This paper introduces Protein Calculus, a special modeling language designed for encoding and calculating the behaviors of compartmentilized biological systems. The formalism combines, in a unified framework, two successful computational paradigms - process algebras and membrane systems. The goal of Protein Calculus is to provide a formal tool for transforming collected information from in vivo experiments into coded definition of the different types of proteins, complexes of proteins, and membrane-organized systems of such entities. Using this encoded information as input, our calculus computes, in silico, the possible behaviors of a living system. This is the preliminary version of a paper that was published in Proceedings of International Conference of Computational Methods in Sciences and Engineering (ICCMSE), American Institute of Physics, AIP Proceedings, N 2: 642-646, 2007 (http://scitation.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=963&Issue=2)

Ihekwaba, Adaoha

Mardare, Radu

English

Unitn-eprints Research

Technical Report CoSBi 16/2007A Process Algebraical Approach to ModellingCompartmentalized Biological SystemsRadu MardareThe Microsoft Research-University of Trento Centre for Computational and Systems Biology,Trento, Italymardare@cosbi.euAdaoha IhekwabaThe Microsoft Research-University of Trento Centre for Computational and Systems Biology,Trento, Italyihekwaba@cosbi.euThis is the preliminary version of a paper that will appear inIn Proceedings of International Conference of Computational Methods in Sciences andEngineering (ICCMSE), American Institute of Physics, AIP Proceedings, N 2: 642-646, 2007.A Process Algebraical Approach to ModellingCompartmentalized Biological SystemsRadu Mardare, Adaoha IhekwabaThe Microsoft Research-University of Trento, ItalySeptember 30, 2009AbstractThis paper introduces Protein Calculus, a special modeling language designed for en-coding and calculating the behaviors of compartmentilized biological systems. The for-malism combines, in a unified framework, two successful computational paradigms - pro-cess algebras and membrane systems. The goal of Protein Calculus is to provide a formaltool for transforming collected information from in vivo experiments into coded definitionof the different types of proteins, complexes of proteins, and membrane-organized systemsof such entities. Using this encoded information as input, our calculus computes, in silico,the possible behaviors of a living system.1 The computational challenge in Systems BiologyMany of the activities of the biological cells are regulated by proteins which carry signals thatmodify the expression of different genes at a given time, but how these signals do so is notknown. The use of Systems Biology is therefore essential if we are ever to make sense of thebiological complexity, as intuitive ’conceptual’models.Systems Biology is a multi-disciplinary approach that studies the relationships and inter-actions of various cellular mechanisms and cellular components. It encompases the growingnumber of mechanistic but largely isolated insights and increasingly, high-throughput ’omics’data sets which tends to be semi-quantitative and specific to its experimental system and inte-grates them into a conceptual modelling framework that is holistic, quantitative and predictive.By abstracting biological systems on the level of their behaviours, models are obtained shar-ing many characteristics with computational systems. Consequently, on all levels of organisa-tion (i.e. from the atomic to full organisms) concurrent behaviours - be it event-driven, causal,time-dependent or distributed - is attained. Owing to these similarities, modelling (reverse-engineering, simulating and analysing) a biological system has been considered in a similarway to the engineering of complex artificial systems [2, 4]. This has resulted in questions suchas - Is the structure of a cell much more complex than the structure of a jet plane? - to beraised. In answering the question, the challenge therefore is to construct, in silico, reliable‘copies’ of biological systems that would allow the simulation of various experiments, which1will then assist in proposing and verifying hypotheses. Recent small-scale modelling efforts inthis direction have had encouraging results [14, 6, 5].Traditionally, mathematical modelling of biological systems have always involved the useof ordinary differential equations (ODEs) for the reason that they offer a description of net-works as continuous time dynamical systems as well as the added advantage of being a maturetechnique with many existing mathematical and computational resources available. Howeverin the context of this work - use of ODE is limited - due to the highly non-linear behavioursexhibited by bio-systems, in addition to the lack of compositionality and systemic view ofODE-based techniques. ODEs simply model the bio-system as a whole and compute its over-all behaviour, making it impossible for the description of the model to be disassembled into itsconstituent components. As a consequence, updating or extension of the model is not easilydone when new information on the system becomes available.For obtaining reliable models in silico, we expect to be able to combine small-scale modelsof molecular or cellular networks in large-scale models that might be used as valid images ofcomplex biological phenomena. We also expect it to be possible to integrate new informationinto the models already developed. The need for small-scale modelling as a first step in design-ing complex models emerges from the methods of Molecular Biology - typically a molecularbiologist spends years of research in studying a particular molecule. This information can beeventually used for designing a model of that molecule, but for further analysis, the ability tointegrate such small models in order to define big scenarios as the behaviours of an organ or ofan organism is required. Therefore, if the behaviour of a molecule is described as a program(possibly involving a high level of indeterminism) then by running the program in a concurrentenvironment bigger scenarios can expect to be obtained.Following this intuition, alternative approaches inspired by different paradigms from Com-puter Science have been recently proposed to complement the ODE approach while ensuringcompositionality and extendability of the models. An example of such approach is processalgebra.Process algebras [1] are developed for designing and analysing complex systems of agentsorganised in a modular way and able to interact, collaborate, communicate, move and com-pose/decompose composite agents. In this paradigm, agents are understood as spatially lo-calised and independently observable units of behaviour and computation (e.g. programs orprocessors running in parallel). They are successfully used in modelling distributed systemslike computer networks. Recently the paradigm has been demonstrated to be particularly ap-propriate for modelling molecular and cellular networks [2, 14, 10, 11]. In this approach, abiological system is modelled as a network of modules (subsystems) that run in a