore, although cooperative binding is not ruled out,
this lack of evidence raises the general question of
what contribution could be expected from the mul-
tiple binding sites reported in the promoter of seg-
mentation clock cyclic genes. Here we use theory to
study how multiple binding sites affect nonlinearity
and biochemical oscillations in a generic description
of a genetic oscillator.

II. A promoter with multiple bind-
ing sites

We first evaluate the regulatory function of a pro-
moter that contains N binding sites for a transcrip-
tional repressor, see Fig. 2. We consider transcrip-
tional repressors although the results in this section
are more general and would apply to other types of
transcription factors. We focus on a single tran-
scriptional repressor for the sake of simplicity, and
assume that all binding sites are identical. That
is, binding and unbinding of repressors to the dif-
ferent sites occur at the same rates for all sites.
Moreover, we assume that there is no cooperativity
in repressor binding. This means that binding and
unbinding rates are not affected by the presence of
bound factors to any of the other sites.

A

B

1 2 3 4 N

1 2 3 4 N

k
�

k
✁

Figure 2: Schematic representation of a promoter
with multiple binding sites, numbered from 1 to
N. Transcriptional repressors (orange squares)
bind and unbind from binding sites (numbered
platforms) at the promoter of a gene (light blue
stretch). (A) and (B) illustrate two equivalent con-
figurations of the promoter with identical number
of bound transcriptional repressors having the same
inhibiting strength.

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

The state Pi of the promoter at any time can
be characterized by the number i of bound factors,
which goes from 0 to N. With a rate k+, a free
repressor binds to an empty site at the promoter,
which steps from state Pi to state Pi+1; with a rate
k−, a bound repressor falls off the promoter, which
steps from state Pi to state Pi−1:

P0
k+−−⇀↽−−
k−
· · · −⇀↽−Pi−1

k+−−⇀↽−−
k−
Pi

k+−−⇀↽−−
k−
Pi+1 −⇀↽−

···
k+−−⇀↽−−
k−
PN. (1)

Denoting by Pi the promoter occupation probabil-
ities which describe the fraction of time that the
promoter spends at state Pi in the thermodynamic
limit [46], the kinetics of binding and unbinding
of transcriptional repressors to the binding sites is
given by the set of ordinary differential equations

Ṗ0 = −k+NxP0 + k−P1
...

Ṗi = −k+(N − i)xPi −k−iPi (2)
+k+(N − i + 1)xPi−1 + k−(i + 1)Pi+1

...

ṖN = −k−NPN + k+xPN−1 ,

together with the conservation law

P =

N∑
i=0

Pi , (3)

where x is the repressor concentration and P is pro-
portional to the number of gene copies.

We assume here that binding and unbinding
of repressors to the promoter occur much faster
than other processes like transcription, translation,
transport and decay of molecules. This means that
the promoter occupation probabilities quickly reach
equilibrium with a given concentration of transcrip-
tional repressors [46,47]. This situation is described
by Ṗi = 0 for all i in Eq. (3), and the resulting
algebraic equations can be solved by induction to
obtain

Pi =

(
N

i

)(
x

x0

)i
P0 , (4)

where x0 = k
−/k+ is the equilibrium constant for

binding of factors. Using the constraint Eq. (3) we
express P0 in terms of P

P0 = P

(
1 +

x

x0

)−N
. (5)

Equation (4) and Eq. (5) describe the equilibrium
occupation of the promoter in terms of the concen-
tration x of the transcriptional repressor.

III. Abrupt inhibition

In the previous section, we evaluated the kinetics of
noncooperative binding to a promoter with multi-
ple binding sites. How does the presence of bound
transcriptional repressors affect the transcription
rate of the gene downstream of the promoter? In
general, the strength of inhibition will depend on
the number of bound repressors, and the regulatory
function f(x) will have the form

f(x) =

N∑
i=0

aiPi , (6)

where a0 = b is the basal transcription rate in the
absence of bound repressors and ai is the transcrip-
tion rate in the presence of i bound repressors. Here
we assume, for the sake of simplicity, that transcrip-
tion proceeds at its basal rate b while there are M
or less sites occupied by the repressors, and drops
to zero when the number of occupied sites is larger
than M, see Fig. 3. We shall consider an alternative
scenario below.

