Process Calculi for Biological Processes

Transcription

1 a joint work with A. Bernini, P. Degano, D. Hermith, M. Falaschi to appear in Natural Computing, Springer-Verlag, 2018 Siena, May 2018

2 Outline 1 Specification languages: Process Algebras 2 Biology 3 Modelling Systems Biology 4 Evolution of the Process Algebras for Systems Biology 5 Case studies

3 Process Algebras... introduced to model systems of communicating agents (computers). in Agent out in a Agent outb Communication channels in a outb in b A1 A2 inc A1 and A2 can communicate on channel b A2 and A3 can communicate on channel c A3 outc

4 Process Algebras: basic actions input a x a channel - x variable output ab a,b channels (channels as names) A1 and A2 share a common communication channel communication between A1 and A2 A1 P A2 Q[b/x] after communication A2 gets name b

5 Process algebras: basic operators sequential non deterministic choice parallel new name generator a x.p P+Q P Q (ν c).p Specification models are built from the composition of basic actions and operators; new actions and operators can be introduced to closely describe different system rules (and structures).

6 Operational semantics Semantic rules formalized as inference rules. The rule for communication: A 1 ab A 1, A a b 2 Q[b/x] τ A 1 A 2 A 1 (Q[b/x]) A1 A2 A1' A2' a b.p a<x>.q τ P Q[b/x]

7 Forcing interactions (communications)... A 1 and A 2 can also proceed asynchronously: A 1 = ab.p ab P A 2 = a x.q a<x> Q If you want to prevent this, the new operator (ν) should be applied on the channel name: (νa)(ab.p a x.q) τ (P Q[b/x])

8 bibliography Communication and Concurrency, Robin Milner Prentice-Hall Process algebra CCS Communicating and Concurrent Systems actions (input) & co-actions (output)

9 Some words on Biology How to classify life: Prokariotes: are single-celled organisms where the DNA is not contained within a membrane Eukariotes: show a well-defined nucleus surrounded by a membranous nuclear envelope. Prokarya and Eukarya both share the same genetic material, the biomolecule DeoxyriboNucleic Acid (DNA) that acts as the carrier of genetic information. Eukarya include both unicellular and pluricellular organisms: protista, fungi, plantae, and animalia. observation This kind of classification does not help us too much!

10 Some words on Biology (different areas) Molecular Biology: concerns the study of nucleic acids (DNA and RNA), polysaccharides (starches and cellulose, made of sugars), lipids (constituting the membranes, usually in the form of bilayers), water, and proteins. Genetics: deals with genes, genetic variations and heredity. How DNA stores an encoding of the traits of an organism, it is therefore strictly linked to information systems Cytology: studies structure and function of cell that is organized as an outer cytoplasmic membrane containing molecules, organelles and other structures (the cytoplasm). Biochemistry: studies how living organisms cooperate by mean of chemical reactions. Biochemical processes are both responsible for the flow of chemical energy (the metabolism) and for molecular signaling. In living matter, chemical reactions occur continuously within cells and build molecules that are crucial for the functioning of a cell, by breaking atomic bonds within a molecule and generating new ones.

11 Biochemistry reactions Chemical behaviour Different kinds of chemical reactions are occurring simultaneously. Both the extent to which chemical reactions can proceed and the kinetic rate at which they take place determines the chemical composition of cells. reactants products it is specified the quantities of each species to make the reaction possible, and the quantities of the resulting products: A + B 2C

12 Elementary reactions Reversible reaction, and, with rates k 1 and k 2 : A + B k 1 k2 products, Caveat a rate represent the instant velocity of a kind of reaction. molecules of species A and B mix together the actual reaction rate is computed as rate = k 1 [A] [B] [A] and [B] express the actual concentrations of elements A and B. The kinetic rate associated with each reaction depends on many physical factors of the systems, such as the temperature, the volume occupied by the whole system, etc..

