miRNA regulatory circuits in ES cells differentiation: a chemical kinetics modeling approach.

Abstract

MicroRNAs (miRNAs) play an important role in gene regulation for Embryonic Stem cells (ES cells), where they either down-regulate target mRNA genes by degradation or repress protein expression of these mRNA genes by inhibiting translation. Well known tables TargetScan and miRanda may predict quite long lists of potential miRNAs inhibitors for each mRNA gene, and one of our goals was to strongly narrow down the list of mRNA targets potentially repressed by a known large list of 400 miRNAs. Our paper focuses on algorithmic analysis of ES cells microarray data to reliably detect repressive interactions between miRNAs and mRNAs. We model, by chemical kinetics equations, the interaction architectures implementing the two basic silencing processes of miRNAs, namely "direct degradation" or "translation inhibition" of targeted mRNAs. For each pair (M,G) of potentially interacting miRMA gene M and mRNA gene G, we parameterize our associated kinetic equations by optimizing their fit with microarray data. When this fit is high enough, we validate the pair (M,G) as a highly probable repressive interaction. This approach leads to the computation of a highly selective and drastically reduced list of repressive pairs (M,G) involved in ES cells differentiation.

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miRNA Regulatory Circuits in ES Cells Differentiation: A
Chemical Kinetics Modeling Approach
Zijun Luo1*, Xuping Xu2, Peili Gu2, David Lonard2, Preethi H. Gunaratne3, Austin J. Cooney2, Robert
Azencott1
1Department of Mathematics, University of Houston, Houston, Texas, United States of America, 2Department of Molecular and Cellular Biology, Baylor College of
Medicine, Houston, Texas, United States of America, 3Biology and Biochemistry Department, University of Houston, Houston, Texas, United States of America
Abstract
MicroRNAs (miRNAs) play an important role in gene regulation for Embryonic Stem cells (ES cells), where they either down-
regulate target mRNA genes by degradation or repress protein expression of these mRNA genes by inhibiting translation.
Well known tables TargetScan and miRanda may predict quite long lists of potential miRNAs inhibitors for each mRNA gene,
and one of our goals was to strongly narrow down the list of mRNA targets potentially repressed by a known large list of
400 miRNAs. Our paper focuses on algorithmic analysis of ES cells microarray data to reliably detect repressive interactions
between miRNAs and mRNAs. We model, by chemical kinetics equations, the interaction architectures implementing the
two basic silencing processes of miRNAs, namely ‘‘direct degradation’’ or ‘‘translation inhibition’’ of targeted mRNAs. For
each pair (M,G) of potentially interacting miRMA gene M and mRNA gene G, we parameterize our associated kinetic
equations by optimizing their fit with microarray data. When this fit is high enough, we validate the pair (M,G) as a highly
probable repressive interaction. This approach leads to the computation of a highly selective and drastically reduced list of
repressive pairs (M,G) involved in ES cells differentiation.
Citation: Luo Z, Xu X, Gu P, Lonard D, Gunaratne PH, et al. (2011) miRNA Regulatory Circuits in ES Cells Differentiation: A Chemical Kinetics Modeling
Approach. PLoS ONE 6(10): e23263. doi:10.1371/journal.pone.0023263
Editor: Vladimir N. Uversky, University of South Florida College of Medicine, United States of America
Received December 13, 2010; Accepted July 14, 2011; Published October 19, 2011
Copyright: ? 2011 Luo et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by a grant from the National Institute for General Medicine (P01- GM081627-01). The funders had no role in study design,
data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: boluomiduo1@gmail.com
Introduction
MicroRNAs (miRNAs) are small non-coding RNAs, *22
nucleotides in length that are able to bind and repress protein
coding mRNAs through complementary base pairing. The mini-
mum requirement for this interaction is six consecutive nucleotides,
which undergo base pairing to establish a miRNA-mRNA duplex.
The only constraints being that the six nucleotides must be localized
in the 59seed sequence (between nucleotides 2–8) of the miRNA and
the complementary binding sites, which are largely located in the 39-
untranslated regions (39-UTRs) of target mRNAs.
Because of this very minimal binding requirement, a given
miRNA can potentially bind and silence hundreds of mRNAs
across a number of signaling pathways to integrate multiple genes
into biologically meaningful networks regulating a variety of
cellular processes [1–3]. In animals, miRNAs regulate gene
expression post-transcriptionally by either down-regulating their
target mRNAs or by inhibiting their translation [4]. MiRNAs have
two types of effects on their target mRNAs. When a miRNA M
binds to its target mRNA gene G with partial complementarity,
then the translation of gene G is inhibited; however, when M binds
to its target G with near-perfect complementarity, then gene G is
cleaved, resulting in its degradation. Thus, when we ectopically
over-express a miRNA we expect to see a decrease in the target
genes at the protein level but not at the gene level if the miRNA-
mRNA duplex is formed through imperfect complementarity. In
contrast, we expect both mRNA and protein levels to change
when the miRNA-mRNA duplex binds with near perfect
complementarity.
Expression of miRNA genes is ultimately controlled by the same
transcription factors which regulate the expression of protein
coding genes. The expression of these same transcription factors
can in turn be regulated by miRNAs, leading to positive and
negative feedback loops [5–7]. Thus transcription factors such as
Oct4, Sox2 and Nanog, which regulate gene networks controlling
key properties of ES cells, are closely linked with miRNAs that are
enriched in ES cells in both mice and humans [5,8,9].
Genome-wide studies using microarray and sequencing tech-
nologies have significantly expanded our knowledge of the
complex regulatory networks underpinning the key properties of
ES cells, namely self-renewal and pluripotency. Classical methods
like sequence analysis, correlation analysis and other statistical
inference techniques, have often yielded very large lists of
potentially interacting miRNA-mRNA pairs, so that experimental
testing of all possible interactions would be too costly.
In previous work on ES cells regulatory network, ES cells
microarray data recorded during differentiation were mainly
studied by linear correlation analysis, focused on simultaneity of
high miRNA levels and low mRNA levels or vise versa. But
correlation analysis cannot tell whether miRNAs and their target
genes/proteins interact directly or indirectly, nor give clear
indication about the interaction mechanisms.
In this paper, we deepen the analysis of several ES cells
microarray data, by parameterized chemical kinetics modeling of
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miRNA-mRNA interactions, involving associated protein prod-
ucts. Our goal was to drastically narrow down the list of potential
repressive miRNA-mRNA links. We define two specific chemical
kinetic models underlying the two basic repressive actions of a
typical miRNA on a targeted mRNA gene G, namely by direct
degrading of G or by inhibiting the translation of the protein
generated by G.
