Stochastic modelling of biological systems: membrane systems in Systems Biology

1
Stochastic modelling of biological systems
membrane systems in Systems Biology
Giancarlo Mauri Università di Milano-Bicocca
(Italy)
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Lab. of Bioinformatics and Natural Computing -
Activities

• Bioinformatics
• Tools for sequence analysis
• Alternative splicing prediction
• Approximation algorithms (Fingerprint Clustering
  with Bounded Number of Missing Values Maximum
  Isomorphic Agreement Subtree LCS)
• DNA Computing
• Splicing systems and formal languages
• DNA word design
• Membrane systems
• Computational power of P-systems
• Modeling of biological systems
• Stochastic software simulators (Gillespie, )
• Brane calculi vs P-systems

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Outline

• Membrane systems (P-systems)
• Definition
• Computational power of P-systems
• Modeling biological processes with P-systems
• Stochasticity in nature
• Stochastic P-systems
• Stochastic simulation algorithms
• Simulations with stochastic P-systems
• Ras/cAMP/PKA pathway in yeast

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Structure of living cells
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Abstracting features

• A hierarchical arrangement of organelles, with
  separating membranes
• Each membrane delimits a region
• Each region contains a multiset of elements
  (simple molecules, DNA sequences, other regions)
• The chemicals/bio-elements evolve in time
  according to some (rewriting/combination) rules
  specific to each region or may be moved across
  the membranes
• The rules may also dissolve/create/move regions

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Membrane (P-)systems G. Paun, 1998

• Distributed, parallel and nondeterministic
  computing model
• Basic ingredients
• membrane structure (finite string of well
  matching parentheses)
• (multisets of) objects (symbols, strings, etc.,)
  in each membrane
• evolution rules associated to each membrane
  objects can evolve within membranes, move into
  neighboring membranes membranes can be
  dissolved/divided/created/merged.
• communication of objects among membranes

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Membrane systems - an example
environment
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Membrane systems

• The membrane structure
• separation and communication
• identification of relevant spaces
• The rules
• object evolution
• object communication
• membrane evolution
• Peculiar roles of objects
• catalysts, promoters, inhibitors

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Computing with Membrane systems

• Initial configuration, identified by
• membrane structure
• initial multisets of objects
• sets of evolution rules
• Nondeterministic and maximal parallel application
  of evolution rules
• universal clock (synchronous)
• all membranes are simultaneously processed
• all applicable rules are (nondeterministically)
  applied
• Successful computation
• no rule can be further applied
• output ? (multi)sets of objects collected in a
  prescribed membrane or outside the system

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Computing with membranes the TCS perspective

• Computability aspects
• Generated languages
• Computational power, Turing completeness,
  universality
• Complexity classes
• Comparison with other models
• Polynomial solutions to hard problems (time-space
  trade-off)
• Applications biology, bio-medicine, economics,
  linguistics, computer science, optimization
• Software and simulations

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Variants of the basic model

• Catalytic membrane systems
• Symport/antiport membrane systems
• Membrane systems with string objects
• Active membranes
• Polarized membranes of variable thickness
• Spiking neural membrane systems
• .

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Example of Symport/Antiport System
1
2
(oo, out) (oo, in)
3
(o, in)
oooo
(oo, out o, in)

• Types of rules (no transformation of objects)
• symport (u, out) or (u, in)
• antiport (u, out v, in) where u, v are
  multisets
• Symbols (above example is unary) available
  abundantly in the environment.

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Example 1-Object / 3-Membranes Acceptor
1

• Types of rules (no transformation of objects)
• symport (u, out) or (u, in)
• antiport (u, out v, in) where u, v are
  multisets
• Symbols available abundantly in the environment.
• System accepts
• L o2n n 0.
• V o

2
3
(oo, out) (oo, in)
(o, in)
oooo
(oo, out o, in)
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Example (cont.)
1
2
3

• Step 1
• Correct computation uses 2 instances of rule
• (oo, out o, in).

(oo, out) (oo, in)
(o, in)
oooo
(oo, out o, in)
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Example (cont.)
1
2
3

• Step 2
• Correct computation uses 1 instance of rule
• (oo, out o, in).