In this situation, Eq. (6) becomes

f(x) = b

M∑
i=0

Pi . (7)

Using Eq. (4) and Eq. (5), the resulting regula-
tory function for a promoter with N binding sites
is

fN,M (x) = bP

(
1 +

x

x0

)−N
×

M∑
i=0

(
N

i

)(
x

x0

)i
. (8)

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

0 1 2 3
0

1

x

re
g
u
la

to
ry

 f
u
n
c
ti
o
n
 f

1
2
,M

0 N

b

0
M

number of occupied sites

tr
a
n
s
c
ri
p
ti
o
n
 r

a
te

A B

Figure 3: Abrupt inhibition. Full inhibition occurs
when more than M sites are bound by transcrip-
tional repressors. (A) Transcription rate as a func-
tion of the number of bound transcriptional repres-
sors. (B) Normalized regulatory function, Eq. (8),
as a function of repressor concentration x, with
N = 12 and M = 0, . . . , 11 (dark blue to dark
red).

The zebrafish segmentation clock may be an in-
teresting system to evaluate these results. In the
her1/her7 locus of zebrafish, an estimated number
of about 12 binding sites has been reported [32,45].
The regulatory functions resulting from Eq. (8)
for N = 12 are displayed in Fig. 3B. Although
it is clear that inhibition of the promoter for a
given level of repressors shifts to the right as M
increases, it is less obvious how the value of M af-
fects the steepness —that is the nonlinearity— of
the negative feedback. We use Hill functions to
parametrize the regulatory functions Eq. (8) in a
more transparent form, see Appendix. Hill func-
tions are parametrized by a Hill coefficient h char-
acterizing the steepness of the curve, and an inhibi-
tion threshold K that describes the concentration
of repressors that halves the production rate. We
fit Hill functions to fN,M and obtain an effective
Hill coefficient h and effective inhibition threshold
K, for each value of N and M, Fig. 4A,B. Increas-
ing the number of binding sites N while keeping
M fixed can increase the Hill coefficient. For fixed
N, increasing M changes the Hill coefficient in a
nonmonotonic way: there is an optimal value of M
that maximizes the Hill coefficient and therefore
nonlinearity. The effective inhibition threshold K
changes in a simpler form, increasing both with N
and M. In conclusion, multiple binding sites can
effectively increase the nonlinearity of the feedback
via the regulatory function.

As discussed above, nonlinearity is an essen-

1.5

2.0

2.5

3.0

M
0 5 10 15

N

4

6

8

10

12

14

A

1

2

3

4

M
0 5 10 15

N

4

6

8

10

12

14

B

C D

M
0 5 10 15

N

4

6

8

10

12

14

N

4

6

8

10

12

14

M
0 5 10 15

3.1

3.2

0.1

0.5

1.0

Figure 4: Multiple binding sites can increase non-
linearity and enhance oscillations. (A) and (B) Ef-
fective Hill parameters for the regulatory function
fN,M , Eq. (8), with bP = 1 and x0 = 1. (A) Ef-
fective Hill coefficient h. (B) Effective inhibition
threshold K. (C) and (D) Oscillations described
by Eq. (9) with bP = 2, τ = 1, c = 1 and x0 = 1.
(C) Amplitude of oscillations. (D) Period of oscil-
lations. In C and D, the white region represents the
nonoscillatory regime, in which the system settles
to a fixed point. Color bar labels indicate values in
each panel.

tial ingredient in a theory of biochemical oscilla-
tions [18]. We therefore ask how multiple binding
sites in the promoter affect a biochemical oscilla-
tor. We use the regulatory function Eq. (8) in a
generic model for genetic oscillations. We consider
a gene that encodes a protein that forms dimers,
and these dimers can bind to multiple binding sites
at the promoter to inhibit transcription, Fig. 1A.
We introduce an explicit delay τ to account for
transcription, translation, splicing, and other pro-
cesses involved in the assembly of the gene product
and its dimerization. We assume that dimerization
is a fast reaction, with a separation of timescales
from other processes. Therefore, at any time the
dimer concentration can be approximated by that
of the monomers squared. The dynamics of the
product concentration x(t) is given by

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

dx

dt
= bP

M∑
i=0

(
N
i

)(
x(t−τ)
x0

)2i
(

1 +
(
x(t−τ)
x0

)2)N − cx(t), (9)
where b is the basal production rate, c is the decay
rate of products, P relates to the number of gene
copies, τ is the total delay for product assembly,
and x0 is the product concentration that halves the
basal transcription rate. The regulatory function
Eq. (8) is parametrized by N and M. This genetic
oscillator is a reduction of the models proposed by
Lewis [19], and other authors [48, 49]. It describes
the protein concentration, but does not include the
mRNA; the duration of transcription and transla-
tion are both included in the delay τ. Furthermore,
it does not describe effects present in theories that
include more than one regulator [32, 50, 51], but
here it is enough to illustrate the effects of multiple
binding sites in a simpler context.