We implement fast parameter estimation algorithms to
adequately fit these chemical kinetics models to microarray data
from ES cells undergoing retinoic acid (RA) induced differentia-
tion and compute a precise quality of fit between models and data.
We have thus generated, parameterized, and tested more than
10,000 models, to evaluate as many potential instances of miRNA-
mRNA interactions. By thresholding the ‘‘quality of fit’’ of these
models, we then accept or reject the validity of the associated
miRNA-mRNA interaction.
Our presentation here is focused on 10 key regulatory genes for
ES cells differentiation, namely Oct4, Nanog, Sox2, Klf4, Esrrb,
cMyc, Tbx3, Ezh1, Ezh2, Eed, and on the main miRNAs which
may target these 10 key genes, according to the in silico target
prediction databases TargetScan (version 5.0) and/or miRanda.
Our approach radically narrows down the lists of potentially
interacting miRNA-mRNA pairs predicted by TargetScan or
miRanda, and for each validated miRNA-mRNA pair, we identify
wether target mRNA repression occurs by direct degradation or
by translation inhibition.
Materials and Methods
Microarray Data Description
We have centered our miRNA-mRNA interactions study on
microarray data of mouse ES cells undergoing RA-induced
differentiation. This dataset had been previously analyzed by
classical techniques in Gu et al. [5]. The miRNAS microarray was
provided by LC Science Inc. Each microarray data file gathers
genes expressions recordings from two ES cells differentiation
experiments: Wild Type (WT) and GCNF- Knock-Out (KO).
In both experiments, expression levels were recorded for 266
well characterized miRNAs on days 0, 1, 3, 6, based on 6 probe
replicates for each miRNA prediction (MCEMIR, Cand, MIR)
and 8 probe replicates for miRNAs (mmu-miRs). Simultaneously,
the expression profiles of 30,000 mRNAs were recorded on days 0,
3, 6 using an Affymetrix mouse 430 2 array, based on three
biological replicates per time point. Expression levels range from 1
to 46,559 for the miRNAs, and from 1 to 21,845 for mRNAs.
For both WT and KO data, and each day, we have several
arrays (chips) recording expression levels for our 266 miRNAs and
30,000 mRNAs. For each miRNA and each mRNA, and for each
day, we synthesize the replicate recordings by simply averaging the
available multiple values of their expression levels.
For each miRNA and each mRNA, we can then interpolate the
available profile data to generate interpolated expression levels
values at 19 intermediary time points (t=0, 1/3, 2/3, …, 17/3,
18/3), by a Piecewise Cubic Hermite Interpolation (PCHIP)
technique, which is well known to preserve monotonicity and the
basic qualitative features of expression profiles.
Analysis of Western Blot Data for Four Pluripotency
Proteins
By Western blots analysis, we have also recorded protein
expression profiles For
GCNF,Oct4,Nanog,Sox2
f
cell differentiation (see Figure 1), for both WT and GCNF-KO, at
time points (0, 1.5, 3, 6).
The raw image data provided by Western blotting were
converted into numerical values by standard image analysis
software tools [10]. We have then normalized the image intensities
by the corresponding recorded actin intensity (which functions as
an internal loading control). These normalized proteins intensity
data were then interpolated as above to compute proteins
expressions at the same 19 intermediary time points.
g during ES
Previous Results Linking miRNAs and Regulatory Loops
for ES Cells Differentiation
Several publications indicate that miRNAs have important
functions in post-transcriptional silencing and are involved in the
regulation of self-renewal and of differentiation for ES cells. In the
ES cells differentiation study [5], the authors classified miRNAs
into three classes: for class HL, expression levels are High on days
0–1 and Low on day 6; for class LH, expression levels are Low on
days 0–1 and High on day 6; class TR gathers all other
‘‘Transient’’ expression profiles.
Figure 1. Western blots for 4 proteins and actins. Oct4 and Nanog levels exhibit quite strong decrease for WT cells and very slow decrease for
GCNF-KO cells. Sox2 levels vanish after 1.5 days for WT cells and slowly decrease for GCNF-KO cells. GCNF levels are initially low, peak at day 3 and fall
back on day 6 for WT cells. For GCNF-KO cells, GCNF levels naturally vanish.
doi:10.1371/journal.pone.0023263.g001
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Our microarray recordings included 46 miRNAs in class TR,
105 miRNAs in class HL, and 78 miRNAs in class LH. According
to TargetScan and/or miRanda, pluripotency regulatory genes in
ES cells are targeted by 26 of the 105 miRNAs in class HL, and by
23 of the 78 miRNAs in class LH.
To detect interacting miRNA-mRNA pairs [5], used qualitative
correlation of expression profiles, mostly for miRMNA classes HL
and LH, without any conclusions for miRNA class TR. For Wild
Type ES cells differentiation [5], outlined a regulatory network
(see Figure 2 [5]) involving the orphan nuclear receptor GCNF
(NR6A1), which is a transcriptional repressor of Oct4 and Nanog.
Both Oct4 protein and Nanog protein are transcriptional
regulators for two groups of mRNAs: the Self-Renewal Regulators
SRR (Sox2, Klf4, Esrrb, Tbx3, cMyc), and the Differentiation
Inhibitors DI such as the Polycomb complex (Ezh1, Ezh2, Eed). In
the Figure 2 network, the miRNAs of class HL target
Oct4,Nanog,SRR,DI
½
class LH target GCNF,SRR,DI
½
Our goal was to deepen this analysis, by nonlinear modeling
techniques, to validate more precisely if any miRNA ‘‘M’’ actually
represses an ES cell regulatory gene G belonging to the list
TARG(M) of all mRNAs targeted by M according to miRanda or
to TargetScan 5.0. Our approach was to first select several basic
small interaction motifs including the pair (M,G), and then to
parametrize a chemical kinetic model for each such motif, in order
to fit the expression profiles recorded in our microarray data sets.
We now introduce the two basic interaction architectures we have
systematically modeled.
? and the Hox cluster, while miRNAs of
?.
Transcription-Degradation (Transcr.Degr.) Architecture
Linking miRNAs to ES Cells Regulatory Networks
Basic Transcription-Degradation architecture.
basic interaction architecture for any miRNA-mRNA pair (M,G)
deals with situations where the miRNA ‘‘M’’ directly degrades the
transcription of its mRNA target G by direct binding with perfect
or near-perfect complementarity. For a fixed ‘downstream’’
Our first
mRNA G, we assume that few miRNAs may simultaneously bind
with G with near-perfect complementarity.