13 Reaching an equilibrium When a biochemical reaction occurs reactants transform themselves into products, their concentration decreases and so does the forward reaction rate eventually, the reaction rates of the forward and reverse reactions become equal, the concentrations of reactants and products stop changing. the system is then said to be in chemical equilibrium. Equilibrium reactions are crucial for biological systems as they allow for a process rate and direction to be tuned according to the metabolic activities of the cell (respiration, synthesis, degra- dation,...).

14 Pathways inside cells most reactions are linked in pathways a product of one reaction serves as a reactant in another regulation of the reaction rates is therefore crucial it is usually implemented as a feedback inhibition: the raise in concentration of the final product makes the reaction rate of a previous step to slow down

15 Biochemical networks Biochemical reactions are numerous, interconnected and cross-regulated in the biochemical networks. Metabolic networks describe proteins, genes, metabolites, reactions, etc., based on the way in which the matter flows in cells Regulatory networks the focus is on the different controls (catalysis of a reaction, regulation of the gene expression, inhibition, etc.) that are coupled and that affect the expression of the information within cells Signaling pathways are focused on the flow of biological information in the cell and with its environment, i.e., how cells interact with the surrounding environment in different domains and levels

16 Stoichiometric equations Consider again the equation: A + B 2C reactants and products are balanced (law of conservation of mass): it need 1 unit of A and 1 of B, that will disappear, and 2 quantities of C are generated. The quantity of the whole molecules is balanced. What can be modeled a wide variety of biological entities molecules, membranes, cells and also... populations, then ecosystems.

17 Mathematical modelling In the traditional approach, a biochemical system is represented by a set of ordinary differential equations (ODEs). As an example, if we have the stoichiometric equation: X + Y k1 k2 Y, then, its corresponding ODE system is: dx dt = k 1xy + k 2 z dy dt = k 1xy + k 2 z dz dt = k 1xy k 2 z where x, y, z are the concentrations of species X, Y, Z. The solution of the differential equations provides the average concentration of each species as a time function. When kinetic data is available, these equations can be solved via numerical integration.

18 Alternative to ODEs hypothesis When biological elements are present in small numbers in a cell (e.g., genes, mrnas, signalling and regulatory proteins, etc.) seems that significant stochastic fluctuations take place. In these conditions, approach of the Chemical Master Equation (CME) represents an alternative to the classical deterministic ODE approach for modelling nonlinear biochemical reaction networks. CME Is a differential equation system describing the time evolution of the probability distribution over the possible states of a biochemical system. Its solution gives the probability for a biochemical reaction network to be in a certain state at time t.

19 ODEs vs. CME Important to note In ODEs the reaction rate represents the instantaneous velocity of a reaction and it is expressed as a rate constant, when it is known. In the stochastic model the reaction rate is derived by the rate constant and represents the probability that the associated reaction will happen in the next time unit, i.e., it represents the reaction propensity.

20 Drawback of CME solution of CME Unfortunately, CME can be rarely solved analytically, especially for large systems, while numerical solutions suffer from the curse-of-dimensionality and therefore need to be approximated. Alternative Developing discrete stochastic models for biological systems. The evolution of a system is driven by a stochastic algorithm that computes the probability of state transitions according to a given probability density function. Each state records the concentration of each species.... S1 S2 S3...

21 Gillespie s Stochastic Simulation Algorithm (SSA) It is the most widespread algorithm used for implementing discrete stochastic models of biological systems based on the CME (Gillespie 1976) As any stochastic simulation, a computed trajectory represents an (accurate) approximation of the continuous-time Markov chain described by the CME. Gillespie and other authors have also contributed to optimizations of the SSA (Gibson and Bruck 2000; Cao et al. 2004; Thanh et al. 2017)

22 Optimizations for particular cases If a prespecified finite number of molecules may be spontaneously created, it is possible to only consider the states reachable from a given initial state (Cao and Liang 2008; Cao et al. 2010) The finite state projection introduced by Munsky and Khammash (2006) and the sliding window abstraction of Henzinger et al. (2009) compute approximations of the solution of CME rather than solving a truncated version of the original Markov process. The moment closure defined in Gómez-Uribe and Verghese (2007), the linear noise approximation of Kampen (2007), chemical Langevin equation by Gillespie (2000), replace the discrete description of the population counts with a continuous one. however these methods perform poorly when small population species are in the system.

