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

mailto:[email protected]

mailto:[email protected]

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.


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


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 ]


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 ]


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 ]


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 ]


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 ]

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


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


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]


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 ]


[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.


[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





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


Overview:

• Networks in Biology

• Reverse-engineering gene networks of unknown topology(de novo)

• Parametrisation of network with known topology


Gene Networks

cell membrane

metabolites

proteins

RNA

genes

transcriptnetworks

proteinnetworks

metabolicnetworks

Our focus: methods to decode transcription regulation networks


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


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


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


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?


What info can we gain? protein-protein interaction network (yeast S. cerevisiae)


Reverse engineering (or inference) gene networks:

?Unknown network Inferred network


“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)

?


INPUT(S) OUTPUT(S)

To infer a network means to find what is inside the “black box”


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…


Reverse engineering gene networks

Goal: Learn structure and function from expression data


Reverse-engineering networks can help in understanding thedisease:

?

Unknown network Inferred “healthy” network Inferred “disease” network


Methods to reverse-engineer gene networks:

• Given the experimental data, how can we reverse-engineerthe network?


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:


Reverse-engineering strategy:


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:


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


Reverse-engineering strategy: ODEs

dX1/dt = 0 = a2 X2 + a6 X6 + a9 X9 + a12 X12

promoters

RNAs

Directed graph

+ u

u


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:


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:


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


Pilot study: E. coli DNA-damage repair pathway (SOS pathway)

DNA-damage repairpotentially involves 100s ofgenes

Applied NIR to 9 transcriptsubnetwork


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:


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)


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.)


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



• 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



• 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



-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


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)

Mathematical Models for Synthetic Biology...ODE Models: Solution Existence and uniqueness of...

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Transcript of Mathematical Models for Synthetic Biology...ODE Models: Solution Existence and uniqueness of...