Arithmetical Metabolic P Systems - Foundations on Natural and Artificial Computation - page 285

S ,if R − ( x ) is the subset of
reactions consuming x ,then x verifies the x -availability constraint with respect
to Φ when

Definition 1. ( x -availability constraint) Let x ∈

r∈R − ( x ) |

r − | x ϕ r ( X ) ≤ x.

r − | x is the multiplicity of x in r − using the Parikh notation. Where |

Definition 2. (Blocked rule) Let R be a set of reactions with regulators Φ .If,
for a given substance x ,the x -availability constraint is not satisfied with respect
to Φ , then the reactions of R − ( x ) are said to be blocked on x with respect to Φ .

Definition 3. (Simple regulators) A set of regulators Φ = { ϕ r | r ∈ R } is

simple when, given the state X =( x 1 ,...,x n ) ,

⎨

k ∈ N (constant)

∀ r ∈ R, ϕ r ( X )= or

x j

⎩

| 1 ≤ j ≤ n (projection) .

Definition 4. (Arithmetical MP System) An Arithmetical MP System of type
( n, m, k ) having n substances, m reactions and k parameters, is specified by the
construct:

M =( S, R, H, Φ, In , Out , Start , Halt )

where:

S is a set of n substances, considered in a conventional order, determining the
metabolic state X

n of the system.
R is a set of m reactions ( r 1 ,r 1 ) ,..., ( r m ,r m ) , whose stoichiometric balances
are the columns of the stoichiometric matrix

∈ N

=( r 1 ,...,r m ) . A

k providing, at each step i

H is a function H : N → N ∈ N , the vector H [ i ] of

parameters.
Φ is the set of simple regulators

n each reaction

{ ϕ r | r ∈ R } such that, ∀ X ∈ N

r ∈ R has associated the regulator:

ϕ r ( X )= 0

if r is blocked w.r.t. Φ

ϕ r ( X ) otherwise .

In , Out are a set of substances with In ⊆ S , Out ⊆ S .

Start , Halt are two substances with Start ∈ S , Halt ∈ S .

n be the initial state of the system M , its dynamics is specified by
the vector recurrent equation, called Equational Matabolic Algorithm, defined in
MP Systems [7]:

Let X [0] ∈ N

A × U [ i ]+ X [ i ]

X [ i +1]= (1)

providing in X [ i +1] the metabolic state of the system by means of the vector of
fluxes

U [ i ]=(

ϕ r ( x 1 [ i ] ,...,x n [ i ] ,p 1 [ i ] ,...,p k [ i ])
are computed upon the components of the status X [ i ] , H [ i ] .

u r [ i ] | r ∈ R ) where u r [ i ]=