Cells as Machines: towards Deciphering Biochemical Programs in the Cell

Cells as Machines:

towards Deciphering

Biochemical Programs in the Cell

François Fages

Inria Saclay-Ile de France http://lifeware.inria.fr

To master the complexity of cell processes, investigate:

• Programming theory concepts

• Formal methods of circuit and program verification

• Parameter search and optimization algorithms

• Theory of analog computation (new in this course)

Context: Systems Biology

Gain system-level understanding of multi-scale biological processes in terms of their elementary interactions at the molecular level. [Kitano 1999]

Cell mitosis [Lodish et al. 03] controlled by protein complex cdk1-cycB

Follow-up of Human Genome Project (90s), beyond genomic data:

à Creation of protein-protein interaction databases, RNAs, …

à Model repositories of cell processes : e.g. biomodels.net1000 models à Systems Biology Markup Language (SBML): model exchange format

à Modeling software (Cell designer, Cytoscape, Copasi, KaSim, BIOCHAM,…) à Simulation of a whole-cell mycoplasma genitalium [Karr Covert et al 12]

Biochemical reactions

• Binding, complexation: 𝐴 + 𝐵 → 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 → 𝐵 + 𝐶

+ ↔

Biochemical reactions

• Binding, complexation: 𝐴 + 𝐵 → 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 → 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 → 𝐵 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

→

Biochemical reactions

• Binding, complexation: 𝐴 + 𝐵 → 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 → 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 → 𝐵 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

• Gene expression, synthesis: 𝐴 → 𝐴 + 𝐵

𝐸2𝐹𝑎 → 𝐸2𝐹𝑎 + 𝑅𝑁𝐴𝑐𝑦𝑐𝐴

How to Program with Biochemical Reactions?

• Binding, complexation: 𝐴 + 𝐵 → 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 → 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 → 𝐵 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

• Gene expression, synthesis: 𝐴 → 𝐴 + 𝐵

𝐸2𝐹𝑎 → 𝐸2𝐹𝑎 + 𝑅𝑁𝐴𝑐𝑦𝑐𝐴

• Degradation: 𝐴 → ^_

Biochemical reaction rates

• Binding, complexation: 𝐴 + 𝐵 ^4.6.7 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 ^4.6 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 ^8.6/(4;6) 𝐵

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

• Gene expression, synthesis: 𝐴 ^8.6=/(4;6=) 𝐴 + 𝐵

𝐸2𝐹𝑎 → 𝐸2𝐹𝑎 + 𝑅𝑁𝐴𝑐𝑦𝑐𝐴

• Degradation: 𝐴 ^{4.6 _}

Biochemical reaction rates

• Binding, complexation: 𝐴 + 𝐵 ^4.6.7 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 ^4.6 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 ^8.6/(4;6) 𝐵

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

• Gene expression, synthesis: 𝐴 ^8.6=/(4;6=) 𝐴 + 𝐵

𝐸2𝐹𝑎 → 𝐸2𝐹𝑎 + 𝑅𝑁𝐴𝑐𝑦𝑐𝐴

• Degradation: 𝐴 ^{4.6 _}

Biochemical reaction rates

• Binding, complexation: 𝐴 + 𝐵 ^4.6.7 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 ^4.6 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 ^8.6/(4;6) 𝐵 (𝐴 + 𝐸 → 𝐶 → 𝐵 + 𝐸)

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

• Gene expression, synthesis: 𝐴 ^8.6=/(4;6=) 𝐴 + 𝐵

𝐸2𝐹𝑎 → 𝐸2𝐹𝑎 + 𝑅𝑁𝐴𝑐𝑦𝑐𝐴

• Degradation: 𝐴 ^{4.6 _}

Biochemical reaction rates

• Binding, complexation: 𝐴 + 𝐵 ^4.6.7 𝐶

𝑐𝑑𝑘1 + 𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

• Unbinding, decomplexation: 𝐴 ^4.6 𝐵 + 𝐶

• Transformation, phosphorylation, transport: 𝐴 ^8.6/(4;6) 𝐵

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 → 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

