Javier Campos, Jorge Julvez´ University of...

Systems Biology Population Dynamics Example Formal Models Stochasticity

Modelos Formales en Bioinformatica

Javier Campos, Jorge JulvezUniversity of Zaragoza

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Outline

1 Systems Biology

2 Population Dynamics Example

3 Formal Models

4 Stochasticity

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Outline

1 Systems Biology

3 Formal Models

4 Stochasticity

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Formal Models in Bioinformatics

What is systems biology?

• Systems biology is the study of all the elements in abiological system (all genes, mRNAs, proteins, etc) andtheir relationships one to another in response toperturbations.

• Systems approaches attempt to study the behaviour of allthe elements in a system and relate these behaviours tothe systems or emergent properties.

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Bioinformatics Basics

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Systems Biology: Interaction in Networks

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Systems Biology: Interaction in Networks

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What is systems biology?

• ...systematic study of complex interactions in biologicalsystems, thus using a new perspective (integration insteadof reduction) to study them... one of the goals of systemsbiology is to discover new emergent properties (Wikipedia)

• Systems biology is the study of an organism, viewed as anintegrated and interacting network of genes, proteins andbiochemical reactions... systems biologists focus on all thecomponents and the interactions among them, all as partof one system (Institute for Systems Biology, Washington)

• To understand complex biological systems requires theintegration of experimental and computational research –in other words a systems biology approach (Kitano, 2002)

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NetworksGene regulation

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Metabolic Pathway

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Signalling

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Protein-protein interaction

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Metabolic Pathway

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Systems Biology is a multidisciplinar field

Which is the volume of a cow?

An informatics perspective: for cow=0; cow+1 end

Where is the cow???

A physicist perspective: Consider a spherical cow with negligiblemass…

Even E. coli is not that

spherical !!

• A chemist perspective: Dissolve the cow in H2SO4, weightthe result and measure the volume.

• Sometimes it is important not to destroy the cow.

• An engineering perspective: Immerse the cow in a tank ofwater and measure the volume.

• Some cows cannot swim. Water pressure might changevolume.

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Where is the cow???

spherical !!

• A chemist perspective: Dissolve the cow in H2SO4, weightthe result and measure the volume.

• Sometimes it is important not to destroy the cow.

• An engineering perspective: Immerse the cow in a tank ofwater and measure the volume.

• Some cows cannot swim. Water pressure might changevolume.

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Where is the cow???

spherical !!

• A mathematician perspective: Cut the cow into pieces andsum up the pieces.

• The total might not be equal to the sum of its parts→emergent properties.

• A physicist perspective: Consider a spherical cow withnegligible mass...

• Even E. coli is not that spherical

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Where is the cow???

spherical !!

• A mathematician perspective: Cut the cow into pieces andsum up the pieces.

• The total might not be equal to the sum of its parts→emergent properties.

• A physicist perspective: Consider a spherical cow withnegligible mass...

• Even E. coli is not that spherical

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Outline

1 Systems Biology

3 Formal Models

4 Stochasticity

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Mathematics to model the time evolution of a population

Theoretical immunology

Modelling a population

Deterministic birth process

Figure: A lot of penguins!

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From penguins to rabbits..

First attemt at combining mathematics and biology

From penguins to rabbits

Figure: A lot of rabbits!

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Modelling the time evolution of a population

Early attempts to make use of mathematics in biologyLeonardo de Pisa or Fibonacci (1202)• At month 0 there is a pair of rabbits (one female and one

male).• Every pair of rabbits (one female and one male) can mate

at the age of one month.• The female rabbit always produces a new pair of rabbits

(one female and one male) every month from the secondmonth on.

• As there is no death, all rabbits survive.What is the number of pairs of rabbits in month n?

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Rabbits population

First attemt at combining mathematics and biology

From penguins to rabbits

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Leonardo de Pisa or Fibonacci (1202)

• At month 0 there is a pair of rabbits (one female and one male).

• Every pair of rabbits (one female and one male) can mate at the age ofone month.

• The female rabbit always produces a new pair of rabbits (one femaleand one male) every month from the second month on.

• As there is no death, all rabbits survive.

The number of pairs in month n, Rn, satisfies:

Rn+1 = Rn + Rn−1

R0 = 1R1 = 1R2 = 1 + 1 = 2R3 = 2 + 1 = 3R4 = 3 + 2 = 5R5 = 5 + 3 = 8 18 / 51

Deterministic modelling

• Let N(t) be the population of those penguins at time t .• N(t) is the number of individuals in the population at time t .• The change in the number of penguins in a small time

interval, from t to t + ∆t , is given by:

N(t + ∆) = N(t) + births − deaths + migration

• This equation is a conservation equation for the number ofindividuals of the population.