(oo, out) (oo, in)
(o, in)
oo
(oo, out o, in)
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Example (cont.)
1
2
3

• Step 3
• Correct computation uses 1 instance of rule
• (o, in).

(oo, out) (oo, in)
(o, in)
o
(oo, out o, in)
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Example (cont.)
1

• The system halts and accepts o4.
• Interesting question
• Is halting decidable for S/A acceptors over one
  symbol?
• At least as hard as VAS reachability.
• Known 2 symbols is universal

2
3
(oo, out) (oo, in)
o
(o, in)
(oo, out o, in)
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References

• Paun, Gh. Membrane Computing. An Introduction.
  Springer, Berlin (2002)
• The P Systems Web Page http//psystems.disco.unim
  ib.it/
• bibliography
• download papers
• download simulators

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Back to biology

• Use of the framework of P-systems to give a
  formal description of specific cellular phenomena
  or cellular structures
• Software tools for dynamical analysis of
  biological systems

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P-systems and biology
P systems as computing devices
P systems as bio-simulators
from in-vivo to in-silico
returning meaningful information
Looking for bioimplementations
cells
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Systems biology

• The omics era
• holistic approach to the interpretation and the
  analysis of biological systems (Ideker et al.,
  Ann. Rev. Gen. Hum. Gen., 2001 Kitano, MIT
  Press, 2001)
• investigation of the system at a global scale
  (components and interactions ! global behaviour)
• integration and representation of quantitative
  and qualitative data
• different types of perturbations
• Modeling biological systems
• identification of structure and parameters
• analysis of system dynamics via software
  simulations methods for system control and design

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From structures (syntax) to functions (semantics)
in Biology

• Bio-components as information and computational
  devices
• Millions of simultaneous computational threads
  active (e.g., metabolic networks, gene regulatory
  networks, signaling pathways)
• Components interaction changes the future
  behavior
• Interactions occur only if components are
  correctly located (e.g., they are close enough or
  they are not divided by membranes)

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Membrane systems and Systems Biology

• Advantages
• parallel and stochastic processing
• discreteness
• cellular localizations
• easily comprehensible
• scalability

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Applications of membrane systems

• Membrane systems as modelling tools at molecular
  and cellular scale
• transport proteins
• Na/K pump, Ca2 channels, mechanosensitive
  channels
• chemical reactions
• Belousov-Zhabotinsky, Michaelis-Menten
• cellular signaling pathways
• EGFR, Ras/cAMP/PKA
• bacterial colonies
• Vibrio fischeri, Pseudomonas aeruginosa

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Applications of membrane systems

• Membrane systems as modelling tools at ecological
  scale
• Lotka-Volterra equation
• population dynamics
• tritrophic systems
• plantsherbivorescarnivores
• metapopulations
• populations living in fragmented habitats

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Other approaches

• ODE (global, monolithic, difficult)
• Petri nets
• The pi-calculus (Milner, Parrow, Walker 1992)
• The stochastic pi-calculus (Priami 1995)
• (Mem)Brane calculus (Cardelli 2004)

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Stochasticity in biological systems

• Noise in nature ? many experimental evidences
  of stochasticity in living systems
• observation of genes expression has shown the
  stochastic nature of transcription and
  translation Abkowitz et al, 1996 Ozbudak et al,
  2002
• the mRNA production is quantal Hume, 2000 and
  occurs in random pulses Ross et al, 1994
  Walters et al, 1995
• the protein production occurs in short bursts and
  at random time intervals Yarchuk et al, 1992
  Chapon, 1982
• lysis/lysogeny switch in the ?-phage Oppenheim
  et al, 2005

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Stochasticity in biological systems

• 2 kinds of noise
• intrinsic noise - due to the inherent nature of
  the biochemical interactions
• extrinsic noise - due to the external
  environmental conditions
• Complex systems such as the biological ones are
  extremely non-linear and often exhibits many
  steady states, bifurcations or chaotic behavior

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Stochasticity in biological systems

• Stochastic models are suitable in this framework
  because
• take into consideration discrete quantities of
  components,
• are in accordance with the stochastic processes,
• are appropriate to describe "small systems" and
  instability phenomena.
• Stochastic simulation is the probe to access the
  different evolutions

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Stochastic simulation algorithm

