A couple of approaches to modelling and analysis of biochemical networks

”Biomodelling” seminar, October 2006

Matúš Kalaš

more an inspiration for a discussion than a talk ...

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Contents

1. The variety of modelling paradigms

2. An example of systematic approach(M. Heiner & D. Gilbert)

3. Another example(GOALIE; B. Mishra, M. Antoniotti et al.)

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Models of biochemical networks

How do various modelling paradigms differ?

entities concentrations

individuals

qualitative

continuous

discrete

AMOUNTS OF SPECIES

PRESENCE/ABSENCE, HIGH/LOW/MEDIUM, ACTIVE/INACTIVE,

HIGH-LEVEL STATES

WITH ID, WITH INTERNAL STATE

. . .

WITH SHAPE

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space

divided into homogeneous compartments

continuous

homogeneous WELL-STIRRED

discrete space points

containing non-reacting entities

Models of biochemical networks (cnt.)

”unspaced” HIGH-LEVEL STATES

AFFECTING MOVEMENT OF THE ENTITIES

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time timed

hybrid

untimed

discrete

continuous

QUANTITATIVE TIME

EVENTS, QUALITATIVE TIME

TIMED EVOLUTION + EVENTS

Models of biochemical networks (cnt.)

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progression non-deterministic

stochastic

deterministic

synchronous

asynchronous

APPROXIMATION, MORE REACTIONS IN 1 STEP

IDEAL CASE, AVERAGE CASE

MORE CASES, ”ALL” CASES, ALL CASES

INDIVIDUAL REACTIONS, CONCURRENT & COMPETITIVE

Models of biochemical networks (cnt.)

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Models of biochemical networks (cnt.)

Example models ?

entities concentrations

individuals

qualitative

continuous

discrete

space

divided into homogeneous compartments

continuous

homogeneous

discrete space points

containing non-reacting entities

time timed

hybrid

untimed

discrete

continuous

progression non-deterministic

stochastic

deterministic

synchronous

asynchronous

unspaced

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Prevalent paradigms / buzz words :

ODEs continuous concentrations

homogeneous space or compartments

continuous time

deterministic

Petri Nets qualitative, discrete or continuous concentrations

homogeneous space (or compartments)

untimed, discrete or continuous time

non-determistic, deterministic, stochastic synch. or asych.

Hybrid Automata continuous concentrations

homogeneous space or compartments

hybrid

non-determistic, deterministic, . . .

Gillespie’s Algorithm and alternatives

discrete or continuous concentrations

homogeneous space or compartments

continuous time

stochastic asynchronous or synchronous

Process Algebras and Logics

qualitative, . . .

homogeneous, compartments, . . .

untimed, timed, . . .

non-deterministic or stochastic

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Now an instant introduction to Petri Nets . . .

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An example of systematic modelling:

Step-wise modelling

David Gilbert, Monika Heiner:

From Petri Nets to Differential Equations – An Integrative Approach for Biochemical Network Analysis

ICATPN 2006, TR 2005

. . . a tutorial example of

• different useful features of different modelling paradigms

• step-wise modelling

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Step-wise modelling

REACTIONS IDENTIFICATION

QUALITATIVE MODEL

QUALITATIVE ANALYSIS

CONTINUOUS MODEL

QUANTITATIVE ANALYSIS

STRUCTURAL PROPERTIES

DYNAMIC PROPERTIES (PREDICTION/SIMULATION,

STEADY STATES...)

”debugging”

qualitative model (i.e. model structure) validated

adjusting constants

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REACTIONS IDENTIFICATION

a simple signalling system: ERK/RKIP pathway

Raf-1* + RKIP Raf-1*_RKIP

Raf-1*_RKIP + ERK-PP Raf-1*_RKIP_ERK-PP

Raf-1*_RKIP_ERK-PP Raf-1* + ERK + RKIP-P

MEK-PP + ERK MEK-PP_ERK

MEK-PP_ERK MEK-PP + ERK-PP

RKIP-P + RP RKIP-P_RP

RKIP-P_RP RP + RKIP

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QUALITATIVE MODEL

a standard place/transition Petri Net (discrete, untimed, non-deterministic)

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Static analysis of marking-independent properties

QUALITATIVE ANALYSIS – automated tool-supported checking of properties

- in the example there are 5 minimal P-invariants

(Raf-1* , Raf-1*_RKIP , Raf-1*_RKIP_ERK-PP)

(MEK-PP , MEK-PP_ERK)

(RP , RKIP-P_RP)

(ERK , ERK-PP , MEK-PP_ERK , Raf-1*_RKIP_ERK-PP)

(RKIP , Raf-1*_RKIP , Raf-1*_RKIP_ERK-PP , RKIP-P_RP , RKIP-P)

- these cover the whole net (thus, net is bounded)

- Biological meaning: P-invariants correspond to several states of a given species

• P-invariants (sets of places, over which the weighted sum of tokens is constant during operation)

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- example net is covered by T-invariants

- only 1 non-trivial minimal T-invariant: (k1; k3; k5; (k6; k8), (k9; k11))

QUALITATIVE ANALYSIS (cnt.)

Static analysis of marking-independent properties (cnt.)

