Hybrid systems methods for biochemical networks

1
Hybrid systems methods for biochemical networks

• Adam Halasz

2
Outline

• Hybrid systems, reachability
• Piecewise affine approximations of biochemical
  systems
• Example I Glucose-lactose
• Example II Tetracyclin resistance

3
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

4
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

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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

6
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.

7
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

8
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

9
Hybrid systems

• much of the underlying dynamics is continuous,
  but..
• complexity and lack of detailed kinetic
  information require the use of hybrid abstractions

10
Hybrid systems

• Two topics to be addressed
• How to build a good hybrid abstraction
• How to analyze a network that includes hybrid
  abstractions

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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 xM,B,A,L,P

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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!

13
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
Question 1 which ones end up in a viable final
state?
Question 2 which ones survive?
Final
Irreversible damage
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Hybrid systems

• Hybrid systems discrete continuous
• We use piecewise affine, continuous like linear
  approximations
• Example Michaelis-Menten
• Piecewise affine properites
• Reachability with piecewise affine
• Toolshybridizersimulatorreachability

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Hybrid Systems Combining Continuous with the
Discrete

• Example Transcription

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HybridModels
Approximation
rate
concentration
Abstraction
rate
concentration
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Hybrid systems

• Collection of modes
• Each mode has continuous dynamics
• Transitions possible between modes

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Hybrid Systems Use Cases in BioSPICE

• Quorum sensing in Vibrio fischeri (2002)
• Stringent Response in M.tuberculosis (2003)
• Lactose Induction in the lac Operon (2004)
• Lactose-Glucose system (2005)
• Tetracycline resistance in E.coli (2005)

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Systems BiologyModeling
Logical, Graph
Savageau, Hood, Kitano, Arkin,
Discrete, Static
Continuous, Dynamic
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Systems BiologyModeling
Logical, Graph
Savageau, Hood, Kitano, Arkin,
Discrete, Static
Hybrid, Dynamic
Continuous, Dynamic
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Challenges

• Analysis
• Validation
• Prediction
• Conventional
• Currently, only simulation
• Not scalable
• Dont incorporate uncertainty

Reachability Analysis - one alternative approach
Hybrid dynamic models offer alternative
approaches for validation and prediction
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Reachability

• Direct simulation of ODEs can be expensive
• Alternative approaches focus on what we want to
  know
• Example a population of cells, with significant
  variability

Initial
Question 1 which ones end up in a viable final
state?
Question 2 which ones survive?
Final
Irreversible damage
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Using piecewise affine approximations to
investigate reachability the plan

• Construct a faithful piecewise multi-affine
  approximation of the full ODE model
• Formulate the desired switching behavior as a
  reachability problem
• Study the reachability problem on the
  multi-affine system

24

• Hybrid systems, reachability
• Piecewise affine approximations of biochemical
  systems
• Example I Glucose-lactose
• Example II Tetracyclin resistance

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Kinetics 1

• Dynamic models have a special structure!
• More generally,

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Kinetics 1 (continued)
y
x
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Kinetics 1 (continued)
y
y2
x2
x1
x
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Kinetics 1 (continued)
The vector field is a unique function of the
vectors at the vertices
Belta, Habets, Kumar 2002
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Kinetics (2)
Transcription rate
Concentration of repressor

• Hybrid System
• Rectangular partitions
• Affine dynamics

Transcription rate
Concentration of allolactose
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Analysis Multiaffine, Rectangular
x3
x2
x1
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Reachability

• Anything can be calculated given enough time and
  resources
• say, a population of 105 bacteria whose sizes,
  growth rates, etc. are evenly spread over an
  interval
• and we only know their kinetic parameters to
  within a factor of 10.
• About 1/5 of them survived antibiotic treatment.
  Which ones?

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Mass action rate laws are multi-affine
Bi-linear
Linear
Affine f(x) a bx Multi-affine f(x,y,..)
a bx cy dxy
33
(Not) everything is mass action
a
b

• QSSA
• 
  (Michaelis-Menten)
• Approximate but more economical

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Piecewise affine approximation
Simplest approximation with two affine
pieces Can use any number, to achieve any
desired precision
35
Piecewise is hybrid
Piecewise approximation has different equations
in each interval Transitions occur as the
variable switches intervals
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Several substrates that saturate
Piecewise approximation has different equations
in each interval Transitions occur as the
variable switches intervals Can continue in many
dimensions
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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

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Reachability analysis

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

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Analysis
x3
x2
x1
40
Analysis
x3






x2
Initial


x1
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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

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Hybrid systems and reachability

• Any ODE system is approximated with piecewise
  affine, continuous functions
• Multi-affine hybrid systems are more easily
  amenable to many types of analysis
• Software tools to
• Build piecewise multi-affine approximations to
  continuous ODE models -- HSMB
• Simulate hybrid systems -- Charon
• Perform reachability analysis -- Charon

43

• Hybrid systems, reachability
• Piecewise affine approximations of biochemical
  systems
• Example I Glucose-lactose
• Example II Tetracyclin resistance

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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

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Lac system in E.coli
mRNA
ß-gal
perm
repressor
External Lactose
Lactose
Allo- Lactose
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Lac system in E.coli
Crucial switching property, sensitive to basal
rate Can be framed in terms of reachability
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Lac system in E.coli
Hybrid model constructed using a fine grained
linearization of the nonlinear rate
laws Predictions of the two models are very
similar Hybrid model within 5 uncertainty of
model parameters
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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