23 Stochastic process algebras molecule-as-computation abstraction A system of interacting molecules is described and modelled by a system of interacting computational processes. Abstract computer languages can simulate the behavior of biomolecular systems, supporting qualitative and quanti- tative reasoning on these systems properties. Regev, A. and Shapiro, E.. Cellular abstractions: cells as computation. Nature, Regev, A. and Shapiro, E.. The π-calculus as an abstraction for biomolecular systems. Modelling in Molecular Biology

24 The idea Process calculi focus on the visible behaviour (i.e., of interest to observers) of the components of the system. For example, let consider: two molecules of different types that participate in the formation and the breakage of a bimolecular complex. each molecule is e represented as a process, Mol 1 and Mol 2, whose behaviour is specified as CCS processes.

25 modelling molecules with CCS Formal description of the system: Mol 1 bind; BMol 1 BMol 1 unbind; Mol 1 Mol 2 bind; BMol 2 BMol 2 unbind; Mol 2 The whole system: System Mol 1 Mol 2 The computation: System τ BMol 1 BMol 2 τ System Mol 1 Mol 2 τ BMol1 BMol 2 τ Mol1 Mol 2

26 Non determinism where System 1 Mol 1 Mol 2 Mol 3 Mol 1 bind 2 ; unbind 2 ; Mol 1 + bind 3 ; unbind 3 ; Mol 1 Mol 2 bind 2 ; unbind 2 ; Mol 2 Mol 3 bind 3 ; unbind 3 ; Mol 3 three processes are forced to interact on channels bind 2, and bind 3. The process Mol 1 can non-deterministically interact either on channel bind 2 to form a complex with Mol 2 or on channel bind 3 with Mol 3.

27 species-as-processes execution of stochastic system may also be governed by algorithms such as the Gillespie s Stochastic Simulation Algorithm (SSA) X (t) is a vector of the quantities of n species in the system X (t) = (x 1 (t),..., x n (t)), x i (t) is the number of elements of species S i at time t the reaction j is represented by action j, with an associated rate constant r j, derived by biological observations. unimolecular reaction/action j has propensity a j (X ) = r j x i x i is the number of elements of the species S i r j a bimolecular reaction j : S i + S h Products, has propensity aj (X ) = r j x i x h, where x i, and x h are the number of elements of the species S i and S h

28 selecting the next reaction probability Gillespie has shown that the P(t, j) = a j (X ) a 0(X )t is the probability P(t, j)δt that the reaction j will be the next one and will happen in the system within the time interval [t 0 + t, t 0 + t + δt]. t 0 is the current time a 0 (X ) = M k=1 a k(x ) stands for the sum of the probabilities of all possible reactions. The number of reactions M is derived by biological observations, and is part of our knowledge of the considered system.

29 stochastic simulation SSA consists of generating stochastic realisations of the probability function thus identifying the next reaction j to be executed and its time duration, updating the number of molecules and the time instant, recomputing the values of all a k (X ) and a 0 (X ). This iterative step is repeated until the reagents finish or a predetermined maximum number of iterations is reached. j... P(t,j) k h i S1 S2 S3 P(t',k) P(t'',h) P(t''',i)

30 our example Enriching the model with stochastic values: (bind 1, r 1 ), (bind 2, r2), (unbind 1, r 3 ), (unbind 2, r 4 ) also, the number of copies for the three types of molecules have to be considered :[Mol i ], for i {1, 2, 3}. (bind1,r1) (bind1,r1) (bind2,r2) Mol1 Mol2 Mol3 (bind2,r2) The synchronisation between Mol 1 and Mol 2 on channel bind 2 has propensity a bind2 (X ) = r 1 [Mol 1 ] [Mol 2 ] to be executed, where X = ([Mol 1 ], [Mol 2 ], [Mol 3 ])

