Mathematical Models for Synthetic Biology...ODE Models: Solution Existence and uniqueness of...
Transcript of Mathematical Models for Synthetic Biology...ODE Models: Solution Existence and uniqueness of...
Mathematical Models for Synthetic Biology
Diego di Bernardo, TIGEM Naples / I, [email protected]
Jörg Stelling, ETH Zurich / CH, [email protected]
Synthetic Biology 3.0, Zurich, June 2007
Synthetic Biology Vision
Rational forward-
engineering design of ...
... robust / reliable
biology-based parts and
modules with
standardized interfaces
allowing plug-and-play ...
... and their combination
into complex systems.
Engineering Design & Synthetic Biology
Novel design methods / tools because of 'sloppyness', sto-
chasticity, and limited insulation of components in biology.
NAND Gate
Circuits: Model-based Design Process
Possible design alternatives → Qualitative behavior
Design for quantitative performance specification
Design for reliable function → Robustness
Formalization of the design problem & goals
Steps in Model Development
Steps in Model Development
Level of detail for the mathematical descriptions ?
Modeling approach (qualitative / mechanistic / ...) ?
Experimental data for identification & validation ?
Most important aspects of the system ?
Complete knowledge on components / interactions ?
Exact mechanisms of interactions ?
Modeling Approaches
Interaction-based
Constraint-based
Mechanism-based
A + B C
stoichiometry
A + B Ck
-1
k1
biochemistry
A BC
topology Graph theory
Structural analysis
Dynamic analysis
Modeling Approaches: Comparison
Usefulness for design
Interaction-based
Constraint-based
Mechanism-based
Level of detail / accuracy
Net
wor
k co
mpl
exity
Dynamic Systems Analysis: Approach
Analyze engineered circuits as dynamic (bio)chemical reaction networks → Description of reaction kinetics.
Based on first principles: Conservation of mass (and energy and possibly other constraints).
Theoretical background: Chemical kinetic theory.
(Ordinary) differential equations / Stochastic processes.
Reaction Kinetics: Law of Mass Action
Law of mass action → Concentrations of reacting molecules in thermodynamic equilibrium.
Product of concentrations taken to the power of the stoichiometric factors (reaction order) equals a constant (dependent on temperature, pressure, ...).
Example: 1 C1 A + 2 B
[ A ]⋅[ B ]2
[C ]= k T , p
Reaction Kinetics: Dynamic Systems
Reaction network → System of elementary reactions:
Law of mass action → System of differential equations:
Equivalence to: d c t dt
= N⋅r t
1, j⋅X 1n , j⋅X n
k j
1, j⋅X 1n , j⋅X n
dci t dt
= ∑j=1
q
k j⋅i , j−i , j⋅∏l
c l t l , j
Reaction Kinetics: Dynamic Models
Reactand concentrations c(t) → To be determined.
Stoichiometric matrix N → Systems invariant.
Reaction rates r → Time- and state-dependent:
Kinetic rate law r(∙) → From reaction structure.
Parameters (kinetic constants) p → Identification.
Inputs u(t) → Additional (time-varying) influences.
d c t dt
= N⋅r c t ,u t , p , t
ODE Models: General Form
System of ordinary, first-order, linear or nonlinear differential equations (ODEs) characterized by:
Right hand sides f(x(t),u(t),p) = function in .
System states x(t) = nx x 1 state vector.
Parameters p = np x 1 parameter set.
Inputs u(t) = nu x 1 input vector.
ℝnx
d x t dt
= f x t ,u t , p , t
ODE Models: Solution
Existence and uniqueness of solution to the initial value
problem (IVP) of finding x(t) with given x0 guaranteed.
Three possible ''solution'' methods:
Analytical → Only applicable for simple systems.
Numerical → Always possible for well-posed IVPs.
Graphical → Qualitative analysis methods.
d x t dt
= f x t , p , t , x t0=x0
Example: Two-step Repressor Cascade
Zas P2
R2
M. Kaern & R. Weiss, in Szallasi / Periwal / Stelling (eds.) Systems modelling in cell biology, MIT Press (2006).
Signal-response characteristics → Promoter selection.