• Stochastic Simulation Algorithm Gillespie, 1977
• N chemical species Si in a single fixed volume V,
  with Xi current number of molecules of Si
• M chemical reactions Rµ, with reaction parameters
  cµ
• 2 questions when will the next reaction occur?
  Which reaction will it be?
• SSA computes the probability that the next
  reaction in V will occur in the differential time
  interval (t?, t?d?) and will be Rµ
• huge computational time needed it increases with
  the number of reactant species (and of reaction
  channels)

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Stochastic membrane systems DPP (Pescini et al.,
2006)

• Rules are applied according to the probability
  associated with them via a known function
• At each step the probability changes dynamically
  looking after the systems variations
• To have a parallel step the probability
  distribution is
• fixed by the actual system state
• kept constant for the whole parallel step

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Stochastic simulation algorithm

• Tau Leaping stochastic simulation method
  Gillespie et al, 2006
• used to speed up SSA
• during every leap (?) several reactions are
  executed
• the rules execution order, within each step, does
  not matter
• the complexity of the algorithm increases
  linearly with the number of reactant species

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Stochastic membrane systems ?-DPP

• The membrane system structure is exploited to
  extend the Tau Leaping procedure to multiple
  volumes systems
• tracing the time of each membrane as well as the
  time of the whole system
• qualitative and quantitative evolution
• communication of objects among membranes as in
  standard membrane systems
• computational time increases with the number of
  reactant species and with the number of volumes

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?-DPP How it works

• There is a ?-leaping engine in every membrane.
• Every membrane generates a ? value based on its
  internal state
• The system evolves according to the smallest ? of
  the system.
• In each membrane the probability distribution is
  generated according to
• ?
• the underlying process
• the system status.

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?-DPP test case
Consecutive reactions system
starting from a population of 1000 individuals of
species A
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Resources

• Cluster with Beowulf architecture, with 30 nodes
• AMD Athlon(TM) XP 2800 processors
• Linux OS
• Stochastic simulator
• C language code
• MPI library (http//www.mcs.anl.gov/mpi/index.html
  )
• mpicc compiler

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Stochastic Modeling and Simulations of the
Ras/cAMP/PKA Pathway in Budding Yeast

• This is a signalling
• transduction pathway.
• In yeast, it plays a major role in the
• control of cell growth, stress resistance and
  proliferation, in relation to the available
  nutrients.

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Ras/cAMP/PKA Pathway
Cyclic AMP plays a key regulatory role in yeast
growth High cAMP Required for growth and cell
cycle
progression. Low stress resistance. Low
cAMP Arrest of growth and cell cycle in
G1/G0, accumulation of trehalose and
glicogen. Stress resistance
A moderate PKA activity is required for
growth. PKA activates protein synthesis,
ribosomes and rRNA synthesis and synthesis of
Cln3.
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Ras/cAMP/PKA Pathway

• The Ras/cAMP/PKA pathway in yeast trasduces two
  different signals
• Allows the G1/S transition at START through a
  nutrient sensing mechanism ( still not
  defined..), likely through a modulation of the
  activity of Cdc25 or Ira proteins, and also
  regulates Ps (cell size required for entry in S
  phase)
• 2) Generates a peak of PKA activity (mediated by
  a fast cAMP increase) in response to addition of
  fermentable sugars (glucose and fructose). This
  signalling regulates the transition between
  respiratory and fermentative metabolism.

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Cycle of Ras proteins activation
Cdc25, Sdc25
Adenylate Cyclase
Ira1, Ira2
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Proposed model
H
Glucose
Glucose
Cdc25
Ira1,2
Gpr1
Hxt
Ras
Gpa2
Hxk 1,2
Adenylate cyclase
Glk1
ATP cAMP
PKA
GTP/GDP
Downstream signalling
Colombo et al. 2004
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Modeling cAMP pathway in a single cell
Input module
Initial set (molecules/cell)
glucose, H, GTP/GDP
Ras2.GDP 20000 Ira2 400 Cdc25
300 GDP 1.5 x 106 GTP 5 x 106 ATP
2.4 x 107 Cyr1 200 Pde1
1400 Pde2 6500 PKA 2500 Ppa2
4000 Gpa2 4000 Gpr1 200
Signalling
CDC25/Ras/Ira
Gpr1/Gpa2
P-Pde1 P-Cdc25 P-Ira ?
Feed-back
Adenylate cyclase
cAMP
The amount of different molecular species, that
is, the discrete number of molecules per cell,
were estimated either from literature data, or
through our experimental determinations (assuming
the approximate cell volume V 310-14 L).
Effector
PKA
S. Ghaemmaghami, et al. Global analysis of
protein expression in yeast, Nature, 425
737-741, 2003
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Methodology
4 main logical modules have been identified in
the model

