Calculi for Systems Biologyhomepages.inf.ed.ac.uk/jeh/TALKS/pisa.pdfCalculi for Systems Biology Jane...

Introduction to Systems Biology Stochastic Process Algebra Case Studies Summary

Calculi for Systems Biology

Jane Hillston.LFCS, University of Edinburgh

18th October 2007

Joint work with Federica Ciocchetta

Jane Hillston. LFCS, University of Edinburgh.

The PEPA project

I The PEPA project started in Edinburgh in 1991.

I It was motivated by problems encountered when carrying outperformance analysis of large computer and communicationsystems, based on numerical analysis of Markov processes.

I Process algebras offered a compositional descriptiontechnique supported by apparatus for formal reasoning.

I Performance Evaluation Process Algebra (PEPA) sought toaddress these problems by the introduction of a suitableprocess algebra.

I The project has sought to investigate and exploit the interplaybetween the process algebra and the continuous time Markovchain (CTMC).

The PEPA project

I The PEPA project started in Edinburgh in 1991.I It was motivated by problems encountered when carrying out

performance analysis of large computer and communicationsystems, based on numerical analysis of Markov processes.

The PEPA project

PEPA Case Studies (1)

I Multiprocessor access-contention protocols (Gilmore, Hillstonand Ribaudo, Edinburgh and Turin)

I Protocols for fault-tolerant systems (Clark, Gilmore, Hillstonand Ribaudo, Edinburgh and Turin)

I Multimedia traffic characteristics (Bowman et al, Kent)I Database systems (The STEADY group, Heriot-Watt

University)I Software Architectures (Pooley, Bradley and Thomas,

Heriot-Watt and Durham)I Switch behaviour in active networks (Hillston, Kloul and

Mokhtari, Edinburgh and Versailles)

I Locks and movable bridges ininland shipping in Belgium(Knapen, Hasselt)

I Robotic workcells (Holton,Gilmore and Hillston, Bradfordand Edinburgh)

I Cellular telephone networks(Kloul, Fourneau and Valois,Versailles)

I Automotive diagnostic expertsystems (Console, Picardi andRibaudo, Turin)

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OutlineIntroduction to Systems Biology

MotivationChallenges

Stochastic Process AlgebraAbstract ModellingCase StudyBio-PEPA

Case StudiesSimple genetic networkGoldbeter’s modelExtended model

Summary

Motivation

Systems Biology

I Biological advances mean that much more is now knownabout the components of cells and the interactions betweenthem.

I Systems biology aims to develop a better understanding ofthe processes involved.

I It involves taking a systems theoretic view of biologicalprocesses — analysing inputs and outputs and therelationships between them.

I A radical shift from earlier reductionist approaches, systemsbiology aims to provide a conceptual basis and amethodology for reasoning about biological phenomena.

Motivation

Systems Biology

Motivation

Systems Biology

Motivation

Systems Biology

Motivation

Systems Biology Methodology

Explanation

ExplanationInterpretation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

Biological Phenomena

-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

Biological Phenomena-

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

Biological Phenomena-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

� Deduction

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Systems Analysis?

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

� Deduction

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Systems Analysis?

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Induction

Modelling

Formal System

-Measurement

Measurement

Observation

Deduction

Inference

Motivation

Explanation

Interpretation

Natural System

Systems Analysis

Systems Analysis?

Induction

Modelling

Formal System

Biological Phenomena-Measurement

Measurement

Observation

� Deduction

Deduction

Inference

Motivation

Formal Systems

There are two alternative approaches to contructing dynamicmodels of biochemical pathways commonly used by biologists:

I Ordinary Differential Equations:I continuous time,I continuous behaviour (concentrations),I deterministic.

I Stochastic Simulation:I continuous time,I discrete behaviour (no. of molecules),I stochastic.