Hence for each key ES regulatory gene G, we have selected a
family Transcr.Degr(G) of small Transcription-Degradation (Tran-
scr.Degr.) architectures (see Figure 3) potentially involving transcrip-
tion-degradation of G by one or several miRNAs (M1,M2,...) as
well as the interactions of G with the main proteins acting as
transcriptional factors of G. Combinatorial considerations show
that the size of Transcr.Degr.(G) can be quite large (see below).
Here is an example of a small network of Transcr.Degr. type,
involving 5 molecules: GCNF, mRNA gene Oct4, Oct4 protein,
Nanog protein, and miRNA mmu-miR-186.
Indeed, in view of Figure 2, GCNF is a transcriptional repressor
of the mRNA gene Oct4, and miRNAs of class HL may target and
degrade Oct4. According to [8,11], the Oct4 protein and the
Nanog protein are potential transcriptional activators of gene
Oct4. Finally, by miRanda, the miRNA mmu-miR-186 may
target Oct4.
KeyFamiliesof smallnetworksofTranscr.Degr.type.
now construct the family of all small networks of Transcr.Degr. type
involving arbitrary downregulating pairs (M,G) where M is any one
of our 266 miRNAs and G is any one of the 10 key ES regulatory
factors Oct4, Nanog, Sox2, Klf4, Esrrb, Tbx3, cMyc, Ezh1, Ezh2,
Eed. Naturally we require the mRNA gene G to belong to the target
list TARG(M), which reduces the initial set of 2660 pairs (M,G) to
only 238 pairs where M may target G, including for instance 19 pairs
(miRNA, Oct4), 2 pairs (miRNA, Nanog), 29 pairs (miRNA, Sox2),
etc.
Based on [5,11], transcription of Oct4 and Nanog is repressed
by GCNF, and activated by Oct4, Nanog, Sox2. So for each one
of the 19 potentially downregulating pairs (M,Oct4), we study
seven Transcr.Degr. architectures, ‘‘combining the molecules M
and Oct4 with each one of the following 7 proteins combinations:
(Oct4), (Nanog), (Sox2), (Oct4,Nanog), (Oct4,Sox2), (Nanog,
Sox2), (Oct4, Nanog, Sox2). This generates 7|19~133 potential
networks of Transcr.Degr. type downregulating Oct4 via an
miRNA.
By a similar construction, the only 2 pairs (M, Nanog) retained
above are associated to 7|2~14 networks of Transcr.Degr. type
potentially downregulating Nanog via an miRNA.
When the downstream gene G is in the list L=(Sox2, Klf4,
Esrrb, Tbx3, cMyc, Ezh1, Ezh2, Eed) [5], suggests, as seen in
Figure 2), that Oct4 and Nanog are transcriptional activators or
repressors of G. For each one of the 217 pairs (M,G) retained
above with gene G belonging to the list L, we then generate one
network of Transcr.Degr. type including M,G, as well as the Oct4
and Nanog proteins. This defines 217 corresponding networks of
Transcr.Degr. type.
Thus we have determined a set of 217+147=364 potential
Transcr.Degr. architectures to be studied below by chemical
kinetics model fitting.
We
Translation-Inhibition Architectures (Transl.Inhib.) Linking
miRMNAs to key ES Cell Regulatory genes
Basic Translation-Inhibition Architecture.
basic interaction architecture for any miRNA-mRNA pair (M,G)
models the cases where the upstream miRNA ‘‘M’’ inhibits the
translation of the downstream mRNA gene G, and thus represses
the expression of the protein P generated by G. For a fixed mRNA
G, we may have several upstream mRNAs M1,M2,:::Mk
inhibiting the translation of G. The molecules G,P,M1,M2,:::Mk
then define a Translation-Inhibition (Transl.Inhib.) architecture
(Figure 3).
Our second
Figure 2. A few key regulatory loops for ES cells according to
[5]. Arrows indicate ‘‘activation’’ while bars ending with a hash indicate
‘‘repression’’.
doi:10.1371/journal.pone.0023263.g002
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We have generated Western blot data for the 4 proteins GCNF,
Oct4, Nanog, Sox2. Hence in this paper we restrict the study of
Transl.Inhib. architectures to the cases where P is one of these 4
proteins. For instance among the 19 miRNAs M potentially
targeting gene Oct4, as listed by miRanda or Targetscan, we can
select the 3 miRNAs (mmu-mir-290, mmu-mir-296, mmu-mir-
138) and study the Transl.Inhib. architecture defined by (gene
Oct4, protein Oct4) and these 3 mRNAs.
Key Families of small gene networks of Transl.Inhib.
type.
To restrict somewhat the combinatorial complexity, we
have limited to 3 the number of upstream miRNAs involved in any
Transl.Inhib. architecture repressing each one of the 3 proteins
Oct4, Nanog, Sox2. For the Transl.Inhib. architectures repressing
gene GCNF, we have restricted the number of upstream miRNAs
to 1. Indeed, since GCNF is not expressed in the KO context, the
number of profile points available from our WT microarray is too
small to correctly parametrize the Transl.Inhib. architectures
repressing GCNF as soom as they involve 2 or more upstream
miRNAs.
With these restrictions we have thus defined and studied 5337
Transl.Inhib. architectures potentially repressing one of the 4
proteins GCNF, Oct4, Nanog, Sox2.
Chemical Kinetic Equations for Transcr.Degr. and
Transl.Inhib. Architectures
Select any miRNA-mRNA pair (M,G). Call P the protein
generated by gene G. The two main modalities of interaction
within the triplet of molecules [G, P, M] were formalized above by
the Transcr.Degr. and the Transl.Inhib. architectures. We now
model these two types of interactions by chemical kinetic equations
linking the expression levels of these molecules.
We point out an essential mathematical property of the two
chemical kinetic models introduced below. Under arbitrary scale
changes affecting the numerical expression levels of mRNAs,
proteins and miRNAs, the chemical kinetic equations for both
architectures Transcr.Degr. and Transl.Inhib. will still keep the
same mathematical form but with corresponding nonlinear
changes in the equations parameters. This is a crucial ‘‘model
invariance’’ property for model fitting since raw microarray data
and Western blot data are at best roughly proportional to the
absolute expression levels of the molecules of interest, and the
corresponding constants of proportionality are fundamentally
unknown.
