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
ExternalLactoseLactoseAllo-
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
FinalIrreversible
damage
Question 1: which ones end up in a viable final state?
Question 2: which ones survive?
Kinetics 1Dynamic 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)
hxygyfxeydxycybxax
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
Tran
scrip
tion
rate
Concentration of repressor
Tran
scrip
tion
rate
Concentration of allolactose
Piecewise affine approximation][
][])([ max
SKSVSr
m
Simplest approximation with two affine pieces
Can use any number, to achieve any desired precision
m
mm
KSV
KSKSV
Sr2][
2][2
][])([
max
max
Piecewise is hybrid
m
mm
KSV
KSKSV
Sr2][
2][2
][])([
max
max
Piecewise approximation has different equations in each interval Transitions occur as the variable switches intervals
][2
][ max SKVP
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
AbstractionModel 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 analysisCan 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
ExternalLactoseLactoseAllo-
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
ExternalLactoseLactoseAllo-
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 gridInduction 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
TetATc
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
Model SBMLEditor
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 responseThe 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 diagramtRNAc,u
proteins
(p)ppGppreactions
Stalled complexes
TranscriptionTranslation
Ribosomeassembly
Upregulated mRNA
Downegulated mRNA
Ribosomal RNARibosomes
(p)ppGpp
Model block diagramModel block diagramtRNAc,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