Modeling sRNA-dependent circuits

QUANTITATIVE MODELING OF GENETIC CIRCUITS
INTEGRATING TRANSCRIPTIONAL AND SRNA
MEDIATED REGULATIONS
Matteo Brilli
INRIA - RHONE-ALPES
LBBE
UMR CNRS 5558
UNIV-LYON1
Trento November 27, 2012
(INRIA, CNRS, UNIV-LYON1) MODELING INTEGRATED GENETIC CIRCUITS NOVEMBER 27, 2012 1 / 31

TOC
1 INTRODUCTION
2 EXAMPLES
3 BASICS OF MATHEMATICAL MODELING
4 NETWORK MOTIFS AND THEIR DYNAMICAL PROPERTIES
5 MODELING SRNA REGULATION
Dynamical properties of sRNA-transcription integrated circuits

INTRODUCTION
GENERAL FEATURES
1 sRNAs are today recognized as pivotal post-transcriptional regulators;
2 size ranges from 50 to a few hundreds nucleotides;
3 the majority modulate gene expression by direct base-pairing with target
mRNA;
4 regulation is predominantly negative;
5 increasing evidence of multiple targets per sRNA.

WIDESPREAD OCCURRENCE

FUNDAMENTAL ROLES IN PATHOGENESIS
FIGURE 1: Gopel2011a

MAIN ROLES IN E. coli
FIGURE 2: Predictions from Modi2011.
A ROLE IN..
Mainly stress and
environmental related
functions.

FULLY INTEGRATED WITHIN THE GENE REGULATORY
NETWORK
FIGURE 3: sRNA are often regulated by speciﬁc transcription factors (TF) and often regulate TFs
Storz2011

MOST COMMON MECHANISMS OF ACTION
FIGURE 4: Waters2009
DIFFERENT MODES OF ACTION
1 Block translation (often bind the
Shine-Dalgarno) and increase
degradation;
2 Increase mRNA degradation;
3 Promote transcription termination;
4 Increase translation rate by removing
inhibitory secondary structures;
5 Act in stoichiometric fashion
(degraded with target);
6 Often in conjunction with Hfq.

HFQ
FIGURE 5: Storz2011
HFQ
1 Interacts with both the
sRNA and the target
mRNA;
2 Interacts with the RNA
degradosome;
3 Affects the translation
and turnover rates of
speciﬁc transcripts;
4 Distant homologues in
Archaea and Eukaryotes.

HFQ
FIGURE 6: Chao2010

SHORT SUMMARY FOR EUKARYOTES
FIGURE 7: RNA regulation in Eukaryotes Kim2005
• Different types of
small RNAs;
• Different helper
proteins/protein
complexes;
• Pre-processing;
• Act catalytically.

RYHB AND IRON HOMEOSTASIS
FIGURE 8: Ferrous iron (Fe2+) is essential but it
becomes toxic in the presence of normal respiratory
by-products (H2O2): → ﬁnely controlled
homeostasis; Salvail2012
UNDER IRON STARVATION...
RyhB is a master regulator of iron
homestasis:
1 stimulates the degradation
of ∼ 18 mRNAs encoding
Fe-proteins;
2 feedbacks on Fur;
3 promotes siderophore
production e.g. activating
shiA mRNA translation;

QRR AND QUORUM SENSING REGULATION IN Vibrio
Quorum-sensing: regulation of gene
expression in response to cell density;
it allows to track population density,
synchronize gene expression on a
population-wide scale, and thereby
carry out collective activities.
FIGURE 9: Fenley2011
DIFFERENT ARRANGEMENTS → DIFFERENT
PHENOTYPES
1 V. harveyi produces and monitors the
concentrations of 3 autoinducers (AI), V.
cholerae produces and monitors 2 AIs;
2 AI-1 and AI-2 act additively in V. harvey, but
redundantly in V. cholerae;
3 ∆luxU: always bright (density-independent)
in V. harveyi but not in V. cholerae.
4 ∆ sensor kinases (e.g. cqsS and luxQ)
changed the luminescence phenotype in V.
harveyi but not in V. cholerae.

QRR AND QUORUM SENSING REGULATION IN Vibrio
harvey
FIGURE 10: Input-output relation for the WT and mutated genetic circuits of
quorum-sensing in V. harvey. [Teng2011].
Different strains with one or more
regulatory feedback destroyed and single
cell ﬂuorescence measurements as a
function of AI-1 and AI-2 concentrations.
RESULTS
1 Feedback into LuxN allows V.
harvey to actively adjust its relative
sensitivity to AI signals as cells
transition from low to high cell
densities;
2 The other feedback loops control
the input and output dynamic
ranges and the noise in the circuit.

VIRULENCE REGULATION IN Clostridium perfringens
FIGURE 11: [Frandi2010]
HERE COMES THE QUESTION...
How to study the different regulatory schemes?