decentralisedmanner, where they interact and change their relative positions which results in a global be-haviour change of the systems generated. As a result, the overall model of the system caneasily be obtained by mixing the models of its components. What’s more, this paradigm wasadapted to include quantitative and stochastic information [7, 13] - which are essential featuresof bio-chemical processes - and used together with stochastic simulators, built on the Gille-spie method, for assisting in model validation and hypothesis forming. A stochastic simulatoruses data from actual measurements and computes the expected behaviour of the system. Inthis manner, the high cost of running in vivo experiments is reduced, due to the derivation andrunning of in silico models. This method has proved to be effective for systems on small-2scale levels, such as cellular systems and has been integrated in BetaWB [15], Cyto-Sim [16],BioSPI [13], which are simulators that have been successfully used in modelling and analysingsignalling pathways.Yet while in modelling biochemical soups, the straight concurrent environment is success-ful, it is still not sufficient for describing the complex behaviour of living systems. Biologi-cal systems involve compartmentalisation [3], which plays a major role in the functioning ofliving systems. They are not just unstructured, heterogeneous chemical soups, but spatiallydistributed hierarchies of bio-compartments organising and isolating chemical reactions andtheir products. On the level of cellular and molecular systems we can identify membrane-bound compartments including cells, organelles and vesicles each enclosed in a membrane,which isolates the bio-chemical components of a compartment from the external environment.These membranes organise a hierarchy of locations in the system. Then we have also molecu-lar compartments, including stable or transient multi-molecular complexes, which insulate thecomponent molecules from the environment. All these compartments have the role of a keyregulatory mechanism for the biological system (in order to perform its function, a moleculemust be present in the right location), organising it in a modular way.The biological compartments can be extremely dynamic. The evolution of a biological sys-tem can involve movements of molecules between compartments (e.g. Golgi apparatus) anddynamic rearrangements of cellular compartments and molecular complexes (i.e. moleculesmay join a molecular compartment by binding to one of its members). There are also com-partments that move, as a whole, with respect to the topology of the system. A typical exam-ple in this respect is phagocytosis (a cell is engulfed by another cell, e.g. macrophages ‘eat’viruses or contaminated cells) or entry of a complex molecule into an organelle. Sometimestwo membrane-bound compartments merge, forming one compartment. An approach knownas membrane computing would be adventageous.Membrane computing [12] was introduced as a new paradigm in computer science aimingto abstract computing models from the structure and the functioning of living cells. It focuseson the idea of dividing the space of computation by means of “membranes”. Accordingly,membrane systems are computational devices designed for modelling membrane-distributedand parallel computational scenarios. The membrane has the role of a separator of the spaceof computation. In between membranes there are distributed objects that can interact or movefrom one membrane to others following some laws (similar with the bio-chemical laws). Thus,membrane computing is a computational framework, inspired from biological reality, and de-signed to model compartmentalized phenomena. It has been proved useful for applications incomputer science and essential in modelling biological systems [4, 8, 9] due to the role of thecompartments in generating complex behaviours.2 Protein CalculusIn this report, we introduce Protein Calculus, a calculus that combines the process algebraand membrane systems paradigms in a unified framework designed for modelling compart-mentalized systems of molecules. The calculus focuses on bimolecular interactions betweenmolecules which are modelled as objects spatially organized by a network of membranes.3With respect to the membrane, a protein can be placed inside the area enclosed by themembrane, outside it or on the membrane. A protein situated on a membrane can interactwith internal or external proteins, but the proteins enclosed in the inner space delimited by amembrane cannot interact with outside proteins. The bimolecular interactions are of two types:complexation and decomplexation. The complexation is the phenomenon of binding proteinsby means of their active sites. The result, a complex of proteins, which is a complex object thatcan be further involved in bimolecular interactions. Decomplexation however is the reversephenomenon and consists of splitting a complex of proteins into the constituent parts. Thefollowing section gives a description of the syntax used.2.1 SyntaxThe syntax of Protein Calculus is presented by a grammar with non-terminals {P,C, S} and isshown in Table 1 - where a desired system is represented by two disjoint alphabets, Ω and Σ.For the system, the elements of Ω is used to name proteins (i.e. each element a ∈ Ω appears inthe syntax of a protein to identify it), whilst the elements of Σ is used for the interaction sitesof the protein. With this in mind - α.a : β.a -denotes a protein “a” with two active sites α andβ. Note - that the operator “:” is used to express the coexistence of different active sites.The same operator can also be used to represent complexes of proteins. For instance,α.b : β.c : γ.c represents a complex formed by two proteins b and c, where, after complexation,b has only one active site α whilst c has two active sites, β and γ. By using these sites thecomplex can be further involved in other complexations.However, we know that proteins and complexes tend to float in the same compartment /soup, in making allowances for this - the operator “‖” is used. In other words, in a givensystem, when protein α.a : β.a shares the same solution with complex α.b : β.c : γ.c, thesolution α.a : β.a ‖ α.b : β.c : γ.c is given.Anywhere structure SM is given, this will be used to express a membrane enclosed system,where the fraction line represents a membrane and encodes the way in which the space isseparated. The structure below the fraction line, M , represents the structure enclosed by themembrane, while the structure above the fraction line, S, represents the soup of molecules thatfloats on the membrane.The use of the fraction line for representing the division of space has the following advan-tage; in that SM ‖M ′ can be used to represent a cell having the membrane composed by S, thecontent M and the environment in which it evolves M ′. Obviously M and M ′ can be furtherdescribed as composed by other membranes.P ::= a | α.P | P : P a ∈ Ω, α ∈ Σ (proteins)C ::= P | C.