Computations with mobile membranes can be defined over specific configurations (like process calculi), while they represent also a rule-based formalism (like P systems).
The model is characterized by two essential features: The computations are performed in the following way: starting from an initial structure, the system evolves by applying the rules in a nondeterministic and maximally parallel manner.
A rule is applicable when all the involved objects and membranes appearing in its left hand side are available.
A specific feature of the mobile membranes is that this new rule-based model is appropriate to prove computability results in terms of Turing machines rather by reduction to the lambda calculus as in the case of process calculi with mobility.
In this section are defined four classes of membranes inspired from biological facts, and it is shown that their computational power depends on the initial configuration and on the set of rules used.
The systems of simple mobile membranes (SM) are defined over the set of configurations
denotes the family of sets of vectors of natural numbers computed by using at most $n$ membranes.
denoted the family of Turing computable sets of vectors generated by arbitrary grammars.
The research line initiated in membrane computing is to find membrane systems with a minimal set of ingredients which are powerful enough to achieve the full power of Turing machines.
The operations governing the mobility of the systems of enhanced mobile membranes are endocytosis (endo), exocytosis (exo), forced endocytosis (fendo), forced exocytosis (fexo).The evolution from a configuration to another is made using rules from the set of rules
When proving the result of the previous theorem the authors have not used an optimal construction of a membrane system.
Membrane systems [24] and brane calculus [10] start from the same observations; however, they are built having in mind different goals: membrane systems investigate formally the computational nature and power of various features of membranes, while the brane calculus is capable to give a faithful and intuitive representation of the biological reality.
In [12] the initiators of these two formalisms describe the goals they had in mind: "While membrane computing is a branch of natural computing which tries to abstract computing models, in the Turing sense, from the structure and the functioning of the cell, making use especially of automata, language, and complexity theoretic tools, brane calculi pay more attention to the fidelity to the biological reality, have as a primary target systems biology, and use especially the framework of process~algebra."
Objects and co-objects are used in phago and exo rules in order to illustrate the fact that both involved membranes agree on the movement.
The family of all sets generated by systems of mutual membranes with objects on surface using at any moment during a halting computation at most
It is proven in [19] that systems of eight membranes with objects on surface and using pino and exo operations of weight four and three are universal.
This means that in order to construct a universal system of mobile membranes with objects on surface by using pino and exo operations, one needs to decide either he wants to minimize the number of membranes, or the weights of the operations.
It is proven in [19] that systems of nine membranes with objects on surface and using phago and exo operations of weight four and three (or five and two) are universal.
The number of membranes can be reduced from nine to four, but in order to do this the weight of the phago and exo operations are increased from four and three (or five and two) to six and three.
A short description of pure safe ambients (SA) is given below; more information can be found in [22,23].
The operational semantics of pure ambient safe calculus are defined in terms of a reduction relation
A simulation of the PEP fragment of brane calculus by using mutual membranes with objects on surface is presented.
This approach is related to some other papers trying to show the relationship between membrane systems and brane calculus [8,9,14,18,19].
By defining an encoding of the PEP fragment of brane calculus into mutual membranes with objects on surface, it is shown that the difference between the two models is not significant.
Another difference regarding the semantics of the two formalisms is expressed in [8]: "whereas brane calculi are usually equipped with an interleaving, sequential semantics (each computational step consists of the execution of a single instruction), the usual semantics in membrane computing is based on maximal parallelism (a computational step is composed of a maximal set of independent interactions)".
The operations of the two basic brane calculi, pino, exo, phago (PEP) and mate, drip, bud (MBD) are directly inspired by biologic processes such as endocytosis, exocytosis and mitosis.
Cardelli motivates that the replication operator is used to model the notion of a "multitude" of components of the same kind, which is in fact a standard situation in biology [10].
The structural congruence relation is a way of rearranging the system such that the interacting parts come together, as illustrated in what follows: In what follows the reduction rules of the calculus are presented: The action
action resides; imagine that the original membrane buckles towards inside and pinches off.
The rules of the systems of mutual membranes with objects on surface (MMOS) are presented in what follows.