process algebras for quantitative analysis: lecture 10...
TRANSCRIPT
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process algebras for quantitative analysis:Lecture 10 — Modelling circadian rhythms using
Bio-PEPA
Stephen Gilmore
School of InformaticsThe University of Edinburgh
Scotland
Joint work with Jane Hillston and Maria Luisa Guerriero
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Outline
1 Process Algebras for Systems BiologyProcess AlgebrasProcess Algebras for Systems Biology
2 Bio-PEPALevels of Abstraction
3 Modelling Circadian RhythmsOstreococcus TauriStochastic modelling and model checkingInvestigating mutational effects
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Current Hypothesis
Some of the techniques we have developed over the last thirtyyears for modelling complex software systems can be beneficiallyapplied to the modelling aspects of systems biology.
In particular formalisms which encompass support for
Abstraction
Modularity and
Reasoning
have a key role to play.
Process algebras have mechanisms for each of these, andstochastic extensions which allow dynamic properties to beanalysed.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Current Hypothesis
Some of the techniques we have developed over the last thirtyyears for modelling complex software systems can be beneficiallyapplied to the modelling aspects of systems biology.
In particular formalisms which encompass support for
Abstraction
Modularity and
Reasoning
have a key role to play.
Process algebras have mechanisms for each of these, andstochastic extensions which allow dynamic properties to beanalysed.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Current Hypothesis
Some of the techniques we have developed over the last thirtyyears for modelling complex software systems can be beneficiallyapplied to the modelling aspects of systems biology.
In particular formalisms which encompass support for
Abstraction
Modularity and
Reasoning
have a key role to play.
Process algebras have mechanisms for each of these, andstochastic extensions which allow dynamic properties to beanalysed.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Process Algebra
Models consist of agents which engage in actions and bydoing so change state.
Agents then interact, sharing some actions and beingindependent on others.
P
cad
b
f
e
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Process Algebra
Models consist of agents which engage in actions and bydoing so change state.Agents then interact, sharing some actions and beingindependent on others.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Process Algebra
Models consist of agents which engage in actions and bydoing so change state.Agents then interact, sharing some actions and beingindependent on others.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Process Algebra
Models consist of agents which engage in actions and bydoing so change state.Agents then interact, sharing some actions and beingindependent on others.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Process Algebra
Models consist of agents which engage in actions and bydoing so change state.Agents then interact, sharing some actions and beingindependent on others.
Stochastic process algebra model in the same style butassume that each action has a delay governed by a randomvariable.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Deriving quantitative data
SPA models can be analysed for quantified dynamic behaviour in anumber of different ways.
The language may be used to generate a Markov Process(CTMC).
The language also may be used to generate a stochasticsimulation.
The language may be used to generate a system of ordinarydifferential equations (ODEs).
Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Deriving quantitative data
SPA models can be analysed for quantified dynamic behaviour in anumber of different ways.
The language may be used to generate a Markov Process(CTMC).
The language also may be used to generate a stochasticsimulation.
The language may be used to generate a system of ordinarydifferential equations (ODEs).
Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Deriving quantitative data
SPA models can be analysed for quantified dynamic behaviour in anumber of different ways.
The language may be used to generate a Markov Process(CTMC).
The language also may be used to generate a stochasticsimulation.
The language may be used to generate a system of ordinarydifferential equations (ODEs).
Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Deriving quantitative data
SPA models can be analysed for quantified dynamic behaviour in anumber of different ways.
The language may be used to generate a Markov Process(CTMC).
The language also may be used to generate a stochasticsimulation.
The language may be used to generate a system of ordinarydifferential equations (ODEs).
Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras
Deriving quantitative data
SPA models can be analysed for quantified dynamic behaviour in anumber of different ways.
The language may be used to generate a Markov Process(CTMC).
The language also may be used to generate a stochasticsimulation.
The language may be used to generate a system of ordinarydifferential equations (ODEs).
Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Process Algebras for Systems Biology
Process algebraic formulations are compositional and makeinteractions/constraints explicit — not the case with classicalordinary differential equation models.
Structure can also be apparent.
Abstraction can be used to mask complexity or incompleteknowledge.
