Hybrid systems methods for biochemical networks

Adam Halasz

Outline

• Hybrid systems, reachability

• Piecewise affine approximations of biochemical systems

• Example I: Glucose-lactose

• Example II: Tetracyclin resistance

Biomolecular networks as hybrid systems

Networks of chemical and molecular processes State = {values of all concentrations} Rates of each process are continuous functions of the state Several layers of processes, different timescales State space can be huge (O(103) variables for one cell)

Lots of truly discrete behavior: Genes on/off Discrete variables

Lots of apparent discrete behavior Nontrivial continuous dynamics produces multistability, bifurcations Abstractions – commonly used and/or required for simplification

Biomolecular dynamical systems

Central dogma of molecular biology DNA encodes genes; it replicates Genes are transcribed into mRNA mRNA is translated into proteins

Proteins may: act as enzymes that catalyze metabolic reactions act as transcription factors

Metabolic reactions big network that converts incoming nutrients into useful

substances and by-products reactions proceed much faster when the right enzymes are

available

The Central Dogma•DNA replicates during cell division

•Transcription performed by RNA polymerase

•Requires a promoter site

•Several genes bundled to one promoter = operon

•In higher organisms, mRNA is spliced

•Translation performed by Ribosomes

•Protein synthesis needs raw material

Genes to proteins• Proteins are synthesized

as chains of elementary proteins, amino-acids

• They fold, giving rise to complicated 3d structures

• Several molecules may be assembled into more complicated ‘machines’, such as RNAP, ribosomes, etc.

Metabolic network•Very complex

•Structured

•Stoichiometry is more easily identified than rate laws

•Many networks available in databases, e.g. Kegg

•Reactions linked to individual genes

•Lots of feedback

Metabolic network has a lot of control

• Feedback between Metabolites Genes and proteins

• Continuous adjustment to external conditions

• Signaling networks

• Control is through rate laws, but also through stochastic mechanisms

• much of the underlying dynamics is continuous, but..

• complexity and lack of detailed kinetic information require the use of hybrid abstractions

Hybrid systems

Hybrid systems

Two topics to be addressed:

1. How to build a good hybrid abstraction

2. How to analyze a network that includes hybrid abstractions

Using hybrid systems abstractions to build hybrid systems abstractions

•The lac operon is a bistable genetic switch Multiple positive feedback bistable Input: external lactose State: x={M,B,A,L,P}

β-gal

mRNA

perm

ExternalLactose

LactoseAllo-

Lactose

repressor

)(xfx

Using hybrid systems abstractions to build hybrid systems abstractions

•May be abstracted to an automaton:

Input: external lactose State: {I}

The characteristic still depends on the underlying kinetic parameters!

HIGHI=1

LOWI=0

highee LL

lowee LL

(...)(...); gLfL highe

lowe

Reachability•The full lac model can be simulated to investigate induction, but that can be expensive•The question of whether induction is possible may be framed as a reachability problem•Many other situations with discrete outcomes are amenable to reachability

Initial

Final

Irreversible damage

Question 1: which ones end up in a viable final state?

Question 2: which ones survive?

Kinetics 1

Dynamic models have a special structure!

More generally,

}1,0{,

1,,

1

1

1)(

N

N

Nii

iN

iliil xxcxf

)(xfx l

CBA

CkBAkdtd

BAkCkdtBd

BAkCkdtAd

rf

fr

fr

C

Example

Kinetics 1 (continued)

hxygyfxey

dxycybxax

y1

xhyfgyey

xdybcyax

11

11

xhyfgyey 11

x

y

The vector field is a unique (affine) function of the vectors at the end points

Kinetics 1 (continued)

The vector field is a unique function of the vectors at the vertices

)(xfx

[Belta, Habets, Kumar 2002]

Kinetics (2)

Hybrid System Rectangular

partitions Affine dynamics

Tra

nsc

ripti

on

rate

Concentration of repressor

Tra

nsc

ripti

on

rate

Concentration of allolactose

Piecewise affine approximation

][

][])([ max

SK

SVSr

m

Simplest approximation with two affine pieces

Can use any number, to achieve any desired precision

m

mm

KSV

KSK

SVSr

2][

2][2

][])([

max

max

Piecewise is hybrid

m

mm

KSV

KSK

SVSr

2][

2][2

][])([

max

max

Piecewise approximation has different equations in each interval Transitions occur as the variable switches intervals

][2

][ max SK

VP

m

mKS 2][

mKS 2][

max][ VP

Several substrates that saturate

][][

][][

22

11

SDP

SBP

CPAP ][;][ 21

CPSBP ][;][][ 211

Piecewise approximation has different equations in each interval Transitions occur as the variable switches intervals