Low-pass filter: High I levels – low Z synthesis rate.
Example: Two-step Repressor Cascade
d [R2 ]dt
= a1⋅k1k1⋅[ I 1 ] /K 1
n1
1[ I 1 ]/K 1n1−d 1⋅[R2 ]
d [Z ]dt = a2⋅k 2
k2
1[R2] /K 2n2−d 2⋅[ Z ]
d [R2 ]dt
= a1⋅k1k1⋅[ I 1 ] /K1
n1
1[ I 1 ]/K 1n1−d 1⋅[R2 ]
d [Z ]dt = a2⋅k 2
k2
1[R2] /K 2n2−d 2⋅[ Z ]
Design Cycle
Example: Two-step Repressor Cascade
M. K
aern
& R
. Wei
ss, i
n S
zalla
si /
Per
iwal
/ S
tellin
g (e
ds.)
Sys
tem
s m
odel
ling
in c
ell
biol
ogy,
MIT
Pre
ss (2
006)
.
d [R2 ]dt
= a1⋅k1k1⋅[ I 1 ]/K 1
n1
1[ I 1 ]/K1n1−d 1⋅[ R2 ]
d [Z ]dt
= a2⋅k 2k 2
1[ R2 ]/K 2n2−d 2⋅[Z ]
Example: Two-step Repressor Cascade
Low constitutive activity of P1 and P
2.
M. K
aern
& R
. Wei
ss, i
n S
zalla
si /
Per
iwal
/ S
tellin
g (e
ds.)
Sys
tem
s m
odel
ling
in c
ell
biol
ogy,
MIT
Pre
ss (2
006)
.
d [R2 ]dt
= a1⋅k 1k 1⋅[ I 1 ]/K 1
n1
1[ I 1 ]/K 1n1−d 1⋅[R2 ]
d [Z ]dt = a2⋅k 2
k 2
1[R2 ]/K 2n2−d 2⋅[Z ]
Example: Two-step Repressor Cascade
Constitutive degradation of all proteins.
M. K
aern
& R
. Wei
ss, i
n S
zalla
si /
Per
iwal
/ S
tellin
g (e
ds.)
Sys
tem
s m
odel
ling
in c
ell
biol
ogy,
MIT
Pre
ss (2
006)
.
d [R2 ]dt
= a1⋅k 1k 1⋅[ I 1 ]/K 1
n1
1[ I 1 ]/K 1n1−d 1⋅[R2 ]
d [Z ]dt = a2⋅k 2
k 2
1[R2 ]/K 2n2−d 2⋅[Z ]
Example: Two-step Repressor Cascade
Binding of R1 and I
1 → Cooperative transcriptional activation.
M. K
aern
& R
. Wei
ss, i
n S
zalla
si /
Per
iwal
/ S
tellin
g (e
ds.)
Sys
tem
s m
odel
ling
in c
ell
biol
ogy,
MIT
Pre
ss (2
006)
.
d [R2 ]dt
= a1⋅k 1k 1⋅[ I 1 ]/K 1
n1
1[ I 1 ]/K 1n1−d 1⋅[R2 ]
d [Z ]dt = a2⋅k 2
k 2
1[R2 ]/K 2n2−d 2⋅[Z ]
Example: Two-step Repressor Cascade
Cooperative transcriptional repression of P2 by R
2.
M. K
aern
& R
. Wei
ss, i
n S
zalla
si /
Per
iwal
/ S
tellin
g (e
ds.)
Sys
tem
s m
odel
ling
in c
ell
biol
ogy,
MIT
Pre
ss (2
006)
.
d [R2 ]dt
= a1⋅k 1k 1⋅[ I 1 ]/K 1
n1
1[ I 1 ]/K 1n1−d 1⋅[R2 ]
d [Z ]dt = a2⋅k 2
k 2
1[R2 ]/K 2n2−d 2⋅[Z ]
Circuit Models: Generalizations
Derivation of rate laws or equilibrium binding concen-trations for structurally similar reaction networks yields similar basic functional terms.