31 until now, only CCS exchanging and sharing some biological information between the partners of a reaction The π-calculus Mol 1 (bind) (ν bound) bind bound ; BMol 1 Mol 2 (bind) bind(x); BMol 2 Sys (ν bind) ( Mol 1 (bind) Mol 1 (bind) Mol 2 (bind) Mol 2 (bind) ) τ Sys (ν bind)(bmol 1 [bound/y] Mol 1 (bind) BMol 2 [bound/x] Mol 2 (bind)) τ (ν bind)(bmol 1 [bound/y] BMol 1 [bound /y] BMol 2 [bound/x] BMol 2 [bound /x])

32 another example APK phosph Tyr ; Kinase TBD phosph(x); x b ; TBD APK TBD τ Kinase Tyr b ; TBD The molecule TBD is shaded to show its new phosphorylated state. After phosphorization, TBD can use the Tyr channel and send the message b on it.

33 implementation of π-calculus the π-calculus Priami, C., Regev, A., Shapiro, E. and Silvermann, W..Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Theor. Comput. Sci its implementation the Stochastic Pi Machine (SPiM) Cardelli, L. and Phillips, A..Spim: Stochastic pi machine microsoft.com/en-us/research/project/ stochastic-pi-machine/ Paulevè, L., Youssef, S., Lakin, M. and Phillips, A., A generic abstract machine for stochastic process calculi, in CMSB 10, Phillips, A., A visual process calculus for biology, Symbolic Systems Biology: Theory and Methods

34 dealing with membranes environments The membrane of a cell acts as a border between the cytoplasm and the external environment and can also host some biological entities. The Ambient Calculus deals with environments Cardelli, L. and Gordon, A., Mobile ambients, Theoretical Computer Science 240(1), ,2000. BioAmbients Regev, A., Panina, E., Silverman, W., Cardelli, L. and Shapiro, E., Bioambients: an abstraction for biological compartments, Theoretical Computer Science 325(1), , Phillips, A. An abstract machine for the stochastic bioambient calculus, Electronic Notes in Theoretical Computer Science

35 BioAmbients Sys = [[DNA] nucleus accept n; Mol1] cell [enter n; expel n; 0 [exit n; RNA] nucap ] virus Sys τ [[DNA] nucleus Mol 1 [expel n; 0 [exit n; RNA] nucap ] virus ] cell τ [[DNA] nucleus Mol 1 [0] virus [RNA] nucap ] cell

36 Bioambients nucleus cell DNA accept n; Mol 1 nucap virus exit n;rna enter n; expel n; 0 enter nucleus DNA cell virus nucap Mol 1 exit n;rna expel n; 0 exit cell nucleus nucap virus DNA Mol 1 RNA 0

37 BioPepa different roles each biological element plays a role: reactant, inhibitor, activator, or product J. Hillston. A Compositional Approach to Performance Modelling, PEPA. Cambridge University Press, Ciocchetta, F. and Hillston, J.. Bio-PEPA: a framework for the modelling and analysis of biochemical networks, Theoretical Computer Science, Until now we have only spoken about reactanct and products. Bio-PEPA differentiates: reactant ( ), product ( ), activator ( ), inhibitor ( ), or generic modifier ( ).

38 BioPepa An example 2X + Y r 3Z r is the rate constant [X ] and [Y ] are the concentrations of the biological species as usual, under the mass-action kinetic law, the reaction-rate is computed as = r [X ] [X ] [Y ] = r [X ] 2 [Y ]

39 BioPepa The Bio-PEPA specification of the roles X = (α, 2) X Y = (α, 1) Y Z = (α, 3) Z where 2, 1, and 3 are the stoichiometric values for the species X, Y and Z. Finally, the model of the complete system is: (X (x 0 ) {α} Y (y 0)) {α} Z(z 0) where x 0, y 0, z 0 represent the initial concentrations of the three components level of model precision Concentrations are continuous values bounded from above by a maximum M i that varies for each species. Stochastic simulations work on discrete values, the range [0... M i ] of concentrations is then divided in N sub-intervals. The choice on N, the granularity, gives the level of precision of a model: the bigger, the more precise.