Chemical Kinetics for Transcr.Degr. Architectures.
transcription-degradation architecture Transcr.Degr. is a small
size interaction model formalizing how the rate of change for the
expression of a downstream mRNA ‘‘G’’ depends on the
expression levels of its main upstream factors. These factors
The
include the post-transcriptional repressor miRNA ‘‘M’’, and two
sets of proteins: the set rep(G)~fR1,R2,...,Rkg of transcriptional
repressors for G, and the set act(G)~fA1,A2,...Aqg of
transcriptional activators for G.
Denote by g(t),p(t),m(t),r1(t),...rk(t),a1(t),...,aq(t) the ex-
pression levelsof molecules G,P,M, R1,...Rk, A1,...,Aqat time t.
To model the transcription of downstream mRNA gene G by
interaction with transcription repressors rep(G) and activators
act(G), we introduce a nonlinear chemical kinetic equation (CKE)
similar to equations proposed in [12,13], but with a complemen-
tary term encoding the repressive influence of miRNA M on its
target mRNA G, as follows (see [14]).
We first define the individual impacts of proteinic repressors Ri
and activators Ajon the rate of change dg(t)=dt by
represi(t)~
1
(1zuiri(t))BSRi
for 1ƒiƒk
ð1Þ
activj(t)~
1
(1zwjaj(t))BSAj
for 1ƒjƒq
ð2Þ
where BSRiw0,uiw0 and BSAjw0,wjw0 are respectively the
number of binding sites and the affinity constants with G for the
transcriptional factors Riand Aj.
The synthetic impacts REP(t) and ACT(t) of repressors rep(G)
and activators act(G) on dg(t)=dt are then (see [12–14]) given by
REP(t)~repres1(t)|repres2(t)|...|represk(t)
ð3Þ
ACT(t)~activ1(t)|activ2(t)|...|activq(t)
ð4Þ
By a probabilistic analysis detailed in [12–14], the fraction F(t) of
DNA templates committed to the transcription of G can then be estimated
by
F(t)~REP(t) 1{ACT(t)
½?ð5Þ
Let bw0 be the degradation rate of G, kw0 the transcription rate
of G, and vw0 be the reaction rate between G and M. During the
small time interval ½t,tzdt?, the concentration of new G molecules
synthesized by transcription is equal to kF(t)dt, the repressive
interaction of molecules M and G eliminates vg(t)m(t)dt
molecules of G, and natural decay destructs bg(t)dt molecules of
G. Hence the expression level g(t) of G verifies the following CKE,
Figure 3. Transcription-Degradation (Transcr.Degr.) architectures and Translation-Inhibition (Transl.Inhib.) architectures.
doi:10.1371/journal.pone.0023263.g003
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characteristic of Transcr.Degr. architectures,
dg(t)
dt
~{bg(t){vg(t)m(t)zkF(t)
ð6Þ
Note that this CKE is parametrized by the (3z2kz2q) unknown
positive parameters v,b,k, BSRi,ui, BSAj,wj.
Chemical Kinetics for Transl.Inhib. Architectures.
translation-inhibition architectures Transl.Inhib. involves one
downstream mRNA gene G, the protein P produced by G, and
a selected set MIR(G) of n upstream miRNAs ½M1,M2,:::Mn?
repressing the translation of G. The concentrations at time t of
P,G,M1,...,Mn, are denoted by p(t),g(t),m1(t),...,mn(t). We
have modeled Transl.Inhib. architecture by the following chemical
kinetics equation [12,14]. For each miRNA Mi, call BSMiw0 the
number of binding sites with G and and biw0 the affinity constant
between Mi and G. The individual repressive impact of Mi on
dg(t)=dt is as above defined by
The
represi(t)~
1
(1zbimi(t))BSMi
and the global repressive impact of all the miRNAs in MIR(G) on
the rate of change dp(t)=dt of protein P is given as above by
H(t)~repres1(t)|repres2(t)|...|represk(t)
ð7Þ
Let cw0 and lw0 be the degradation rate and the translation
rate for protein P. During the small time interval ½t,tzdt?, the
concentraion of P molecules destroyed by natural degradation of P
is cp(t), the concentration of G molecules committed to the
translation of G is equal to g(t)H(t)dt, and hence the
concentration of P molecules generated by translation of G
molecules is lg(t)H(t). Thus the concentration p(t) of protein P is
driven by the following CKE, characteristic of the Transl.Inhib.
architectures
dp
dt~{cp(t)zlg(t)H(t)
ð8Þ
A probabilistic analysis justifying the preceding arguments is
outlined in [12,14]. Since c can be estimated from protein
evolution recordings such as Western blots data, the CKE just
derived depends on the (1z2n) unknown positive parameters
l,(BSM1,u1),...,(BSMn,un).
Parameter Estimation for Chemical Kinetics Models
For any given Transcr.Degr. or Transl.Inhib. architecture, an
immediate challenge is to determine which parameter values must
be injected in the corresponding CKE 6 or 8 in order to best fit
our given sets of microarray data. We have developed and
implemented innovative algorithms to compute these optimal
parametric values.
For instance, consider the simplest Transcr.Degr. architecture
model where there is only 1 transcription factor, which can either
be a repressor or an activator. The number of parameters to be
evaluated is 5 and after extrapolating the concentration data
recorded by microarrays for both WT and GCNF-KO experi-
ments on ES cells differentiation.
With the preceding notations, for Transcr.Degr. architectures,
g(t),p(t),m(t),ri(t),aj(t) At a finite number of time values ‘‘t’’,
which we have extended to 19 instants by specific extrapolation,
each one of our two microarray data sets (WT and GCNF-KO)
provides observed values proportional to the concentrations of all
the molecules involved in each one of our 5701 instances of
Transcr.Degr. or Transl.Inhib. architectures. Hence after dis-
cretization of CKE 6 or of CKE 8 at 19 time instants, our two
microarray data sets provide us with 38~2|19 nonlinear
algebraic equations (of high degree) involving the unknown
parameters of the corresponding CKE.
For each Transcr.Degr. networks, the number (3z2kz2q) of
unknown parameters remains between 5 and 15, since we have
imposed kƒ3 and qƒ3.
For each Transl.Inhib. networks, the number (1z2n) of
unknown parameters remains between 3 and 7 since we have
imposed nƒ3.