αC | C : C α ∈ Σ (complexes)S ::= C | S ‖ S (soups)M ::= S | SM |M ‖M (membrane systems)Table 1: Syntax of Protein CalculusConsider, for example, the structure in Figure1. We have a membrane system composed by4a main membrane that enclose other four membranes between which the fourth one encloses afifth membrane. On the membrane and between membranes we have proteins and complexesof proteins. Using our syntax we can represent this system with the following equationA ‖ U ‖ (A : B : D) ‖ B ‖ AVN‖ F ‖ GF : E‖ L ‖ H ‖ D(A : B) ‖ K ‖M ‖ A ‖ T(V : Q : R) ‖ N ‖ DE.In the above representationA,B,D,E, F,G,H,K,L,N,Q,R, U, V are uninterpreted proteinsthat can be further defined using our syntax, and (A : B : D), (F : E), (A : B) and(V : Q : R) are complexes of proteins.This syntax therefore, could be used to describe a membrane system in any state. But youask what of the behaviour? The operational semantics of our calculus will be used to computethis.2.2 Operational semanticsThe behaviour of a membrane system is encoded by some basic axioms shown in Table2.“φ −→ φ′” is used to mark the evolution of the system from a state φ to the state φ′ inany contextual situation, while “φ =⇒ φ′” marks the evolution governed by side conditionsyet to be defined in the lab. (C1) is a complexation axiom that establishes the type of pro-tein or complexes that binds when floating in the same environment, while (D1) is the cor-responding decomplexation. Similarly (C1) and (C2) dually with (D1), (D2) are complexa-tion/decomplexation axioms, but are used to represent conditions of bimolecular interactionscase where one of the molecules is anchored on the membrane and the other is floating freelyin the inner respectively outer regions. Note - that (D1) and (D2) have identical premises,but their products differ depending on some contextual conditions to be defined in the lab forparticular systems of proteins.(C1) : α.p ‖ α.p′ ‖ s −→ p.αp′ : p′.αp ‖ s (D1) : p.αp′ : p′.αp ‖ s −→ α.p ‖ α.p′ ‖ s(C2) :α.p ‖ sα.p′ ‖ m −→p.αp′ : p′.αp ‖ sm (D2) :p.αp′ : p′.αp ‖ sm =⇒α.p ‖ sα.p′ ‖ m(C3) :α.p ‖ sm ‖ α.p′ −→p.αp′ : p′.αp ‖ sm (D3) :p.αp′ : p′.αp ‖ sm =⇒α.p ‖ sm ‖ α.p′(S) :s ‖ s′m ‖ m′ =⇒sm ‖ s′m′(J) : sm ‖ s′m′=⇒ s ‖ s′m ‖ m′(P ) :s ‖ s′m ‖ m′ =⇒ sm ‖ s′m′(E) : sm ‖ s′m′=⇒ s ‖ s′m ‖ m′Table 2: Operational semantics(S) and (J) are the split and join axioms that model the division of a cell or merging of twomembranes. (P ) and (E) encodes the phagocitosys and exocitosys phenomena respectively.5UAABVEGDHDMATFLNNKA:B:DF:EA:BV:Q:RFigure 1: A membrane system3 Concluding remarksProcess calculus is an invaluable technique that can be used for describing the molecular sys-tems and their behaviours; for instance in modelling trans-membrane interactions (i.e. proteinmovement between cytoplasm and the nucleus) - examples of which include the translocationof the NF-kappa B proteins into the nucleus. Protein Calculus is particularly useful in trans-forming collected information from in vivo experiments into coded definition of the differenttypes of proteins and complexes of proteins, and establishing positions of the coded objectswith respect to the membranes of the system. In addition, the syntax of the calculus allows theuse of the operational semantics for calculating future behaviours.References[1] J. A. Bergstra et al. (ed.), Handbook of Process Algebra, Elsevier, 2001[2] L. Cardelli, Abstract Machines of Systems Biology, TCSB, III-3737, 2005[3] L. Cardelli, Brane Calculi - Interactions of Biological Membranes, In Proc. CMSB’04,LNCS 3082, 2005[4] G. Ciobanu and G. Rozenberg Ed., Modelling in Molecular Biology, Springer-Verlag,Berlin, 2004[5] S. Efroni et al., Toward rigorous comprehension of biological complexity: modeling, exe-cution, and visualization of thymic T-cell maturation, Genome Res., vol.13, 2003[6] N. Kam et al., Formal Modeling of C. elegans Development: A Scenario-Based Approach,LNCS-2602, 20036[7] P. Lecca and C. Priami, Cell cycle control in eukaryotes: a BioSpi model, In Proc. Bio-Concur’03, ENTCS, To appear[8] S. Marcus, Language at the crossroad of Computation and Biology, In G. Paun (ed.), Com-puting with Biomolecules; Theory and Experiment, Springer, 1998[9] S. Marcus, Membranes versus DNA, Fundamenta Informaticae, 49(1-3), 2002[10] R. Mardare, et. al., Model Checking Biological Systems Described Using Ambient Cal-culus, In Proc. CMSB’04, LNCS 3082, 2005[11] R. Mardare, C. Priami, Logical Analysis of Biological Systems, Fundamenta Informati-cae, 64(1-4), 2005[12] Gh. Pa˘un, Membrane Computing. An Introduction, Springer-Verlag, Berlin, NaturalComputing Series, 2002[13] A. Phillips and L. Cardelli, A correct abstract machine for the stochastic pi-calculus, InProc. BioConcur’04, ENTCS, To appear[14] A. Regev and E. Shapiro, Cells as Computation, Nature, vol.419, 2000[15] BetaWB simulator, http://www.cosbi.eu[16] Cyto-Sim simulator, http://www.cosbi.eu7

A Process Algebraical Approach to Modelling Compartmentalized Biological Systems

University of Trento : UNITN-eprints

Technical Report CoSBi 16/2007
A Process Algebraical Approach to Modelling
Compartmentalized Biological Systems
Radu Mardare
The Microsoft Research-University of Trento Centre for Computational and Systems Biology,
Trento, Italy
mardare@cosbi.eu
Adaoha Ihekwaba
The Microsoft Research-University of Trento Centre for Computational and Systems Biology,
Trento, Italy
ihekwaba@cosbi.eu
This is the preliminary version of a paper that will appear in
In Proceedings of International Conference of Computational Methods in Sciences and
Engineering (ICCMSE), American Institute of Physics, AIP Proceedings, N 2: 642-646, 2007.
A Process Algebraical Approach to Modelling
Compartmentalized Biological Systems
Radu Mardare, Adaoha Ihekwaba
The Microsoft Research-University of Trento, Italy
September 30, 2009
Abstract
This paper introduces Protein Calculus, a special modeling language designed for en-
coding and calculating the behaviors of compartmentilized biological systems. The for-
malism combines, in a unified framework, two successful computational paradigms - pro-
cess algebras and membrane systems. The goal of Protein Calculus is to provide a formal
tool for transforming collected information from in vivo experiments into coded definition
of the different types of proteins, complexes of proteins, and membrane-organized systems
of such entities. Using this encoded information as input, our calculus computes, in silico,
the possible behaviors of a living system.
1 The computational challenge in Systems Biology
Many of the activities of the biological cells are regulated by proteins which carry signals that
modify the expression of different genes at a given time, but how these signals do so is not
known. The use of Systems Biology is therefore essential if we are ever to make sense of the
biological complexity, as intuitive ’conceptual’models.
Systems Biology is a multi-disciplinary approach that studies the relationships and inter-
actions of various cellular mechanisms and cellular components. It encompases the growing
number of mechanistic but largely isolated insights and increasingly, high-throughput ’omics’
data sets which tends to be semi-quantitative and specific to its experimental system and inte-
grates them into a conceptual modelling framework that is holistic, quantitative and predictive.
