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Signalling pathways and causal depth

Generating deep signalling pathways with refinements

Yvan Sraka & Younesse Kaddar

I. Biological context

II. Theoretical Model

III. Refinements

IV. Implementation

I. Biological context

Why does evolution favor deep signalling pathways?

I. Biological context

Why does evolution favor deep signalling pathways?

Why does evolution favor deep signalling pathways?

Example: the MAPK/ERK pathway

Animation courtsey of Ribosome Studio Youtube channel

MAPK/ERK Signalling Pathway - Simplified overview

$⇓ \;$ in Kappa

Simplified MAPK/ERK causal flow - KaFlow-generated story

Why does evolution favor deep signalling pathways?

Natural selection: Deeper stories $⟹$ Improving biological fitness

?   BUT counter-intuitive

  • energetically

  • probabilistically: Indeed: if

    $$ \begin{align*} & 𝔼(\max_i \depth(P_i)) \\ & ≤ 𝔼\Big(\ln\Big(\underbrace{\sum\limits_{ i=0 }^n \exp(\depth(P_i))}_{\; ≝ \; α_n}\Big)\Big) \\ & ≤ \ln \underbrace{𝔼(α_n)}_{\rlap{\substack{= \; 𝔼(α_{n-1}) + 𝔼(\exp(\depth(P_n))) \\ = \, (1 + \frac e i) \, 𝔼(α_{n-1})}}}\\ & = \ln \Big(\prod\limits_{ i=1 }^n \Big(\underbrace{1 + \frac e i}_{≤\, \exp (\frac e i)}\Big) \Big) = e H_n \sim \boxed{e \ln n} \end{align*} $$

Possible explanations:

Greater depth:
$$\, \\ \\ \\ \begin{cases} \text{ enhances stability } \\ \text{ deletions make pathways non-functional more often than additions do } \end{cases}$$

II. Theoretical model

Does refinement increase depth?

Expectation - Time evolution

Genetic algorithm

An heuristic optimisation that loop on two steps:

  • MUTATION: map ancesters to $n$ descendent with random mutations
  • SELECTION: filter descendent on their fitness (ERK quantity)

Model biais: biological clock assumption, spontaneous discrete generation, etc …

III. Refinements

$$\MEK(x_{pp}), \MAPK(x_{p}) ⟶_{τ_b} \MEK(x_{pp}^{\color{DeepPink}{1}}), \MAPK(x_{p}^{\color{DeepPink}{1}})\\ \color{DarkCyan}{\MEK(x_{pp}^1), \MAPK(x_{p}^1) ⟶_{τ_u} \MEK(x_{pp}), \MAPK(x_{p})}\\ \MEK(x_{pp}^1), \MAPK(x_{p}^1) ⟶_{τ_p} \MEK(x_{pp}^1), \color{OrangeRed}{\MAPK(x_{pp}^1)}\\$$

  • $\color{DarkCyan}{τ_u}$ too big: unbind before phosphorylation (too liquid)
  • $\color{DarkCyan}{τ_u}$ too small: never unbind (too sticky)

Category of site graphs: $\SGph$

Objects $(V, λ, σ, μ)$:
  • $V$: set of nodes
  • $λ: V ⟶ 𝒜$: name assignment
  • $σ: V ⟶ 𝒫(𝒮)$: site assignment
  • $μ: \underbrace{\text{ matching}}_{\rlap{\text{irreflexive symmetric binary relation}}}$ over $\sum\limits_{ v ∈ V } σ(v)$

Morphism $f: (V, λ, σ, μ) ⟶ (V', λ', σ', μ')$:
Name/site/edge-preserving and edge reflecting monomorphism $f: V ⟶ V'$

Signature $Σ: 𝒜 ⟶ 𝒮$:
  • $x ≤ Σ \quad ⟺ \quad σ_x(V_x) ⊆ Σ(λ_x(V_x))$
  • $Σ ≤ x \quad ⟺ \quad Σ(λ_x(V_x)) ⊆ σ_x(V_x)$


Forgetful functor to the category of graphs:
$$U: \SGph ⟶ \Gph$$

Epimorphisms from $x$ to $y$:
$$[x, y]^e \; ≝ \; \big\lbrace h ∈ \underbrace{\Hom[\SGph]{x, y}}_{\text{denoted by } [x,y]} \; \mid \; h \text{ is an epi (i.e. right-cancellable)} \big\rbrace$$


$$h ∈ \overbrace{\Hom[\SGph]{x, y}}^{\text{denoted by } [x,y]} \text{ is an epi } \\ \; ⟺ \; ∀ c_y ⊆ y \text{ connected component}, \, h^{-1}(c_y) ≠ ∅$$