40 Bio-PEPA workbench Duguid, A., Gilmore, S., Guerriero, M., Hillston, J. and Loewe, L.. Design and development of software tools for Bio-PEPA, WSC 09, Ciocchetta, F. and Hillston, J.. Bio-PEPA: a frame- work for the modelling and analysis of biochemical net- works, Theoretical Computer Science (2009). stochastic simulations transformation into the associated Continuos Time Markov Chain, transformation into an equivalent ODE

41 Beta-binders locations the location where a phenomenon takes place: it may occur inside or outside or on the surface of a specific membrane of a cell. Priami, C. and Quaglia, P.. Beta binders for biological interactions. Computational Methods in Systems Biology Guerriero, M. L., Priami, C. and Romanel, A.. Mod-eling static biological compartments with beta-binders. Algebraic Biology A binder syntactically: β(x : ) when it is active, β h (x : ) when it is hidden, where x is the subject of the binder and is the type of x.

42 Two molecules: enzyme, substrate (E and S). The two interfaces: (w, DE) and (y, DS) are compatible because of (DS, DE, K ES, K 1, K EP )

43 BlenX Beta-binders has given rise to the full-fledged programming language BlenX. Dematt`,L.,Priami,C.andRomanel,A.. The BlenX Language: A tutorial. Formal Methods for Computational Systems Biology, The Beta Workbench, freely available at research/prototypes/betawb Demattè, L., Priami, C. and Romanel, A.. The Beta Workbench: A computational tool to study the dynamics of biological systems. Briefings Bioinformatics, Demattè, L., Larcher, R., Palmisano, A., Priami, C. and Romanel, A.. Programming biology in BlenX. Systems Biology for Signaling Networks, the semantic-based interpreter incorporates an efficient variant of Gillespies algorithm it includes a graphical editor to specify models in a friendly way, a tool to generate the associated Continuous Time Markov Chain.

44 BlenX4Bio Allows to describe spatial information: boxes can be nested and can move, changing their location from one box to another one. (Caveat: similar features of the BioAmbients). Priami, C., Ballarini, P. and Quaglia, P.. Blenx4bio - blenx for biologists. CMSB 2009.

45 membrane calculus Cardelli, L.. Brane Calculi, CMSB 2005.

46 Brane calculus A virus enters a cell where it frees its nucleocaspis membrane i.e., the membrane that contains the virus RNA. Sys = phago n (exo n ; δ).ζ[ρ[dna] nucleus Mol 1 ] cell phago n ; exo n [σ[rna] nucap ] virus Sys Sys = ζ[ρ[dna] nucleus exo n ; δ[exo n [σ[rna] nucap ] virus ] new Mol 1 ] cell In Sys, the virus is wrapped in the membrane new, which offers the complementary action exo n to allow the virus to exit and dissolve. As a result, the nucap membrane lays in the membrane cell and can proceed with the virus DNA replication Sys ζ[ρ[dna] nucleus Mol 1 δ[] new σ[rna] nucap ] cell

47 Shape calculus The basic components of the Shape calculus are 3-D shapes. They expose a behavior coded as a process that associates a channel with each interacting site on its surface. Channels a, X, where a is a channel and X is a portion of the surface of the shape, called active surface. S[ b, Y ; B + a, X ; B ] It can non-deterministically interact with one of the two open interacting channels: one channel is b on site Y, and the other one is a on site X.

48 Shape calculus Cacciagrano, D., Corradini, F., Merelli, E. and Tesei, L.. Uniformity in multiscale models: From complex automata to bioshape, Journal of Cellular Automata, the movements of shapes are ruled by physical functions accordingly to their velocity and direction. These can change over time according to a general motion law or to a collision occurring between two or more shapes. When shapes collide they form compounds only if they share an active surface containing compatible channels. If an elastic collision occurs and no compound is formed. Two forms of de-complexation : weak splitting, not urgent so it can be delayed strong splitting, executed instantaneously being urgent. It is important to remark that the Shape calculus takes also into account time!

