Francisco J. Romero-Campero, Natalio Krasnogor. Automated Scheduling, Optimisation and Planning Research Group. School of Computer Science, Jubilee ...
An Approach to BioModel Engineering Based On P Systems Francisco J. Romero-Campero, Natalio Krasnogor Automated Scheduling, Optimisation and Planning Research Group School of Computer Science, Jubilee Campus, University of Nottingham Nottingham NG8 1BB, United Kingdom Email: {fxc,nxk}@cs.nott.ac.uk
Living cells in colonies or tissues communicate through complex systems consisting in the interaction between many molecular species distributed over many compartments. Different cellular processes are used by cells to monitor their environment and to respond accordingly. Among these processes, gene regulatory networks, rather than individual genes, are responsible for the information processing and orchestration of the appropriate response [15]. In this respect, synthetic biology has emerged recently as a novel discipline aiming at unravelling the design principles in gene regulatory systems by synthetically engineering transcriptional networks performing a specific and prefixed task [2]. Formal modelling and analysis are key methodologies used in the field to forward engineer, assess and compare different genetic designs or devices. In order to model faithfully cellular systems in colonies or tissues one requires a formalism able to represent the following relevant features: – Single cells should be described as the elementary unit in the system. Nevertheless, they cannot be represented as a homogeneous point as they exhibit complex structures containing different compartments where specific molecular species interact according to particular reactions. – The molecular interactions taking place in cellular systems are inherently discrete and stochastic processes which are key points to take into account when describing their dynamics [8]. – Gene regulatory networks seem to be organised in a modular manner in such a way that cellular processes arise from the orchestrated interaction between different genetic transcriptional units that can be considered separable modules [1]. – Spatial and geometric information must be represented in the system in order to describe processes involving pattern formation. In this work we review recent advances in the use of the computational paradigm termed membrane computing or P systems as a formal methodology in synthetic biology for the specification and analysis on cellular system models according to the previously presented points. P systems were introduced by G. P˘aun in 2000 as maximally parallel and nondeterministic rewriting systems distributed over a hierarchically compartmentalised structure inspired by the structure of the living cells and the biochemical interactions between molecular species [10, 11]. The main area of research within
the field focuses on the study of the computational universality and efficiency of the different proposed variants. Nevertheless, there is an emerging application of P systems as a framework for the specification and analysis of cellular systems [3, 5, 6, 9, 12–14, 16–20]. In our specific case we review the use of stochastic P systems. In our formalism single cells are considered to be the elementary units and are representing using the following definition. Definition 1. (Stochastic P System) A Stochastic P System is a construct Π = (O, µ, M1 , . . . , Mn , R1 , . . . , Rn ) where: – O is the finite alphabet of objects specifying molecular species; – µ is the membrane structure, a set of hierarchically embedded membranes describing compartments; – Mi , 1 ≤ i ≤ n, is the initial multiset of membrane i which represent the initial number of molecular species in compartment i; – Ri , 1 ≤ i ≤ n, is the finite set of rewriting rules on multisets of objects representing molecular interactions. These are called boundary rules [4], and are of the following form: c
r : u [ v ]i −→ u0 [ v 0 ]i where u, v, u0 , v 0 are multisets of objects that are replaced simultaneously outside and inside the corresponding compartment i. A kinetic stochastic constant c is associated with each rule and it is used to determine the probability of applying a rule and the time elapsed between rule applications. The original maximally parallel and non-deterministic semantics for the evolution of P systems was proved not suitable for reproducing the dynamics of cellular systems. In order to solve this problem stochastic semantics based on Gillespie’s theory of stochastic kinetics [7] were introduced [13]. Modularity is explicitly represented in P systems by allowing the specification of the sets of rewriting rules associated with compartments as a combination of instantiated P system modules [19, 20]. A P system module is a set of rewriting rules over multisets of variables representing objects, V arO , whose stochastic kinetic constants are also variables V arC . A module is identified with a name, M od and is specified as M od(V arO , V arC ). Then the instantiation of such module M od with specific objects O0 , constants C0 is specified as M od(O0 , C0 ) and its rules are obtained from the rules in M od(V arO , V arC ) by replacing all the occurrences of the variables V arO and V arC with the specific objects O0 and stochastic kinetic constants C0 . Finally, a spatially distributed colony or tissue of cells can be represented by specifying each cell as a stochastic P system and locating them in a geometrical lattice according to the following definition of lattice population P systems introduced in [20]. Definition 2 (Lattice Population P system).