Hence optimal parametrization of our 5701 genes interaction
networks NETi of types Transcr.Degr. or Transl.Inhib. was a
numerical challenge since to parametrize each small network, we
had to solve an overdetermined system of 38 algebraic equations of
quite high degree involving between 3 and 15 unknowns. A
natural mathematical approach is to solve the associated highly non
linear least squares problem involving 38 equations and a number of
unknowns inferior to 15.
There are no explicit solutions for such problems; moreover fast
computing was essential here, since we had to implement the
solutions to 5701 such nonlinear least squares problems.
This is a nonlinear optimization problem, since we want to
select up to 15 positive parameters minimizing the sum of squares of
38 ‘‘residuals’’, in order to optimize the quality of fit of the model
under 38 equality constraints. We have of course tested several
generic optimization approaches (see [15–18]) such as ‘‘gradient
descent’’ and ‘‘genetic algorithms’’. These last two well known
optimization techniques turned out to require far too much
computing time and were often unreliable due to their high
dependence on initialization values.
So we had to develop and implement specific efficient
algorithms dedicated to the parameterization of equations CKE
6 (or CKE 8) based on a combination of multi-scale searches for
the ui,BSRiand the uj,BSAj(or for the ui,BSMi) combined with
constrained linear programming determination of the other
parameters once the affinity constants and the numbers of binding
sites have been tentatively fixed.
For CKE6, thefactors
tj~1=(1zwjaj(t)) always lie within the range ½0,1?, so we
perform a grid search on the interval ½0,1? to explore the values
of these factors at fixed key instants t, and then derive the
associated potential values for ui,wj. We also impose a moderate
bound S on the numbers of binding sites BSRiand BSAj. This of
course generates a large set of potential values for the 2k
parameters ui,BSRi and the 2q parameters wj,BSAj. Once we
fix tentative values for these (2kz2q) parameters, we can estimate
the decay rate b, reaction rate v, and transcription rate k by
solving a constrained linear programming problem in dimension 3
(see [14]).
For the parametrization of CKE 8, we have developed an
analogous algorithm.
For both CKEs 6 and 8, our parametrization algorithm is quite
fast and easily implemented. For each one of the 5701 small
network selected above, the parametrization generated by our
algorithm provides a good optimization for the quality of fit
between CKE model and our two microarray data sets. To reach
robust conclusions, our parametric modeling applies a ‘‘parameter
hi~1=(1zuiri(t))
and
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parsimony’’ principle, selecting among models with high quality of
fit to data, the models having the smallest number of parameters.
Our parametrized chemical kinetic modeling approach handles
microarray data through nonlinear parameterization algorithmics, and
clearly goes further than linear techniques such as Principal
Components Analysis (PCA) or profiles correlation analysis. Our
parsimonious modeling of small interaction networks, combined
with the quality of fit evaluations outlined below, provides a
powerful tool for microarray data analysis, complementing
classical linear data mining techniques. Moreover, since our
network models are either of Transcr.Degr. or Transl.Inhib. type,
the miRNA-mRNA pairs we validate below are automatically
assigned to one of the two basic repressive modalities Tran-
scr.Degr. or Transl.Inhib.
Validation of parametrized Transcr.Degr. and Transl.Inhib.
architectures by Quality of Fit to data
To model the potentially interacting pairs (M,G) where M
belong to our set of 266 miRNAs and G is one of the 10 key ES
cell regulatory genes listed above, we have generated, as outlined
above, a group of 5701 Small Interaction Networks which we
denote SIN~½NET1,...,NET5701?. The group SIN comprises
364 Transcr.Degr. motifs and 5337 Transl.Inhib. motifs.
Each miRNA-mRNA pair (M,G) considered above was only
retained when M targets G according to miRanda or Targetscan.
Then we consider the family Net(M,G) of all the networks which
belong to the group SIN and include the pair (M,G). We will
validate the pair (M,G) as a repressive miRNA-mRNA pair if and
only if at least one of the networks in the subset Net(M,G) has been
modeled by a parametrized CKE having a high enough quality of fit
to our microarray data. We now need to precisely define this
quality of fit.
Each specific network NETibelonging to Net(M,G) has been
modeled by a chemical kinetic equation CKEi of type Tran-
scr.Degr. or Transl.Inhib., and the parameters of CKEi were
computed to optimize the fit with our WT and GCNF-KO
microarray data sets.
Call D the downstream target of CKEi. The molecule D is the gene
G if NETi is of Transcr.Degr. type; if NETi is of Transl.Inhib.
type, D is the protein P produced by G. The concentrations D(t) of
D are separately recorded by microarrays for WT ES cells and for
GCNF-KO ES cells.
For each such experiment, the parametrized equation CKEi
clearly generates model predicted values^D D(t) for the concentra-
tions D(t) of the downstream target, by numerically solving the
Ordinary Differential Equation (ODE) specified by CKEi. More
precisely, the ODE 6 can be solved for the concentration
g(t)~^D D(t) of D~G, and the ODE 8 can be solved for the
concentration p(t)~^D D(t) of P, since the recorded data provide
values for all the m(t),r1(t),...,rk(t),a1(t),...,aq(t) involved in the
ODE 6 and for all the g(t),m1(t),...,mn(t) involved in the ODE
8.
To assess the quality of fit between the model predictions^D D(t)
and the recorded microarray data D(t), a natural criterion is the
Relative Error of Prediction jD(t){^D D(t)j=D(t). However this
relative error of prediction becomes meaninglessly large whenever
D(t) is close to zero. To avoid these spurious large values, we
introduce the mean value?D D of D(t) over all t, and we compute the
Smoothed Relative Error of Prediction by
SRER(t)~jD(t){^D D(t)j=D(t) if D(t)§0:15:?D D
ð9Þ
SRER(t)~jD(t){^D D(t)j=?D D
if D(t)v0:15:?D D
ð10Þ
Then we define the global Error of Prediction ErrPred of the CKEi
model by
ErrPred~max
t
(SRER(t))
The error of prediction ‘‘ErrPred’’ is percentage valued and
quantifies the quality of fit between the model CKEi and the
microarray data. Note that small values of ErrPred correspond to
high quality of fit between model and data. Hence our parameters
estimation algorithms were actually implemented to select parameters
minimizing ErrPred for both WT and the GCNF-KO microarray
data.
We will consider the network NETias ‘‘validated’ if, for both the
WT and the GCNF-KO microarray data, the global ‘‘error of
prediction’’ ErrPred of the model CKEiis less than 10%.
