Biochemical Abstract Machine

new teacher: François Fages

Two types of bioinformatics:

  1. tools to process cell data (suited data structures, etc…)
  2. science of data processing in cells ⟶ very hard, but systems biology tries to have a go at it

Molecular systems biology ⟹ synthetic biology: e.g. possible to distinguish different types of diabetes by implementing molecular boolean circuits inside the cell

Protein computing = Turing complete ⟶ we can compile functions in chemical reaction networks.

Molecular programming/DNA computing: IGEM competition (competition about using genes to simulate)

Chemical Reaction Networks (CRN)

Can be interpreted with

  • differential equations
  • continuous-time Markov chains
  • Petri Nets
  • Boolean nets

Interplay between the CRN theory (computability/Turing completeness, computaitonal complexity, static analysis, quasi-steady states) and the CRN dynamics

Kappa: site graph rewriting language ⟹ more expressive than CRN (you can encode polymerization), it’s Turing complete.

CRN: more restrictive but decidable for some key tasks: reachability, model-checking (even the number of states is tremendous: with SAT-solvers/Boolean constraints, you tell many things about your system), …

BIOCHAM: Biochemical Abstract Machine

Several languages:

  • one to describe CRNs
  • one based on temporal logic

Elementary reactions: un/binding, de/complexation, transformation, phosphrylation, transport, gene expression, synthesis, degradation…

Maschke action law:

\[A + B \overset{k.A.B}{⟶} C\]

Several interpretation of $A + B \overset{k.A.B}{⟶} C$

Continuous Semantics

\[A + B \overset{k.A.B}{⟶} C\]

⟹

\[\frac {dA} {dt} = -k.A.B \\ \frac {dB} {dt} = -k.A.B \\ \frac {dC} {dt} = k.A.B\]

Stochastic Semantics

Continuous-Time Markov Chain (CTMC):

\[A, B \overset{p(S_i), t(S_i)}{⟶} C++, A--, B--\]

Instrinstic vs. Extrinsic sensitivity (sensitivity to parameter changes)

Multi-Agent Simulation

Random walk/Diffusion in 3D space (Hsim simulator)

Petri net semantics

You forget the probabilities: multi-set rewriting system (Chemical Abstract Machine (CHAM) by Gérard Béry! ⟶ $π$-calculus, operational semantics)

\[A, B ⟶ C++, A--, B--\]

Boolean semantics

About the presence or absence of molecule:

\[A ∧ B ⟶ C ∧ ¬ A ∧ ¬ B\\ A ∧ B ⟶ C ∧ A ∧ ¬ B\\ A ∧ B ⟶ C ∧ ¬ A ∧ B\\ A ∧ B ⟶ C ∧ A ∧ B\\\]

Abstract interpretation theorem: There are Galois connections between these.

Chemical Master Equation (CME)

Assumption: infinite/perfect dilute solution

\[\frac d {dt} p^{(t)}(x) = \sum\limits_{ j: x - v_j ≥ 0} α(x-v_j) p^{(t)}(x-v_j) - \sum\limits_{ j} α(x) p^{(t)}(x)\]

where

  • $p^{(t)}(x)$: probability of being in state $x$ at time $t$
  • $α_j(x)$: propensity of reaction $j$
  • $v_j$: change vector of reaction $j$

Evolution of the mean:

\[\frac d {dt} μ(t) = \frac d {dt} E[X(t)] = \frac d {dt} \sum\limits_{ x } x p^{(t)}(x)\\ = \sum\limits_{ j } v_j \sum\limits_{ x } α_j(x) p^{(t)}(x) = \sum\limits_{ j } v_j E[α_j(X(t))]\]

Michaelis-Menten Example

\[E + S \underset{c_2}{\overset{c_1}{\rightleftarrows}} C \overset{c_3}{⟶} E+P\] \[X = (X_E, X_S, X_C, X_P)\]

Continuous semantics = First-order approximation of the stochastic semantics

(not good for Lotka-Volterra, but better for enzyme kinetics for instance)

Turing completeness of CRNs

Hierarchy of semantics:

  digraph {
    rankdir=BT;
    "Reaction Set" -> "Stochastic sem.", "ODE Semantics";
    "Stochastic sem." -> "Discrete semantics" -> "Boolean"
  }

Reaction Set/Petri nets: not Turing complete, except if you add:

  • test of absence in Petri net

  • Polymerization reactions, that we have in Kappa but not Biocham. E.g.:

    \[A + A^n -> A^{n+1}\]

    In Biocham/Petri nets: no formalism for $A^n$, we have a finite number of species

  • Non uniform computability: for each integer function, for each input there exists a circuit computing the result

MAPK Signalling Cascade: huge class of receptors (7-TMR receptors, targeted by lots of drugs in the pharmaceutic industry).

Ex: RTK floats on the membrane, binds to several ligands and triggers a signal.

\[MM(v,k) = \frac{v \cdot A}{k+A}\]
  • $A$ in $volume^{-1}$
  • $MM$ in $time^{-1} volume^{-1}$

Abstract Interpretation in Systems Biology

Stocheometric Influence Graph:

\[A + B ⟹ C\] 
  digraph {
    rankdir=LR;
    A, B -> C;
    A -> B[label="inhibition"];
    B -> A[label="inhibition"];
    A -> A[label="inhibition"];
    B -> B[label="inhibition"];
  }

Example:

\[p53 + Benzopyrene => M\]
  • $p53$: supposed to control the cell cycle
  • $Benzopyrene$: involved in cigarettes

Differential Influence Graph

\[A ⟶ C \qquad \text{ if } \frac {\partial \dot C}{\partial A} > 0\\ A \overset{\text{inhibition}}{⟶} B \qquad \text{ if } \frac {\partial \dot B}{\partial A} < 0\\\]

e.g.:

\[\dot A = -k A \cdot B\\ \dot B = -k A \cdot B\\\]

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