• Gene expression, synthesis: 𝐴 ^8.6=/(4;6=) 𝐴 + 𝐵

𝐸2𝐹𝑎 → 𝐸2𝐹𝑎 + 𝑅𝑁𝐴𝑐𝑦𝑐𝐴

• Degradation: 𝐴 ^{4.6 _}

Boolean Semantics: presence-absence of molecules A Ù B → C Ù A/¬A Ù B/¬B Asynchronous Transition System

Semantics of Reactions ^{A+B 𝒇(𝑨,𝑩) C}

Boolean Semantics: presence-absence of molecules A Ù B → C Ù A/¬A Ù B/¬B Asynchronous Transition System

Petri Net Semantics: numbers of molecules A , B → C++, A--, B--

Multiset rewriting CHAM [Berry Boudol 90] [Banatre Le Metayer 86]

Semantics of Reactions ^{A+B 𝒇(𝑨,𝑩) C}

Boolean Semantics: presence-absence of molecules A Ù B → C Ù A/¬A Ù B/¬B Asynchronous Transition System

Petri Net Semantics: numbers of molecules A , B → C++, A--, B--

Multiset rewriting CHAM [Berry Boudol 90] [Banatre Le Metayer 86]

Stochastic Semantics: reaction probabilities, time of next reaction in a given state S_i

Continuous Time Markov Chain (CTMC) A , B^{𝒑 𝑺𝒊 , 𝒕(𝑺𝒊)} C++, A--, B--

Semantics of Reactions ^{A+B 𝒇(𝑨,𝑩) C}

Boolean Semantics: presence-absence of molecules A Ù B → C Ù A/¬A Ù B/¬B Asynchronous Transition System

Petri Net Semantics: numbers of molecules A , B → C++, A--, B--

Multiset rewriting CHAM [Berry Boudol 90] [Banatre Le Metayer 86]

Stochastic Semantics: reaction probabilities, time of next reaction in a given state S_i

Continuous Time Markov Chain (CTMC) A , B^{𝒑 𝑺𝒊 , 𝒕(𝑺𝒊)} C++, A--, B--

Semantics of Reactions ^{A+B 𝒇(𝑨,𝑩) C}

Boolean Semantics: presence-absence of molecules A Ù B → C Ù A/¬A Ù B/¬B Asynchronous Transition System

Petri Net Semantics: numbers of molecules A , B → C++, A--, B--

Multiset rewriting CHAM [Berry Boudol 90] [Banatre Le Metayer 86]

Stochastic Semantics: reaction probabilities, time of next reaction in a given state S_i

Continuous Time Markov Chain (CTMC) A , B^{𝒑 𝑺𝒊 , 𝒕(𝑺𝒊)} C++, A--, B--

Continuous Semantics: concentrations, continuous evolution

Ordinary Differential Equations (ODE)

𝐴 ̇ = ∑

f

𝗑 δ

(𝐴𝑖)

Semantics of Reactions ^{A+B 𝒇(𝑨,𝑩) C}

Abstraction Relationships

Stochastic traces Petri net traces abstract

concrete

Boolean traces

Theory of abstract Interpretation Abstractions as Galois connections

[Cousot Cousot POPL’77]

Thm. Galois connections between the syntactical, stochastic, Petri Net and Boolean semantics

[FF Soliman CMSB’06,TCS’08]

Reaction sets

ODE traces

Abstraction Relationships

Stochastic traces Petri net traces abstract

concrete

Boolean traces

Theory of abstract Interpretation Abstractions as Galois connections

[Cousot Cousot POPL’77]

Thm. Galois connections between the syntactical, stochastic, Petri Net and Boolean semantics

[FF Soliman CMSB’06,TCS’08]

Reactions

ODE traces

Abstraction Relationships

Stochastic traces

ODE traces Petri net traces

abstract

concrete

Boolean traces

Thm. Under large number conditions the ODE semantics approximates

the mean stochastic behavior

[Gillespie 71]