Can continue in many dimensions

][][;][][ 2211 SDPSBP

11][ TS

][][;][ 221 SDPAP

11][ TS 11][ TS

22 ][ TS

22 ][ TS 11][ TS

1S

2S

1T

2T

22 ][ TS

22 ][ TS

Abstraction

Model the biochemical network as a switched system with continuous multi-affine dynamics

Each mode has simple dynamics More insight Approximation may be refined as needed Partition may be refined independently of dynamics No additional computational difficulties Traditional simulations are easier Efficient reachability algorithms can be applied

Reachability analysis

Can the system reach a set of states starting from a set of initial conditions?

Analysis

13swx

23swx

11swx

21swx

12swx 2

2swx

x2

x1

x3

Analysis

13swx

23swx

11swx

21swx

12swx 2

2swx

x2

x1

x3

Initial

Reachable

Unreachable

Hybrid System Analysis

Reachability Cell A is reachable from

cell B if there is at least one trajectory from B to A

Cell A is not reachable from cell B if there are no trajectories from B to A

Glucose-lactose system• The lactose metabolism is self-nourishing:

The cell needs enzymes for: Inbound lactose transport (permease) Lactose processing (ß-galactosidase)

Permease and ß-galactosidase are gene products of the lac operon

Lac operon is repressed in the absence of allolactose

Allolactose is produced when lactose is processed

• Bistability: a low and a high lactose metabolism state induction needed to move into the high state

Lac system in E.coli

mRNA

β-gal perm

ExternalLactose

LactoseAllo-

Lactose

repressor

Lac system in E.coliCrucial switching property, sensitive to basal rateCan be framed in terms of reachability

Lac system in E.coliHybrid model constructed using a fine grained linearization of the nonlinear rate lawsPredictions of the two models are very similarHybrid model within 5% uncertainty of model parameters

Glucose-lactose system• Lactose is an alternative energy source

• Glucose is the preferred nutrient; bacteria also grow on lactose, but only when glucose is absent

• There are two mechanisms that ensure this: Inducer exclusion Catabolite repression

mRNA

-gal perm

ExternalLactose

LactoseAllo-

Lactose

Lacrepressor

ExternalGlucose

CAPcAMP

cAMP is produced when glucose is

absent

CAP competes with lac repressor, enhancing

transcription

Glucose inhibits the influx of

lactose

Steady states

• For a given Glucose (Ge) value, the steady state line is S-shaped• The bistable section increases as Ge increases• The upper threshold for Lactose (Le) is higher if Ge is present

Induction and reachabilityExpect the vicinity of zero to be confined when system is bi-

stable

Suppose initially the system is at zero allolactose. Then it will have to

settle on the lower sheet..

… unless it is induced by increasing Le, decreasing Ge, or both

… unless it is induced by increasing Le, decreasing Ge, or both

Induction and reachabilityUp-switching possible if (Le,Ge) outside the bistable

region for some time

Initial state, close to zero

Upward switching trajectories

Final, induced state

Induction and reachability

A

B Follow trajectories in state spaceInduced trajectories leave the vicinity of the

initial state

Cover the area of interest with a grid

Induction and reachability

A

B

Induction and reachability

A

B Induced trajectories leave the vicinity of the initial state

For reachability, only need to cover the vicinityVerify those configurations that do not leave the

grid

Discretization

Reachability results

Bistable regions are non-inducible, hence they reach only the lower A values

• Calculate highest Allolactose (A) reached• Sweep for (Le,Ge)

Reachability resultsNon-inducible region should match the

footprint of bi-stability

Reachability results

Analyzing networks of hybrid abstractions

• The lac switch is one piece in a potentially huge circuit, which has both discontinuous and continuous elements

• A “true” of hybrid system:

Discontinuous dynamics

Different state variables

Filippov states!

Hierarchy of modes!