Equivalence relations allow formal comparison of high-leveldescriptions.
There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Process Algebras for Systems Biology
Process algebraic formulations are compositional and makeinteractions/constraints explicit — not the case with classicalordinary differential equation models.
Structure can also be apparent.
Abstraction can be used to mask complexity or incompleteknowledge.
Equivalence relations allow formal comparison of high-leveldescriptions.
There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Process Algebras for Systems Biology
Process algebraic formulations are compositional and makeinteractions/constraints explicit — not the case with classicalordinary differential equation models.
Structure can also be apparent.
Abstraction can be used to mask complexity or incompleteknowledge.
Equivalence relations allow formal comparison of high-leveldescriptions.
There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Process Algebras for Systems Biology
Process algebraic formulations are compositional and makeinteractions/constraints explicit — not the case with classicalordinary differential equation models.
Structure can also be apparent.
Abstraction can be used to mask complexity or incompleteknowledge.
Equivalence relations allow formal comparison of high-leveldescriptions.
There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Process Algebras for Systems Biology
Process algebraic formulations are compositional and makeinteractions/constraints explicit — not the case with classicalordinary differential equation models.
Structure can also be apparent.
Abstraction can be used to mask complexity or incompleteknowledge.
Equivalence relations allow formal comparison of high-leveldescriptions.
There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Systems Biology Methodology
ExplanationInterpretation
6
Natural System
Systems Analysis
?
InductionModelling
Formal System
Biological Phenomena-MeasurementObservation
DeductionInference
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Formal Systems
There are two alternative approaches to constructing dynamicmodels of biochemical pathways commonly used by biologists:
Ordinary Differential Equations:continuous time,continuous behaviour (concentrations),deterministic.
Stochastic Simulation:continuous time,discrete behaviour (no. of molecules),stochastic.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Formal Systems
There are two alternative approaches to constructing dynamicmodels of biochemical pathways commonly used by biologists:
Ordinary Differential Equations:continuous time,continuous behaviour (concentrations),deterministic.
Stochastic Simulation:continuous time,discrete behaviour (no. of molecules),stochastic.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Formal Systems
There are two alternative approaches to constructing dynamicmodels of biochemical pathways commonly used by biologists:
Ordinary Differential Equations:continuous time,continuous behaviour (concentrations),deterministic.
Stochastic Simulation:continuous time,discrete behaviour (no. of molecules),stochastic.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Systems Biology Modelling
In most current work mathematics is being used directly asthe formal system.
Previous experience in computer performance modelling hasshown us that there can be benefits to interposing a formalprocess algebra model between the system and theunderlying mathematical model.
Moreover taking this “high-level programming” style approachoffers the possibility of different “compilations” to differentmathematical models: integrated modelling and analysis.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Systems Biology Modelling
In most current work mathematics is being used directly asthe formal system.
Previous experience in computer performance modelling hasshown us that there can be benefits to interposing a formalprocess algebra model between the system and theunderlying mathematical model.
Moreover taking this “high-level programming” style approachoffers the possibility of different “compilations” to differentmathematical models: integrated modelling and analysis.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Systems Biology Modelling
In most current work mathematics is being used directly asthe formal system.
Previous experience in computer performance modelling hasshown us that there can be benefits to interposing a formalprocess algebra model between the system and theunderlying mathematical model.