We integrate Eq. (9) numerically and evaluate
the resulting dynamics by calculating the ampli-
tude and period of oscillations. In all numerical
simulations, we use the function dde23 from MAT-
LAB [52]. Scanning the values of N and M, we
determine whether the system oscillates in steady
state: when the difference in the maxima over the
last ten cycles falls below 0.01, the simulation is
stopped and we record the output.

The amplitude of oscillations grows with the
number of binding sites N, Fig. 4C. The range
where the system oscillates grows with the num-
ber of binding sites N. The amplitude is nonmono-
tonic in M: for a fixed number of binding sites
N, there is an optimal value for M that maximizes
the amplitude of oscillations, Fig.4C. The period
of oscillations grows with N and decreases with M.
These results show how the change in nonlinear-
ity observed in the regulatory functions is reflected
in the oscillations as the number of binding sites
change.

IV. Gradual inhibition

There is strong evidence for the products of some
cyclic genes of the zebrafish segmentation clock
binding their own promoters and acting as tran-
scriptional repressors [23,27,32,33,35,44]. However,

we do not have detailed knowledge of how these
transcriptional repressors affect transcription rates
when bound to the promoters of cyclic genes. There
is evidence from transcriptional analysis of the Hes1
gene in mouse indicating that inhibition is grad-
ual [30]. While the wildtype Hes1 promoter con-
taining all three N Box elements that are bound by
Hes1 proteins showed a 30-fold inhibition of tran-
scription in the presence of Hes1, mutations in one,
two, and three of the N Box elements showed im-
paired inhibition with 14-, 7-, and 2-fold inhibition,
respectively [30].

Motivated by these results, we consider here a
scenario where additional bound repressors gradu-
ally reduce transcription rate until it drops to zero,
Fig. 5A. For the sake of simplicity, we assume that
transcription rate drops linearly as a function of
bound repressors, from a basal rate b, to zero for
k + 1 occupied sites

ai =

{
b (1 − i/(k + 1)) if i ≤ k + 1
0 if i > k + 1 ,

(10)

see Fig. 5A. Using this gradual inhibition in Eq. (6)
together with Eq. (4) and (5), we obtain regulatory
functions fN,k(x)

fN,k(x) = bP

(
1 +

x

x0

)−N
×
k+1∑
i=0

(
1 −

i

k + 1

)(
N

i

)(
x

x0

)i
, (11)

see Fig. 5B. As in the previous case, it is clear that
the effective inhibition threshold shifts to the right
as k increases, but it is not so clear if the steepness
of the regulatory function changes and, if so, how.
Performing fits to Hill functions, we find that the
nonmonotonic behavior of the effective Hill expo-
nent h is observed again as k increases, Fig. 6A,B.
Oscillations are similarly affected by noncoopera-
tive binding with gradual inhibition, Fig. 6C,D.
These results show that the prediction of an opti-
mal value for the number of bound repressors that
fully inhibits the promoter is robust with respect to
the details of how multiple bound repressors reduce
the transcription rate.

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

0 N

b

0
k 0 1 2 3

0

1

x

A B

number of occupied sites

re
g
u
la

to
ry

 f
u
n
c
ti
o
n
 f

1
2
,k

tr
a
n
s
c
ri
p
ti
o
n
 r

a
te

Figure 5: Gradual inhibition. Binding of multiple
transcriptional repressors gradually inhibits tran-
scription. (A) Transcription rate as a function
of the number of bound transcriptional repressors.
Transcription occurs at the basal rate b in the ab-
sence of bound repressors, and decreases linearly
to zero for more than k bound repressors. (B)
Normalized regulatory function Eq. (11) as a func-
tion of repressor concentration x, with N = 12 and
k = 0, . . . , 11 (dark blue to dark red).

