computational methods for systems and synthetic...

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Computational Methods for

Systems and Synthetic Biology

François Fages

The French National Institute for Research in

Computer Science and Control

INRIA Paris-Rocquencourt

Constraint Programming Group

http://contraintes.inria.fr/

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Cell Molecules

• Small molecules: covalent bonds 50-200 kcal/mol

– 70% water

– 1% ions

– 6% amino acids (20), nucleotides (5),

– fats, sugars, ATP, ADP, …

• Macromolecules: hydrogen bonds, ionic, hydrophobic, Waals 1-5 kcal/mol

Stability and bindings determined by the number of weak bonds: 3D shape

– 20% proteins (50-104 amino acids)

– RNA (102-104 nucleotides AGCU)

– DNA (102-106 nucleotides AGCT)

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DNA Deoxyribonucleic Acid

1) Primary structure: word over 4 nucleotides

Adenine, Guanine, Cytosine, Thymine

2) Secondary structure:

double helix of pairs A--T and C---G

stabilized by hydrogen bonds

Nobel Prizes Watson and Crick (1956)

Size of DNA = number of pairs

A gene is a sequence of DNA pairs coding for a protein or an RNA having a function in the organism

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DNA: Genome Size

Species Genome size Chromosomes Coding DNA

E. Coli (bacteria) 5 Mb 1 circular 100 %

12 Mb

… 3 Gb

… 15 Gb

… 140 Gb

Artificial life by Craig Venter:

fully synthetic bacteria genome (0,39 $/b)

implemented in a bacteria without DNA

still living and proliferating!

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DNA: Genome Size

Species Genome size Chromosomes Coding DNA

E. Coli (bacteria) 5 Mb 1 circular 100 %

S. Cerevisae (yeast) 12 Mb 16 70 %

… 3 Gb

… 15 Gb

… 140 Gb

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DNA: Genome Size

Species Genome size Chromosomes Coding DNA

E. Coli (bacteria) 5 Mb 1 circular 100 %

S. Cerevisae (yeast) 12 Mb 16 70 %

Mouse, Human 3 Gb 20, 23 15 %

… 15 Gb

… 140 Gb

3,200,000,000 pairs of nucleotides

single nucleotide polymorphism 1 / 2kb

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Genome Size

Species Genome size Chromosomes Coding DNA

E. Coli (bacteria) 4 Mb 1 100 %

S. Cerevisae (yeast) 12 Mb 16 70 %

Mouse, Human 3 Gb 20, 23 15 %

Onion 15 Gb 8 1 %

… 140 Gb

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Genome Size

Species Genome size Chromosomes Coding DNA

E. Coli (bacteria) 4 Mb 1 100 %

S. Cerevisae (yeast) 12 Mb 16 70 %

Mouse, Human 3 Gb 20, 23 15 %

Onion 15 Gb 8 1 %

Lungfish 140 Gb 0.7 %

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DNA Replication

1. Separation of the two helices

2. Production of one complementary strand for each copy

(from one or several starting points of replication)

3. Segregation

4. Mitosis (cell division)

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Gene Transcription and Translation

1. Activation (Inhibition): Nobel prize Jakob and Monod (1965)

transcription factors bind to the regulatory region of the gene

2. Transcription:

RNA polymerase copies the DNA from start to stop positions

into a single stranded pre-mature messenger pRNA

3. (Alternative) splicing:

non coding regions of pRNA are removed giving mature messenger mRNA

4. Translation:

mRNA moves to cytoplasm and binds to ribosome to assemble a protein

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Formal Genes: Syntax

• Part of DNA #E2

• Activation

binding of promotion factor #E2-(E2f13-DP12)

• Repression (inhibition) Genes and signals [Ptashne Gann 01]

binding of another molecule #E2-Rep

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Transcription and Translation Rules

Activation

#E2 + E2f13-DP12 => #E2-E2f13-DP12

Repression

#E2 + Rep => #E2-Rep

Genes and signals [Ptashne Gann 01]

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Transcription and Translation Rules

Activation

#E2 + E2f13-DP12 => #E2-E2f13-DP12

Repression

#E2 + Rep => #E2-Rep

Transcription

_ =[#E2-E2F13-DP12]=> pRNAcycA

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Transcription and Translation Rules

Activation

#E2 + E2f13-DP12 => #E2-E2f13-DP12

Repression

#E2 + Rep => #E2-Rep

Transcription

_ =[#E2-E2F13-DP12]=> pRNAcycA

(Alternative) Splicing

pRNAcycA => mRNAcycA (pRNAcycA => mRNAcycA2)

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Transcription and Translation Rules

Activation

#E2 + E2f13-DP12 => #E2-E2f13-DP12

Repression

#E2 + Rep => #E2-Rep

Transcription

_ =[#E2-E2F13-DP12]=> pRNAcycA

(Alternative) Splicing

pRNAcycA => mRNAcycA (pRNAcycA => mRNAcycA2)