Motivation

Formal Systems

There are two alternative approaches to contructing dynamicmodels of biochemical pathways commonly used by biologists:I Ordinary Differential Equations:

I continuous time,I continuous behaviour (concentrations),I deterministic.

Motivation

Formal Systems

There are two alternative approaches to contructing dynamicmodels of biochemical pathways commonly used by biologists:I Ordinary Differential Equations:

I continuous time,I continuous behaviour (concentrations),I deterministic.

Motivation

Limitations of Ordinary Differential Equations

I Given knowledge of initial molecular concentrations, the law ofmass action provides a complete picture of the componentconcentrations at all future time points.

I This is based on the assumption that chemical reactions to bemacroscopic under convective or diffusive stirring, continuousand deterministic.

I This is a simplification, because in reality chemical reactionsinvolve discrete, random collisions between individualmolecules.

I As we consider smaller and smaller systems, the validity of acontinuous approach becomes ever more tenuous.

Motivation

Stochastic simulation algorithms

Gillespie’s Stochastic Simulation Algorithm (SSA) gives anumerical simulation of the time evolution of a well-stirredchemically reacting system by taking proper account of therandomness inherent in such a system.

It is derived from the chemical master equation and gives a morerealistic representation of a system’s evolution than thedeterministic reaction rate equation (RRE) representedmathematically by ODEs.

Since each molecule is represented explicitly the number ofgenerated states can be extremely large.

Motivation

Systems Analysis

I In biochemical signalling pathways the events of interests are:I when reagent concentrations start to increase;I when concentrations pass certain thresholds;I when a peak of concentration is reached.

I E.g. in a gene network the delay from the activation of onegene until the next promoter reaches an effective level toactivate the next gene depends on the rate of proteinaccumulation.

I The accumulation of protein is a stochastic process affectedby several factors in the cell (temperature, pH, etc.).

I Thus it is a distribution rather than a deterministic time.I Models should match wet lab experimental data.

Motivation

Systems Analysis

Motivation

Systems Analysis

Motivation

Systems Analysis

I Thus it is a distribution rather than a deterministic time.

I Models should match wet lab experimental data.

Motivation

Systems Analysis

Challenges

Individual vs. Population behaviourI Biochemistry is concerned with the reactions between

individual molecules and so it is often more natural to modelat this level.

I However experimental data is usually more readily available interms of populations rather than individual molecules cf.average reaction rates rather than the forces at play on anindividual molecule in a particular physical context.

I These should be regarded as alternatives, each beingappropriate for some models. The challenge then becomeswhen to use which approach.

I Note that given a large enough number of molecules an“individuals” model will (in many circumstances) beindistinguishable from the a “population” level model.

Challenges

Noise vs. Determinism

I With perfect knowledge the behaviour of a biochemicalreaction would be deterministic.

I However, in general, we do not have the requisite knowledgeof thermodynamic forces, exact relative positions,temperature, velocity etc.

I Thus a reaction appears to display stochastic behaviour.I When a large number of such reactions occur, the

randomness of the individual reactions can cancel each otherout and the apparent behaviour exhibits less variability.

I However, in some systems the variability in the stochasticbehaviour plays a crucial role in the dynamics of the system.

Challenges

I Thus a reaction appears to display stochastic behaviour.

I When a large number of such reactions occur, therandomness of the individual reactions can cancel each otherout and the apparent behaviour exhibits less variability.

Challenges

Modularity vs. Infinite Regress

As computer scientists we are firm believers in modularity andcompositionality. When it comes to biochemical pathways opinionamongst biologists is divided about whether is makes sense totake a modular view of cellular pathways.

Some biologists (e.g. Leibler) argue that there is modularity,naturally occuring, where they define a module relative to abiological function.

Others such as Cornish-Bowden are much more skeptical and citethe problem of infinite regress as being insurmountable.

Challenges

The problem of Infinite Regress

Challenges

Dealing with the Unknown

There is a fundamental challenge when modelling cellularpathways that little is known about some aspects of cellularprocesses.