V. Discussion

We studied the effects of multiple noncooperative
binding sites in the promoter of a genetic oscillator.
We evaluated the behavior of a promoter with mul-
tiple binding sites when binding of transcriptional
repressors is noncooperative, Fig. 2. We considered
two different hypotheses for how bound transcrip-
tional repressors affect transcription rates, Figs. 3
and 5. In both cases, we calculated how the number
of binding sites and the number of bound repres-
sors required to produce full inhibition affect the
nonlinearity of regulatory functions. We showed
that there is an optimal value of the number of re-
pressors needed to fully inhibit transcription that
maximizes nonlinearity, Figs. 4AB and 6AB. This
increased nonlinearity is reflected in the behavior
of a genetic oscillator controlled by such regulatory
functions, Figs. 4CD and 6CD.

Cooperative binding is a well known means to
increase the nonlinearity of a biological dynamical
system [21, 39]. Here we show that the nonlinearity
of a regulatory function can be increased by multi-
ple binding sites, even if binding is noncooperative.
This idea may have application in other biological
control systems as well. Using the same formu-
lation as we did here, one can also describe non-
linearity in the transcriptional activation of gene
expression, thereby creating effective on-switches

1.0

1.2

1.4

1.6

k

N

0 5 10

4

6

8

10

12

14

15

0.3

0.4

0.5

0.6

0.7

0.8

0.9

k

N

0 5 10

4

6

8

10

12

14

15

A B

C D

k

N

0 5 10

4

6

8

10

12

14

15

0.08

0.24

0.4

0.56

k

N

 
0 5 10

4

6

8

10

12

14

15

3.12

3.15

3.18

3.21

Figure 6: Multiple binding sites can increase non-
linearity and enhance oscillations in the presence of
gradual inhibition. (A) and (B) Effective Hill pa-
rameters for the regulatory function fN,k, Eq. (11),
with bP = 1 and x0 = 1. (A) Effective Hill co-
efficient h. (B) Effective inhibition threshold K.
(C) and (D) Oscillations obtained using regulatory
functions Eq. (11) with bP = 2, τ = 1, c = 1 and
x0 = 1. (C) Amplitude of oscillations. (D) Pe-
riod of oscillations. In C and D the white region
represents the nonoscillatory regime, in which the
system settles to a fixed point. Color bar labels
indicate values in each panel.

in developmental and physiological regulatory net-
works. A similar effect has been reported in a theo-
retical study of enzymes with multiple phosphory-
lation sites [53–55]. It was found that nonessential
phosphorylation sites give rise to an increase in ef-
fective Hill coefficients, enhancing ultrasensitivity
in signal transduction [55].

Previous work on the segmentation clock ad-
dressed the case of three binding sites [40], mo-
tivated by experimental observations of the Hes7
mouse promoter [31]. The authors assumed that
a single bound dimer inhibits transcription com-
pletely, corresponding to the particular case M = 1
of the theory developed here. They considered
both noncooperative and cooperative binding, and
showed that cooperativity increases the effective
Hill coefficient as expected. In a follow-up [56],
the authors addressed the case of Hes1 regulation
based on the report of four binding sites in the Hes1
mouse promoter [30]. Again assuming that a sin-

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

gle bound dimer inhibits transcription completely,
they used the data from the transcriptional analysis
of the Hes1 gene to estimate an effective Hill coef-
ficient for the Mouse Hes1 oscillator, obtaining an
upper bound of about 3. More recently, the effect
of two binding sites as compared to a single binding
site was discussed together with differential decay
of the monomers [57].

Our theory predicts how changing the number
of binding sites N, and the number M of bound
repressors that produce full inhibition affects a sin-
gle cell oscillator. Although an experiment that
changes the value of M may currently be challeng-
ing in a cellular system, the number of binding sites
N is more amenable to experimental manipulation.
For example, binding sites could be mutated [30], or
deleted from the promoter, or they could be inter-
fered with using genome editing strategies such as
TALEN [58] or CRISPR [59] to alter or delete spe-
cific binding sites. To assess the effects of these per-
turbations in experiments may also pose some chal-
lenges. Dropping the number of binding sites from
N = 12 to N = 6 introduces a period change of
about 2.5%, while amplitude halves over the same
range, Fig. 4C,D. Experiments will require at least
such precision to reliably detect changes.