Moreover taking this “high-level programming” style approachoffers the possibility of different “compilations” to differentmathematical models: integrated modelling and analysis.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Integrated Analysis
System
MathematicalModel
e.g ODE or SSA
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Integrated Analysis
ODE
System
Stochastic Process Algebra
Model
SSAModel Checking
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Integrated Analysis
ODE
Stochastic Process Algebra
Model
SSAModel Checking
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Process Algebras for Systems Biology
Integrated Analysis
ODE
Stochastic Process Algebra
Model
Analysis
SSAModel Checking
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Molecular processes as concurrent computations
Concurrency MolecularBiology
SignalTransduction
Concurrentcomputational processes
MoleculesInteractingproteins
Synchronouscommunication
Molecularinteraction
Binding andcatalysis
Transition or mobility Biochemicalmodificationor relocation
Protein binding,modification orsequestration
[Regev et al 2000]
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Molecular processes as concurrent computations
Concurrency MolecularBiology
SignalTransduction
Concurrentcomputational processes
MoleculesInteractingproteins
Synchronouscommunication
Molecularinteraction
Binding andcatalysis
Transition or mobility Biochemicalmodificationor relocation
Protein binding,modification orsequestration
[Regev et al 2000]
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Molecular processes as concurrent computations
Concurrency MolecularBiology
SignalTransduction
Concurrentcomputational processes
MoleculesInteractingproteins
Synchronouscommunication
Molecularinteraction
Binding andcatalysis
Transition or mobility Biochemicalmodificationor relocation
Protein binding,modification orsequestration
[Regev et al 2000]
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Molecular processes as concurrent computations
Concurrency MolecularBiology
SignalTransduction
Concurrentcomputational processes
MoleculesInteractingproteins
Synchronouscommunication
Molecularinteraction
Binding andcatalysis
Transition or mobility Biochemicalmodificationor relocation
Protein binding,modification orsequestration
[Regev et al 2000]
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA: motivations
Work on the stochastic π-calculus and related calculi, is typicallybased on Regev’s mapping, meaning that a molecule maps to aprocess.
This is an inherently individuals-based view of the system andanalysis will generally then be via stochastic simulation.
With PEPA and Bio-PEPA we have been experimenting with moreabstract mappings between elements of signalling pathways andprocess algebra constructs.
Abstract models are more amenable to integrated analysis.
We also wanted to be able to capture more of the biologicalfeatures expressed in the models such as those found in theBioModels database.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA: motivations
Work on the stochastic π-calculus and related calculi, is typicallybased on Regev’s mapping, meaning that a molecule maps to aprocess.
This is an inherently individuals-based view of the system andanalysis will generally then be via stochastic simulation.
With PEPA and Bio-PEPA we have been experimenting with moreabstract mappings between elements of signalling pathways andprocess algebra constructs.
Abstract models are more amenable to integrated analysis.
We also wanted to be able to capture more of the biologicalfeatures expressed in the models such as those found in theBioModels database.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA: motivations
Work on the stochastic π-calculus and related calculi, is typicallybased on Regev’s mapping, meaning that a molecule maps to aprocess.
This is an inherently individuals-based view of the system andanalysis will generally then be via stochastic simulation.
With PEPA and Bio-PEPA we have been experimenting with moreabstract mappings between elements of signalling pathways andprocess algebra constructs.
Abstract models are more amenable to integrated analysis.
We also wanted to be able to capture more of the biologicalfeatures expressed in the models such as those found in theBioModels database.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA: motivations
Work on the stochastic π-calculus and related calculi, is typicallybased on Regev’s mapping, meaning that a molecule maps to aprocess.
This is an inherently individuals-based view of the system andanalysis will generally then be via stochastic simulation.
With PEPA and Bio-PEPA we have been experimenting with moreabstract mappings between elements of signalling pathways andprocess algebra constructs.
Abstract models are more amenable to integrated analysis.
We also wanted to be able to capture more of the biologicalfeatures expressed in the models such as those found in theBioModels database.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA: motivations
Work on the stochastic π-calculus and related calculi, is typicallybased on Regev’s mapping, meaning that a molecule maps to aprocess.
This is an inherently individuals-based view of the system andanalysis will generally then be via stochastic simulation.
With PEPA and Bio-PEPA we have been experimenting with moreabstract mappings between elements of signalling pathways andprocess algebra constructs.
Abstract models are more amenable to integrated analysis.
We also wanted to be able to capture more of the biologicalfeatures expressed in the models such as those found in theBioModels database.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Motivations for Abstraction
Our motivations for seeking more abstraction in process algebramodels for systems biology are:
Process algebra-based analyses such as comparing models(e.g. for equivalence or simulation) and model checking areonly possible is the state space is not prohibitively large.
The data that we have available to parameterise models issometimes speculative rather than precise.