By abstracting biological systems on the level of their behaviours, models are obtained shar-
ing many characteristics with computational systems. Consequently, on all levels of organisa-
tion (i.e. from the atomic to full organisms) concurrent behaviours - be it event-driven, causal,
time-dependent or distributed - is attained. Owing to these similarities, modelling (reverse-
engineering, simulating and analysing) a biological system has been considered in a similar
way to the engineering of complex artificial systems [2, 4]. This has resulted in questions such
as - Is the structure of a cell much more complex than the structure of a jet plane? - to be
raised. In answering the question, the challenge therefore is to construct, in silico, reliable
‘copies’ of biological systems that would allow the simulation of various experiments, which
1
will then assist in proposing and verifying hypotheses. Recent small-scale modelling efforts in
this direction have had encouraging results [14, 6, 5].
Traditionally, mathematical modelling of biological systems have always involved the use
of ordinary differential equations (ODEs) for the reason that they offer a description of net-
works as continuous time dynamical systems as well as the added advantage of being a mature
technique with many existing mathematical and computational resources available. However
in the context of this work - use of ODE is limited - due to the highly non-linear behaviours
exhibited by bio-systems, in addition to the lack of compositionality and systemic view of
ODE-based techniques. ODEs simply model the bio-system as a whole and compute its over-
all behaviour, making it impossible for the description of the model to be disassembled into its
constituent components. As a consequence, updating or extension of the model is not easily
done when new information on the system becomes available.
For obtaining reliable models in silico, we expect to be able to combine small-scale models
of molecular or cellular networks in large-scale models that might be used as valid images of
complex biological phenomena. We also expect it to be possible to integrate new information
into the models already developed. The need for small-scale modelling as a first step in design-
ing complex models emerges from the methods of Molecular Biology - typically a molecular
biologist spends years of research in studying a particular molecule. This information can be
eventually used for designing a model of that molecule, but for further analysis, the ability to
integrate such small models in order to define big scenarios as the behaviours of an organ or of
an organism is required. Therefore, if the behaviour of a molecule is described as a program
(possibly involving a high level of indeterminism) then by running the program in a concurrent
environment bigger scenarios can expect to be obtained.
Following this intuition, alternative approaches inspired by different paradigms from Com-
puter Science have been recently proposed to complement the ODE approach while ensuring
compositionality and extendability of the models. An example of such approach is process
algebra.
Process algebras [1] are developed for designing and analysing complex systems of agents
organised in a modular way and able to interact, collaborate, communicate, move and com-
pose/decompose composite agents. In this paradigm, agents are understood as spatially lo-
calised and independently observable units of behaviour and computation (e.g. programs or
processors running in parallel). They are successfully used in modelling distributed systems
like computer networks. Recently the paradigm has been demonstrated to be particularly ap-
propriate for modelling molecular and cellular networks [2, 14, 10, 11]. In this approach, a
biological system is modelled as a network of modules (subsystems) that run in a decentralised
manner, where they interact and change their relative positions which results in a global be-
haviour change of the systems generated. As a result, the overall model of the system can
easily be obtained by mixing the models of its components. What’s more, this paradigm was
adapted to include quantitative and stochastic information [7, 13] - which are essential features
of bio-chemical processes - and used together with stochastic simulators, built on the Gille-
spie method, for assisting in model validation and hypothesis forming. A stochastic simulator
uses data from actual measurements and computes the expected behaviour of the system. In
this manner, the high cost of running in vivo experiments is reduced, due to the derivation and
running of in silico models. This method has proved to be effective for systems on small-
2
scale levels, such as cellular systems and has been integrated in BetaWB [15], Cyto-Sim [16],
BioSPI [13], which are simulators that have been successfully used in modelling and analysing
signalling pathways.
Yet while in modelling biochemical soups, the straight concurrent environment is success-
ful, it is still not sufficient for describing the complex behaviour of living systems. Biologi-
cal systems involve compartmentalisation [3], which plays a major role in the functioning of
living systems. They are not just unstructured, heterogeneous chemical soups, but spatially
distributed hierarchies of bio-compartments organising and isolating chemical reactions and
their products. On the level of cellular and molecular systems we can identify membrane-
bound compartments including cells, organelles and vesicles each enclosed in a membrane,
which isolates the bio-chemical components of a compartment from the external environment.
These membranes organise a hierarchy of locations in the system. Then we have also molecu-
lar compartments, including stable or transient multi-molecular complexes, which insulate the
component molecules from the environment. All these compartments have the role of a key
regulatory mechanism for the biological system (in order to perform its function, a molecule
must be present in the right location), organising it in a modular way.
The biological compartments can be extremely dynamic. The evolution of a biological sys-
tem can involve movements of molecules between compartments (e.g. Golgi apparatus) and
dynamic rearrangements of cellular compartments and molecular complexes (i.e. molecules
may join a molecular compartment by binding to one of its members). There are also com-
partments that move, as a whole, with respect to the topology of the system. A typical exam-
ple in this respect is phagocytosis (a cell is engulfed by another cell, e.g. macrophages ‘eat’
viruses or contaminated cells) or entry of a complex molecule into an organelle. Sometimes
two membrane-bound compartments merge, forming one compartment. An approach known
as membrane computing would be adventageous.
Membrane computing [12] was introduced as a new paradigm in computer science aiming
to abstract computing models from the structure and the functioning of living cells. It focuses
on the idea of dividing the space of computation by means of “membranes”. Accordingly,
membrane systems are computational devices designed for modelling membrane-distributed
and parallel computational scenarios. The membrane has the role of a separator of the space
of computation. In between membranes there are distributed objects that can interact or move
from one membrane to others following some laws (similar with the bio-chemical laws). Thus,
membrane computing is a computational framework, inspired from biological reality, and de-
signed to model compartmentalized phenomena. It has been proved useful for applications in
computer science and essential in modelling biological systems [4, 8, 9] due to the role of the
compartments in generating complex behaviours.
2 Protein Calculus
In this report, we introduce Protein Calculus, a calculus that combines the process algebra
and membrane systems paradigms in a unified framework designed for modelling compart-
mentalized systems of molecules. The calculus focuses on bimolecular interactions between
molecules which are modelled as objects spatially organized by a network of membranes.