Example: Gene G bound by transcription factor T:
Without repression:
Competitive repressor R:
Cooperative binding:
[G⋅T ] =[G ]T [T ]
[T ]K 1[T ]/K I
[G⋅T ] =[G ]T [T ][T ]K
[G⋅T ] = [G ]T [T ]n
[T ]nK n
Circuit Models: Generalizations
General model structure for (simple) genetic circuits:
■Activation of expression of Xi by X
j → μ = 1.
■Repression of expression of Xi by X
j → μ = 0.
■Always: Basal expression / constitutive degradation.
d [ X i ]dt
= ai⋅k i k i⋅[ X j ]
n/K i
n
1[ X j ]n/K i
n− d i⋅[ X i ]
Possible design alternatives → Qualitative behavior
Design for quantitative performance specification
Design for reliable function → Robustness
Formalization of the design problem & goals
Feedback Systems
Feedback of module's output signal on the input signal.
Main categories: Positive feedback / negative feedback.
Essential for: Controllers, switches, oscillators, ...
OutputBranch 1Signal
+/-
Feedback Systems: Simple Types
Patterns of interactions between two components:
Positive or negative (net) effect of interactions:
X Y
+
+
X Y
_
----
_
X Y
_
+
--X Y
_
+
--
Positive
Feedback
Mutual
Antagonism
Negative
Feedback
d x t dt
= f x t ,ut , p , t ⇒∂ f i x t ,ut , p , t
∂ x j≠0
Example Switch: System
Component X: Inactivates component Y → YP.
Component Y: Degrades component X.
Input signal u: Control of production rate for X.
X Y
_
----
_
Abstraction
YP Y
E
u
X
Example Switch: ODE Model
E
u
X
Assuming constant total concentration of Y → YT:
d [ X ]dt
= k 1⋅u−k 2 'k 2⋅[Y ] [ X ]
d [Y ]dt
=k 3⋅[ E ] [Y ]T−[Y ]K M3[Y ]T−[Y ]
−k 4 [ X ] [Y ]K M4[Y ]
R4
R3
R2
R1
YP Y
Example Switch: Numerical Solution
Assume: Different initial concentrations of X / Y.
Convergence to qualitatively different solutions.
Example Switch: Graphical 'Solution'
Derivatives dx(t)/dt define vector field in state space.
Qualitative analysis for two-dimensional systems:
Nullclines: Zero velocity in one dimension.
Steady states: Zero velocity in both dimensions.
x t0=x 0
d x t dt
= f x t , p , t x2
x1
xx0
x(t)
Example Switch: Nullclines
States with zero velocity in one of the directions (nullclines):
d [ X ]dt
= 0 ⇒ [Y ] =k 1⋅u−k 2 '⋅[ X ]
k 2⋅[ X ]
d [Y ]dt
= 0 ⇒k 3⋅[E ] [Y ]T−[Y ]K M3[Y ]T−[Y ]
=k 4 [ X ] [Y ]K M4[Y ]
E
u
Xd [ X ]
dt= k 1⋅u−k 2 'k 2⋅[Y ] [ X ]
d [Y ]dt
=k 3⋅[ E ] [Y ]T−[Y ]K M3[Y ]T−[Y ]
−k 4 [ X ] [Y ]K M4[Y ]R
4
R3
R2
R1
YP Y
Example Switch: Y-Nullcline
Y-nullcline in original variables:
Introduction of new variables:
Rescaled equation for Y-nullcline:
v11− yJ 2 y = v2⋅y J 11− y
k 3⋅[E ] [Y ]T−[Y ]K M3[Y ]T−[Y ]
=k 4 [ X ] [Y ]K M4[Y ]
y =[Y ][Y ]T
, v1 = k 3⋅[E ] , v2 = k 4⋅[ X ]
J 1 =K M3
[Y ]T, J 2 =
K M4
[Y ]TE
u
X
R4
R3
R2
R1
YP Y
Example Switch: Y-Nullcline
Rescaled equation for Y-nullcline:
Solution in new variables → Goldbeter-Koshland function:
B = v2−v1v2 J 1v1 J 2
y = [Y ][Y ]T
, v1 = k3⋅[ E ] , v2 = k4⋅[ X ]
J 1 =K M3
[Y ]T, J 2 =
K M4
[Y ]T
v11− yJ 2 y = v2⋅y J 11− y
y = G v1 , v2 , J 1 , J 2 =2 v1 J 2
BB2−4 v2−v1v1 J 2
E
u
X
R4
R3
R2
R1
YP Y
Example Switch: Y-Nullcline
Sigmoidal function of input X → Switch-like for 0 < J1,J
2 << 1.