In some cases this is because no experimental data is available, orthat the experimental data that is available is inconsistent.

In other cases the data is unknowable because experimentaltechniques do not yet exist to collect the data, or those that doinvolve modification to the system.

Even when data exists the quality is often very poor.

Challenges

Formal Systems Revisited

I In most current work mathematics is being used directly asthe formal system.

I Previous experience in the performance arena has shown usthat there can be benefits to interposing a formal modelbetween the system and the underlying mathematical model.

I Moreover taking this “high-level programming” style approachoffers the possibility of different “compilations” to differentmathematical models.

Challenges

Summary

Using Stochastic Process Algebras

Process algebras have several attractive features which could beuseful for modelling and understanding biological systems:

I Process algebraic formulations are compositional and makeinteractions/constraints explicit.

I Structure can also be apparent.I Equivalence relations allow formal comparison of high-level

descriptions.I There are well-established techniques for reasoning about the

behaviours and properties of models, supported by software.These include qualitative and quantitative analysis, and modelchecking.

Process algebras have several attractive features which could beuseful for modelling and understanding biological systems:I Process algebraic formulations are compositional and make

interactions/constraints explicit.

I Structure can also be apparent.I Equivalence relations allow formal comparison of high-level

interactions/constraints explicit.I Structure can also be apparent.

I Equivalence relations allow formal comparison of high-leveldescriptions.

I There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.These include qualitative and quantitative analysis, and modelchecking.

interactions/constraints explicit.I Structure can also be apparent.I Equivalence relations allow formal comparison of high-level

descriptions.

I There are well-established techniques for reasoning about thebehaviours and properties of models, supported by software.These include qualitative and quantitative analysis, and modelchecking.

interactions/constraints explicit.I Structure can also be apparent.I Equivalence relations allow formal comparison of high-level

Molecular processes as concurrent computations

ConcurrencyMolecularBiology Metabolism Signal

Transduction

Concurrentcomputational processes

MoleculesEnzymesandmetabolites

Interactingproteins

Synchronous communica-tion

Molecularinteraction

Binding andcatalysis

Transition or mobilityBiochemicalmodification orrelocation

Metabolitesynthesis

Protein binding,modification orsequestration

[Regev et al 2000]

Molecular processes as concurrent computations

ConcurrencyMolecularBiology Metabolism Signal

Transduction

Concurrentcomputational processes

MoleculesEnzymesandmetabolites

Interactingproteins

Synchronous communica-tion

Molecularinteraction

Binding andcatalysis

Transition or mobilityBiochemicalmodification orrelocation

Metabolitesynthesis

Protein binding,modification orsequestration

[Regev et al 2000]

Abstract Modelling

Mapping biological systems to process algebraThe work using the stochastic π-calculus and related calculi, mapsa molecule to a process in the process algebra description.

This is an inherently individuals-based view of the system andanalysis will generally be via stochastic simulation.

In the PEPA modelling we have been doing we have experimentedwith more abstract mappings between process algebra constructsand elements of signalling pathways.

In our mapping we focus on species (c.f. a type rather than aninstance, or a class rather than an object).

Alternative mappings from the process algebra to underlyingmathematics are then readily available.

Abstract Modelling

Motivations for Abstraction

Our motivations for seeking more abstraction in process algebramodels for systems biology are:

I 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.

I The data that we have available to parameterise models issometimes speculative rather than precise.

This suggests thatit can be useful to use semiquantitative models rather thanquantitative ones.

Abstract Modelling

Our motivations for seeking more abstraction in process algebramodels for systems biology are:I 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.

Abstract Modelling

I The data that we have available to parameterise models issometimes speculative rather than precise. This suggests thatit can be useful to use semiquantitative models rather thanquantitative ones.

