Robustness Analysis and Tuning of Synthetic Gene Networks

Grégory Batt1‡ Boyan Yordanov1 Calin Belta1 Ron Weiss2

1 Centers for Information and Systems Engineering and for BioDynamics

Boston University (‡ now at )

2 Departments of Molecular Biology and of Electrical Engineering

Princeton University

Towards Systems Biology 2007

Synthetic biology

Synthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks

Synthetic biology

Synthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks

banana-smelling bacteria

Synthetic biology

Synthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks

banana-smelling bacteria

Synthetic biology Synthetic biology: application of engineering approaches to produce

novel artificial devices using biological building blocks

Numerous potential engineering and medical applications biofuel production, environment depollution, . . .

biochemical synthesis, tumor cell destruction, . . .

banana-smelling bacteria

Synthetic gene networks

Gene networks are networks of genes, proteins, small molecules and their regulatory interactions

Ultrasensitive I/O response at steady-state

Transcriptional cascade [Hooshangi et al, PNAS, 05]

Need for rational design Gene networks are networks of genes, proteins, small molecules and

their regulatory interactions

Network design: analysis of non-linear dynamical system with parameter uncertainties

current limitations in experimental techniques

fluctuating extra and intracellular environments

Problem: most newly-created networks are non-functioning and need tuning

Robustness analysis and tuning

Two problems of interest: robustness analysis: check whether dynamical properties are satisfied for

all parameters in a set

tuning: find parameter sets such that dynamical properties are satisfied for

all parameters in the sets

Approach:

unknown parameters, initial conditions and inputs given by intervals

piecewise-multiaffine differential equations models of gene networks

dynamical properties specified in temporal logic (LTL)

adapt techniques from hybrid systems theory and model checking

Hybrid systems approach Analysis of dynamical systems

Traditional view: fixed initial condition and fixed parameter More interesting: set of initial conditions and set of parameters

x0

p1

p2

X0

P1

P2

Hybrid systems approach Analysis of dynamical systems

Traditional view: fixed initial condition and fixed parameter More interesting: set of initial conditions and set of parameters

How to reason with infinite number of parameters and initial conditions ?

x0

p1

p2

X0

P1

P2

Hybrid systems approach Analysis of dynamical systems

Traditional view: fixed initial condition and fixed parameter More interesting: set of initial conditions and set of parameters

How to reason with infinite number of parameters and initial conditions ? direct vs indirect approaches

x0

p1

p2

X0

P1

P2

X0

P1

P2

X0

P1

P2

Hybrid systems approach Analysis of dynamical systems

Traditional view: fixed initial condition and fixed parameter More interesting: set of initial conditions and set of parameters

How to reason with infinite number of parameters and initial conditions ? direct vs indirect approaches

x0

p1

p2

X0

P1

P2

X0

P1

P2

X0

P1

P2

model checking possible

Overview

I. Introduction

II. Problem definition

III. Robust design of gene networks

IV. Application: tuning a synthetic transcriptional cascade

V. Discussion and conclusions

Overview

I. Introduction

II. Problem definition

III. Robust design of gene networks

IV. Application: tuning a synthetic transcriptional cascade

V. Discussion and conclusions

Gene network models

cross-inhibition network

Gene network models

cross-inhibition network

x: protein concentration

, : rate parameters

: threshold concentration

Gene network models

cross-inhibition network

x: protein concentration

, : rate parameters

: threshold concentration

x0

1

Hill functionx

0

1

step functionx

0

1

ramp function

Hill-type models PMA models PA models

regulation functions:

Gene network models

cross-inhibition network

x: protein concentration

, : rate parameters

: threshold concentration

Gene network models

cross-inhibition network

x: protein concentration

, : rate parameters

: threshold concentration

Gene network models

Partition of the state space: rectangles

Gene network models

Specifications of dynamical properties

Dynamical properties expressed in temporal logic (LTL) set of atomic proposition

usual logical operators

temporal operators ,

Specifications of dynamical properties

Dynamical properties expressed in temporal logic (LTL) set of atomic proposition

usual logical operators

temporal operators ,

Semantics of LTL formulas defined over executions of transition systems

...

...

...

q q q qq

qq q q q

qqqp , qp , qp ,

:Fq

:Gq

:Uqp

Specifications of dynamical properties

Dynamical properties expressed in temporal logic (LTL) set of atomic proposition

usual logical operators

temporal operators ,

Semantics of LTL formulas defined over executions of transition systems

Solution trajectories of PMA models are associated with executions of

embedding transition system

...

...