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mRNA
perm
b-gal
Lac repressor
External Lactose
Lactose
Allo- Lactose
Glucose inhibits the influx of lactose
CAP
cAMP
External Glucose
CAP competes with lac repressor, enhancing
transcription
cAMP is produced when glucose is absent
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Lactose Transport

• Lactose crosses the cell membrane in the presence
  of permease
• Inbound transport is inhibited by glucose
• Both inbound and outbound transport characterized
  by the same velocity and saturation constant

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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

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New Use Case (2005)
CHARON
SBML2Charon Penn
Jig Cell
Reachability Tools
Hybrid Model (SBML)
Equilibrium Point Analyzer
Stanford Toolset Stanford
Hybrid System Model Builder (HSMB) Penn
Full Model (SBML, annotated)
Simulator
Hybrid SAL SRI
Penn
Decimator
Charon Simulator
Validation of hybrid system abstractions
Decimator
SBML2Charon Penn
Simpathica Toolset NYU
Time-dependent parameter traces
Decimator
Investigation of Hysteresis properties
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Use Case Workflow
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Induction and reachability

• Expect the vicinity of zero to be confined when
  system is bi-stable

unless it is induced by increasing Le,
decreasing Ge, or both
unless it is induced by increasing Le,
decreasing Ge, or both
Suppose initially the system is at zero
allolactose. Then it will have to settle on the
lower sheet..
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Induction and reachability

• Up-switching possible if (Le,Ge) outside the
  bistable region for some time

Upward switching trajectories
Final, induced state
Initial state, close to zero
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Induction and reachability

• Follow trajectories in state space
• Induced trajectories leave the vicinity of the
  initial state

B
A
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Induction and reachability

• Cover the area of interest with a grid

B
A
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Induction and reachability

• Induced trajectories leave the vicinity of the
  initial state
• For reachability, only need to cover the vicinity
• Verify those configurations that do not leave the
  grid

B
A
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Discretization
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Reachability results

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

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

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Reachability results

• Non-inducible region should match the footprint
  of bi-stability

62
Reachability results
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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!

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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

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Networks of hybrid abstractions

• Partition of continuous part of state space along
  threshold values
• Boundaries treated as separate modes
• Discrete transition system
• Model checking

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Networks of hybrid abstractions

• Can analyze complex interconnections
• Elucidate roles of genes

67
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

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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

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Reachability results

• Coarse partition shows a lot of sideways
  induction

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Reachability results

• Coarse partition shows a lot of sideways
  induction

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Reachability in BioCharon

• Reachability results in Charon match expectations

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Switching
Non-Switching
Switching
A
A
L
L
Time min
Time min
73
Switching driven by Le or Ge
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Trace Analysis with Simpathica

• Simpathica uses temporal logic to analyze traces
• Do two traces converge to the same steady state?
• Do two traces pass through different state space
  regions
• Decimator allows Sympathica operate on arbitrary
  traces
• Sympathica expects same sample points in both
  traces
• Decimator performs interpolation

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Stanford Hybrid Analysis Tool
Hybrid Automaton File Parser
Reachable Set Computation and Visualization
Hybrid Automaton Transition Computation And
Abstraction
Hybrid Automaton Model Generator
76
Stanford Analysis Tool

• Model order reduced by taking advantage of
  quasi-steady states of two proteins (Adam Halasz)
• Model analyzed using Stanford tool using initial
  (coarse) grid (or partitions)
• Results correspond to those obtained using
  Bio-Charon, computation grid needs to be refined
  to be useful
• Reachability computation on finer grid being
  done initial results promising and show initial
  conditions unique to each of three equilibria
• More work is being done to refine the computation
  and interpret the resulting reachable sets

77

• Hybrid systems, reachability
• Piecewise affine approximations of biochemical
  systems
• Example I Glucose-lactose
• Example II Tetracyclin resistance

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Tetracycline resistance via TetA efflux
Diffusion

• TetA

TetR
TetRTc
Hillen1994
79
Tetracycline resistance via TetA efflux
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Tc0
periplasm
efflux
diffusion
cytoplasm
TetA
Mg
Tc
TcMg
tetR
tetA
O2
O1
TetR
TcMgTetR
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Tet Model Analysis

• Model describes a bacterial defense mechanism
  against attack with an antibiotic (tetracycline,
  Tc)
• Tc destroys the cells 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
• Assess the actual model parameters
• Identify parameter modifications that disrupt the
  mechanism

82
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

83
Tet Model Analysis

• Measures of the defense mechanisms
  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

84
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

85
TetModel Use Case (2005)
CHARON
SBML2Charon UPenn
SBMLEditor
Model
Reachability Tools
Parameter ranges
Equilibrium Point Analyzer
Hybrid System Model Builder (HSMB) UPenn
Hybrid Model (SBML)
Full Model (SBML, annotated)
Simulator
UPenn
Stochastic Simulator UTenn
ODE Simulator UPenn
Hybrid SAL SRI
Simpathica Toolset NYU
Experimental Traces
Validation of hybrid system abstractions
86
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

87
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 decreased
• Promoters for stable RNA are downregulated
• Promoters for amino-acids are upregulated

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Stringent response
89
Stringent response
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Stringent response

• Hybrid model with 9 variables, 2 modes
• One outside control (amino-acid availability)
• Negative feedback, only one steady state for
  given conditions

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Stringent response

• Steady-state calculations
• Signaling substance increases with parameter r
• Transcription initiation rate decreases

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Stringent response

• Dynamic calculations
• Surge of signaling substance indicates
  potentially lethal condition excessive
  accumulation of stalled transcriptional complexes
• Reachability analysis can constrain the peak value