Abstract Modelling

Alternative Representations

population view

StochasticSimulation

individual view

AbstractSPA model

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HHHHHj

Abstract Modelling

ODEs population view

StochasticSimulation individual view

AbstractSPA model

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��*

HHHHHj

Abstract Modelling

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.

Abstract Modelling

population view

CTMC withM levels

abstract view

individual view

AbstractPEPA model

-��

��*

HHHHHj

Model checking andMarkovian analysis

Abstract Modelling

ODEs population view

CTMC withM levels abstract view

StochasticSimulation individual view

AbstractPEPA model

-��

��*

HHHHHj

Abstract Modelling

population view

CTMC withM levels

abstract view

individual view

AbstractPEPA model

-��

��*

HHHHHj

Abstract Modelling

SPA Languages

SPASPA ��

��

integrated timeintegrated timeintegrated time

orthogonal timeorthogonal time

��

exponential onlyexponential onlyexponential onlyPEPA, Stochastic π-calculusPEPA, Stochastic π-calculus

exponential + instantaneousexponential + instantaneousEMPA, Markovian TIPPEMPA, Markovian TIPP

general distributionsgeneral distributionsTIPP, SPADES, GSMPATIPP, SPADES, GSMPA

exponential onlyexponential onlyIMCIMC

general distributionsgeneral distributionsIGSMP, ModestIGSMP, Modest

Abstract Modelling

SPA Languages

��

integrated time

integrated timeintegrated time

orthogonal time

��

exponential onlyexponential onlyexponential onlyPEPA, Stochastic π-calculusPEPA, Stochastic π-calculus

exponential + instantaneousexponential + instantaneousEMPA, Markovian TIPPEMPA, Markovian TIPP

general distributionsgeneral distributionsTIPP, SPADES, GSMPATIPP, SPADES, GSMPA

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

exponential onlyexponential onlyPEPA, Stochastic π-calculusPEPA, Stochastic π-calculus

exponential + instantaneous

exponential + instantaneousEMPA, Markovian TIPPEMPA, Markovian TIPP

general distributions

general distributionsTIPP, SPADES, GSMPATIPP, SPADES, GSMPA

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

exponential onlyexponential onlyPEPA, Stochastic π-calculusPEPA, Stochastic π-calculus

exponential + instantaneousEMPA, Markovian TIPPEMPA, Markovian TIPP

general distributionsTIPP, SPADES, GSMPATIPP, SPADES, GSMPA

exponential only

exponential onlyIMCIMC

general distributionsIGSMP, ModestIGSMP, Modest

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

PEPA, Stochastic π-calculus

EMPA, Markovian TIPPEMPA, Markovian TIPP

TIPP, SPADES, GSMPATIPP, SPADES, GSMPA

exponential only

IMCIMC

IGSMP, ModestIGSMP, Modest

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

exponential + instantaneousEMPA, Markovian TIPP

EMPA, Markovian TIPP

TIPP, SPADES, GSMPATIPP, SPADES, GSMPA

exponential only

IMCIMC

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

general distributionsTIPP, SPADES, GSMPA

TIPP, SPADES, GSMPA

exponential only

IMCIMC

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

TIPP, SPADES, GSMPA

exponential only

exponential onlyIMC

Abstract Modelling

SPA Languages

��

integrated time

orthogonal time

��

exponential only

TIPP, SPADES, GSMPA

exponential only

exponential onlyIMC

general distributionsIGSMP, Modest

IGSMP, Modest

Abstract Modelling

SPA Languages

SPASPA

��

integrated timeintegrated time

integrated time

orthogonal time

��

exponential onlyexponential only

exponential only

TIPP, SPADES, GSMPA

exponential only

IGSMP, Modest

Abstract Modelling

PEPA: Performance Evaluation Process Algebra

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

The language may be used to generate a Markov Process (CTMC).

SPAMODEL

LABELLEDTRANSITION

SYSTEMCTMC Q- -

SOS rules state transition

diagram

Q is the infinitesimal generator matrix characterising the CTMC.Each of these has tool support so that the underlying model isderived automatically according to the predefined rules.