...

q q q qq

qq q q q

qqqp , qp , qp ,

:Fq

:Gq

:Uqp

Specifications of dynamical properties

Dynamical properties expressed in temporal logic (LTL) set of atomic proposition

usual logical operators

temporal operators ,

Semantics of LTL formulas defined over executions of transition systems

bistability property:

Overview

I. Introduction

II. Problem definition

III. Robust design of gene networks

IV. Application: tuning a synthetic transcriptional cascade

V. Discussion and conclusions

PMA model

specifications

gene network

intervals for uncertain parameters

Robust design of gene networks

model checking

PMA model

specifications

synthesis of parameter constraints

gene network

discrete abstractionsconvexity properties

intervals for uncertain parameters

Robust design of gene networks

Valid parameter setNo conclusion[Batt et al., HSCC07]

Computation of discrete abstraction

Multiaffine function: in every rectangular region, the flow is a convex combination of its values at the vertices

[Belta and Habets, Trans. Autom. Contr., 06]

Computation of discrete abstraction

Transition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles

Multiaffine function: in every rectangular region, the flow is a convex combination of its values at the vertices

[Belta and Habets, Trans. Autom. Contr., 06]

Computation of discrete abstraction

Transition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles

Transitions can be computed by polyhedral operations

where

(Because is a piecewise-multiaffine function of x and an affine function of p)

Multiaffine function: in every rectangular region, the flow is a convex combination of its values at the vertices

[Belta and Habets, Trans. Autom. Contr., 06]

RoVerGeNe

Approach implemented in publicly-available tool RoVerGeNe

Written in Matlab, exploits polyhedral operation toolbox MPT and model checker NuSMV

http://iasi.bu.edu/~batt

Overview

I. Introduction

II. Problem definition

III. Robustness design of gene networks

IV. Application: tuning a synthetic transcriptional cascade

V. Discussion and conclusions

Transcriptional cascade: approach

Approach for robust tuning of the cascade: develop a model of the actual cascade

specify expected behavior

tune network by searching for valid parameter sets

verify robustness of tuned network

Transcriptional cascade [Hooshangi et al, PNAS, 05]

Transcriptional cascade: modeling

PMA differential equation model (1 input and 4 state variables)

Parameter identification

Transcriptional cascade: specification

Expected input/output behavior of cascade at steady state and for all initial states

Temporal logic specifications

Liveness property: additional fairness constraints needed

[Batt et al., TACAS’07]

Transcriptional cascade: tuning

Tuning: search for valid parameter sets Let 3 production rate parameters unconstrained

Answer: 15 sets found (<4 h., 1500 rectangles, 18 parameter constraints)

comparison with numerical simulation results

in parameter space and for input/output behavior

[Batt et al., Bioinfo, 07]

Transcriptional cascade: robustness

Robustness: check that tuned network behaves robustly

Let all production and degradation rate parameters range in intervals

centered at their reference values (with ±10% or ±20% variations)

Answer for ±10% parameter variations: Yes (< 4hrs)

proves that specification holds despite ±10% parameter variations

Answer for ±20% parameter variations: No (< 4hrs)

suggests that specification does not hold for some parameters in

the ±20% set (confirmed by manual analysis of counter-example)

11 uncertain parameters:

Overview

I. Introduction

II. Problem definition

III. Analysis for fixed parameters

IV. Analysis for sets of parameters

V. Tuning of a synthetic transcriptional cascade

VI. Discussion and conclusions

Summary

Gene networks modeled as uncertain PMA systems

piecewise-multiaffine differential equations models

unknown parameters, initial conditions and inputs given by intervals

dynamical properties expressed in temporal logic

Use of tailored combination of parameter constraint synthesis, discrete abstractions, and model checking

Method implemented in publicly-available tool RoVerGeNe

Approach can answer non-trivial questions on networks of biological interest

Discussion

First computational approach for tuning synthetic gene networks

Related work: qualitative/discrete approaches (reachability or model checking)

quantitative approaches with fixed parameter values (reachability or MC)

quantitative approaches with uncertain parameters (optimisation-based)

Further work: verification of properties involving timing constraints (post doc, Verimag)

deal with uncertain threshold parameters too

use of compositional verification for design of large modular networks

[de Jong et al., Bull. Math. Biol. 04; Ghosh and Tomlin, Syst.Biol. 04; Batt et al., Bioinfo. 05]

[Bernot et al., J.Theor.Biol. 04; Gonzalez et al., Biosystems 06, Calzone et al., Trans.Comput.Syst.Biol 06]

[Belta et al., CDC’02; Berman et al., HSCC’07; Fages and Rizk, CMSB’07]

[Kuepfer et al., BMC Bioinfo. 07]

AcknowledgementsThanks to Calin Belta, Boyan Yordanov, Ron Weiss…

… and to Ramzi Ben Salah and Oded Maler

References:

• G. Batt, B. Yordanov, C. Belta and R. Weiss (2007) Robustness analysis and tuning of synthetic gene networks. In

Bioinformatics, 23(18):2415-1422

• G. Batt, C. Belta and R. Weiss (2007) Temporal logic analysis of gene networks under parameter uncertainty. Accepted to

Joint Special Issue on Systems Biology of IEEE Trans. Circuits and Systems and IEEE Trans. Automatic Control

Center for BioDynamics Center for Information and Systems Engineering Boston University

Verimag Lab Grenoble Polytechnic Institute