Kinetic Logic - Formulism

Formulism

Following is René Thomas’s formulism for Kinetic Logic :

In a directed graph G = (V, A), we note G− (v) and G+ (v) the set of predecessors and successors of a node v ∈ V respectively.

Definition 1: A biological regulatory network (BRN) is a tuple G = (V, A, l, s, t, K) where
(V, A) is a directed graph denoted by G,
l is a function from V to N,
s is a function from A to {+, −},
t is a function from A to N such that, for all u ∈ V, if G+(u) is not empty then {t(u, v) | v ∈ G+(u)} = { 1, . . ., l(u)}.
K = {Kv | v ∈ V} is a set of maps: for each v ∈ V, Kv is a function from 2G− (v) to {0, . . ., l(v)} such that Kv(ω) ≤ Kv(ω_) for all ω ⊆ ω_ ⊆ G−(v).

The map l describes the domain of each variable v: if l (v) = k, the abstract concentration on v holds its value in {0, 1, . . ., k}. Similarly, the map s represents the sign of the regulation (+ for an activation, − for an inhibition). t (u, v) is the threshold of the regulation from u to v: this regulation takes place iff the abstract concentration of u is above t(u, v), in such a case the regulation is said active. The condition on these thresholds states that each variation of the level of u induces a modification of the set of active regulations starting from u. For all x ∈, the set of active regulations of u, when the discrete expression level of u is x, differs from the set when the discrete expression level is x + 1. Finally, the map Kv allows us to define what is the effect of a set of regulators on the specific target v. If this set is ω ⊆ G− (v), then, the target v is subject to a set of regulations which makes it to evolve towards a particular level Kv(ω).

Definition 2 (States):
A state μ of a BRN G = (V, A, l, s, t, K) is a function from V to N such that μ (v) ∈ {0 .., l (v)} for all variables v ∈ V. We denote EG the set of states of G.
When μ (u) ≥ t (u, v) and s (u, v) = +, we say that u is a resource of v since the activation takes place. Similarly when μ (u) < t (u, v) and s (u, v) = −, u is also a resource of v since the inhibition does not take place (the absence of the inhibition is treated as an activation).

Definition 3 (Resource function):
Let G = (V, A, l, s, t, K) be a BRN. For each v ∈ V we define the resource function ωv: EG → 2G− (v) by: ωv (μ) = {u ∈ G−(v) | (μ(u) ≥ t(u, v) and s(u, v) = +) or (μ (u) < t (u, v) and s (u, v) = −)}.
As said before, at state μ, Kv (ωv(μ)) gives the level towards which the variable v tends to evolve. We consider three cases,

• if μ(v) < Kv(ωv(μ)) then v can increase by one unit
• if μ(v) > Kv(ωv(μ)) then v can decrease by one unit
• if μ(v) = Kv (ωv (μ)) then v cannot evolve.

Definition 4 (Signs of derivatives):
Let G = (V, A, l, s, t, K) be a BRN and v ∈ V.
We define αv: EG → {+1, 0, −1} by αv(μ) =
+1 if Kv (ωv(μ)) > μ(u)
0 if Kv (ωv(μ)) = μ(u)
−1 if Kv (ωv(μ)) < μ(u)

The signs of derivatives show the tendency of the solution trajectories.

The state graph of BRN represents the set of the states that a BRN can adopt with transitions among them deduced from the previous rules:

Definition 5 (State graph):
Let G = (V, A, b, s, t,K) be a BRN. The state graph of G is a directed graph G = (EG, T) with (μ, μ_) ∈ T if there exists v ∈ V such that:
αv (μ) ≠ 0 and μ’ (v) = μ (v) + αv (μ) and μ (u) = μ’ (u), ∀u ∈ V \ {v}.