Networks of hybrid abstractions

•Continuous part of state space is still a set of concentrations•Dynamics is still given by reaction rates•Reaction rates are given by discontinuous functions of the state variables:

)(xfx

Networks of hybrid abstractions

• Partition of continuous part of state space along threshold values

• Boundaries treated as separate modes• Discrete transition system• Model checking

Networks of hybrid abstractions

•Can analyze complex interconnections

•Elucidate roles of genes

Summary• Molecular biology offers many instances of

‘natural’ hybrid systems• Very large state spaces, thousands of

substances• Complex networks, nonlinear equations• Switching and other discontinuous behavior

Genes on/off Multistability, bifurcation Hybrid abstractions

• Two aspects: Constructing hybrid abstractions Analyzing networks a hybrid systems

• Both directions work towards automated analysis

Reading

• Calin Belta – Boston U.• Hidde de Jong – INRIA Rhone-Alpes, FR/EU• Claire Tomlin – Berkeley• Ashish Tiwari – SRI, Palo Alto, CA• Joao Hespanha – Santa Barbara• V. Kumar, O. Sokolsky, G. Pappas, A. Julius,

A. Halasz – U. Penn

• Hybrid systems, reachability

• Piecewise affine approximations of biochemical systems

• Example I: Glucose-lactose

• Example II: Tetracyclin resistance

Tc0

O2O1

Mg

TetR TetA

Tc

periplasm

cytoplasm

tetR tetA

diffusion efflux

[TcMg]+

[TcMg]+TetR

Tetracycline resistance via TetA efflux

Tc0

O2O1

Mg

TetR

TetA

Tc

periplasm

cytoplasm

tetR tetA

diffusion efflux

[TcMg]+

[TcMg]+TetR

Tet Model Analysis

• Model describes a bacterial defense mechanism against attack with an antibiotic (tetracycline, Tc)

• Tc destroys the cell’s ribosomes, inflicting potentially irreversible damage to the transcription-translation apparatus.

• Objective is to avoid accumulation of Tc inside the cell.

• Our objective: to disrupt the defense mechanism. For this we first have to:1. Assess the actual model parameters2. Identify parameter modifications that disrupt the mechanism

Tet Model Building• Model parameters not fully known

Use existing information on known reactions

Use consistency checks and qualitative arguments

Determine parameters indirectly by comparing model predictions to experimental results

• Perform experiments to verify model

Measures of the defense mechanism’s effectiveness:

•Irreversible damage to transcription-translation apparatus:

Direct investigation would require a greatly expanded model

Use proxies instead•Final Tc concentration

May not tell the whole story

•Maximum transient Tc concentration

May cause irreversible damage

Tet Model Analysis

•We wish to investigate how these efficiency measures depend on model parameters, especially those parameters that are not well known.•We may indirectly pin down their value ranges•We may learn which aspects of this mechanism are the most easy to compromise by targeting with a drug

•Final Tc concentration Computed by a steady state calculation

•Maximum transient Tc Not directly calculable One way: many simulations Other method: reachability

Tet Model Analysis

CHARON

TetModel Use Case (2005)

Hybrid System Model Builder

(HSMB)UPenn

Hybrid Model(SBML)

SBML2CharonUPenn

ReachabilityTools

Equilibrium Point Analyzer

Hybrid SALSRI

UPenn

SimpathicaToolsetNYU

Simulator

StochasticSimulatorUTenn

Validation of hybrid system abstractions

Full Model(SBML,

annotated)

ExperimentalTraces

ModelSBMLEditor

Parameterranges

ODESimulatorUPenn

Summary• Hybrid systems bridge the gap between discrete “big

picture” models and detailed, continuous dynamics

• Piecewise multi-affine approximations are well suited for biochemical networks

• Several software tools apply efficient algorithms for: Model building Simulation Reachability

• Type of problems: Mid-sized networks, focus on one mechanism Analysis of parameter and initial state ranges Prediction of qualitative outcomes

Stringent response

The stringent response is the set of metabolic and regulatory changes that take place in a bacterium as a consequence of a downshift in the availability of nutritional substances, especially amino-acids.Transcription is globally decreasedPromoters for stable RNA are downregulatedPromoters for amino-acids are upregulated

Stringent response

Stringent response

TranscriptionTranslation

Ribosomeassembly

Upregulated mRNA

Downegulated mRNA

Ribosomal RNARibosomes

(p)ppGpp

Model block diagramModel block diagram

tRNAc,u

proteins

(p)ppGppreactions

Stalled complexes

TranscriptionTranslation

Ribosomeassembly

Upregulated mRNA

Downegulated mRNA

Ribosomal RNARibosomes

(p)ppGpp

Model block diagramModel block diagram

tRNAc,u

proteins

(p)ppGppreactions

Stalled complexes

Stringent responseHybrid model with 9 variables, 2 modesOne outside control (amino-acid availability)Negative feedback, only one steady state for given conditions

Stringent responseSteady-state calculations

Signaling substance increases with parameter rTranscription [initiation] rate decreases

Stringent responseDynamic calculations

Surge of signaling substance indicates potentially lethal condition: excessive accumulation of stalled transcriptional complexesReachability analysis can constrain the peak value