• The form of the various terms on the right-hand-siderequires essential feedback from biologists.

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Deterministic birth processLet us assume that:• There are no death events in the population.• There are only birth events in the population.• The birth rate (number of births per unit of time), b, is the

same for all individuals of the population.• We have:

N(t + ∆t) = N(t) + births

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Deterministic birth process• Change in the population is due to birth events.• The births in the time interval [t , t + ∆t ] due to a single

individual is b∆t .

• The births in the time interval [t , t + ∆t ] due to allindividuals is N(t)b∆t .

• N(t+∆t) = N(t)+N(t)b∆t =⇒ N(t + ∆t)− N(t)∆t

= bN(t)

• For a very small time interval, ∆t → 0,

lim∆t→0

N(t + ∆t)− N(t)∆t

=dN(t)

dt= bN(t)

• This equation can be easily solved by integration.

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individual is b∆t .• The births in the time interval [t , t + ∆t ] due to all

individuals is N(t)b∆t .

= bN(t)

lim∆t→0

N(t + ∆t)− N(t)∆t

=dN(t)

dt= bN(t)

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= bN(t)

lim∆t→0

N(t + ∆t)− N(t)∆t

=dN(t)

dt= bN(t)

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= bN(t)

lim∆t→0

N(t + ∆t)− N(t)∆t

=dN(t)

dt= bN(t)

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Deterministic birth processN(t): Number of individuals at time tdN(t)

dt= bN(t)

• If the population at time t = t0 is given by N0, we have:

N(t) = N0eb(t−t0)

• In a deterministic birth process the population size ispredicted at time t with absolute certainty, once the initialsize N0 and birth rate b are given.

• The population size N(t) and time t are both continuousvariables (both take real values) and not discrete (takeinteger values).

• Is this a good mathematical population growth model?

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Deterministic birth processN(t): Number of individuals at time tdN(t)

dt= bN(t)

• If the population at time t = t0 is given by N0, we have:

N(t) = N0eb(t−t0)

• In a deterministic birth process the population size ispredicted at time t with absolute certainty, once the initialsize N0 and birth rate b are given.

• The population size N(t) and time t are both continuousvariables (both take real values) and not discrete (takeinteger values).

• Is this a good mathematical population growth model?22 / 51

Stochastic birth process• Let Xt be the dicrete random variable that describes the

number of individuals of the population at time t .• The stochastic process that describes the population

satisfies:Xt ∈ {1,2, . . .} and t ∈ [0,+∞)

• Denote by pn(t) the probability that at time t the size of thepopulation is n, i.e., the probability that at time t there are nindividuals in the population:

pn(t) = Prob(Xt = n)

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Stochastic birth process

• Consider a small time interval [t , t + ∆t ]• How is Xt+∆t related to X• We have the following rules:

• There are no death events in the population.• There are birth events in the population: the probability that

a birth takes place in ∆t is b∆t .• The probability of more than one birth in a time interval ∆t

is negligible (no twin births allowed).

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Stochastic birth process II

· · · · · ·n ! 1 n n + 1

! The probability that a population of size n ! 1 increases to n in the timeinterval (t, t + !t) is b "!t " (n ! 1).

! The probability that a population of size n increases to n + 1 in the timeinterval (t, t + !t) is b "!t " n.

! If at time t the population has n individuals, the probability that no birthevent takes place in the time interval (t, t + !t) is 1 ! b "!t " n.

! Evolution equation for pn(t):

pn(t + !t) = (n ! 1) b !t pn!1(t) + (1 ! n b !t) pn(t) .

• The probability that a population of size n − 1 increases ton in the time interval (t , t + ∆t) is (n − 1)b∆t .

• The probability that a population of size n increases ton + 1 in the time interval (t , t + ∆t) is nb∆t .

• If at time t the population has n individuals, the probabilitythat no birth event takes place in the time interval(t , t + ∆t) is 1− nb∆t .

• Evolution equation for pn(t):

pn(t + ∆t) = (n − 1)b∆tpn−1(t) + (1− nb∆t)pn(t)

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Stochastic birth process II

· · · · · ·n ! 1 n n + 1

! The probability that a population of size n ! 1 increases to n in the timeinterval (t, t + !t) is b "!t " (n ! 1).