COMPARATIVE SYSTEMS BIOLOGY
Sinorhizobium
meliloti
Caulobacter
crescentus
Rhodobacter
sphaeroides
CckA-RD
CckA-HK~P
CckA-RD~P
CckA-HK
CtrA
ChpT~P
CtrA~P
ChpT
CpdR~PCpdR
proteolysis
(ClpX)
Phosphorelay
DivK~P
DivK
DivJ
DivJ~P
PleC
PleC + Pi
CckA-RD
CckA-HK~P
CckA-RD~P
CckA-HK
CtrA
ChpT~P
CtrA~P
ChpT
CpdR~PCpdR
proteolysis
(ClpX)
Phosphorelay
CtrA GcrA DivK~P
DivK~P
DivK
DivJ
DivJ~P
PleC
PleC + Pi
CckA-RD
CckA-HK~P
CckA-RD~P
CckA-HK
CtrA
ChpT~P
CtrA~P
ChpT
CpdR~PCpdR
proteolysis
(ClpX)
Phosphorelay
CtrA GcrA DivK~P
CtrA GcrA
Modeled circuit Dynamics
FIGURE 12: Studying the phenotypes of different regulatory circuits in
different organisms.
1 Reconstruct circuit in
different species;
2 Build corresponding
mathematical models;
3 Compare dynamical
behaviors;
4 Similarly: mutate the
same circuits and
explore how its
properties change.

MODELING GENE REGULATORY NETWORKS
TF

target

Cons-tu-ve
synthesis

Regulated
synthesis
Degrada-on
and
dilu-on

A) Promoter activity:
• Positive regulation: A = h+
(TF, θ, n) = TFn
θn+TFn ;
• Negative regulation: A = h−
(TF, θ, n) = 1 − h+
(TF, θ, n);
• Combinatorial regulation: e.g.
for an AND logic: AAND
=
N
i=1
Ai;
for OR logic: AOR
=
N
i=1
Ai.
B) Protein degradation rate γ;
C) Dilution µ.
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
x
y=xn
xn+θn
Positive Hill
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
x
y=θn
xn+θn
Negative Hill
n=1
n=2
n=3
n=4
n=5
n=6

NETWORK MOTIFS
FIGURE 13: Feed Forward Loops [Shoval2010].
SUMMARY ON NETWORK MOTIFS
• Transcription regulation and
signaling networks are composed
of recurring patterns called network
motifs;
• Network motifs are much more
abundant in biological networks
than would be expected by their
randomized versions;
• The same small set of network
motifs has been found from
bacteria to plants to humans;

MODELING THE FEEDFORWARD LOOP
-1-1
I-1
SYSTEM OF EQUATIONS FOR THE FFLC-1 AND
dY
dt
= By
Basal
+
Regulated synthesis
κY
Xn
θn
XY + Xn
− (γ + µ)
Degr. and dil.
Y (1)
dZ
dt
= Bz + κZ
Xn
θn
XZ + Xn
Yn
θn
YZ + Yn
AND logic
−(γ + µ)Z. (2)
Let’s put:α = γ + µ.
SYSTEM OF EQUATIONS FOR THE FFLI-1 AND
dY
dt
= By + κY
Xn
θn
XY + Xn
− αY (3)
dZ
dt
= Bz + κZ
Xn
θn
XZ + Xn
θn
YZ
θn
YZ + Yn
− αZ. (4)

NETWORK MOTIFS HAVE SPECIFIC DYNAMICAL
PROPERTIES
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
Z(Y)
FFLC-1 Vs Simple regulation
9.5 10 10.5 11 11.5 12
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
time
Z(Y)
FFLI-1 Vs Simple regulation
9.5 10 10.5 11 11.5 12
0
0.2
0.4
0.6
0.8
1
time
Z FFLC−1
Y FFLC−1
Z Simple
Y Simple
Z FFLI−1
Y FFLI−1
Z Simple
Y Simple
FIGURE 14: [Shen-Orr2002,Mangan2003,Mangan2003a]
FFLC-1
-1-1
• Sign-sensitive delay;
• Persistence detector;
• Noise reduction;
FFLI-1
I-1
• Sign-sensitive
accelerator;
• other incoherent FFL are
good pulsers;

INTRODUCING SRNA-MEDIATED REGULATION
FIGURE 15:
• sRNAs act stoichiometrically;
• sRNAs affect mRNA
stability/translation;
• most regulations are negative;
EQUATIONS OF THE SYSTEM
ds
dt
= Ss −
Degr. and dil.
αs − kms
Degr.complex
(5)
dm
dt
= Sm − αm − kms. (6)
Note that this is the basic modeling framework for all works published on sRNA regulation e.g. from [Levine2007] on.