Epi-mono factorisation

$f ∈ [s, x]$ is said to be factored by $t$ if $f = γ ϕ$ for $ϕ ∈ [s, t]^e$ and $γ ∈ [t, x]$

Every iso $α ∈ [t, t']$ conjugates the factorisations $ϕ, γ$ and $ϕ', γ'$:

⟹ Equivalence relation:

$$ϕ, γ \; ≃_{tt'} \; ϕ', γ'$$

The group $[t, t]$ acts freely on $[s, t]^e × [t, x]$ (as we have epis and monos), so by Burnside’s lemma:

$$\vert \underbrace{[s, t]^e × [t, x] / [t, t]}_{\text{denoted by } [s, t]^e ×_{[t, t]} [t, x]} \vert = \frac 1 {\vert [t,t] \vert} \sum\limits_{ (ϕ, γ) ∈ [s, t]^e × [t, x]} \underbrace{\vert Stab_{[t, t]}((ϕ, γ)) \vert}_{= 1} \\ = \vert [s, t]^e × [t, x] \vert/\vert [t,t] \vert$$

Object refinement

A refinement $Σ(s)$ of $s ≤ Σ$ under $Σ$:
is a set comprised of one element per isomorphism class in $\lbrace t : Σ \; \mid \; [s, t]^e ≠ ∅ \rbrace$

Theorem: if $s ≤ Σ$ and $x : Σ$:

$$[s,x] \; ≅ \; \sum\limits_{ t ∈ Σ(s) } [s, t]^e ×_{[t,t]} [t, x]$$

Rule refinement

Well defined atomic action (in a rewriting rule):

edge addition/deletion (when permitted), agent addition/deletion

Labelled transition:

$$x \quad ⟶_f^R \quad f(α) \cdot x$$


  • the rule $R \; ≝ \; \underbrace{s}_{\text{object}}, \underbrace{α}_{\text{action}}, \underbrace{τ}_{\text{rate}}$, whose activity $a(x, r) \; ≝ \; τ \vert [s, x] \vert$
  • $f ∈ [s, x]$

If $θ$ is an iso: $θ(r) \; ≝ \; θ(s), θ(α), τ$ satisfies:

$$x \, ⟶_f^R \, f(α) \cdot x \qquad ⟺ \qquad x \, ⟶_{fθ^{-1}}^{θ(R)} \, fθ^{-1}(θ(α)) \cdot x$$

Rule refinement:
If $s ≤ Σ$, the refinement of the rule $R \; ≝ \; s, α, τ$ under $Σ$ is:

$$Σ(s, α, τ) \; ≝ \; (t, ϕ(α), τ)_{t ∈ Σ(s), ϕ ∈ [s,t]^e/[t,t]}$$

IV. Methods

Our naive code implementation was split between:

  • KaSimir: Genetic algorithm implementation, using KaSim to compute fitness
  • KaStorama: wrapper around KaStor to retreive causal story depth from KaSim traces
  • KaRapuce: Slitghly change rates of refined rules as mutation event


Demo time!


  • P. Boutillier, J. Feret, J. Krivine, and W. Fontana, “The Kappa Language
    and Kappa Tools,” p. 52.

  • “Signaling Pathways,” Tocris Bioscience. https://www.tocris.com/signaling-pathways.

  • V. Danos, J. Feret, W. Fontana, R. Harmer, and J. Krivine, “Rule-Based
    Modelling, Symmetries, Refinements,” in Formal Methods in Systems
    , vol. 5054, J. Fisher, Ed. Berlin, Heidelberg: Springer Berlin
    Heidelberg, 2008, pp. 103–122.

  • E. Murphy, V. Danos, J. Féret, J. Krivine, and R. Harmer, “Rule-Based
    Modeling and Model Refinement,” in Elements of Computational Systems
    , H. M. Lodhi and S. H. Muggleton, Eds. Hoboken, NJ, USA: John
    Wiley & Sons, Inc., 2010, pp. 83–114.

  • “List of signalling pathways,” Wikipedia. 30-Nov-2016.