y =[Y ][Y ]T
, v1 = k3⋅[ E ] , v2 = k4⋅[ X ] , J 1 =K M3
[Y ]T, J 2 =
K M4
[Y ]T
y = G v1 , v2 , J 1 , J 2 =2 v1 J 2
BB2−4v2−v1v1 J 2
, B = v2−v1v2 J 1v1 J 2
J1,J
2 large
J1,J
2 small
[Y]/[
Y]T
[X]
Example Switch: Y-Nullcline
General: Switch-like functions using reversible reactions.
Necessary: High affinities and / or excess of total regulator.
y =[Y ][Y ]T
, v1 = k3⋅[ E ] , v2 = k4⋅[ X ] , J 1 =K M3
[Y ]T, J 2 =
K M4
[Y ]T
J1,J
2 large
J1,J
2 small
YP Y
v2
v1
R4
R3
[Y]/[
Y]T
[X]
Example Switch: Qualitative Behavior
Exampletrajectory
X-Nullcline
Y-Nullcline
Stablesteadystate
Unstablesteadystate
Stablesteadystate
Classification of steady states (nodes) according to directions of the vector field:
unstable node stable node saddle point (unstable)
Stability: Global versus local (w.r.t. 'small' perturbations).
Example Switch: Stability
Example Switch: Response to Input
X-Nullcline:
Bifurcation: Change of the number of attractors in a (nonlinear) dynamic system upon parameter changes.
[X
] ss
ucrit1
ucrit2u
YP Y
E
u
Xstable
unstable
stable
d [ X ]dt
= 0 ⇒ [Y ] =k 1⋅u−k 2 '⋅[ X ]
k 2⋅[ X ]
Example Switch: Response to Input
[X
] ss
ucrit1
ucrit2u
YP Y
E
u
Xstable
unstable
stable
For u < ucrit1
and u > ucrit2
: Globally monostable system.
For ucrit1
≤ u ≤ ucrit2
: Bistable system → Switch possible.
History dependence of the system's state (here with respect to changes in the input): Hysteresis.
Functional implication for circuit behavior: Memory.
Example Switch: Response to Input
[X
] ss
ucrit1
ucrit2u
YP Y
E
u
X
Switches: Generalization
Analysis of alternative designs for biological switches.
Phase plane analysis, multiplicity of steady states.
Mechanisms: Cooperativity (at least in one branch).
J. C
herr
y &
F. A
dler
, J. t
heor
. Bio
l. 20
3:11
7 (2
000)
.
X Y
_
----
_[Y ]
[ X ]
Feedback Systems
Main categories: Positive feedback / negative feedback.
Essential for: Controllers, switches, oscillators, ...
And beyond switches relying on mutual repression ... ?
OutputBranch 1Signal
+/-
Positive Feedback: Functions
Simple positive feedback systems:
Multiple (stable / unstable) steady states possible.
Phenomenon in nonlinear systems: Hysteresis.
Functions in biological networks:
Discrete decisions from continuous signals.
Irreversibility of decisions, e.g. in development.