To each miRNA-mRNA pair (M,G) considered above, we have
associated the family Net(M,G) of all the networks NETiwhich
include the pair (M,G) and belong to our group SIN of small
networks. We will validate the pair (M,G) as a repressive miRNA-mRNA
pair if and only if at least one of the networks NETibelonging to
Net(M,G) has been validated, as just outlined, by exhibiting small
enough global errors of prediction.
Once an miRNA-mRNA pair (M,G) is actually validated, we
can then rank all the validated networks NETi belonging to
Net(M,G) in decreasing order of reliability, which is equivalent to
the increasing order for their errors of prediction ErrPred. To
break ties between networks models with comparable ErrPred
values, we apply a parameter parsimony principle and give priority
to models with smaller numbers of parameters.
Results
Validated miRNA-mRNA Pairs of Transcr.Degr. Type
Our main results on validated miRNA-mRNA pairs of
Transcr.Degr. type are summarized in Tables 1 and 2.
We had initially constructed 364 small networks of Tran-
scr.Degr. type, for which the mRNA downstream targets belonged
to one of the 3 following sets of ES cell regulatory genes: the self-
renewal regulators (Sox2, Klf4, cMyc, Tbx3, Esrrb), the
differentiation inhibitors (Ezh1, Ezh2, Eed), and the differentiation
regulators Oct4 and Nanog.
For the 2 downstream targets Oct4 and Nanog, as seen in
Figure 2, the proteins produced by Oct4, Nanog, Sox2 are
transcription activators [8,11], GCNF is a key transcription
repressor, and the potential miRNAs transcription repressors
included 19 miRNAs for Oct4 and 2 miRNAs for Nanog. We had
generated potential lists of upstream transcription repressors for
Oct4 and for Nanog by combining each one of these 21~(19z2)
miRNAs with GCNF and with one of the 7 short lists (Oct4),
(Nanog), (Sox2), (Oct4, Nanog), (Oct4, Sox2), (Nanog, Sox2),
(Oct4, Nanog, Sox2).
After parametrization and validation of these Transcr.Degr.
models repressing the downstream target Oct4, only 5 repressing
pairs (miRNA,Oct4) were validated by high quality of fit to data,
and they involved the five miRNAs (miR-24, miR-103, miR-107,
miR-186, miR-466). Each one of the 5 corresponding validated
Transcr.Degr. architectures combined one of these 5 miRNA
repressors of Oct4 with the 3 transcriptional repressors (GCNF,
Oct4, Nanog).
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Figure 4 displays expression profiles corresponding to one of
these 5 validated Transcr.Degr. model, with upstream transcrip-
tion repressors mmu-miR-186, GCNF, Oct4, Nanog, and with
transcription activators (protein Oct4, protein Nanog).
For the 2 downstream targets Nanog and Esrrb, as reported in
Table 1, there were no validated repressing pair (miRNA, Nanog)
or (miRNA, Esrrb) of Transcr.Degr. type. However, our datasets
of recorded microarray data contained only 266 known miRNAs,
so our modeling could not include a small group of miRNAs
known to be potentially targeting Nanog or Esrrb, but for which
we had no recorded microarray data.
We omit details (see [14]) and refer to Tables 1 and 2 for the
other 6 downstream targets (Klf4, cMyc, Tbx3, Ezh1, Ezh2, Eed).
Over all our initial 364 models of Transcr.Degr. type, only 88
models were validated after parametric modeling. In Table 2, we
list all the 85 validated miRNA-mRNA pairs of Transcr.Degr. type
repressing one of the 10 downstream targets (Oct4, Nanog, Sox2,
Klf4, cMyc, Tbx3, Esrrb, Ezh1, Ezh2, Eed).
Validated miRNA-mRNA Pairs of Transl.Inhib. Type
We had initially generated a list of 5337 small networks of
Transl.Inhib. type potentially inhibiting the translation of one of the 4
downstream targets (Oct4, Nanog, Sox2, GCNF). Each one of these
5337 networks involved at least 1 and at most 3 of the 133 miRNAs
targetingoneofthe4downstreamgenes(Oct4,Nanog,Sox2,GCNF).
After parametric modeling of these 5337 networks, and
validation by requesting high quality of fit to data, we have
validated only 24 miRNAs as translation inhibitors repressing one
of these 4 downstream proteins Oct4, Nanog, Sox2, GCNF. These
results are summarized in In Tables 3 and 4.
Among the 19 miRNAs targeting Oct4, only 13 were validated
as inhibiting the translation of Oct4. Each one of these 13 miRNA
inhibitors of Oct4 was included in several of the 51 validated
Transl.Inhib. architectures repressing Oct4. Each one of these 51
validated architectures involved a group of 3 miRNAs inhibiting
the translation of Oct4.
Figure 5 displays the expression profiles for one example of
validated network inhibiting the translation of Oct4 through 3
upstream miRNA repressors (mmu-miR-542-3p, mmu-miR-484,
mmu-miR-138).
Among the 11 validated miRNAs inhibiting the translation of
GCNF, we find no miRNA in the high-low class HL, 4 miRNAs in
the low-high class LH, and 7 miRNAs in the transient class TR.
This result agrees quite well with [5], which indicates that
miRNAs of class HL do not repress GCNF but that some miRNAs
of class LH may repress GCNF.
Among the 83 miRNAs targeting GCNF, we have validated
only 11 miRNAs inhibiting the translation of GCNF. For the
GCNF protein, we naturally only have profile data for the WT ES
cells since GCNF vanishes in GCNF-KO data. The number of
data points for GCNF data hence half of the data points available
for other proteins. The Transl.Inhib. architectures potentially
repressing GCNF were thus restricted to include only one miRNA
targeting GCNF, in order to restrict the number of model
parameters.
Finally, among the 31~2z29 miRNAs potentially targeting
Nanog or Sox2, none could be validated within a translation
inhibiting pair of the form (miRNA, Nanog) or (miRNA, Sox2).