• T-invariants

- can be also read as the relative firing rates of transitions (reactions/phases in sysbio)

(this corresponds to the steady-state behaviour)

- minimal T-invariants characterise minimal self-contained subnetworks

with an enclosed biological meaning

- useful to comprehend the network if it is very complex {not in this tutorial example}

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QUALITATIVE ANALYSIS (cnt.)

Static analysis of marking-independent properties (cnt.)

• reasonable initial marking constructed with a help of identified invariants

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QUALITATIVE ANALYSIS (cnt.)

Static analysis of marking-dependent properties

• example net is boolean / 1-bounded / safe

• the net is live

Dynamic analysis of marking-dependent properties

• example net is reversible

• MODEL CHECKING of any interesting properties formulated in CTL (Computational Tree Logic)

- e.g.: ”the phosphorylation of ERK does not depend on a phosphorylated state of RKIP”

EG [ERK E (~(RKIP-P \/ RKIP-P_RP) U ERK-PP) ]

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QUALITATIVE ANALYSIS (cnt.)

VALIDATION OF THE QUALITATIVE MODEL (i.e. structure of the system)

all expected structural and general behavioural properties hold

covered by P-invariants

no minimal P-invariant without biological interpretation

covered by T-invariants

no minimal T-invariant without biological interpretation

no known biological behaviour without corresponding T-invariant

all expected logic-formulated properties hold

a break?

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- within this step, all we need is to find suitable rate constants(e.g. to fit in-vivo or in-vitro quantitative experiments)

CONTINUOUS QUANTITATIVE MODEL

Continuous Petri Net- tokens: real numbers- transitions associated with a rate- semantics: a set of ODEs (e.g. reaction-rate equation)- thus a continuous, timed (continuously) and deterministic model

- basically a set of ODEs enhanced with a graphical representation

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QUANTITATIVE ANALYSIS

• Prediction (easy)

- both qualitative and quantitative

• Steady-state properties, oscillations, sensibility, ... (hard)

(... you know better ...)

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Discussion before the next example?

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Marco Antoniotti, Naren Ramakrishnan, Bud Mishra:

GOALIE, A Common Lisp Application to Discover Kripke Models: Redescribing Biological Processes from Time-Course Data

ILC 2005

Another example: Automated modelling

Samantha Kleinberg, Marco Antoniotti, Satish Tadepalli,

Naren Ramakrishnan, Bud Mishra:

Remembrance of Experiments Past: A redescription based tool for discovery in complex systems

ICCS 2006

. . . building a model in order to understand very complex processes ...

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GOALIE approach / software system

GENOMIC MICROARRAY TIME-COURSE DATASET

SYSTEM MODEL EXPRESSED IN GENE ONTOLOGY TERMS

SYSTEM MODEL ANALYSIS BY FORMAL REASONING

GOALIE = Gene Ontology Algorithmic Logic for Invariant Extraction

MODEL OF THE SYSTEM /PROCESS

DYNAMIC QUALITATIVE PROPERTIES

=

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- qualitative, high-level, untimed and non-deterministic model with clear biological meaning

GOALIE approach / software system (cnt.)

Model: Kripke Structure

- called also ”Hidden Kripke Model” in GOALIE

- annotated by Gene Ontology terms (propositional logic)

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Controlled vocabulary: Gene Ontology

GOALIE approach / software system (cnt.)

8517 possible GO process ontology terms

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Example: yeast cell cycle

GOALIE approach / software system (cnt.)

(a small part of the whole model)

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Techniques used to automatically build a model:

- time-windowed clustering (k-means)- data-to-GO association done by GoMiner software- Fisher exact test (p-values)- empirical Bayes approach (Benjamini-Hochberg test)- information bottleneck principle (generalised Shannon-Kolmogorov’s rate-distortion theory)- connecting annotated clusters (Jaccard’s coefficient)

GOALIE approach / software system (cnt.)

Analysis:

- propositional temporal-logic reasoning (model checking of temporal invariants (CTL))

- graph rewriting rules for projection and collapsing, preserving ”bisimulation-like” relations getting higher-level clusters

- process / dataset alignment (similarity of cellular processes)

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A couple of diverse systematic approaches:

• C. Wiggins, I. Nemenman: Process Pathway Inference via Time Series Analysis, 2006

• M. Calder, S. Gilmore, J. Hillston: Automatically deriving ODEs from process algebra models of signalling pathways, CMSB 2005

• N. Chabrier-Rivier, M. Chiaverini, V. Danos, F. Fages, V. Schächter: Modeling and Querying Biomolecular Interaction Networks, TCS 2004

• A. Arkin, P. Shen, J. Ross: A Test Case of Correlation Metric Construction of a Reaction Pathway from Measurements, Science 1997

• M. Chen, R. Hofestädt: A medical bioinformatics approach for metabolic disorders: Biomedical data prediction, modeling, and systematic analysis, JBMI 2006

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”Clearly, if the truth must be found, it will need formal methods that no amount of simulation can deliver.”

DISCUSSION?

THANK YOU!

Carla Piazza & Bud Mishrain ’Stability of Hybrid Systems and Related

Questions from Systems Biology’, 2005