Positive Feedback: Realizations
Feedback: γ
M. Kaern & R. Weiss, in Szallasi / Periwal / Stelling (eds.) Systems modelling in cell biology, MIT Press (2006).
Negative Feedback: Functions
Simple negative feedback systems:
Approaching steady state (transient dynamics).
Existence of a unique steady state.
Functions in biological networks:
Set point regulation → Homeostasis.
Rejection of external or internal perturbations.
Negative Feedback: Realization
From: Becskei & Serrano (2000) Nature 405: 591-593.
Feedforward Systems
Common input and output, propagation via separate paths.
Behavior depends on signs and timing for the branches.
Branch 2
Output
Branch 1
Signal
+/-
+/-
Feedforward Systems: Functions
Positive branch OR delayed negative branch: Pulse generator. Negative low-pass NOR negative high-pass: Bandpass filter. Positive branch AND positive branch: Low-pass frequency filter. Many others: Speed-up of signaling, signal filtering, ...
Output
Branch 2Branch 1
Signal
From: Shen-Orr et al. (2002) Nat. Genetics 31: 64-68.
Complex Circuits: Basic Approaches
Alternative #1: Augmentations at the module level:
Additional feedback / feedforward loops.
Aim: More complicated systems dynamics.
Alternative #2: Combination of modules:
Modules with defined input / output behavior.
More complicated circuits through linking basic elements (cascades, switches, oscillators, ...).
Example: Pattern Generator
Combination of simple standard building blocks: Genetic filters.
Design: Modularization and specific interconnections.
Signal processing device
High-pass filter
Low-pass filterOutput device
Input device
Example: Repressilator
Proof-of-principle for oscillator design, yet:
Stable oscillations not achieved.
High sensitivity to molecular noise.
M. Elowitz & S. Leibler, Nature 403:335 (2000).
Challenges: Models & Reality
Possible design alternatives → Qualitative behavior
Design for quantitative performance specification
Design for reliable function → Robustness
Formalization of the design problem & goals
How to analyze performance?
How to obtain parameters?
How to deal with noise?
Further Reading
M. Kaern & R. Weiss. Synthetic gene regulatory systems. In:
Szallasi / Periwal / Stelling (eds.) System modeling in cell biology.
(MIT Press, Cambridge / MA) (2006).
J.J. Tyson, K.C. Chen & B. Novak. Sniffers, buzzers, toggles and
blinkers: dynamics of regulatory and signaling pathways in the cell.
Curr Opin Cell Biol. 15, 221 – 231 (2003).
J.L. Cherry & F.R. Adler. How to make a biological switch. J. theor.
Biol. 203: 117 – 133 (2000).
E. Andrianantoandro, S. Basu, D. K. Karig & R. Weiss. Synthetic
biology: new engineering rules for an emerging discipline.
Molecular Systems Biology 2: 0028 (2006).
dibernardo.tigem.it1
Synthetic Biology 3.0
Reverse-engineering gene networks
Diego di BernardoTIGEM
Telethon Institute of GEnetics and Medicine
www.tigem.it
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Overview:
• Networks in Biology
• Reverse-engineering gene networks of unknown topology(de novo)
• Parametrisation of network with known topology
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Gene Networks
cell membrane
metabolites
proteins
RNA
genes
transcriptnetworks
proteinnetworks
metabolicnetworks
Our focus: methods to decode transcription regulation networks
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How can we describe gene interactions: Network theory
• The cell is the result of many sub-components working together• Graph (network) theory is useful to describe such systems• Definitions:
– graph G={V,E} where V is a set of verteces ornodes, and E is a set of edges
– degree k: number of edges connected to a node– digraph: the edges have a direction– P(k) degree distributin: probability that a node has
degree k: P(k)=N(k)/N– C(k) clustering: if node A is connected to node B, and
B to C, are A and C connected?