3
With respect to the membrane, a protein can be placed inside the area enclosed by the
membrane, outside it or on the membrane. A protein situated on a membrane can interact
with internal or external proteins, but the proteins enclosed in the inner space delimited by a
membrane cannot interact with outside proteins. The bimolecular interactions are of two types:
complexation and decomplexation. The complexation is the phenomenon of binding proteins
by means of their active sites. The result, a complex of proteins, which is a complex object that
can be further involved in bimolecular interactions. Decomplexation however is the reverse
phenomenon and consists of splitting a complex of proteins into the constituent parts. The
following section gives a description of the syntax used.
2.1 Syntax
The syntax of Protein Calculus is presented by a grammar with non-terminals {P,C, S} and is
shown in Table 1 - where a desired system is represented by two disjoint alphabets, Ω and Σ.
For the system, the elements of Ω is used to name proteins (i.e. each element a ∈ Ω appears in
the syntax of a protein to identify it), whilst the elements of Σ is used for the interaction sites
of the protein. With this in mind - α.a : β.a -denotes a protein “a” with two active sites α and
β. Note - that the operator “:” is used to express the coexistence of different active sites.
The same operator can also be used to represent complexes of proteins. For instance,
α.b : β.c : γ.c represents a complex formed by two proteins b and c, where, after complexation,
b has only one active site α whilst c has two active sites, β and γ. By using these sites the
complex can be further involved in other complexations.
However, we know that proteins and complexes tend to float in the same compartment /
soup, in making allowances for this - the operator “‖” is used. In other words, in a given
system, when protein α.a : β.a shares the same solution with complex α.b : β.c : γ.c, the
solution α.a : β.a ‖ α.b : β.c : γ.c is given.
Anywhere structure SM is given, this will be used to express a membrane enclosed system,
where the fraction line represents a membrane and encodes the way in which the space is
separated. The structure below the fraction line, M , represents the structure enclosed by the
membrane, while the structure above the fraction line, S, represents the soup of molecules that
floats on the membrane.
The use of the fraction line for representing the division of space has the following advan-
tage; in that SM ‖M ′ can be used to represent a cell having the membrane composed by S, the
content M and the environment in which it evolves M ′. Obviously M and M ′ can be further
described as composed by other membranes.
P ::= a | α.P | P : P a ∈ Ω, α ∈ Σ (proteins)
C ::= P | C.αC | C : C α ∈ Σ (complexes)
S ::= C | S ‖ S (soups)
M ::= S | SM |M ‖M (membrane systems)
Table 1: Syntax of Protein Calculus
Consider, for example, the structure in Figure1. We have a membrane system composed by
4
a main membrane that enclose other four membranes between which the fourth one encloses a
fifth membrane. On the membrane and between membranes we have proteins and complexes
of proteins. Using our syntax we can represent this system with the following equation
A ‖ U ‖ (A : B : D) ‖ B ‖ A
V
N
‖ F ‖ G
F : E
‖ L ‖ H ‖ D
(A : B) ‖ K ‖
M ‖ A ‖ T
(V : Q : R) ‖ N ‖ DE
.
In the above representationA,B,D,E, F,G,H,K,L,N,Q,R, U, V are uninterpreted proteins
that can be further defined using our syntax, and (A : B : D), (F : E), (A : B) and
(V : Q : R) are complexes of proteins.
This syntax therefore, could be used to describe a membrane system in any state. But you
ask what of the behaviour? The operational semantics of our calculus will be used to compute
this.
2.2 Operational semantics
The behaviour of a membrane system is encoded by some basic axioms shown in Table2.
“φ −→ φ′” is used to mark the evolution of the system from a state φ to the state φ′ in
any contextual situation, while “φ =⇒ φ′” marks the evolution governed by side conditions
yet to be defined in the lab. (C1) is a complexation axiom that establishes the type of pro-
tein or complexes that binds when floating in the same environment, while (D1) is the cor-
responding decomplexation. Similarly (C1) and (C2) dually with (D1), (D2) are complexa-
tion/decomplexation axioms, but are used to represent conditions of bimolecular interactions
case where one of the molecules is anchored on the membrane and the other is floating freely
in the inner respectively outer regions. Note - that (D1) and (D2) have identical premises,
but their products differ depending on some contextual conditions to be defined in the lab for
particular systems of proteins.
(C1) : α.p ‖ α.p′ ‖ s −→ p.αp′ : p′.αp ‖ s (D1) : p.αp′ : p′.αp ‖ s −→ α.p ‖ α.p′ ‖ s
(C2) :
α.p ‖ s
α.p′ ‖ m −→
p.αp′ : p
′.αp ‖ s
m (D2) :
p.αp′ : p
′.αp ‖ s
m =⇒
α.p ‖ s
α.p′ ‖ m
(C3) :
α.p ‖ s
m ‖ α.p′ −→
p.αp′ : p
′.αp ‖ s
m (D3) :
p.αp′ : p
′.αp ‖ s
m =⇒
α.p ‖ s
m ‖ α.p′
(S) :
s ‖ s′
m ‖ m′ =⇒
s
m ‖ s
′
m′
(J) : sm ‖ s
′
m′
=⇒ s ‖ s
′
m ‖ m′
(P ) :
s ‖ s′
m ‖ m′ =⇒ s
m ‖ s
′
m′
(E) : s
m ‖ s
′
m′
=⇒ s ‖ s
′
m ‖ m′
Table 2: Operational semantics
(S) and (J) are the split and join axioms that model the division of a cell or merging of two
membranes. (P ) and (E) encodes the phagocitosys and exocitosys phenomena respectively.
5
UA
A
B
V
E
G
D
H
D
M
A
T
F
L
N
N
K
A:B:D
F:E
A:B
V:Q:R
Figure 1: A membrane system
3 Concluding remarks
Process calculus is an invaluable technique that can be used for describing the molecular sys-
tems and their behaviours; for instance in modelling trans-membrane interactions (i.e. protein
movement between cytoplasm and the nucleus) - examples of which include the translocation
of the NF-kappa B proteins into the nucleus. Protein Calculus is particularly useful in trans-
forming collected information from in vivo experiments into coded definition of the different
types of proteins and complexes of proteins, and establishing positions of the coded objects
with respect to the membranes of the system. In addition, the syntax of the calculus allows the
use of the operational semantics for calculating future behaviours.