! The probability that a population of size n increases to n + 1 in the timeinterval (t, t + !t) is b "!t " n.

! If at time t the population has n individuals, the probability that no birthevent takes place in the time interval (t, t + !t) is 1 ! b "!t " n.

! Evolution equation for pn(t):

pn(t + !t) = (n ! 1) b !t pn!1(t) + (1 ! n b !t) pn(t) .

• The probability that a population of size n − 1 increases ton in the time interval (t , t + ∆t) is (n − 1)b∆t .

• The probability that a population of size n increases ton + 1 in the time interval (t , t + ∆t) is nb∆t .

• If at time t the population has n individuals, the probabilitythat no birth event takes place in the time interval(t , t + ∆t) is 1− nb∆t .

• Evolution equation for pn(t):

pn(t + ∆t) = (n − 1)b∆tpn−1(t) + (1− nb∆t)pn(t)25 / 51

T cell

Immunology

T cell

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Cell division

Immunology

Cell division

Birth event

! At time t there are n cells.

! During the time interval !t there is a single birth event.

! At time t + !t there are n + 1 cells.

Birth event:• At time t there are n cells.• During the time interval ∆t there is a single birth event.• At time t + ∆t there are n + 1 cells.

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Cell death

Immunology

Cell death

Death event

! At time t there are n cells.

! During the time interval !t there is a single death event.

! At time t + !t there are n ! 1 cells.

Death event:

• At time t there are n cells.• During the time interval ∆t there is a single death event.• At time t + ∆t there are n − 1 cells.

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Cell-cell interactions lead to events

Immunology

Cell-cell interactions lead to events

Figure: T cell-dendritic cell interaction.29 / 51

T cell and Tumour cell

Immunology

T cell and tumour cell

Figure: T cell and tumour cell.30 / 51

Outline

1 Systems Biology

3 Formal Models

4 Stochasticity

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Modelling in systems biology

Modelling: Design and construction of models of existingbiological systems, which explain observed properties andpredict the response to experimental interventions.

systems biology: modelling as formal knowledge representation

synthetic biology: modelling for system construction

biosystemnatural

biosystemsynthetic

observedbehaviour

predictedbehaviour

model(blueprint)

desiredbehaviour

design construction

verification verification

observedbehaviour

predictedbehaviour

wetlab

model-basedexperiment design

experiments

formalizingunderstanding

wetlab experiments

model(knowledge)

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Model:• Formal representation of the real

world.• Simplified abstract view of the

complex reality

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Why model?

• A model can generate new insights• A model can make testable predictions

• E.g., predict the effect of drugs on an organism• E.g., predict the effect on an inhibitor on a pathway

• A model can test conditions that may be difficult to study inthe laboratory

• A model can rule out particular explanations for anexperimental observation

• A model can help you identify what’s right and wrong withyour hypotheses

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Textbook view of the cell and reality

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Three D EM image of a pancreatic

Beta cell

Campbell, Reece & Mitchell (1998) Biology, 5th

Edition

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In silico humans - spatial & temporal scalesIn silico humans-spatial & temporal scales

• 1 m person

• 1 mm electrical length scale of cardiac tissue

• 1 mm cardiac sarcomere spacing

• 1 nm pore diameter in a membrane protein

Range = 109

• 109 s (70 yrs) human lifetime

• 106 s (10 days) protein turnover

Requires a hierarchy of inter-related models

pathway

modelsODEs

stochastic

modelsPDEs (continuum models)gene reg.

networks

• 106 s (10 days) protein turnover

• 103 s (1 hour) digest food

• 1 s heart beat

• 1 ms ion channel HH gating

• 1 ms Brownian motion

Range = 1015

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Models must be validated by experimental data: Simulationsmust be accurate representations of the real world.

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A Framework for Modelling• Define all the components of the system• Systematically perturb and monitor components of the

system• Reconcile the experimentally observed responses with

those predicted by the model• Design and perform new perturbation experiments to

distinguish between multiple or competing modelhypotheses.

(Ideker, Galitski & Hood, 2001)

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Models should be:• Readable.• Unambiguous.• Analysable.• Executable.

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Suggestions:• Occam’s razor: Don’t overcomplicate things.• Einstein: Everything should be made as simple as

possible, but not simpler.

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Models should be:• Readable.• Unambiguous.• Analysable.• Executable.

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Suggestions:• Occam’s razor: Don’t overcomplicate things.• Einstein: Everything should be made as simple as

possible, but not simpler.

