Process algebras for quantitative analysis: Lecture 10...

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

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

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.

Current Hypothesis

Abstraction

Modularity and

Reasoning

Current Hypothesis

Abstraction

Modularity and

Reasoning

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

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

Process Algebra

Process Algebras

Process Algebra

Process Algebras

Process Algebra

Stochastic process algebra model in the same style butassume that each action has a delay governed by a randomvariable.

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

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.

Systems Biology Methodology

ExplanationInterpretation

Natural System

Systems Analysis

InductionModelling

Formal System

Biological Phenomena-MeasurementObservation

DeductionInference

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.

Formal Systems

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.

Integrated Analysis

System

MathematicalModel

e.g ODE or SSA

Integrated Analysis

System

Stochastic Process Algebra

SSAModel Checking

Integrated Analysis

SSAModel Checking

Integrated Analysis

Analysis

SSAModel Checking

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]

SignalTransduction

[Regev et al 2000]

SignalTransduction

[Regev et al 2000]

SignalTransduction

[Regev et al 2000]

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.

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.

Alternative Representations

population view

StochasticSimulation

individual view

AbstractSPA model

@@@@@@@@R

ODEspopulation view

individual view

AbstractSPA model

@@@@@@@@R

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.

population view

individual view

AbstractSPA model

@@@@@@@@R

CTMC withM levels

abstract view

ODEspopulation view

individual view

AbstractSPA model

@@@@@@@@R

CTMC withM levels

abstract view

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.

Alternative models

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.

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.

Bio-PEPA

In Bio-PEPA:

Bio-PEPA

In Bio-PEPA:

Bio-PEPA

In Bio-PEPA:

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).

LP | S(l)

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.

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.

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.

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.

Ostreococcus Tauri

The biological model

Feedback loop between the two main genes (TOC1 and LHY).

Effect of light on several parts of the network.

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.

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

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

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)

Simulations – constant light (LL)

0 50 100 150 200 2500

Time (hours)

LL −− ODEs

LHY mRNA TOC1 mRNATotal LHYTotal TOC1

0 50 100 150 200 2500

Time (hours)

LL −− SSA (10000 runs)

0 50 100 150 200 2500

Time (hours)

LL −− SSA (10 runs)

0 50 100 150 200 2500

Time (hours)

LL −− SSA (1 run)

Simulations – light/dark (LD 12:12)

0 50 100 150 200 2500

Time (hours)

LD 12:12 −− ODEs

0 50 100 150 200 2500

Time (hours)

LD 12:12 −− SSA (10000 runs)

0 50 100 150 200 2500

Time (hours)

LD 12:12 −− SSA (10 runs)

0 50 100 150 200 2500

Time (hours)

LD 12:12 −− SSA (1 run)

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.

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

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960

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

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

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960

0 2 4 6 8 10 12 14 16 18 20 22 240

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

Sensitivity of clock phase to mRNA degradation

Ø : Degradation : Light input Carl Troein

TOCmRNA TOC1

LHYmRNA LHYc LHYn

TOC2 Ø

Previous work suggested that the peak of the total amount of TOCis a good indicator of the clock phase.

Distribution of phase in wildtype (10,000 runs)

Phase of peak of Total TOC1 [hour of day]

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.

Distribution of phase in mutant (10,000 runs)

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.

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.

Distribution of phase in wildtype(10,000 runs), large Ω

0 3 6 9 12 15 18 21 24

duskdawn

Distribution of phase in wildtype(10,000 runs), large Ω

21.38 21.40 21.42 21.44

Distribution of phase in mutant(10,000 runs), large Ω

0 3 6 9 12 15 18 21 24

duskdawn

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

wildtype

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

wildtype

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

0.6 0.8 1.0 1.2 1.4

0 3 6 9 12 15 18 21 24

duskdawn010

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.

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).

Process algebras for quantitative analysis: Lecture 10...

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