References
[1] J. A. Bergstra et al. (ed.), Handbook of Process Algebra, Elsevier, 2001
[2] L. Cardelli, Abstract Machines of Systems Biology, TCSB, III-3737, 2005
[3] L. Cardelli, Brane Calculi - Interactions of Biological Membranes, In Proc. CMSB’04,
LNCS 3082, 2005
[4] G. Ciobanu and G. Rozenberg Ed., Modelling in Molecular Biology, Springer-Verlag,
Berlin, 2004
[5] S. Efroni et al., Toward rigorous comprehension of biological complexity: modeling, exe-
cution, and visualization of thymic T-cell maturation, Genome Res., vol.13, 2003
[6] N. Kam et al., Formal Modeling of C. elegans Development: A Scenario-Based Approach,
LNCS-2602, 2003
6
[7] P. Lecca and C. Priami, Cell cycle control in eukaryotes: a BioSpi model, In Proc. Bio-
Concur’03, ENTCS, To appear
[8] S. Marcus, Language at the crossroad of Computation and Biology, In G. Paun (ed.), Com-
puting with Biomolecules; Theory and Experiment, Springer, 1998
[9] S. Marcus, Membranes versus DNA, Fundamenta Informaticae, 49(1-3), 2002
[10] R. Mardare, et. al., Model Checking Biological Systems Described Using Ambient Cal-
culus, In Proc. CMSB’04, LNCS 3082, 2005
[11] R. Mardare, C. Priami, Logical Analysis of Biological Systems, Fundamenta Informati-
cae, 64(1-4), 2005
[12] Gh. Pa˘un, Membrane Computing. An Introduction, Springer-Verlag, Berlin, Natural
Computing Series, 2002
[13] A. Phillips and L. Cardelli, A correct abstract machine for the stochastic pi-calculus, In
Proc. BioConcur’04, ENTCS, To appear
[14] A. Regev and E. Shapiro, Cells as Computation, Nature, vol.419, 2000
[15] BetaWB simulator, http://www.cosbi.eu
[16] Cyto-Sim simulator, http://www.cosbi.eu
7


Radu Mardare

Adaoha Ihekwaba

CiteSeerX

Technical Report CoSBi 16/2007
A Process Algebraical Approach to Modelling
Compartmentalized Biological Systems
Radu Mardare
The Microsoft Research-University of Trento Centre for Computational and Systems Biology,
Trento, Italy
mardare@cosbi.eu
Adaoha Ihekwaba
The Microsoft Research-University of Trento Centre for Computational and Systems Biology,
Trento, Italy
ihekwaba@cosbi.eu
This is the preliminary version of a paper that will appear in
In Proceedings of International Conference of Computational Methods in Sciences and
Engineering (ICCMSE), American Institute of Physics, AIP Proceedings, N 2: 642-646, 2007.
A Process Algebraical Approach to Modelling
Compartmentalized Biological Systems
Radu Mardare, Adaoha Ihekwaba
The Microsoft Research-University of Trento, Italy
September 30, 2009
Abstract
This paper introduces Protein Calculus, a special modeling language designed for en-
coding and calculating the behaviors of compartmentilized biological systems. The for-
malism combines, in a unified framework, two successful computational paradigms - pro-
cess algebras and membrane systems. The goal of Protein Calculus is to provide a formal
tool for transforming collected information from in vivo experiments into coded definition
of the different types of proteins, complexes of proteins, and membrane-organized systems
of such entities. Using this encoded information as input, our calculus computes, in silico,
the possible behaviors of a living system.
1 The computational challenge in Systems Biology
Many of the activities of the biological cells are regulated by proteins which carry signals that
modify the expression of different genes at a given time, but how these signals do so is not
known. The use of Systems Biology is therefore essential if we are ever to make sense of the
biological complexity, as intuitive ’conceptual’models.
Systems Biology is a multi-disciplinary approach that studies the relationships and inter-
actions of various cellular mechanisms and cellular components. It encompases the growing
number of mechanistic but largely isolated insights and increasingly, high-throughput ’omics’
data sets which tends to be semi-quantitative and specific to its experimental system and inte-
grates them into a conceptual modelling framework that is holistic, quantitative and predictive.
By abstracting biological systems on the level of their behaviours, models are obtained shar-
ing many characteristics with computational systems. Consequently, on all levels of organisa-
tion (i.e. from the atomic to full organisms) concurrent behaviours - be it event-driven, causal,
time-dependent or distributed - is attained. Owing to these similarities, modelling (reverse-
engineering, simulating and analysing) a biological system has been considered in a similar
way to the engineering of complex artificial systems [2, 4]. This has resulted in questions such
as - Is the structure of a cell much more complex than the structure of a jet plane? - to be
raised. In answering the question, the challenge therefore is to construct, in silico, reliable
‘copies’ of biological systems that would allow the simulation of various experiments, which
1
will then assist in proposing and verifying hypotheses. Recent small-scale modelling efforts in
this direction have had encouraging results [14, 6, 5].
Traditionally, mathematical modelling of biological systems have always involved the use
of ordinary differential equations (ODEs) for the reason that they offer a description of net-
works as continuous time dynamical systems as well as the added advantage of being a mature
technique with many existing mathematical and computational resources available. However
in the context of this work - use of ODE is limited - due to the highly non-linear behaviours
exhibited by bio-systems, in addition to the lack of compositionality and systemic view of
ODE-based techniques. ODEs simply model the bio-system as a whole and compute its over-
all behaviour, making it impossible for the description of the model to be disassembled into its
constituent components. As a consequence, updating or extension of the model is not easily
done when new information on the system becomes available.
For obtaining reliable models in silico, we expect to be able to combine small-scale models
of molecular or cellular networks in large-scale models that might be used as valid images of
complex biological phenomena. We also expect it to be possible to integrate new information
into the models already developed. The need for small-scale modelling as a first step in design-
ing complex models emerges from the methods of Molecular Biology - typically a molecular
biologist spends years of research in studying a particular molecule. This information can be
eventually used for designing a model of that molecule, but for further analysis, the ability to
integrate such small models in order to define big scenarios as the behaviours of an organ or of
an organism is required. Therefore, if the behaviour of a molecule is described as a program
(possibly involving a high level of indeterminism) then by running the program in a concurrent
environment bigger scenarios can expect to be obtained.