Let us clarify further this situation. Since the Nanog protein is
potentially repressed by only 2 miRNAs (see [4]), we could include
Table 1. Transcr.Degr. architectures: numbers of validated miRNA-mRNA pairs.
mRNA gene GOct4NanogSox2Klf4EsrrbcMycTbx3Ezh1Ezh2 Eed
# of miRNAs targeting G19229 44 10121851 2925
# of validated miRNAs50920 1218 20127
doi:10.1371/journal.pone.0023263.t001
Table 2. Transcr.Degr. architectures: the 85 validated miRNA-mRNA pairs.
miR-24; miR-103; miR-107; miR-186; miR-466Oct4
miR-19a; miR-19b; miR-21; miR-129-3p; miR-182; miR-290; miR-292-5p;Sox2
miR-339; miR-431
miR-19a; miR-29b Klf4
let-7b; let-7c; let-7f; let-7i; miR-96; miR-98; miR-135a; miR-135b;cMyc
miR-182; miR-212; miR-340; miR-451
miR-17-3p; miR-17-5p; miR-20a; miR-20b; miR-26a; miR-26b; miR-93;Tbx3
miR-106b; miR-126-5p; miR-142-3p; miR-142-5p; miR-146; miR-146;
miR-106a; miR-338; miR-448; miR-466; miR-469
miR-15a; miR-15a; miR-15b; miR-16; miR-16; miR-22; miR-28;Ezh1
miR-195; miR-195; miR-291a-3p; miR-301; miR-301; miR-302c; miR-323;
miR-145; miR-183; miR-329; miR-342; miR-345; miR-449
let-7a; let-7b; let-7c; let-7d; let-7e; let-7f; let-7g; let-7i;Ezh2
miR-26a; miR-26b; miR-98; miR-98
miR-1; miR-30a-3p; miR-101a; miR-301; miR-323; miR-337; miR-34bEed
doi:10.1371/journal.pone.0023263.t002
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Page 8

one or both of these miRNAs in only 3 potential Transl.Degr.
architectures. The fact that none of these 3 networks could be
validated by high quality of fit to data is not surprising, since the
rate of attrition due to model validation was fairly high in general.
For the Sox2 protein, we had a choice of 29 miRNAs potentially
targeting Sox2, and hence a large set of 1653 Transl.Inhib. models
to parametrize. For all these models, after parametric fitting to
both WT and GCNF-KO data,the estimated translation rate c was
almost zero. So no matter what upstream miRNAs were selected,
the modeling prediction for Sox2 remained the same. We have
traced this effect to the modeling assumption that the degradation
rates of the downstream protein remains the same for both WT
and GCNF-KO experiments. But our Western Blots data indicate
that the Sox2 protein degradation rate in WT ES cells seems to be
larger than in GCNF-KO ES cells.
Note that the half-life of a protein like Sox2 may quite possibly
change in different cell cultures; for instance, protein Oct4 has a
half-life of 90 minutes in undifferentiated P19 cells [19] and of 6 to
8 hours in NIH3T3 cells transfected with wild type Oct4 [20].
If we now generate parameterized models of the same
translation-inhibition architectures, but with the new assumption
that the half-life of protein Sox2 is different for WT cells and for
GCNF-KO cells, then we obtain much better model predictions.
The estimated half-life of protein Sox2 is then 20.5 hours for WT
cells, and 64.5 hours for GCNF-KO cells. We have left this
question opened for the moment, until further experiments help us
generate a concrete conclusion on the possibly distinct values for
the half-life of protein Sox2 in WT cells and in GCNF-KO cells.
Discussion
Our goal was to analyze the regulatory role of miRNAs in ES
cells differentiation, on the basis of two sets of microarray data,
recorded for WT and for GCNF-KO ES cells.
We have modeled, by 2 basic chemical kinetic equations 6 and
6, the 2 main functions of miRNAs in post-transcriptional down-
regulation of genes expression. These CKEs correspond to 2
Figure 4. Example of small network of Transcr.Degr. type repressing mRNA Oct4. All expression profiles are over days 0–6. Top profiles:
miRNA miR-186 for WT and GCNF-KO data. Middle profiles: transcription factors of Oct4 for WT and GCNF-KO data. Blue solid line=protein Oct4.
Green dash line=protein Nanog. Red dotted-solid line=protein GCNF. Bottom profiles: mRNA Oct4 for WT and GCNF-KO data. Blue line=recorded
levels. Red dash line=predicted levels. ‘‘prediction error’’ is the model global relative error of prediction; std is the relative standard deviation of
recorded levels.
doi:10.1371/journal.pone.0023263.g004
Table 3. Transl.Inhib. architectures: numbers of validated
miRNA-mRNA pairs.
mRNA gene G Oct4Nanog Sox2GCNF
# of miRNAs targeting G 19229 83
# of validated miRNAs 130011
doi:10.1371/journal.pone.0023263.t003
Table 4. Transl.Inhib. architectures: the 24 validated miRNA-
mRNA pairs.
miR-103; miR-107; miR-138; miR-186; miR-218; miR-24; miR-324-5p; Oct4
miR-337; miR-338; miR-369-5p; miR-466; miR-484; miR-542-3p
let-7e; let-7g; miR-10a; miR-10b; miR-23b; miR-30c;GCNF
miR-124a; miR-181b; miR-214; miR-351; miR-382
doi:10.1371/journal.pone.0023263.t004
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Page 9

formal types (Transcr.Degr. and Transl.Inhib.) of interaction
architectures between miRNA, mRNA, and associated proteins.
Starting with the 266 miRNAs recorded in our microarray data,
we have focused on their potential repressive impact on 11 key
regulatory mRNA genes of ES cells differention, namely (Oct4,
Nanog, Sox2, Klf4, cMyc, Tbx3, Esrrb, Ezh1, Ezh2, Eed) for the
Transcription Degradation modality and the (Oct4, Nanog, Sox2,
GCNF) proteins for the Translation Inhibition modality.
We have retained as potentially interacting only the miRNA-
mRNA pairs (M,G) in which M targets G according to miRanda
or Targetscan. Each such miRNA-mRNA pair was naturally
imbedded within one of 5701 small genes interaction networks,
namely 364 networks of Transcr.Degr. type and 5337 networks of
Transl.Inhib. type. We have developed an innovative algorithmic
approach to parametrize the corresponding 5701 chemical kinetics
equations by optimizing their quality of fit to our two sets of
microarray data. Our numerical algorithm solves efficiently a
nonlinear least squares fitting of 38 high degree algebraic
equations involving between 3 and 15 unknown parameters. Each
one of the 5701 parametrized CKE we thus obtained was then
tested for the accuracy with which the CKE could predict the
expression profiles of its downstream mRNA gene target (or
protein target), for comparison with our WT and GCNF-KO
microarray data. The corresponding small network was considered
as valid if and only if the predicted and recorded expression
profiles of these down stream target were well matched, with a
relative error of prediction inferior to 10%.