Reactions

Abstraction Relationships

Stochastic traces

ODE traces Petri net traces

abstract

concrete

Boolean traces

Thm. Under large number conditions the ODE semantics approximates

the mean stochastic behavior

[Gillespie 71]

Reactions

TD 1: Lotka-Volterra Prey-Predator Model

1. Connect to http://lifeware.inria.fr/biocham4

2. Open examples/MPRI

3. Run the notebook TD1_lotka_volterra.ipynb

4. Change parameters using %slider k1 k2 k3

5. Add immigration and emigration reactions

Hybrid Models

Stochastic traces

ODE traces Petri net traces

abstract

concrete

Boolean traces

• Hybrid Boolean-continuous models (hybrid automata)

Boolean gene expression + continuous protein activation

• Hybrid stochastic-continuous models (CTMC+ODE)

Stochastic gene expression + continuous protein activation Specification of hybrid simulators

with reactions+events in SBML

[Chiang FF Huang Soliman 13 cmsb]

Reactions

Hybrid Models

Stochastic traces

ODE traces Petri net traces

abstract

concrete

Boolean traces

• Hybrid Boolean-continuous models (hybrid automata)

Boolean gene expression + continuous protein activation

• Hybrid stochastic-continuous models (CTMC+ODE)

Stochastic gene expression + continuous protein activation Specification of hybrid simulators

with reactions+events in SBML

[Chiang FF Huang Soliman 13 cmsb]

Reactions

Compiling Real Functions in Reaction Systems

biocham: compile_from_expression(cos,time,cos(t)).

present(cos(t)_p,1).

_=[z_p_2]=>cos(t)_p.

_=[z_m_2]=>cos(t)_m.

_=[cos(t)_m]=>z_p_2.

_=[cos(t)_p]=>z_m_2.

biocham: present(x_p, 4).

biocham: compile_from_expression(cos,x,cos(x)).

present(cos(x)_p, 1).

_=[g_m]=>g_p. _=[g_m+cos(x)_m]=>z_p_4.

_=[x_p]=>g_p. _=[g_p+cos(x)_p]=>z_p_4.

_=[g_p]=>g_m. _=[x_p+cos(x)_m]=>z_p_4.

_=[x_m]=>g_m. _=[x_m+cos(x)_p]=>z_p_4.

_=[g_m+z_p_4]=>cos(x)_p. _=[g_m+cos(x)_p]=>z_m_4.

_=[g_p+z_m_4]=>cos(x)_p. _=[g_p+cos(x)_m]=>z_m_4.

_=[x_m+z_m_4]=>cos(x)_p. _=[x_m+cos(x)_m]=>z_m_4.

_=[x_p+z_p_4]=>cos(x)_p. _=[x_p+cos(x)_p]=>z_m_4.

_=[g_m+z_m_4]=>cos(x)_m. _=[x_p+cos(x)_p]=>z_m_4.

Roadmap of the Lectures

19 Jan 2017:

• Introduction to biochemical programming

• Continuous semantics by ordinary differential equations

• Chemical master equation and moment closures

• Exercises: kinetics of enzymatic reactions 26 Jan 2017:

• Hierarchies of model reductions by subgraph epimorphisms

• Hierarchy of semantics and types by abstract interpretation

• Reaction hypergraph and influence graph

• Exercises: analysis of MAPK signaling models 2 Feb 2017:

• Specification of qualitative dynamical behaviors in temporal logic CTL

• Symbolic model-checking and model reductions preserving CTL properties

• Specification of quantitative dynamical behaviors in first-order temporal logic LTL(R_lin)

• Exercises: verification and synthesis of cell cycle models 9 Feb 2017:

• Parameter search in high dimension with LTL(R_lin) constraints

• Robustness of LTL(R_lin) properties

• Compiling programs in biochemical reactions

• Conclusion on hot research topics

• Exercises: designing a biosensor in non-living vesicles