Following this intuition, alternative approaches inspired by different paradigms from Com-
puter Science have been recently proposed to complement the ODE approach while ensuring
compositionality and extendability of the models. An example of such approach is process
algebra.
Process algebras [1] are developed for designing and analysing complex systems of agents
organised in a modular way and able to interact, collaborate, communicate, move and com-
pose/decompose composite agents. In this paradigm, agents are understood as spatially lo-
calised and independently observable units of behaviour and computation (e.g. programs or
processors running in parallel). They are successfully used in modelling distributed systems
like computer networks. Recently the paradigm has been demonstrated to be particularly ap-
propriate for modelling molecular and cellular networks [2, 14, 10, 11]. In this approach, a
biological system is modelled as a network of modules (subsystems) that run in a decentralised
manner, where they interact and change their relative positions which results in a global be-
haviour change of the systems generated. As a result, the overall model of the system can
easily be obtained by mixing the models of its components. What’s more, this paradigm was
adapted to include quantitative and stochastic information [7, 13] - which are essential features
of bio-chemical processes - and used together with stochastic simulators, built on the Gille-
spie method, for assisting in model validation and hypothesis forming. A stochastic simulator
uses data from actual measurements and computes the expected behaviour of the system. In
this manner, the high cost of running in vivo experiments is reduced, due to the derivation and
running of in silico models. This method has proved to be effective for systems on small-
2
scale levels, such as cellular systems and has been integrated in BetaWB [15], Cyto-Sim [16],
BioSPI [13], which are simulators that have been successfully used in modelling and analysing
signalling pathways.
Yet while in modelling biochemical soups, the straight concurrent environment is success-
ful, it is still not sufficient for describing the complex behaviour of living systems. Biologi-
cal systems involve compartmentalisation [3], which plays a major role in the functioning of
living systems. They are not just unstructured, heterogeneous chemical soups, but spatially
distributed hierarchies of bio-compartments organising and isolating chemical reactions and
their products. On the level of cellular and molecular systems we can identify membrane-
bound compartments including cells, organelles and vesicles each enclosed in a membrane,
which isolates the bio-chemical components of a compartment from the external environment.
These membranes organise a hierarchy of locations in the system. Then we have also molecu-
lar compartments, including stable or transient multi-molecular complexes, which insulate the
component molecules from the environment. All these compartments have the role of a key
regulatory mechanism for the biological system (in order to perform its function, a molecule
must be present in the right location), organising it in a modular way.
The biological compartments can be extremely dynamic. The evolution of a biological sys-
tem can involve movements of molecules between compartments (e.g. Golgi apparatus) and
dynamic rearrangements of cellular compartments and molecular complexes (i.e. molecules
may join a molecular compartment by binding to one of its members). There are also com-
partments that move, as a whole, with respect to the topology of the system. A typical exam-
ple in this respect is phagocytosis (a cell is engulfed by another cell, e.g. macrophages ‘eat’
viruses or contaminated cells) or entry of a complex molecule into an organelle. Sometimes
two membrane-bound compartments merge, forming one compartment. An approach known
as membrane computing would be adventageous.
Membrane computing [12] was introduced as a new paradigm in computer science aiming
to abstract computing models from the structure and the functioning of living cells. It focuses
on the idea of dividing the space of computation by means of “membranes”. Accordingly,
membrane systems are computational devices designed for modelling membrane-distributed
and parallel computational scenarios. The membrane has the role of a separator of the space
of computation. In between membranes there are distributed objects that can interact or move
from one membrane to others following some laws (similar with the bio-chemical laws). Thus,
membrane computing is a computational framework, inspired from biological reality, and de-
signed to model compartmentalized phenomena. It has been proved useful for applications in
computer science and essential in modelling biological systems [4, 8, 9] due to the role of the
compartments in generating complex behaviours.
2 Protein Calculus
In this report, we introduce Protein Calculus, a calculus that combines the process algebra
and membrane systems paradigms in a unified framework designed for modelling compart-
mentalized systems of molecules. The calculus focuses on bimolecular interactions between
molecules which are modelled as objects spatially organized by a network of membranes.
3
With respect to the membrane, a protein can be placed inside the area enclosed by the
membrane, outside it or on the membrane. A protein situated on a membrane can interact
with internal or external proteins, but the proteins enclosed in the inner space delimited by a
membrane cannot interact with outside proteins. The bimolecular interactions are of two types:
complexation and decomplexation. The complexation is the phenomenon of binding proteins
by means of their active sites. The result, a complex of proteins, which is a complex object that
can be further involved in bimolecular interactions. Decomplexation however is the reverse
phenomenon and consists of splitting a complex of proteins into the constituent parts. The
following section gives a description of the syntax used.
2.1 Syntax
The syntax of Protein Calculus is presented by a grammar with non-terminals {P,C, S} and is
shown in Table 1 - where a desired system is represented by two disjoint alphabets, Ω and Σ.
For the system, the elements of Ω is used to name proteins (i.e. each element a ∈ Ω appears in
the syntax of a protein to identify it), whilst the elements of Σ is used for the interaction sites
of the protein. With this in mind - α.a : β.a -denotes a protein “a” with two active sites α and
β. Note - that the operator “:” is used to express the coexistence of different active sites.
The same operator can also be used to represent complexes of proteins. For instance,
α.b : β.c : γ.c represents a complex formed by two proteins b and c, where, after complexation,
b has only one active site α whilst c has two active sites, β and γ. By using these sites the
complex can be further involved in other complexations.
However, we know that proteins and complexes tend to float in the same compartment /
soup, in making allowances for this - the operator “‖” is used. In other words, in a given
system, when protein α.a : β.a shares the same solution with complex α.b : β.c : γ.c, the
solution α.a : β.a ‖ α.b : β.c : γ.c is given.
Anywhere structure SM is given, this will be used to express a membrane enclosed system,
where the fraction line represents a membrane and encodes the way in which the space is
separated. The structure below the fraction line, M , represents the structure enclosed by the
membrane, while the structure above the fraction line, S, represents the soup of molecules that
floats on the membrane.