BUT...
THIS IS AN APPROXIMATION
The [m] and [s] concentrations are in the same order of magnitude → the complex should not be neglected. We can overcome this limitation
by using a saturation function telling which is the complexed fraction of the total form at steady state ( dx
dt
= 0):
YA =
AB
Atot
=
Atot − Afree
Atot
. (7)
ms = mtot − mfree, (8)
stot = sfree + ms, (9)
and
ms =
mfree · sfree
Kd
, (10)
where Kd is the dissociation constant for the complex formation; the free quantities are unknown, the tot are known. After some math we get
the ﬁnal form of the saturation function:
Ym =
mtot + stot + Kd − (stot − mtot + Kd)2 + 4Kdmtot
2mtot
(11)
Using this approach we can simply model the control by the sRNA in the following way:
dmtot
dt
= Sm −
degr. free form
γ · (mtot − Ym · mtot) −
degr. complex
γ2 · Ym · mtot −µ · mtot. (12)

THRESHOLD LINEAR RESPONSE
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
1
2
3
4
5
6
7
mRNA synthesis rate.
[mRNA]SS
plus: Levine et al., 2007 model
circles: Our modificated model
other parameters are:
m
=0.2
s
=0.1
k=3.5354e 01
Kd=0.01
A THRESHOLD LINEAR
RESPONSE:
• When αm αs,
translatable target
mRNA is very small;
• When αm αs, the
target mRNA starts to be
available for translation.

THRESHOLD LINEAR RESPONSE
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
8
m
X
SS
=0.5
=1.0
=4.0
=2.0
FIGURE 16:
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
m
SS
/µ
m
SS
m
FIGURE 17:
A THRESHOLD LINEAR RESPONSE:
• The threshold depends only on
the two transcription rates;
• The smoothness of the transition
depends on the degradation rate
of the complex (half-life given by
log2
γ
h);
• Around the crossover region
differences in steady state levels
of the target are much more
dependent on the interaction
strength Kd.
• This gives an easy way for
ordering the expression of
different genes by tuning αm and
Kd.

SRNA CAN PROVIDE A SWITCH-LIKE BEHAVIOR
10
−2
10
−1
10
0
10
1
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
s
/ m
[targetmRNA]
Ultrasensitivity
=10
=100
=1000
=10000
FIGURE 18: Ultrasensitive switch Mitarai2009.
SLIGHTLY DIFFERENT...
but comparable model from Mitarai2009:
ds
dt
= α − s − γsm (13)
dm
dt
= 1 −
m
τ
− γsm. (14)
where: α = αs
αm
the relative transcription rate of s with
respect to that of m. γ = δαmτs quantiﬁes the
inactivation of the mRNA via sRNA:mRNA complex
formation, and τ = τm
τs
is the ratio of the mRNA
lifetime to the sRNA lifetime.

TARGET PRIORITIZATION
0 0.5 1 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sRNA synthesis rate
normalizedsteadystate[mRNA]
sRNA mediated regulation
0 0.5 1 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Repressor synthesis rate
normalizedsteadystate[mRNA]
Transcriptional repression
PRIORITIZATION
• Multistep switch-like degradation
upon changing the rate of
synthesis of the sRNA;
• Separation of the level between
the (k + 1)th
mRNA and the kth
mRNA becomes clearer
increasing the difference in the
degradation rate.
• Separation between mRNA
levels following transcriptional
regulation is not as sharp as with
sRNA regulation;
• prioritization through sRNA is
much more effective especially
for small changes in the sRNA
synthesis rate.

BISTABILITY
sRNA

TF

THE MODEL
dx
dt
=
Translation
γy − δx
Degr.
(15)
dy
dt
=
Synthesis
λSy(x, p) −αy − σyz (16)
dz
dt
= µSz(x, p) − βz − σyz. (17)
Where: x=protein TF, y=mRNA TF, z=sRNA. [Liu2011].
• Bistability: the capacity to achieve two alternative internal states in response to different
stimuli;
• ubiquitous in cellular systems;
• bistability is generated by regulatory interactions;
• fundamental biological signiﬁcance e.g. cell differentiation, cell fate decision, adaptive
response to environmental stimuli, regulation of cell cycle oscillations and so on.
• switches involving ncRNA have been recently studied experimentally
[Bumgarner2009,Iliopulos2009] and theoretically [Zhdanov2009].

BISTABILITY
THE MODEL
dx
dt
=
Translation
γy − δx
Degr.
(18)
dy
dt
=
Synthesis
λSy(x, p) −αy − σyz (19)
dz
dt
= µSz(x, p) − βz − σyz. (20)
Where: x=protein TF, y=mRNA TF, z=sRNA.
[Liu2011].
• bistability in this case only for intermediate association rates between sRNA and mRNA;
• In the monostable regimen lower degradation rates correspond to higher protein level and vice versa. On the converse, when the
association rate is between A and B (the saddle points) the opposite can be true, depending on the initial conditions.
• the noise inherent in biological systems may induce switching between the two stable states.

CONCLUSIONS
• sRNAs provide an efﬁcient regulatory mechanism;
• integration of transcriptional and sRNA mediated regulations generates a
wide variety of interesting dynamical behaviors, such as threshold linear
response, prioritization of targets and even bistability (and oscillations);
• modeling genetic circuits may provide information on both functionality
and evolution of genetic circuits.

COLLABORATORS AND ACKNOWLEDGEMENTS
A SPECIAL THANKS TO
Equipe Baobab (MF Sagot),
LBBE, INRIA; Daniel Kahn
(LBBE, INRA).
AND FOR YOUR ATTENTION,
Thank you!
contact me: matteo.brilli@univ-lyon1.fr
By the way...looking for a dog? 6 Labrador Chocolate available, contact me!

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