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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 $Σ: 𝒜 ⟶ 𝒮$: : - $x ≤ Σ \quad ⟺ \quad σ_x(V_x) ⊆ Σ(λ_x(V_x))$ - $Σ ≤ x \quad ⟺ \quad Σ(λ_x(V_x)) ⊆ σ_x(V_x)$ ### Epimorphisms Forgetful functor to the category of graphs: : $$U: \SGph ⟶ \Gph$$
Epimorphisms from $x$ to $y$: : $$[x, y]^e \; ≝ \; \big\lbrace h ∈ \underbrace{\Hom[\SGph]{x, y}}_{\text{denoted by } [x,y]} \; \mid \; h \text{ is an epi (i.e. right-cancellable)} \big\rbrace$$ __________
> **Lemma**: $$h ∈ \overbrace{\Hom[\SGph]{x, y}}^{\text{denoted by } [x,y]} \text{ is an epi } \\ \; ⟺ \; ∀ c_y ⊆ y \text{ connected component}, \, h^{-1}(c_y) ≠ ∅$$ ### Epi-mono factorisation 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: : $$x \quad ⟶_f^R \quad f(α) \cdot x$$ where - 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 <https://github.com/yvan-sraka/KaSimir> Demo time! # Bibliography - 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 Biology*, 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 Biology*, 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.