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Modelling Regimes

Continuous

SimulationAlgorithms

MolecularDynamics

OrdinaryDifferentialEquation

StochasticDifferentialEquation

Randomness

StateSpace

Discrete

Deterministic Stochastic

Stochastic

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Modelling Regimes• Discrete and stochastic: Small numbers of molecules.

Exact description via Stochastic Simulation Algorithm(SSA) - Gillespie. Large computational time.

• Continuous and stochastic: A bridge connecting discreteand continuous models. Described by SDEs ChemicalLangevin Equation.

• Continuous and deterministic: Law of Mass Action. TheReaction Rate equations. Described by ordinarydifferential equations. Not valid if molecular populations ofsome critical reactant species are small.

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Outline

1 Systems Biology

3 Formal Models

4 Stochasticity

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Biological evidence of noise• “Stochasticity is evident in all biological processes the

proliferation of both noise and noise reduction systems is ahallmark of organismal evolution” Federoff et al.(2002).

• “Transcription in higher eukaryotes occurs with a relatively lowfrequency in biologic time and is regulated in a probabilisticmanner” Hume (2000).

• “Gene regulation is a noisy business” Mcadams et al. (1999).

• “Initiation of gene transcription is a discrete process in whichindividual protein-coding genes in an off state can bestochastically switched on, resulting in sporadic pulses of mRNAproduction” Sano 2001.

• “It is essential to study individual cells and to measure the cell tocell variations in biological response, rather than averaging overcell populations” Zatorsky, Rosenfeld et al. 2006.

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Origin of Stochasticity• Intrinsic noise due to small numbers of molecules (e.g.

mRNA, DNA loci, TFs).• Uncertainty of knowing when a reaction occurs and which

reaction it is.• Relative statistical uncertainty is inversely proportional to

the square root of the number of molecules.• Applies equally well to studying channel behaviour via the

concept of channel molecules.• Extrinsic noise due to (external) environmental effects

(extrinsic factors are: stage in cell cycle, number of RNAPor ribosomes, cellular environment).

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Modelling formalisms

Markov chains• Based on the concept of state of the system• Solution techniques:

• Enumerative• Transient and steady-state analysis• Exact and approximate analysis

• Drawbacks:• Low abstraction level• Model size equals number of states of the system• Only in very particular cases aggregation techniques exist

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Queueing networks• High abstraction level

• The number of states characterizing the system growsexponentially on the model size.

• Solution techniques:• Enumerative (based on Markov chains)• Reduction/transformation-based• Structurally based (product-form solution, exact)• Transient and steady-state analysis• Exact, approximate and bounds

• Drawbacks:• Lack of synchronization primitive• Extensions exist but destroying analysis possibilities

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Stochastic Petri nets• Abstraction level similar to queueing networks• With synchronization primitive

• SPN =Petri nets+ timing interpretation=queueing networks+synchronizations

• Wide range of qualitative (logical properties) analysistechniques:

• Enumerative (based on Markov chains)• Reduction/transformation-based• Structurally based

• Petri nets as a formal modelling paradigm• a conceptual framework to obtain specific formalisms based

on common concepts and principles at different life-cyclephases

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Stochastic Petri nets (cont.)• Analysis techniques:

• Exact: mainly enumerative (based on Markov chains)• Bounding techniques (structurally based)• Approximation techniques (reduction/transformation)

• Drawbacks:• Lack of a product-form solution for efficient exact analysis in

most cases

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Contents of the Course

1 Discrete and Continuous Markov chains2 Birth and Death Processes3 Stochastic Simulation4 Hidden Markov Chains5 Stochastic Petri nets

Course info:http://webdiis.unizar.es/asignaturas/SPN/

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Acknowledgments

Much of the material in the course is based on the followingcourses:• Deterministic models in mathematical biology, Magic 042 - Lecture 1

Carmen Molina-Parıs, Department of Applied Mathematics, School ofMathematics, University of Leeds

• Una Introduccion a la Biologıa de SistemasRaul Guantes (UAM), Juan F. Poyatos (CNB)

• A conceptual framework for BioModel Engineering (Systems Biology,Synthetic Biology)Rainer Breitling, Groningen, NL; David Gilbert, Brunel, UK; MonikaHeiner, Cottbus, DE

• A Petri Net Perspective on Systems and Synthetic BiologyMonika Heiner, Brandenburg University of Technology Cottbus, DEDept. of CS

• Systems Biology: Stochastic models and SimulationKevin Burrage, Institute for Molecular Bioscience, The University ofQueensland, Australia.

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