This suggests that it can be useful to use semi-quantitativemodels rather than quantitative ones.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Motivations for Abstraction
Our motivations for seeking more abstraction in process algebramodels for systems biology are:
Process algebra-based analyses such as comparing models(e.g. for equivalence or simulation) and model checking areonly possible is the state space is not prohibitively large.
The data that we have available to parameterise models issometimes speculative rather than precise.
This suggests that it can be useful to use semi-quantitativemodels rather than quantitative ones.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Motivations for Abstraction
Our motivations for seeking more abstraction in process algebramodels for systems biology are:
Process algebra-based analyses such as comparing models(e.g. for equivalence or simulation) and model checking areonly possible is the state space is not prohibitively large.
The data that we have available to parameterise models issometimes speculative rather than precise.
This suggests that it can be useful to use semi-quantitativemodels rather than quantitative ones.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Motivations for Abstraction
Our motivations for seeking more abstraction in process algebramodels for systems biology are:
Process algebra-based analyses such as comparing models(e.g. for equivalence or simulation) and model checking areonly possible is the state space is not prohibitively large.
The data that we have available to parameterise models issometimes speculative rather than precise.
This suggests that it can be useful to use semi-quantitativemodels rather than quantitative ones.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative Representations
ODEs
population view
StochasticSimulation
individual view
AbstractSPA model
@@@@@@@@R
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative Representations
ODEspopulation view
StochasticSimulation
individual view
AbstractSPA model
@@@@@@@@R
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Discretising the population view
We can discretise the continuous range of possible concentrationvalues into a number of distinct states. These form the possiblestates of the component representing the reagent.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative Representations
ODEs
population view
StochasticSimulation
individual view
AbstractSPA model
@@@@@@@@R
CTMC withM levels
abstract view
-
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative Representations
ODEspopulation view
StochasticSimulation
individual view
AbstractSPA model
@@@@@@@@R
CTMC withM levels
abstract view
-
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative models
The ODE model can be regarded as an approximation of aCTMC in which the number of molecules is large enough thatthe randomness averages out and the system is essentiallydeterministic.
In models with levels, each level of granularity gives rise to aCTMC, and the behaviour of this sequence of Markovprocesses converges to the behaviour of the system of ODEs.
Some analyses which can be carried out via numericalsolution of the CTMC are not readily available from ODEs orstochastic simulation.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative models
The ODE model can be regarded as an approximation of aCTMC in which the number of molecules is large enough thatthe randomness averages out and the system is essentiallydeterministic.
In models with levels, each level of granularity gives rise to aCTMC, and the behaviour of this sequence of Markovprocesses converges to the behaviour of the system of ODEs.
Some analyses which can be carried out via numericalsolution of the CTMC are not readily available from ODEs orstochastic simulation.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Alternative models
The ODE model can be regarded as an approximation of aCTMC in which the number of molecules is large enough thatthe randomness averages out and the system is essentiallydeterministic.
In models with levels, each level of granularity gives rise to aCTMC, and the behaviour of this sequence of Markovprocesses converges to the behaviour of the system of ODEs.
Some analyses which can be carried out via numericalsolution of the CTMC are not readily available from ODEs orstochastic simulation.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Modelling biological features
There are some features of biochemical reaction systems whichare not readily captured by many of the stochastic processalgebras that are currently in use.
Particular problems are encountered with:
stoichiometry — the multiplicity in which an entity participatesin a reaction;
general kinetic laws — although mass action is widely usedother kinetics are also commonly employed.
multiway reactions — although thermodynamic argumentscan be made that there are never more than two reagentsinvolved in a reaction, in practice it is often useful to model ata more abstract level.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Modelling biological features
There are some features of biochemical reaction systems whichare not readily captured by many of the stochastic processalgebras that are currently in use.
Particular problems are encountered with:
stoichiometry — the multiplicity in which an entity participatesin a reaction;
general kinetic laws — although mass action is widely usedother kinetics are also commonly employed.
multiway reactions — although thermodynamic argumentscan be made that there are never more than two reagentsinvolved in a reaction, in practice it is often useful to model ata more abstract level.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Modelling biological features
There are some features of biochemical reaction systems whichare not readily captured by many of the stochastic processalgebras that are currently in use.