The use of the fraction line for representing the division of space has the following advan-
tage; in that SM ‖M
′ can be used to represent a cell having the membrane composed by S, the
content M and the environment in which it evolves M ′. Obviously M and M ′ can be further
described as composed by other membranes.
P ::= a | α.P | P : P a ∈ Ω, α ∈ Σ (proteins)
C ::= P | C.αC | C : C α ∈ Σ (complexes)
S ::= C | S ‖ S (soups)
M ::= S | SM |M ‖M (membrane systems)
Table 1: Syntax of Protein Calculus
Consider, for example, the structure in Figure1. We have a membrane system composed by
4
a main membrane that enclose other four membranes between which the fourth one encloses a
fifth membrane. On the membrane and between membranes we have proteins and complexes
of proteins. Using our syntax we can represent this system with the following equation
A ‖ U ‖ (A : B : D) ‖ B ‖ A
V
N
‖ F ‖ G
F : E
‖ L ‖ H ‖ D
(A : B) ‖ K
‖ M ‖ A ‖ T
(V : Q : R) ‖ N ‖ DE
.
In the above representationA,B,D,E, F,G,H,K,L,N,Q,R, U, V are uninterpreted proteins
that can be further defined using our syntax, and (A : B : D), (F : E), (A : B) and
(V : Q : R) are complexes of proteins.
This syntax therefore, could be used to describe a membrane system in any state. But you
ask what of the behaviour? The operational semantics of our calculus will be used to compute
this.
2.2 Operational semantics
The behaviour of a membrane system is encoded by some basic axioms shown in Table2.
“φ −→ φ′” is used to mark the evolution of the system from a state φ to the state φ′ in
any contextual situation, while “φ =⇒ φ′” marks the evolution governed by side conditions
yet to be defined in the lab. (C1) is a complexation axiom that establishes the type of pro-
tein or complexes that binds when floating in the same environment, while (D1) is the cor-
responding decomplexation. Similarly (C1) and (C2) dually with (D1), (D2) are complexa-
tion/decomplexation axioms, but are used to represent conditions of bimolecular interactions
case where one of the molecules is anchored on the membrane and the other is floating freely
in the inner respectively outer regions. Note - that (D1) and (D2) have identical premises,
but their products differ depending on some contextual conditions to be defined in the lab for
particular systems of proteins.
(C1) : α.p ‖ α.p′ ‖ s −→ p.αp′ : p′.αp ‖ s (D1) : p.αp′ : p′.αp ‖ s −→ α.p ‖ α.p′ ‖ s
(C2) :
α.p ‖ s
α.p′ ‖ m −→
p.αp′ : p
′.αp ‖ s
m (D2) :
p.αp′ : p
′.αp ‖ s
m =⇒
α.p ‖ s
α.p′ ‖ m
(C3) :
α.p ‖ s
m ‖ α.p
′ −→ p.αp′ : p
′.αp ‖ s
m (D3) :
p.αp′ : p
′.αp ‖ s
m =⇒
α.p ‖ s
m ‖ α.p
′
(S) :
s ‖ s′
m ‖ m′ =⇒
s
m ‖
s′
m′
(J) : sm ‖
s′
m′
=⇒ s ‖ s
′
m ‖ m′
(P ) :
s ‖ s′
m ‖ m
′ =⇒ s
m ‖ s
′
m′
(E) : s
m ‖ s
′
m′
=⇒ s ‖ s
′
m ‖ m
′
Table 2: Operational semantics
(S) and (J) are the split and join axioms that model the division of a cell or merging of two
membranes. (P ) and (E) encodes the phagocitosys and exocitosys phenomena respectively.
5
U
A
A
B
V
E
G
D
H
D
M
A
T
F
L
N
N
K
A:B:D
F:E
A:B
V:Q:R
Figure 1: A membrane system
3 Concluding remarks
Process calculus is an invaluable technique that can be used for describing the molecular sys-
tems and their behaviours; for instance in modelling trans-membrane interactions (i.e. protein
movement between cytoplasm and the nucleus) - examples of which include the translocation
of the NF-kappa B proteins into the nucleus. Protein Calculus is particularly useful in trans-
forming collected information from in vivo experiments into coded definition of the different
types of proteins and complexes of proteins, and establishing positions of the coded objects
with respect to the membranes of the system. In addition, the syntax of the calculus allows the
use of the operational semantics for calculating future behaviours.
References
[1] J. A. Bergstra et al. (ed.), Handbook of Process Algebra, Elsevier, 2001
[2] L. Cardelli, Abstract Machines of Systems Biology, TCSB, III-3737, 2005
[3] L. Cardelli, Brane Calculi - Interactions of Biological Membranes, In Proc. CMSB’04,
LNCS 3082, 2005
[4] G. Ciobanu and G. Rozenberg Ed., Modelling in Molecular Biology, Springer-Verlag,
Berlin, 2004
[5] S. Efroni et al., Toward rigorous comprehension of biological complexity: modeling, exe-
cution, and visualization of thymic T-cell maturation, Genome Res., vol.13, 2003
[6] N. Kam et al., Formal Modeling of C. elegans Development: A Scenario-Based Approach,
LNCS-2602, 2003
6
[7] P. Lecca and C. Priami, Cell cycle control in eukaryotes: a BioSpi model, In Proc. Bio-
Concur’03, ENTCS, To appear
[8] S. Marcus, Language at the crossroad of Computation and Biology, In G. Paun (ed.), Com-
puting with Biomolecules; Theory and Experiment, Springer, 1998
[9] S. Marcus, Membranes versus DNA, Fundamenta Informaticae, 49(1-3), 2002
[10] R. Mardare, et. al., Model Checking Biological Systems Described Using Ambient Cal-
culus, In Proc. CMSB’04, LNCS 3082, 2005
[11] R. Mardare, C. Priami, Logical Analysis of Biological Systems, Fundamenta Informati-
cae, 64(1-4), 2005
[12] Gh. Păun, Membrane Computing. An Introduction, Springer-Verlag, Berlin, Natural
Computing Series, 2002
[13] A. Phillips and L. Cardelli, A correct abstract machine for the stochastic pi-calculus, In
Proc. BioConcur’04, ENTCS, To appear
[14] A. Regev and E. Shapiro, Cells as Computation, Nature, vol.419, 2000
[15] BetaWB simulator, http://www.cosbi.eu
[16] Cyto-Sim simulator, http://www.cosbi.eu
7


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