Barabasi et al, Nature Review Genetics, 2004, 5:101: http://www.nd.edu/~networks/PDF/Wuchty03_NatureGenetics.pdf
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Types of network
• Random networks:– Node have similar degrees
• Scale-free networks:– P(k)=k-g few nodes have
a lot of edges (hubs)– Internet, gene networks,
social networks• Hierarchical networks
– Modules– Scale-free
Barabasi et al, Nature Review Genetics, 2004, 5:101: http://www.nd.edu/~networks/PDF/Wuchty03_NatureGenetics.pdf
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Biological networks
• Biological processes can be represented as networks:– Transcriptional networks (protein-DNA)=digraph
• Nodes: genes and proteins• Edges: a TF activaes/inhibits a gene
– Protein-protein networks = graph• Nodes: proteins• Edges: the two proteins interact
– Metabolic networks:• Nodes: metabolites• Edges: there is an enzyme transforming the two
products
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Why “de novo”? example of transcriptional network (E. coli):
• From the structure of the network we canlearn its function.
• For synthetic biology: what are the genes thatwe “replace” in the cell doing?
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What info can we gain? protein-protein interaction network (yeast S. cerevisiae)
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Reverse engineering (or inference) gene networks:
?Unknown network Inferred network
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“System Identification” or “reverse engineering”
INPUT(S) OUTPUT(S)
Input: perturbations to the system (i.e. gene overexpression)
Output: measure response to perturbations (40’000 genes)
?
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INPUT(S) OUTPUT(S)
To infer a network means to find what is inside the “black box”
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Measuring cell activity: experimental methods
• We need to measure input and output of the cell to tackle theidentification process:– There are at least 40’000 genes, i.e. 40’000 species of
mRNA and 40’000 species of proteins…and counting– A “revolution” has been the creation of microarrays to
measure mRNAs levels simultaneously for all the genes– This is not yet possible for proteins or metabolites…but
we are almost there…
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Reverse engineering gene networks
Goal: Learn structure and function from expression data
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Reverse-engineering networks can help in understanding thedisease:
?
Unknown network Inferred “healthy” network Inferred “disease” network
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Methods to reverse-engineer gene networks:
• Given the experimental data, how can we reverse-engineerthe network?
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Reverse-engineering strategy:
• Choose a model• Choose a fit criterion (cost function) to measure the fit of the model to the data• Define a strategy to find the parameters that best fit the data (i.e. that minimise
cost function)• Perform appropriate experiments to collect the experimental data:
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Reverse-engineering strategy:
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Reverse-engineering strategy: Information-theoretic approach
• Assume that the joint probability can be computed as acombination of 2nd order probabilities, i.e. look only at pairof genes.
• Compute Mutual Information I(x,y) for a pair of gene:
• The MI can be computed directly as:
• In practice:
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Reverse-engineering strategy: Bayesian Networks
• Using the Markov rule
• Choose a network topology G• Compute joint probability function P(D/G)• Score each network (i.e. BDe)
• Iterate the above steps and choose among thenetworks the one with highest score
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Reverse-engineering strategy: ODEs
dX1/dt = 0 = a2 X2 + a6 X6 + a9 X9 + a12 X12
promoters
RNAs
Directed graph
+ u
u
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Reverse-engineering strategy: ODEs
dX1/dt = 0 = a2 X2 + a6 X6 + a9 X9 + a12 X12
promoters
RNAs
Directed graph
+ u
u
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x’11(t) = a11x11+a12x21+...+a1nxn1 + u1
........................................
x’n1(t) = an1x11+an2x21+...+annxn1 + 0
xij i:gene number j: experiment number
Or in matrix format:
x’=Ax+u
Overexpression of gene 1
Model structure:
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x’11(t) = a11x1n+a12x2n+...+a1nxnn + 0
........................................