49 Shape calculus Corradini, F., Merelli, E., Tesei, L., Cacciagrano, D., Di Bernardi, M., Bartocci, E. and Buti, F. (2011), The bioshape simulator it/bioshape/download.html. BioShape accepts as input an XML file with the specification of the shapes, their functional behavior other simulation-related information. Different physical laws to describe the movements of shapes choosing the suitable law for tuning the movement simulation according to the desired granularity. Quantum mechanics for microscopic biological environments, and the mechanics for the macroscopic biological environments. An interpreter of the process calculus used to specify the functional behaviuor of the biological elements.

50 3π-calculus It considers a 3-D space and movements of biological objects, by enriching the interaction primitives of the π-calculus with affine geometric transformations. The rationale for this new calculus is to capture movements due to biological development, such as expanding, splitting and twisting of tissues. Cardelli, L. and Gardner, P., Processes in space, Theoretical Computer Science

51 Strand-algebra BioInformatics In the classical approach, algorithms have been developed to study the DNA structure, in particular to determine the order of bases in a sequence through DNA sequencing. Strand-algebra for describing the behavior of the DNA strands Cardelli, L.. Strand algebras for dna computing, Natural Computing, 2011.

52 Strand-algebra The interaction between signals and gates is represented by the transition x x.y y where the signal x is the input to the gate x.y. The occurrence of the transition yields the new signal y.

53 Strand-algebra The Strand-algebra has lead to the definition of the DSD language for designing modular DNA circuits using strand displacement. The implementation of the language is equipped with some tools supporting the design and the stochastic simulation of computational DNA devices. Lakin, M. R., Youssef, S., Cardelli, L. and Phillips, A.. Abstractions for dna circuit design, J. R. Soc. Interface https: // project/programming-dna-circuits/

54 Chemtainer calculus in brief It offers an interplay of autonomous molecular computations, within a chemtainer (think of it a box where you put biochemical elements), and you can externally manipulate it. The execution of a feed that adds three molecules of type A to location l, which are then wrapped in the chemtainer (the circle). The execution of a feed tag that adds the address tag represented by the line with name x in location l.

55 Peptide filtering process (stochastic -calculus) Major Histocompatibility Complex class I (MHC, for short) molecules are present in all the nucleated human cells. These molecules bind peptides arising from intracellular protein turnover done within the cell. The role of MHC is to present these peptides at the cell surface, thus providing a snapshot of the cell content that can subsequently be used to trigger an immune response. The selection of the peptides to be presented is known as peptide optimisation.

56 Peptide filtering process (stochastic -calculus) Wet lab experiments show MHC variations differ in their ability to select peptides. MHC B4402 is highly dependent on tapasin for peptide optimization, MHC B2705, MHC B4405 are less tapasin-dependent. in particular, B2705 is associated with long-term non-progression of HIV. Experimental results, obtained by a combination of stochastic simulations and Information Theory, confirm previous results: B2705 and B4405 (without tapasin) rapidly bound their peptide cargo and exhibited good time-dependent optimization; B2705 and B4405 (with tapasin) express a reduced optimization; B4402 (without tapasin) a very low peptide optimization occurred; B4402 (with tapasin) the peptide optimization increased.

57 Peptide filtering process (stochastic -calculus)... moreover, the analysis identifies the mechanisms that determine the extent and rate of peptide optimization. peptide optimization depends on the rate at which MHC complexes containing a peptide can egress to the cell surface, and hence the rate at which a peptide can unbind from a MHC complex. The competition between unbinding and egress rates defines a peptide filtering step that, for the first time, gives a mechanistic explanation for experimental data on peptide optimization. The discovered mechanism of the peptide filtering step has been also applied to predict peptide optimization at the steady state.

58 The Prince William Sound ecosystem (BlenX) The Prince William Sound ecosystem in Alaska has been modeled. The stochastic model of the food-web focus on the predator-prey interactions. Scotti, M.. The role of stochastic simulations to extend food web analyses.systemic Approaches in Bioinformatics and Computational Systems Biology: Recent Advances Scotti, M., Gjata, N., Livi, C. and Jordan, F.. Dynamical effects of weak trophic interactions in a stochastic food web simulation. Community Ecology

59 The Prince William Sound ecosystem (BlenX) Each species is modeled as a BlenX box. The rates of interaction have been derived by the trophic flows extracted from the weighted food web. The death rates have been chosen from a realistic range. The chemical equations describing the predator-prey interaction between two species are: A + B k1 A and A + B k2 2A The rate k 1 represents the propensity of the predator species A to feed on species B. The rate k 2 represents the propensity of the predator species A to feed on species B and to reproduce itself.