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

LABELLEDTRANSITION

SYSTEMCTMC Q- -

diagram

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

LABELLEDTRANSITION

SYSTEMCTMC Q

SOS rules

state transition

diagram

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

LABELLEDTRANSITION

SYSTEM

CTMC Q

SOS rules

state transition

diagram

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

LABELLEDTRANSITION

SYSTEM

CTMC Q

- -SOS rules state transition

diagram

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

LABELLEDTRANSITION

SYSTEMCTMC Q- -

diagram

Q is the infinitesimal generator matrix characterising the CTMC.

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

The language may be used to generate a system of ordinary differ-ential equations (ODEs).

SPAMODEL

ACTIVITYMATRIX ODEs- -

syntactic

analysis

continuous

interpretation

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

syntactic

analysis

continuous

interpretation

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

ACTIVITYMATRIX ODEs

syntactic

analysis

continuous

interpretation

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

ACTIVITYMATRIX

syntactic

analysis

continuous

interpretation

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

ACTIVITYMATRIX

- -syntactic

analysis

continuous

interpretation

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

syntactic

analysis

continuous

interpretation

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

The language also may be used to generate a stochastic simulation.

SPAMODEL

RATEEQUATIONS

STOCHASTICSIMULATION

- -syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

RATEEQUATIONS

- -syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

RATEEQUATIONS

syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

RATEEQUATIONS

syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

RATEEQUATIONS

- -syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

RATEEQUATIONS

- -syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

S ::= (α, r).S | S + S | A

P ::= S | P BCL

P | P/L

SPAMODEL

RATEEQUATIONS

- -syntactic

analysis

Gillespie’s

algorithm

Abstract Modelling

Reagent-centric modelling [CGH04]Reagent role Impact on reagent Impact on reaction rateProducer decreases concentration has a positive impact,

i.e. proportional to cur-rent concentration

Product increases concentration has no impact on therate, except at saturation

Enzyme concentration unchanged has a positive impact,i.e. proportional to cur-rent concentration

Inhibitor concentration unchanged has a negative im-pact, i.e. inverselyproportional to currentconcentration

Abstract Modelling

PEPA reagent-centric example

AHdef= (ab c, α).AL

ALdef= (b a, β).AH+(c a, γ).AH

BHdef= (ab c, α).BL+(b a, β).BL

BLdef= (c b , δ).BH

CHdef= (c a, γ).CL+(c b , δ).CL

CLdef= (ab c, α).CH

(AH BC{ab c,b a}

BH) BC{ab c,c a,c b}

Abstract Modelling

(AH BC{ab c,b a}

Abstract Modelling

(AH BC{ab c,b a}

Abstract Modelling

(AH BC{ab c,b a}

Abstract Modelling

(AH BC{ab c,b a}

Case Study

Example: The Ras/Raf-1/MEK/ERK pathway

m 1 m 2

m 7 m 5 m 6 m 10

MEK−PP ERK RKIP−P RP

RKIP−P/RP

Raf−1*/RKIP

Raf−1*−RKIP/ERK−PP

RKIPRaf−1*

ERK−PP

MEK−PP/ERK−P

MEK/Raf−1*

k12/k13

k9/k10

Case Study

PEPA components of the reagent-centric modelm 3

Raf−1*/RKIP

Raf-1∗/RKIP/ERK-PPHdef=

(k5product , k5).Raf-1∗/RKIP/ERK-PPL

+ (k4react , k4).Raf-1∗/RKIP/ERK-PPL

Raf-1∗/RKIP/ERK-PPLdef=

(k3react , k3).Raf-1∗/RKIP/ERK-PPH

Each reagent gives rise to a pair of PEPA definitions, one for highconcentration and one for low concentration.