Translation

mRNAcycA => mRNAcycA::cyt

mRNAcycA::cyt =[ribosome::cyt]=> cycA::cyt

(mRNAcycA2::cyt =[ribosome::cyt]=> cycA2::cyt)

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Proteins

1) Primary structure: word of n amino acids residues (20n possibilities)

linked with C-N bonds

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Proteins

1) Primary structure: word of n amino acids residues (20n possibilities)

linked with C-N bonds

Example: MPRI

Methionine-Proline-Arginine-Isoleucine

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Proteins

1) Primary structure: word of n amino acids residues (20n possibilities)

linked with C-N bonds

Example: MPRI

Methionine-Proline-Arginine-Isoleucine

2) Secondary: word of m a-helix, b-strands, random coils,… (3m-10m)

stabilized by hydrogen bonds H---O

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Proteins

1) Primary structure: word of n amino acids residues (20n possibilities)

linked with C-N bonds

Example: MPRI

Methionine-Proline-Arginine-Isoleucine

2) Secondary: word of m a-helix, b-strands, random coils,… (3m-10m)

stabilized by hydrogen bonds H---O

3) Tertiary 3D structure: spatial folding

stabilized by hydrophobic interactions

explains the protein interaction capabilities

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Formal Proteins: Syntax

• Cyclin dependent kinase 1 Cdk1

(free, inactive)

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Formal Proteins: Syntax

• Cyclin dependent kinase 1 Cdk1

(free, inactive)

• Complex Cdk1-Cyclin B Cdk1–CycB

(low activity)

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Formal Proteins: Syntax

• Cyclin dependent kinase 1 Cdk1

(free, inactive)

• Complex Cdk1-Cyclin B Cdk1–CycB

(low activity)

• Phosphorylated form Cdk1~{thr161}-CycB

at site threonine 161

(high activity)

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Formal Proteins

• Cyclin dependent kinase 1 Cdk1

(free, inactive)

• Complex Cdk1-Cyclin B Cdk1–CycB

(low activity)

• Phosphorylated form Cdk1~{thr161}-CycB

at site threonine 161

(high activity)

“Mitosis-Promoting Factor”

phosphorylates actin in microtubules nuclear division

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Elementary Rule Schemas

• Complexation: A + B => A-B. Decomplexation A-B => A + B.

cdk1+cycB => cdk1–cycB

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Elementary Rule Schemas

• Complexation: A + B => A-B. Decomplexation A-B => A + B.

cdk1+cycB => cdk1–cycB

• Phosphorylation: A =[C]=> A~{p}. Dephosphorylation A~{p} =[C]=> A.

Cdk1-CycB =[Myt1]=> Cdk1~{thr161}-CycB

Cdk1~{thr14,tyr15}-CycB =[Cdc25~{Nterm}]=> Cdk1-CycB

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Elementary Rule Schemas

• Complexation: A + B => A-B. Decomplexation A-B => A + B.

cdk1+cycB => cdk1–cycB

• Phosphorylation: A =[C]=> A~{p}. Dephosphorylation A~{p} =[C]=> A.

Cdk1-CycB =[Myt1]=> Cdk1~{thr161}-CycB

Cdk1~{thr14,tyr15}-CycB =[Cdc25~{Nterm}]=> Cdk1-CycB

• Synthesis: _ =[C]=> A. Degradation: A =[C]=> _.

_=[#E2-E2f13-Dp12]=>cycA cycE =[@UbiPro]=> _

(not for cycE-cdk2 which is stable)

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Elementary Rule Schemas

• Complexation: A + B => A-B. Decomplexation A-B => A + B.

cdk1+cycB => cdk1–cycB

• Phosphorylation: A =[C]=> A~{p}. Dephosphorylation A~{p} =[C]=> A.

Cdk1-CycB =[Myt1]=> Cdk1~{thr161}-CycB

Cdk1~{thr14,tyr15}-CycB =[Cdc25~{Nterm}]=> Cdk1-CycB

• Synthesis: _ =[C]=> A. Degradation: A =[C]=> _.

_=[#E2-E2f13-Dp12]=>cycA cycE =[@UbiPro]=> _

(not for cycE-cdk2 which is stable)

• Transport: A::L1 => A::L2.

Cdk1~{p}-CycB::cytoplasm=>Cdk1~{p}-CycB::nucleus

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Graphical Representation

Hypergraph of reactions represented by a bipartite graph (S,R,A) with

vertices for species S and for reactions R (Petri net representation)

A+B => A-B

Systems Biology Graphical Notation (SBGN)

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Biocham Syntax of Objects

Entities E == name | #name | E-E | E~{p1,…,pn}

name of molecular compound or gene binding site

- : binding operator for protein complexes, gene binding sites, …

Associative and commutative.

~{…}: modification operator for phosphorylated sites, …

Set of modified sites (Associative, Commutative, Idempotent).