Our results suggest a possible evolutionary mech-
anism to increase nonlinearity in gene regulatory
systems. In this mechanism, point mutations in the
promoter that increase the number of binding sites
for transcription factors may increase the steepness
of regulatory functions. If the resulting steeper reg-
ulation performs some function better, such muta-
tions would have a good chance to be conserved
by natural selection. In the case of the segmenta-
tion clock, the amplitude of oscillations could in-
crease with the number of binding sites, possibly
reducing the signal to noise ratio. Furthermore,
the range for oscillations would be wider, making
the oscillatory regime less sensitive to slow extrin-
sic fluctuations of parameter values. Remarkably,
it may happen that after an increase in N, an in-
crease in M also raises nonlinearity, see Fig. 4A.
This means that weaker repressors would result in
better oscillations. This evolutionary mechanism
would provide a simple way to gradually increase
the nonlinearity of a feedback or other regulatory
function.

The theory for the zebrafish regulatory function
could be refined using the experimentally measured

0 10
0

1

x0 10
0

1

x

 

 

K = 5

A B

h = 3H
il
l 
fu

n
c
ti
o
n
 f

(x
)

H
il
l 
fu

n
c
ti
o
n
 f

(x
)

Figure 7: Hill functions are characterized by two
parameters, the Hill coefficient h and the inhibition
threshold K. (A) Hill functions with K = 5 and
h = 1 (red), h = 3 (green) and h = 7 (blue). (B)
Hill functions with h = 3 and K = 3 (red), K = 5
(green) and K = 7 (blue).

relative affinities of the binding sites at the her1
and her7 promoters [32]. Apart from the effects re-
ported here, the number of binding sites may have
additional roles. For example, it could serve as a
buffer for fluctuations in gene expression [20,60–64],
augmenting the precision of genetic oscillations.
This will be the topic of future work.

Acknowledgements - We thank Saúl Ares, Lu-
ciana Bruno, Ariel Chernomoretz and Hernan
Grecco for helpful comments on an early version of
this work, and James Ferrell for calling our atten-
tion to Ref. [53]. Thanks to E. Aikau for inspira-
tion. LGM acknowledges support from MINCyT
Argentina (PICT 2012 1952). A.C.O. and D.S.
were supported by the Medical Research Coun-
cil UK (MC UP 1202/3) and the Wellcome Trust
(WT098025MA).

Appendix: Effective Hill functions

Hill functions are often used to describe the non-
linearities present in gene regulatory networks [17].
Hill functions are sigmoidal step functions defined
by

fH(x) =
1

1 + (x/K)
2h
, (12)

where the steepnes of the step is characterized by
the exponent 2h, and the inhibition threshold K is
the concentration of repressor that halves the pro-
duction rate, here scaled to unity, Fig. 7. Here

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Papers in Physics, vol. 6, art. 060012 (2014) / I. M. Lengyel et al.

0 0.5 1 1.5
0

1

 

 

R = 0.9993

fit

0 1 2 3
 

 

R = 0.99992

fit

0 50 100 150
 

 

R = 0.99859

fit

0 6 12
0

1

2

3

4

M

e
ff

e
c
ti
v
e
 t

h
re

s
h
o
ld

 K

0 6 12

2

3

M

e
ff

e
c
ti
v
e
 H

il
l 
c
o
e
ff

ic
ie

n
t 

h

A B C

D E

0

1

0

1
f12,1 f12,6 f12,11

x x x

Figure 8: Effective Hill functions Eq. (12) can fit
regulatory functions of the multiple binding site
regulatory function, Eq. (8) for N = 12, with
bP = 1 and x0 = 1. (A-C) Examples of fits of
Hill functions (solid lines) to regulatory functions
(dashed lines) for (A) M = 1 (green), (B) M = 6
(light blue) and (C) M = 11 (purple). (D) Effective
Hill coefficient h as a function of M. (E) Effective
inhibition threshold K as a function of M. Dots in
D and E correspond to fits from panels A, B and
C.

we include an explicit factor 2 in the exponent to
account for the dimerization of the transcriptional
repressors. One advantage of Hill functions is that
they are very simply parametrized. In the main
text, we fit Hill functions to the more complex reg-
ulatory functions Eqs. (8) and (11). Some fits of
Eq. (8) in the case N = 12 are displayed in Fig. 8
for illustration.

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