Particular problems are encountered with:
stoichiometry — the multiplicity in which an entity participatesin a reaction;
general kinetic laws — although mass action is widely usedother kinetics are also commonly employed.
multiway reactions — although thermodynamic argumentscan be made that there are never more than two reagentsinvolved in a reaction, in practice it is often useful to model ata more abstract level.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Modelling biological features
There are some features of biochemical reaction systems whichare not readily captured by many of the stochastic processalgebras that are currently in use.
Particular problems are encountered with:
stoichiometry — the multiplicity in which an entity participatesin a reaction;
general kinetic laws — although mass action is widely usedother kinetics are also commonly employed.
multiway reactions — although thermodynamic argumentscan be made that there are never more than two reagentsinvolved in a reaction, in practice it is often useful to model ata more abstract level.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Modelling biological features
There are some features of biochemical reaction systems whichare not readily captured by many of the stochastic processalgebras that are currently in use.
Particular problems are encountered with:
stoichiometry — the multiplicity in which an entity participatesin a reaction;
general kinetic laws — although mass action is widely usedother kinetics are also commonly employed.
multiway reactions — although thermodynamic argumentscan be made that there are never more than two reagentsinvolved in a reaction, in practice it is often useful to model ata more abstract level.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA
In Bio-PEPA:
Unique rates are associated with each reaction (action) type,separately from the specification of the logical behaviour.These rates may be specified by functions.
The representation of an action within a component (species)records the stoichiometry of that entity with respect to thatreaction. The role of the entity is also distinguished.
The local states of components are quantitative rather thanfunctional, i.e. distinct states of the species are representedas distinct components, not derivatives of a single component.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA
In Bio-PEPA:
Unique rates are associated with each reaction (action) type,separately from the specification of the logical behaviour.These rates may be specified by functions.
The representation of an action within a component (species)records the stoichiometry of that entity with respect to thatreaction. The role of the entity is also distinguished.
The local states of components are quantitative rather thanfunctional, i.e. distinct states of the species are representedas distinct components, not derivatives of a single component.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA
In Bio-PEPA:
Unique rates are associated with each reaction (action) type,separately from the specification of the logical behaviour.These rates may be specified by functions.
The representation of an action within a component (species)records the stoichiometry of that entity with respect to thatreaction. The role of the entity is also distinguished.
The local states of components are quantitative rather thanfunctional, i.e. distinct states of the species are representedas distinct components, not derivatives of a single component.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Bio-PEPA
In Bio-PEPA:
Unique rates are associated with each reaction (action) type,separately from the specification of the logical behaviour.These rates may be specified by functions.
The representation of an action within a component (species)records the stoichiometry of that entity with respect to thatreaction. The role of the entity is also distinguished.
The local states of components are quantitative rather thanfunctional, i.e. distinct states of the species are representedas distinct components, not derivatives of a single component.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
The Bio-PEPA Language
Sequential component (species component)
S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | |
Model componentP ::= P BC
LP | S(l)
The parameter l is abstract, recording quantitative informationabout the species.
The system description records the impact of action on thisquantity which may be
number of molecules (SSA),
concentration (ODE) or
a level within a semi-quantitative model (CTMC).
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
Model-checking
Explicit state model checking (exhaustive exploration for allpossible. states/executions): exact results obtained vianumerical computation
Statistical model-checking (discrete event simulation andsampling over multiple runs): approximate results.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Analysing models of biological processes via probabilisticmodel-checking has considerable appeal.
As with stochastic simulation the answers which are returnedfrom model-checking give a thorough stochastic treatment tothe small-scale phenomena.
However, in contrast to a simulation run which generates justone trajectory, probabilistic model-checking gives a definitiveanswer so it is not necessary to re-run the analysis repeatedlyand compute ensemble averages of the results.
Building a reward structure over the model it is possible toexpress complex analysis questions.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Analysing models of biological processes via probabilisticmodel-checking has considerable appeal.
As with stochastic simulation the answers which are returnedfrom model-checking give a thorough stochastic treatment tothe small-scale phenomena.
However, in contrast to a simulation run which generates justone trajectory, probabilistic model-checking gives a definitiveanswer so it is not necessary to re-run the analysis repeatedlyand compute ensemble averages of the results.