Case Study

PEPA components of the reagent-centric modelm 3

Raf−1*/RKIP

Raf-1∗/RKIP/ERK-PPHdef=

(k5product , k5).Raf-1∗/RKIP/ERK-PPL

+ (k4react , k4).Raf-1∗/RKIP/ERK-PPL

Raf-1∗/RKIP/ERK-PPLdef=

(k3react , k3).Raf-1∗/RKIP/ERK-PPH

Each reagent gives rise to a pair of PEPA definitions, one for highconcentration and one for low concentration.

Case Study

Commentary on the model

I Here we have shown the model with only high and low levelsof concentration.

I In general we would discretise the concentration into morelevels, say 6 or 7 levels. As we add levels we are capturing theconcentration at finer levels of granularity.

I In fact to generate ODE and SSA models we only need twolevels as this is sufficient to record the impact of each reactionon each reagent.

Case Study

The state space

s21 s22 s15 s17

s16s14s20s19

s7 s8 s10 s12

s9 s11 s13

s18s25 s23 s1 s3

s4s2s24s26

s28 s27 s5

k5product

k6react k6react k6react k6reactk7react k7react k7react k7react

k11product

k11productk11product

k11product

k15productk15productk15productk15productk15productk15product

k8product k8product k8product k8product

k9react

k10react

k9react

k9react k1react

k2react

k3react

k4react

k3react

k4react

k1react

k2react

k1react

k2react

k1react

k2react

k1react

k2react

k15productk15productk15product

k12react k12react k12react k12react k12react k12react

k13react k13reactk13reactk13reactk13reactk14product k14productk14product k14productk14product k14product

Case Study

Alternative modelsI When a molecular mapping is used in general a CTMC state

space is too large to permit anything but stochastic simulation.

I 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.

I In reagent PEPA models with levels, each level of granularitygives rise to a CTMC, and the behaviour of this sequence ofMarkov processes converges to the behaviour of the systemof ODEs.

I Some analyses which can be carried out via numericalsolution of the CTMC are not readily available from ODEs orstochastic simulation.

Case Study

space is too large to permit anything but stochastic simulation.I The ODE model can be regarded as an approximation of a

CTMC in which the number of molecules is large enough thatthe randomness averages out and the system is essentiallydeterministic.

Case Study

Markovian analysis

I Analysis of the Markov process can yield quite detailedinformation about the dynamic behaviour of the model.

I A steady state analysis provides statistics for averagebehaviour over a long run of the system, when the biasintroduced by the initial state has been lost.

I A transient analysis provides statistics relating to the evolutionof the model over a fixed period. This will be dependent onthe starting state.

I Stochastic model checking is available via the PRISM modelchecker, assessing the probable validity of propertiesexpressed in CSL (Continuous Stochastic Logic).

Case Study

Markovian analysis

Case Study

Markovian analysis

Case Study

Markovian analysis

Case Study

Quantified analysis – k8productApproximating a variation in the initial concentration of RKIP byvarying the rate constant k1, we can assess the impact on theproduction of ERK-PP.

Throughput of k8product

2 4 6 8 10k1

Case Study

Quantified analysis – k14product

Similarly we can assess the impact on the production of MEK-PP.

Throughput of k14product

2 4 6 8 10k1

Case Study

ODE analysis

Solving a system of ODEs will show how the concentrations ofreagents vary over time.

Solution is (relatively) fast and definitive....

... but no variability is captured, unlike Markovian analyses (andreal systems).

Case Study

ODE analysis

Case Study

ODE analysis

Case Study

ODEs from SPA

There are advantages to be gained by using a process algebramodel as an intermediary to the derivation of the ODEs.

I The ODEs can be automatically generated from thedescriptive process algebra model, thus reducing humanerror.

I The process algebra model allow us to derive properties ofthe model, such as freedom from deadlock, before numericalanalysis is carried out.

I The algebraic formulation of the model emphasisesinteractions between the biochemical entities.