60 The Prince William Sound ecosystem (BlenX) Two main indexes have been considered: quantifies the community-wide response following some perturbation on a particular species i; quantifies how a species i depends on the perturbations of any other species in the ecosystem.

61 The Prince William Sound ecosystem (BlenX) comment both indexes are higher in the middle of the trophic scale, deterministic dynamics assign the highest importance either to top-predators (such as orca or sharks, which are the living components at the extremities of the food web, considering their trophic positions) or producers (in the food web modeled, the first two producers are macroalgae and phytoplankton). also, a species i with a large effect on the rest of the population of other species, is not guaranteed to have an accordingly large effect also on the corresponding index. Results show that the stochastic simulations are a sensible approach for identifying those species that play a major role in energy delivery in the food web and that are the most likely to cause major damage when removed.

62 Circadian clocks (Bio-PEPA) Circadian clocks are gene regulatory networks present in most organisms that adapt them to the twenty-four hours day/night cycle. They are composed of a small number of genes in Arabidopsis thaliana in the absence of external stimuli (e.g., constant light), the amounts of the involved mrnas and proteins oscillate rhythmically with a period of approximately 24 h in the presence of external stimuli (e.g., light on/off), the oscillations entrain to the external stimuli, adjusting their rhythm to it. Guerriero,M.,Pokhilko,A.,Fernández,A.,Halliday,K., Millar, A. and Hillston, J.. Stochastic properties of the plant circadian clock. Journal of The Royal Society Interface

63 Circadian clocks (Bio-PEPA) As said... Bio-PEPA is equipped with both a stochastic semantics, based on Continuous-time Markov Chain, and with a continuous deterministic semantics defined in terms of ODEs. Then, the continuous and the stochastic behavior of the model have been easily compared. The Bio-PEPA model is derived from a continuous one: continuous variable have been discretized by a parameter N. The precise system size of the clock network of Arabidopsis is still unknown, the hypothesis was that the average copy number of the clock components is around several hundreds of molecules per cell. The results of the stochastic simulations suggest that the size of the real system is in the order of a few hundreds of molecules per cell.

64 Circadian clocks (Bio-PEPA) In the constant light setting, the quantity of mrna expressing TOC1 shows regularly persistent oscillations when the deterministic model is adopted whereas the mean stochastic behavior shows a damped oscillation, matching well with the experimental population data. The suggested explanation for this is that the oscillations of single cells are not synchronized, and taking the average smooths them.

65 summarizing I features Process calculi interaction stochastic extension stochastic π-calculus unary binary Bio- Ambients Bio- Pepa Beta-binders BlenX binary multi binary Brane calculi unary binary Shape calculus 3π-calculus Strand Chemtainer algebra calculus binary binary binary binary X X X X X tool SPiM BAM Bio-PEPA Beta workbench workbench space translation to ODEs to CTMC from SBML BioShape membranes 3D-shapes nested membranes membranes nested points in (nested in membranes 3D-space for BlenX) 3D-space to ODEs to CTMC to PRISM to ODEs to SBML fixed locations - nested membranes

66 summarizing II Process calculi stochastic π-calculus Bio levels atoms molecules membranes organs of indivi- population duals X X X BioAmbients X X X X Bio-PEPA X X X X Beta-binders BlenX Brane calculi X X X X X X Shape calculus X X X 3π-calculus X X X Strand algebra Chemtainer calculus DNA DNA X

67 Process Algebras pros and cons Pros Cons narrative description (almost!) compositionality make visible little (relevant) oscillations in the behaviour of the system easily changing quantitative parameters it fits for models with small number of elements not efficient simulation a simulation is only a possible behaviour to approximate the average behaviour hundred of simulation runs have to be performed