# 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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