Building a reward structure over the model it is possible toexpress complex analysis questions.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Analysing models of biological processes via probabilisticmodel-checking has considerable appeal.
As with stochastic simulation the answers which are returnedfrom model-checking give a thorough stochastic treatment tothe small-scale phenomena.
However, in contrast to a simulation run which generates justone trajectory, probabilistic model-checking gives a definitiveanswer so it is not necessary to re-run the analysis repeatedlyand compute ensemble averages of the results.
Building a reward structure over the model it is possible toexpress complex analysis questions.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Analysing models of biological processes via probabilisticmodel-checking has considerable appeal.
As with stochastic simulation the answers which are returnedfrom model-checking give a thorough stochastic treatment tothe small-scale phenomena.
However, in contrast to a simulation run which generates justone trajectory, probabilistic model-checking gives a definitiveanswer so it is not necessary to re-run the analysis repeatedlyand compute ensemble averages of the results.
Building a reward structure over the model it is possible toexpress complex analysis questions.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Probabilistic model checking in PRISM is based on a CTMCand the logic CSL.
Formally the mapping from Bio-PEPA is based on thestructured operational semantics, generating the underlyingCTMC in the usual way.
As with SSA, in practice it is more straightforward to directlymap to the input language of the tool.
PRISM expresses systems as interacting, reactive modules.From a Bio-PEPA description one module is generated foreach species component with an additional module to capturethe functional rate information.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Probabilistic model checking in PRISM is based on a CTMCand the logic CSL.
Formally the mapping from Bio-PEPA is based on thestructured operational semantics, generating the underlyingCTMC in the usual way.
As with SSA, in practice it is more straightforward to directlymap to the input language of the tool.
PRISM expresses systems as interacting, reactive modules.From a Bio-PEPA description one module is generated foreach species component with an additional module to capturethe functional rate information.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Probabilistic model checking in PRISM is based on a CTMCand the logic CSL.
Formally the mapping from Bio-PEPA is based on thestructured operational semantics, generating the underlyingCTMC in the usual way.
As with SSA, in practice it is more straightforward to directlymap to the input language of the tool.
PRISM expresses systems as interacting, reactive modules.From a Bio-PEPA description one module is generated foreach species component with an additional module to capturethe functional rate information.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Levels of Abstraction
PRISM model and model checking
Probabilistic model checking in PRISM is based on a CTMCand the logic CSL.
Formally the mapping from Bio-PEPA is based on thestructured operational semantics, generating the underlyingCTMC in the usual way.
As with SSA, in practice it is more straightforward to directlymap to the input language of the tool.
PRISM expresses systems as interacting, reactive modules.From a Bio-PEPA description one module is generated foreach species component with an additional module to capturethe functional rate information.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Ostreococcus Tauri
Circadian Rhythms in Ostreococcus Tauri
The green alga Ostreococcus Tauri is an ideal model system forunderstanding the function of plant clocks.
In particular we use a Bio-PEPA model to study the variability androbustness of the clock’s functional behaviour with respect to internalstochastic noise, environmental changes and mutational changes.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Ostreococcus Tauri
The biological model
Feedback loop between the two main genes (TOC1 and LHY).
Effect of light on several parts of the network.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Modelling overview
A previous deterministic (ODE) model of the system had beendeveloped by hand.
We developed a Bio-PEPA model and validated that itscontinuous-deterministic interpretation coincided with the originalmodel.
Stochastic simulation was used with the discrete-stochasticinterpretation of the Bio-PEPA model to investigate stochasticfluctuations, focusing on clock phase and sensitivity analysis.
Model-checking was also used to give more detailed analysis ofvariability within the system, via specific questions about modelbehaviour.
