Post on 15-Jul-2020
Mathematical Models for Synthetic Biology
Diego di Bernardo, TIGEM Naples / I, dibernardo@tigem.it
Jörg Stelling, ETH Zurich / CH, joerg.stelling@inf.ethz.ch
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
dibernardo.tigem.it2
Overview:
• Networks in Biology
• Reverse-engineering gene networks of unknown topology(de novo)
• Parametrisation of network with known topology
dibernardo.tigem.it3
Gene Networks
cell membrane
metabolites
proteins
RNA
genes
transcriptnetworks
proteinnetworks
metabolicnetworks
Our focus: methods to decode transcription regulation networks
dibernardo.tigem.it4
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
dibernardo.tigem.it5
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
dibernardo.tigem.it6
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
dibernardo.tigem.it7
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?
dibernardo.tigem.it8
What info can we gain? protein-protein interaction network (yeast S. cerevisiae)
dibernardo.tigem.it9
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)
?
dibernardo.tigem.it11
INPUT(S) OUTPUT(S)
To infer a network means to find what is inside the “black box”
dibernardo.tigem.it12
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…
dibernardo.tigem.it13
Reverse engineering gene networks
Goal: Learn structure and function from expression data
dibernardo.tigem.it14
Reverse-engineering networks can help in understanding thedisease:
?
Unknown network Inferred “healthy” network Inferred “disease” network
dibernardo.tigem.it15
Methods to reverse-engineer gene networks:
• Given the experimental data, how can we reverse-engineerthe network?
dibernardo.tigem.it16
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:
dibernardo.tigem.it17
Reverse-engineering strategy:
dibernardo.tigem.it18
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:
dibernardo.tigem.it19
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
dibernardo.tigem.it20
Reverse-engineering strategy: ODEs
dX1/dt = 0 = a2 X2 + a6 X6 + a9 X9 + a12 X12
promoters
RNAs
Directed graph
+ u
u
dibernardo.tigem.it21
Reverse-engineering strategy: ODEs
dX1/dt = 0 = a2 X2 + a6 X6 + a9 X9 + a12 X12
promoters
RNAs
Directed graph
+ u
u
dibernardo.tigem.it22
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:
dibernardo.tigem.it23
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:
dibernardo.tigem.it24
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
dibernardo.tigem.it25
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:
dibernardo.tigem.it27
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)
dibernardo.tigem.it28
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.)
dibernardo.tigem.it29
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)