The list VAL of interaction networks which were validated by
high quality of fit to data was naturally much smaller than our
initial list of 5701 small networks modeling potential miRNA-
mRNA interactions. We have then considered that a potential
miRNA-mRNA pair was validated if and only if it had been
imbedded in at least one of the networks belonging to VAL. We
have thus determined 109 ‘‘model validated’’ miRNA-mRNA
pairs, namely 85 pairs interacting by Transcription Degradation of
the mRNA target, and 24 pairs interacting by inhibiting the
translation of their gene target into protein. These results,
summarized in Tables 1, 2, 3, 4, should help to circumscribe
further experimental gene expression analyzes on miRNA-mRNA
pairs.
For any given mRNA gene G in our 11 key genes regulating of
ES cells differentiation, our results provide very short lists of model
validated miRNAs repressing G. A typical experimental validation
will be to first pick two such miRNA candidates, Madegrading the
transcription of G, and Mbinhibiting the translation of G. As in
(see Methods in [4]), one could transfect one set of wild-type ES
cells with precursor miRNA (pre-miRNA) oligomers associated to
Ma and similarly transfect another set of ES cells with Mb.
Recording the expression levels of mRNA gene G should enable
the comparison of the two rates of change of G after a short
transfection time, for both sets of ES cells, considering that, in such
a short time, the expression levels for the transcription factors of G
are not yet likely to be influenced to any great extent. Prediction
by parameterized modeling should help the quantitative interpre-
tation of the experimental recordings and enable concrete
conclusions on the interactive parts played by Maand Mb.
When fairly comprehensive knowledge of the transcriptional
factors for a specific mRNA gene G is available, and for any new set
of microarraydata, we canimplement ourautomated modelingand
validation of miRNA-mRNA pairs involved in transcription-
degradation architecture just as above. A key facilitating point
would be the availability of associated proteins expression levels,
Figure 5. Example of Transl.Inhib. architecture repressing Oct4. All expression profiles are over days 0–6. Upper 6 profiles: miRNAs miR-542-
3p, miR-464 and miR-138 for WT and GCNF-KO. Bottom 2 profiles: Blue line=recorded level. Red dash line=predicted levels.
doi:10.1371/journal.pone.0023263.g005
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Page 10

which are difficult to measure simultaneously for a large set of
proteins. But we can still use our Transcr.Degr. CKEs to determine
whether the proteins with actually recorded expression levels are
indeed transcriptional factors for the mRNA gene G.
Author Contributions
Conceived and designed the experiments: AJC PHG XX PG. Performed
the experiments: XX PG. Analyzed the data: ZL RA. Contributed
reagents/materials/analysis tools: ZL RA DL. Wrote the paper: ZL RA.
Reference
1. Ambros V (2004) The functions of animal microRNAs. Nature 431(7006):
350–5.
2. Aravin A, Tuschl T (2005) Identification and characterization of small RNAs
involved in RNA silencing. FEBS Lett 579(26): 5830–40.
3. Bartel DP (2009) MicroRNAs: target recognition and regulatory functions. Cell
136(2): 215–33.
4. Tay Y, Zhang J, Thomson AM, Lim B, Rigoutsos I (2008) MicroRNAs to
Nanog, Oct4 and Sox2 coding regions modulate embryonic stem cell
differentiation. Nature 455: 1124–1128.
5. Gu P, Reid JG, Gao X, Shaw CA, Creighton C, et al. (2008) Novel miRNA
candidates and miRNAmRNA pairs in ES cells. PLoS ONE 3(7): e2548.
6. Tsang J, Zhu J, van Oudenaarden A (2007) MicroRNA-mediated feedback and
feedforward loops are recurrent network motifs in mammals. Mol Cell 26:
753–67.
7. Cohen SM, Brennecke J, Stark A (2006) Denoising feedback loops by
thresholding–a new role for microRNAs. Genes Dev 20: 2769–72.
8. Marson A, Levine SS, Cole MF, Frampton GM, Brambrink T, et al. (2008)
Connecting microRNA genes to the core transcriptional regulatory circuitry of
embryonic stem cells. Cell 10: 1016.
9. Saxe JP, Tomilin A, Scho ¨lerr HR, Plath K, Huang J (2009) Post-translational
regulation of Oct4 transcriptional activity. PLoS ONE 4(2): e4467.
10. Miller L (2010) Quantifying western blots without expensive commercial
quantification software. Available: Stuff: Miscellaneous topics all vaguely related
to science. http://lukemiller.org/journal/2007/08/quantifying-western-blots-
without.html Accessed: 2010 August.
11. Boyer LA, Lee TI, Cole MF, Johnstone SE, Levine SS, et al. (2005) Core
transcriptional regulatory circuitry in human embryonic stem cells. Cell 122(6):
947–56.
12. Goutsias J, Kim S (2004) A nonlinear discrete dynamical model for
transcriptional regulation: construction and properties. Biophysical Journal
86(4): 1922C1945.
13. Goutsias J, Lee NH (2007) Computational and experimental approaches for
modeling gene regulatory networks. Current Pharmaceutical Design 13:
1415–1436.
14. Luo Z (2010) Modeling genes interaction: fitting chemical kinetics ordinary
differential equations to microarray Data, Ph.D. dissertation, Dept of
Mathematics, University of Houston.
15. Periwal V, Chow CC, Bergman RN, Ricks M, Vega GL, et al. (2008) Evaluation
of quantitative models of the effect of insulin on lipolysis and glucose disposal.
Am J Physiol Regul Integr Comp Physiol 295: R1089–R1096.
16. Gregory PC (2005) Bayesian logical data analysis for the physical sciences: a
comparative approach with mathematical support. Cambridge, UK: Cambridge
University Press.
17. Ku ¨gler P, Gaubitzer E, Mu ¨ller S (2009) Parameter identification for chemical
reaction systems using sparsity enforcing regularization: a case study for the
Chlorite-Iodide reaction. J Phys Chem A 113: 2775–2785.
18. Azencott R (2008) Parsimonious ODE parameterization for transcriptional
regulatory networks, in Colloquium ‘‘Modeling and analysis of biological
networks’’. University of Houston and MD Anderson Cancer Center.
19. Saxe JP, Tomilin A, Scho ¨ler HR, Plath K, Huang J (2009) Post-translational
regulation of Oct4 transcriptional activity. PLoS ONE 4(2): e4467.
20. Wei F, Scholer HR, Atchison ML (2007) Sumoylation of Oct4 enhances its
stability, DNA binding, and transactivation. J Biol Chem 282: 21551–21560.
CKE Modeling miRNA Regulatory Circuits
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