An evolutionary systems biology framework was used to investigatethe potential affect low-level mutational effects might have on thephase of the clock.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Model and experimental settings
Lab experiments in different light conditions and photoperiods
DD (24 hours dark)LL (24 hours light)LD 12:12 (12 hours light / 12 hours dark)LD 6:18 (6 hours light / 18 hours dark)LD 18:6 (18 hours light / 6 hours dark)
Lab measurements on amounts of proteins
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Modelling in Bio-PEPA
Biological species⇒ interacting species components
Reactions⇒ actions, with functional rates expressing the kineticrate laws
Light on/off mechanism for entrainment to day/night cycle⇒events switching between day-time and night-time rates
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Extracts from the Bio-PEPA model
TOC1 mRNA = (transc3 , 1) ↑ + (transl5 , 1) ⊕ + (deg7 , 1) ↓TOC1 a = (deg4 , 1) ↓ + (conv6 , 1) ↑ + (transc8 , 1)⊕TOC1 i = (transl5 , 1) ↑ + (conv6 , 1) ↓
LHY mRNA = (transc8 , 1) ↑ + (deg9 , 1) ↓ + (transl10 , 1)⊕LHY c = (transl10 , 1) ↑ + (transp11, 1) ↓ + (deg12 , 1) ↓LHY n = (transc3 , 1) + (transp11, 1) ↑ + (deg13 , 1) ↓
acc = (prod1, 1) ↑ + (deg2 , 1) ↓ + (transc3 , 1)⊕
total TOC1 = TOC1 i + TOC1 atotal LHY = LHY c + LHY n
t dawn = 6, t dusk = 18 (e.g. for LD 12:12 system)
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Simulations – constant light (LL)
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LL −− ODEs
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LL −− SSA (10000 runs)
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LL −− SSA (10 runs)
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LL −− SSA (1 run)
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Simulations – light/dark (LD 12:12)
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LD 12:12 −− ODEs
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LD 12:12 −− SSA (10000 runs)
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LD 12:12 −− SSA (10 runs)
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
0 50 100 150 200 2500
100
200
300
400
500
Time (hours)
Leve
l
LD 12:12 −− SSA (1 run)
LHY mRNA TOC1 mRNATotal LHYTotal TOC1
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Model-checking
Here we use statistical model-checking (discrete eventsimulation and sampling over multiple runs): approximateresults.
The underlying simulation model is enhanced with arepresentation of time so that rates of reactions can changeappropriately at dawn and dusk.
The model checker can then be used to investigate theprobability distribution of the number of molecules of aspecies over time.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Model-checking – probability distribution of LHY valueover 0-96 h
P=?[F[T ,T ] ( LHY c + LHY n = level)],T = 0 : 3 : 96, level = 0 : 1 : 500
Time
Tot
al L
HY
leve
l
0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960
50
100
150
200
250
300
350
400
450
500
0
0.005
0.01
0.015
0.02
0.025
0.03
Time
Tot
al L
HY
leve
l
0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960
50
100
150
200
250
300
350
400
450
500
0
0.005
0.01
0.015
0.02
0.025
0.03
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Probability that the total LHY stays below some thresholde in the long run
P=?[G[96,500] (LHY c + LHY n ≤ 0 + e)]
e 0 1 2 3 4Prob 0.93 0.96 0.96 0.98 0.98
e 5 6 7 8 9 10Prob 0.99 0.99 0.99 0.99 0.99 1.0
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Stochastic modelling and model checking
Probability distribution/coefficient of variation of LHYvalue – LD 12:12
P=?[F[T ,T ] ( LHY c + LHY n = level)],T = 120 : 1 : 144, level = 0 : 1 : 500
Time
Tot
al L
HY
leve
l
0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960
50
100
150
200
250
300
350
400
450
500
0
0.005
0.01
0.015
0.02
0.025
0.03
Time
Tot
al L
HY
leve
l
0 2 4 6 8 10 12 14 16 18 20 22 240
50
100
150
200
250
300
350
400
450
500
0
0.005
0.01
0.015
0.02
Time
Coe
ffici
ent o
f var
iatio
n
0 2 4 6 8 10 12 14 16 18 20 22 240
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
“Evolutionary Systems Biology”
Define system property P for systems biology model S.
P is a computable property with a link to fitness(e.g. energy costs, accuracy of clock, survival rate).
Assume that non-lethal mutations manifest as variations in therates of reactions (e.g. due to conformational changes of proteinsor mRNAs).