Case Study

ODEs from SPA

There are advantages to be gained by using a process algebramodel as an intermediary to the derivation of the ODEs.I The ODEs can be automatically generated from the

descriptive process algebra model, thus reducing humanerror.

Case Study

ODEs from SPA

Case Study

ODEs from SPA

Case Study

ODEs from SPA

Case Study

ODE Analysis of the MAPK example

Case Study

ODE Analysis of the MAPK example

Bio-PEPA

Some drawbacks of PEPA

Not all the features of biological systems can be represented intoPEPA.

I stoichiometry is not represented explicitlyI general kinetic laws different from Mass Action are not

considered.

The latter assumption is restrictive since general kinetic laws arewidely-used in the models.

Bio-PEPA

I stoichiometry is not represented explicitly

I general kinetic laws different from Mass Action are notconsidered.

Bio-PEPA

considered.

Bio-PEPA

considered.

Bio-PEPA

The aim of the work

In order to overcome the drawbacks above, we have definedBio-PEPA.

The main field of application is the one of biochemical networks.

SchemaBiochemical networks −→ Bio-PEPA system −→ Analysis

Bio-PEPA

The aim of the work

Bio-PEPA

The aim of the work

Bio-PEPA

The aim of the work

Bio-PEPA

Bio-PEPA: main features

I it is based on the reagent-centric view

I it considers general kinetic laws and expresses them asfunctional rates

I the PEPA activities are replaced by new ones withstoichiometry and the information about the role of thespecies (enzyme, inhibitor,...)

I parameters represent concentration levelsI it can be mapped for the analysis by means of ODEs,

stochastic simulation, CTMC, model checking (PRISM)

Bio-PEPA

I it is based on the reagent-centric viewI it considers general kinetic laws and expresses them as

functional rates

I the PEPA activities are replaced by new ones withstoichiometry and the information about the role of thespecies (enzyme, inhibitor,...)

Bio-PEPA

functional ratesI the PEPA activities are replaced by new ones with

stoichiometry and the information about the role of thespecies (enzyme, inhibitor,...)

Bio-PEPA

I parameters represent concentration levels

I it can be mapped for the analysis by means of ODEs,stochastic simulation, CTMC, model checking (PRISM)

Bio-PEPA

The syntax

Sequential component (species component)

S def= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | | �

Model component

P def= P BC

LP | S(l)

Each action αj is associated with a rate fαj

The list N contains the numbers of levels/maximum concentrations

Bio-PEPA

The syntax

Model component

P def= P BC

LP | S(l)

Bio-PEPA

The syntax

Model component

P def= P BC

LP | S(l)

Bio-PEPA

The syntax

Model component

P def= P BC

LP | S(l)

Bio-PEPA

The syntax

Model component

P def= P BC

LP | S(l)

Bio-PEPA

The syntax

Model component

P def= P BC

LP | S(l)

Bio-PEPA

Semantics: prefix rules

prefixReac ((α, κ)↓S)(l)(α,[S:↓(l,κ)])−−−−−−−−−−→S(l − 1) 0 < l ≤ N

prefixProd ((α, κ)↑S)(l)(α,[S:↑(l,κ)])−−−−−−−−−−→S(l + 1) 0 ≤ l < N

prefixMod ((α, κ) op S)(l)(α,[S:op(l,κ)])−−−−−−−−−−→S(l) 0 ≤ l ≤ N

with op = �,⊕, or

Bio-PEPA

Semantics: constant and choice rules

Choice1S1(l)

(α,v)−−−→S

1(l′)

(S1 + S2)(l)(α,v)−−−→S

1(l′)

Choice2S2(l)

(α,v)−−−→S

2(l′)

(S1 + S2)(l)(α,v)−−−→S

2(l′)

ConstantS(l)

(α,S′:[op(l,κ))]−−−−−−−−−−→S′(l′)

C(l)(α,C:[op(l,κ))]−−−−−−−−−−→S′(l′)

with Cdef= S

Bio-PEPA

Choice1S1(l)