x’n1(t) = an1x1n+an2x2n+...+annxnn + un
xij i:gene number j: experiment number
Or in matrix format:
x’=Ax+u
Overexpression of gene n
Model structure:
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Fit criterion and search solution strategy:
• Perturb one gene xi at at time and measure the response of the othergenes at steady-state:
• Repeat the experiment overexpressing all of the N genes:
x’(t) = 0 = A x+u
A x = -u
A X=-U => A=-UX-1 not robust to noise
A (N x N), X (N x N), U (NxN)
? known known
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Pilot study: E. coli DNA-damage repair pathway (SOS pathway)
DNA-damage repairpotentially involves 100s ofgenes
Applied NIR to 9 transcriptsubnetwork
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Example perturbation: lexA
7-9 training perturbationsused to recover 9 gene SOS
subnetworkR
elat
ive
Expr
essi
on C
hang
e
recA lexA ssb recF dinI umuDC rpoD rpoH rpoS-0.4
-0.2
0
0.2
0.4
0.6
Pert
urb
ati
on
Gene
Insignificant changes set tozero during data
preprocessing
Data design and collection:
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SOS subnetwork model identified by NIR
lexA
recA
recF
rpoD
rpoS
rpoH dinI
ssb
umuDC
-2.920.67-1.680.22
-0.030.010.10
-0.51-0.17
0.01-0.040.16-1.09
-0.01
0000
00000
0000
0000
0000
000000000
0000
0000
0000
0.08
0.52
0.020.03-0.02
-0.150.20-0.02-0.400.11
0.28
0.030.05-0.28-1.190.04
-0.070.09-0.01-0.670.39
0.10-0.01-0.180.40
Connection strengths
recA
lexA
ssb
recF
dinI
umuDC
rpoD
rpoH
rpoS
rpoSrpoHrpoDumuDdinIrecFssblexArecA
Graphical model Quantitative regulatory model
Majority of previously observed influences discovered despite high noise (68% N/S)
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Methods to find parameters of known networks:
• Given the experimental data, how can we find physical parameters of a knownnetwork?
• Known network means that:– We known the topology– We know the kind of interaction (protein-dna; protein-protein; rna-rna; etc.)
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Parameter fitting strategy: ODEs
• Build network model (known topology)
• Measure mRNA (or protein) levels
• Find parameters of your model:
• For N genes, we have 2N unknownwith M equations, if we chooseM>=2N we can solve the problemwith linear algebra.
• More complex cases (non-linear inthe parameters) require optimisationtechniques like Simulated Annealing
knowns
unknowns
dibernardo.tigem.it30
Parameter fitting strategy: ODEs
• CASE 1 A(t) activity of protein LexA isknown:– For N genes, we have 2N
unknown with M equations, ifwe choose M>=2N we cansolve the problem with linearalgebra.
– More complex cases (non-linearin the parameters) requireoptimisation techniques likeSimulated Annealing
dibernardo.tigem.it31
Parameter fitting strategy: ODEs
• CASE 2 A(t) activity of protein LexA isnot known:– For N genes, we have 2N+M
unknown with M equations– We have an infinity of solutions
of dimension 2N– We choose one using Singular
Value Decomposition
dibernardo.tigem.it32
Parameter fitting strategy: ODEs
-2.920.67-1.680.22
-0.030.010.10
-0.51-0.17
0.01-0.040.16-1.09
-0.01
0000
00000
0000
0000
0000
000000000
0000
0000
0000
0.08
0.52
0.020.03-0.02
-0.150.20-0.02-0.400.11
0.28
0.030.05-0.28-1.190.04
-0.070.09-0.01-0.670.39
0.10-0.01-0.180.40
Connection strengths
recA
lexA
ssb
recF
dinI
umuDC
rpoD
rpoH
rpoS
rpoSrpoHrpoDumuDdinIrecFssblexArecA
dibernardo.tigem.it33
Our lab: TIGEM, Naples, Italy
Diego di Bernardohttp://dibernardo.tigem.it
Mukesh Bansal (physics) Giusy Della Gatta (biology)
Giulia Cuccato, Ph.D. (biology)Francesco Iorio (computer science)
Velia Siciliano (biology)Vincenzo Belcastro (computer science)
Lucia Marucci (mathematics)Mario Lauria, Ph.D. (computer science)