A lattice population P system is a construct P = (O, L, (Π1 , . . . , Πn ), P os, (T1 , . . . , Tn )) where: • O is a the finite alphabet of objects representing the molecular species; • L is a finite geometrical lattice; • Π1 , . . . , Πn are stochastic P systems as in Definition 1 with the same finite alphabet of objects O; • P os : L −→ {Π1 , . . . , Πn } is the position function that associates a specific P system with each point on the lattice; • Tj = {r1j , . . . , rkj }, for each 1 ≤ j ≤ n, is a finite set of special rewriting rules called translocation rules of the following form: v
cj
v
i [ ]l m [ u ] rij : [ u ]c m [ ] −→
where u is a multiset of objects representing the molecular species that are to be translocated, v is a vector and cji is the corresponding kinetic stochastic constant. When rule rij is applied in the P system Πj located in position p, P os(p) = Πj , the objects u are translocated from Πj to the P system Πj 0 located in position p + v, P os(p + v) = Πj 0 . In order to illustrate our approach we use the following example. Our genetic design consists of three colonies of bacteria arranged in a specific spatial distribution. These colonies of bacteria posses especial genetic circuits that produce the temporal expression of a particular gene in a specific colony. The three different types of bacteria forming the three colonies are specified by the P systems Π1 , Π2 and Π3 , see Table 1. These P systems consist of a single membrane or compartment representing a bacterium. The corresponding set of rules associated with each P system is obtained as a combination of the P systems modules in Table 2. These modules represent the constitutive (U nReg), positive (P osReg) and negative (N egReg) regulation of genes as well as protein degradation (Deg) and diffusion (Dif f ). An extra P system Π0 is used to represent a subvolume of the media between the different colonies. The P systems from Table 1 are distributed over a rectangular lattice as shown in Figure 1. The leftmost colony consists of bacteria of type 1 represented by Π1 . These bacteria express constitutively gene1 whose protein product diffuses freely across the whole system. In the rightmost colony, described by Π3 , gene2 is regulated positively by prot1 producing the freely diffusible prot2. Finally, the colony in the centre of the lattice, specified by Π2 , possesses a genetic circuit according to which prot1 regulates positively gene3 whereas prot2 represses it. Figure 2 shows the evolution over time of the molecules of prot1, prot2 and prot3 in the bacterium at position (4, 2) in the lattice presented in Figure 1. It can be observed that at around 50 minutes prot1 is present in the compartment producing the activation of gene3 and the accumulation of prot3. When prot1 reaches the rightmost colony, prot2 starts to be produced and it diffuses across the system. After a delay of around 300 minutes prot2 reaches our bacterium at position (4, 2) and represses the expression of prot3. Therefore our engineered design consisting in specific gene regulatory circuits and in a particular spatial distribution produces a transient expression of around 250 minutes of prot3 in the colony at the center of our system.
Table 1. The four different P systems, Π0 , Π1 , Π2 and Π3 representing a subvolume from the media, a bacterium from colony one, two and three respectively. P System Π0
Π1
Π2
Π3
Modules Dif f ({prot1}, {0.01}) Dif f ({prot2}, {0.01}) U nReg({gene1, prot1}, {10}) Deg({prot1}, {0.0015}) Deg({prot2}, {0.0015}) Dif f ({prot1}, {0.01}) Dif f ({prot2}, {0.01}) P osReg({prot1, gene3}, {1, 5, 5}) N egReg({prot2, gene3}, {1, 0.001}) Deg({prot1}, {0.0015}) Deg({prot2}, {0.0015}) Deg({prot3}, {0.03}) Dif f ({prot1}, {0.01}) Dif f ({prot2}, {0.01}) P osReg({prot1, gene2}, {1, 0.01, 10}) Deg({prot1}, {0.0015}) Deg({prot2}, {0.0015}) Dif f ({prot1}, {0.01}) Dif f ({prot2}, {0.01})
Table 2. The library of modules used in our example consisting in constitutive (U nReg), positive (P osReg) and negative (N egReg) gene regulation as well as protein degradation (Deg) and diffusion (Dif f ). Module
Rules
U nReg({G, P }, {c1 })
1 r1 : [ G ]b −→ [ G + P ]b
P osReg({A, G, P }, {c1 , c2 , c3 })
1 r1 : [ A + G ]b −→ [ A.G ]b c2 r2 : [ A.G ]b −→ [ A + G ]b c3 r3 : [ A.G ]b −→ [ A.G + P ]b
N egReg({R, G}, {c1 , c2 })
1 r1 : [ R + G ]b −→ [ R.G ]b c2 r2 : [ R.G ]b −→ [ R + G ]b
Deg({P }, {c1 })
1 [ ]b r1 : [ P ]b −→
c
c
c
c
(1,0)
r 1 : [ P ]b m Dif f ({P }, {c1 })
(−1,0)
r 2 : [ P ]b
m (0,1)
r 3 : [ P ]b m m
c
1 [ ] −→ [ ]b
c1
(−1,0)
m
[P ]
(0,1)
[ ] −→ [ ]b m [ P ]
(0,−1)
r 4 : [ P ]b
(1,0)
c
1 [ ] −→ [ ]b m [ P ]
c
1 [ ] −→ [ ]b
(0,−1)
m
[P ]
Π1
Π1
Π0
Π0
Π2
Π2
Π0
Π0
Π3
Π3
(0,4)
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
(7,4)
(8,4)
(9,4)
Π1
Π1
Π0
Π0
Π2
Π2
Π0
Π0
Π3
Π3
(0,3)
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
(7,3)
(8,3)
(9,3)
Π1
Π1
Π0
Π0
Π2
Π2
Π0
Π0
Π3
Π3
(0,2)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
(7,2)
(8,2)
(9,2)
Π1
Π1
Π0
Π0
Π2
Π2
Π0
Π0
Π3
Π3
(0,1)
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
(7,1)
(8,1)
(9,1)
Π1
Π1
Π0
Π0
Π2
Π2
Π0
Π0
Π3
Π3
(0,0)
(1,0)
(2,0)
(3,0)
(4,0)
(5,0)
(6,0)
(7,0)
(8,0)
(9,0)
Fig. 1. Spatial distribution of the three different bacterial colonies over a rectangular lattice
Fig. 2. Number of molecules of prot1, prot2 and prot3 in the bacterium at position (4,2) in single simulation (left) and the average over 100 simulations.
Acknowledgements We would like to acknowledge EPSRC grant EP/E017215/1 and BBSRC grant BB/F01855X/1.
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