Induce a congruence on objects

Objects O == E | E::location

location: symbolic compartment (nucleus, cytoplasm, membrane, …)

dynamic volume

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Biocham Syntax of Rules

Solutions S ::= _ | O + S | i*O + S

+ : solution operator (Associative, Commutative, Neutral element _)

Rules R ::= S => S | kinetic-expression for R

Abbreviations for catalytic reactions: A =[C]=> B stands for A+C => B+C

reversible reactions: A <=> B stands for A=>B and B=>A,

Syntax compatible with the Systems Biology Markup Language SBML

Import/export exchange format for reaction models in XML

Biomodels.net: repository of 241 curated SBML models of cell processes

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Biocham Models

A Biocham model is a finite set of Biocham reaction rules over a finite set of

molecular species.

(a Kappa model is a finite set of Kappa rules over an infinite universe of

molecular species)

Questions:

• Compared to kappa, which type of chemical reactions cannot be

represented ?

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Biocham Models

A Biocham model is a finite set of Biocham reaction rules over a finite set of

molecular species.

(a Kappa model is a finite set of Kappa rules over an infinite universe of

molecular species)

Questions:

• Compared to kappa, which type of chemical reactions cannot be

represented ?

• Which type of locations cannot be represented ?

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Biocham Models

A Biocham model is a finite set of Biocham reaction rules over a finite set of

molecular species.

(a Kappa model is a finite set of Kappa rules over an infinite universe of

molecular species)

Questions:

• Compared to kappa, which type of chemical reactions cannot be

represented ?

• Which type of locations cannot be represented ?

• Decidability properties when rules are interpreted as an asynchronuous

Boolean transition system ? As multiset rewriting?

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Kinetic Rate Constants

• Complexation: probability of reaction upon collision (specificity, affinity)

position of matching surfaces

• Decomplexation: total energy of all bonds

(giving dissociation rates)

• Diffusion speeds (small molecules>substrates>enzymes…)

Average travel in one random walk: 1 μm in 1s, 2μm in 4s, 10μm in 100s

(cells are 10-100μm long)

• For one enzyme:

500000 random collisions per second with a substrate concentration of 10-5

50000 random collisions per second with a substrate concentration of 10-6

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Kinetic Expressions

Mass action law kinetics:

k*[A] for A => B

k*[A]*[B] for A+B => C

k*[A]^m*[B]^n for m*A + n*B => R

…

Michaelis Menten kinetics:

Vm*[A]/(Km+[A]) for A => B

Hill kinetics:

Vm*[A]^n/(Km+[A]^n) for A => B

Guldberg and Waage, 1864

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Semantics of Rule-based Models

Reaction rule k*[A]*[B] for A+B => C

1. Differential Semantics: concentrations

– Ordinary Differential Equations dA/dt = -k*[A]*[B]

dB/dt = -k*[A]*[B]

dC/dt = k*[A]*[B]

– Hybrid automata (for kinetics with conditional expressions)

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Semantics of Rule-based Models

Reaction rule k*[A]*[B] for A+B => C

1. Differential Semantics: concentrations

– Ordinary Differential Equations dA/dt = -k*[A]*[B]

dB/dt = -k*[A]*[B]

dC/dt = k*[A]*[B]

– Hybrid automata (for kinetics with conditional expressions)

2. Stochastic Semantics: numbers of molecules

– Continuous time Markov chain A, B p A--, B--, C++

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Semantics of Rule-based Models

Reaction rule k*[A]*[B] for A+B => C

1. Differential Semantics: concentrations

– Ordinary Differential Equations dA/dt = -k*[A]*[B]

dB/dt = -k*[A]*[B]

dC/dt = k*[A]*[B]

– Hybrid automata (for kinetics with conditional expressions)

2. Stochastic Semantics: numbers of molecules

– Continuous time Markov chain A, B p A--, B--, C++

3. Boolean Semantics: presence-absence of molecules

– Asynchronous Transition System A, B C, A/A, B/B

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Hierarchy of Semantics

Stochastic model Differential model

Discrete model

abstraction

concretization

Boolean model

Theory of abstract Interpretation

Abstractions as Galois connections

[Cousot Cousot POPL’77]

[Fages Soliman CMSB’06,TCS’07]

Syntactical

model

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Regulation Graphs as Abstractions

Discrete model

abstraction

concretization

Boolean model

Syntactical

model [Fages Soliman CMSB’06]

Syntactic regulation graph

(pos/neg influences w.r.t.

the stoichiometric coef.

in rules)

Thm. Same graphs for

monotonic kinetics

Jacobian regulation graph

(pos/neg influences w.r.t.

the sign of the coefficients)

Stochastic model Differential model

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Regulation Graph Circuit Analyses

Discrete model

abstraction

concretization

Boolean model

Syntactical

model

Jacobian circuit analysis

Discrete circuit analysis

Boolean circuit analysis

abstraction

abstraction

abstraction

Thm. Positive (resp. negative) circuits are a necessary condition for multistationarity (resp. oscillations)

[Thomas 81] [Snoussi 89] [Soulé 03] [Remy Ruet Thieffry 05] [Richard 07]

Stochastic model Differential model