Compute P in wild type reference
Collect enough samples
Mutate S by changing a biochemical reaction rate
Measure mutational effects by recomputing P
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Sensitivity of clock phase to mRNA degradation
Ø : Degradation : Light input Carl Troein
TOCmRNA TOC1
LHYmRNA LHYc LHYn
TOC2 Ø
Ø
Ø
Ø
Previous work suggested that the peak of the total amount of TOCis a good indicator of the clock phase.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Distribution of phase in wildtype (10,000 runs)
020
040
060
080
010
00
Phase of peak of Total TOC1 [hour of day]
Cou
nts
0 3 6 9 12 15 18 21 24
duskdawn
Phase is measured by the peak of total amount of TOC.
This almost always occurs between 6pm and 1am.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Distribution of phase in mutant (10,000 runs)
020
040
060
080
010
00
Phase of peak of Total TOC1 [hour of day]
Cou
nts
0 3 6 9 12 15 18 21 24
duskdawn
We investigate mutants in which mRNA degradation rates are multipliedby a factor uniformly distributed between 0.5 and 1.5.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Reducing stochastic noise to identify mutational noise
These results are based on 10,000 simulation runs with thesystem scaled to match the experimental data.
This is our best estimate of a single run which corresponds toa single cell.
In this case the scaling factor Ω is set to 50.
However when considering mutational effects we would like tominimise the inherent stochasticity in the system and focus onthe mutational noise.
To achieve this we increase the scaling factor by six orders ofmagnitude.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Distribution of phase in wildtype(10,000 runs), large Ω
010
020
030
040
050
060
0
Phase of peak of Total TOC1 [hour of day]
Cou
nts
0 3 6 9 12 15 18 21 24
duskdawn
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Distribution of phase in wildtype(10,000 runs), large Ω
Phase of peak of Total TOC1 [hour of day]
Cou
nts
21.38 21.40 21.42 21.44
010
020
030
040
050
060
0
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Distribution of phase in mutant(10,000 runs), large Ω
050
010
0015
0020
0025
00
Phase of peak of Total TOC1 [hour of day]
Cou
nts
0 3 6 9 12 15 18 21 24
duskdawn
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Mean time of the peak as degradation rate varies (largeΩ)
0.6 0.8 1.0 1.2 1.4
Factor by which mRNA degradation is changed
Phas
e of
pea
k of
Tot
al T
OC
1 [h
our o
f day
]
0
3
6
9
12
15
18
21
24
dusk
dawn
wildtype
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Distribution of the time of the peak as degradation ratevaries (small Ω)
0.6 0.8 1.0 1.2 1.4
Factor by which mRNA degradation is changed
Phas
e of
pea
k of
Tot
al T
OC
1 [h
our o
f day
]
0
3
6
9
12
15
18
21
24
dusk
dawn
wildtype
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Evolutionary Systems Biology Framework
Loewe, L. 2009 A framework for evolutionary systems biology. BMC Systems Biology 3:27.
Mutants
Mutations
WildtypeFactor by which mRNA degradation is changed
Cou
nts
0.6 0.8 1.0 1.2 1.4
050
100
150
200
050
010
0015
0020
0025
00
Phase of peak of Total TOC1 [hour of day]
Cou
nts
0 3 6 9 12 15 18 21 24
duskdawn010
020
030
040
050
060
0
Phase of peak of Total TOC1 [hour of day]
Cou
nts
0 3 6 9 12 15 18 21 24
duskdawn
Using this evolutionary systems biology framework we caninvestigate how low-level mutational effects affect fitness.
Such small effects are very hard to measure in the lab but becomeaccessible through simulations.
Bio-PEPA provides a sound modelling basis on which theframework can be built.
Process Algebras for Systems Biology Bio-PEPA Modelling Circadian Rhythms
Investigating mutational effects
Thank you!
People involved in the work:
Ozgur Akman
Federica Ciocchetta
Adam Duguid
Stephen Gilmore
Laurence Loewe
Andrew Millar
Carl Troein
Acknowledgements: Engineering and Physical SciencesResearch Council (EPSRC) and Biotechnology and BiologicalSciences Research Council (BBSRC).