page_title: Kinetic logic
section_title: Formalism
hierarchical_section_title: Kinetic logic / Formalism
caption_attribution_description: English: The time delays tx and tx’ are not same as tx depends principally on the rate of synthesis of x, whereas tx’ depends to a large extent on the stability of x. (Thomas and D’Ari, 1990).
context_page_description: Kinetic logic, developed by René Thomas, is a Qualitative Modeling approach feasible to model impact, feedback, and the temporal evolution of the variables. It uses symbolic descriptions and avoids continuous descriptions e.g. differential equations.The derivation of the dynamics from the interaction graphs of systems is not easy. A lot of parameters have to be inferred, for differential description, even if the type of each interaction is known in the graph. Even small modifications in parameters can lead to a strong change in the dynamics. Kinetic Logic is used to build discrete models, in which such details of the systems are not required. The information required can be derived directly from the graph of interactions or from a sufficiently explicit verbal description. It only considers the thresholds of the elements and uses logical equations to construct state tables. Through this procedure, it is a straightforward matter to determine the behavior of the system.
context_section_description: Following is René Thomas’s formalism 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 ∈ [0, . . ., l(u) − 1], 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}.
caption: H. Time Delays
exif: {}