(α,v)−−−→S

1(l′)

(S1 + S2)(l)(α,v)−−−→S

1(l′)

Choice2S2(l)

(α,v)−−−→S

2(l′)

(S1 + S2)(l)(α,v)−−−→S

2(l′)

ConstantS(l)

(α,S′:[op(l,κ))]−−−−−−−−−−→S′(l′)

C(l)(α,C:[op(l,κ))]−−−−−−−−−−→S′(l′)

with Cdef= S

Bio-PEPA

Choice1S1(l)

(α,v)−−−→S

1(l′)

(S1 + S2)(l)(α,v)−−−→S

1(l′)

Choice2S2(l)

(α,v)−−−→S

2(l′)

(S1 + S2)(l)(α,v)−−−→S

2(l′)

ConstantS(l)

(α,S′:[op(l,κ))]−−−−−−−−−−→S′(l′)

C(l)(α,C:[op(l,κ))]−−−−−−−−−−→S′(l′)

with Cdef= S

Bio-PEPA

Choice1S1(l)

(α,v)−−−→S

1(l′)

(S1 + S2)(l)(α,v)−−−→S

1(l′)

Choice2S2(l)

(α,v)−−−→S

2(l′)

(S1 + S2)(l)(α,v)−−−→S

2(l′)

ConstantS(l)

(α,S′:[op(l,κ))]−−−−−−−−−−→S′(l′)

C(l)(α,C:[op(l,κ))]−−−−−−−−−−→S′(l′)

with Cdef= S

Bio-PEPA

Semantics: cooperation rules

coop1P1

(α,v)−−−→P′1

P1 BCL

P2(α,v)−−−→P′1 BC

with α < L

coop2P2

(α,v)−−−→P′2

P1 BCL

P2(α,v)−−−→P1 BC

LP′2

with α < L

coopFinalP1

(α,v1)−−−−→P′1 P2

(α,v2)−−−−→P′2

P1 BCL

P2(α,v1@v2)−−−−−−−→P′1 BC

LP′2

with α ∈ L

Bio-PEPA

coop1P1

(α,v)−−−→P′1

P1 BCL

P2(α,v)−−−→P′1 BC

with α < L

coop2P2

(α,v)−−−→P′2

P1 BCL

P2(α,v)−−−→P1 BC

LP′2

with α < L

coopFinalP1

(α,v1)−−−−→P′1 P2

(α,v2)−−−−→P′2

P1 BCL

P2(α,v1@v2)−−−−−−−→P′1 BC

LP′2

with α ∈ L

Bio-PEPA

coop1P1

(α,v)−−−→P′1

P1 BCL

P2(α,v)−−−→P′1 BC

with α < L

coop2P2

(α,v)−−−→P′2

P1 BCL

P2(α,v)−−−→P1 BC

LP′2

with α < L

coopFinalP1

(α,v1)−−−−→P′1 P2

(α,v2)−−−−→P′2

P1 BCL

P2(α,v1@v2)−−−−−−−→P′1 BC

LP′2

with α ∈ L

Bio-PEPA

coop1P1

(α,v)−−−→P′1

P1 BCL

P2(α,v)−−−→P′1 BC

with α < L

coop2P2

(α,v)−−−→P′2

P1 BCL

P2(α,v)−−−→P1 BC

LP′2

with α < L

coopFinalP1

(α,v1)−−−−→P′1 P2

(α,v2)−−−−→P′2

P1 BCL

P2(α,v1@v2)−−−−−−−→P′1 BC

LP′2

with α ∈ L

Bio-PEPA

Semantics: rates and transition system

In order to associate the rates we consider a new relation7−→ ⊆ C × Γ × C, with γ ∈ Γ := (α, r) and r ∈ R+.

The relation is defined in terms of the previous one:

FinalP

(αj ,v)−−−−→P′