• the switch cycle of Ras2 protein, involving the
  GEF Cdc25 and the GAP Ira2
• the synthesis of cAMP, via activation of
  adenylate cyclase Cyr1
• the activation of PKA, via the reversible binding
  between its regulatory subunits R and cAMP, and
  subsequent dissociation of the tetramer PKA
• the activity of phosphodiesterases Pde1 and Pde2,
  which determines the feedback mechanisms for cAMP
  degradation

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Ras/cAMP/PKA signaling pathway

• Using tau-DPPs, we can simulate systems
  structured by several volumes, tracing the
  simulated time of the compartments as well as
  time line of the whole system. This gives us the
  possibility to quantitatively and qualitatively
  describe biological systems.
• Our model was able to simulate properly the Ras
  protein cycle, the activation of adenylate
  cyclase, the production of cyclic AMP and the
  activation of cAMP-dependent protein kinase in a
  single yeast cell. The results are compared with
  the experimental data and give information on the
  key regulatory elements of this signalling
  network.

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Ras/cAMP/PKA signaling pathway

• The model involves
• 34 rules
• 30 molecular species
• 1 major feedback
• many Michaelis Menten schemes

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Ras/cAMP/PKA signaling pathway

• Reaction Reagents Products
  Constant
• r1 Ras2GDPCdc25 Ras2GDPCdc25 1
• r2 Ras2GDPCdc25 Ras2GDPCdc25 1
• r3 Ras2GDPCdc25 Ras2Cdc25GDP 1.5
• r4 Ras2Cdc25GDP Ras2GDPCdc25 1
• r5 Ras2Cdc25GTP Ras2GTPCdc25 1
• r6 Ras2GTPCdc25 Ras2Cdc25GTP 1
• r7 Ras2GTPCdc25 Ras2GTPCdc25 1
• r8 Ras2GTPCdc25 Ras2GTPCdc25 1
• r9 Ras2GTP ra2 Ras2GTPIra2 310-2
• r10 Ras2GTPIra2 Ras2GDPIra2 0.7
• r11 Ras2GTPCYR Ras2GTPCYR1 10-3
• r12 Ras2GTPCYR1ATP Ras2GTPCYR1cAMP 0.2110-
  5
• r13 Ras2GTPCYR Ira2 Ras2GDPCYR1Ira2 10-3

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Simulation results
Response of Ras2?GTP to a step decrease of GTP
amount (from 5?106 to 1.5?106 molecules) at time
1500.
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Simulation results
Sensitivity of Ras2?GTP. Left dependence on the
rate of dissociation of Cdc25, (a) k7 0.2 (b)
k7 0.5 (c) k7 1.0 (d) k7 1.5 (e) k7
2.0. Right dependence on the activity of Ira2,
(a) k10 0.25 (b) k10 0.5 (c) k10 0.7 (d)
k10 0.9 (e) k10 1.1.
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Simulation results
Sensitivity of Ras2?GTP module.
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Simulation results
Effect of Pde2 activity on cAMP accumulation, (a)
k32 1.6 (b) k32 1.7 (c) k32 1.8 (d) k32
1.9 (e) k32 2.0.
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Simulation results
Effect of Pde1 activity on cAMP accumulation, (a)
k28 1.7 (b) k28 2.0 (c) k28 3.0 (d) k28
4.0 (e) k28 5.0.
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Simulation results
Variation of the catalytic subunit of PKA (left)
and cAMP (right) dependent on the affinity
between PKA and cAMP (rules r14, , r21). Values
of reaction constants (a) ki 10-4, kj 0.1
(b) ki 10-5, kj 0.1(c) ki 5?10-6, kj
0.1 (d) ki 10-6, kj 0.1, where i 14, 15,
16, 17 j 18, 19, 20, 21.
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Simulation results
Response of C (left) and of cAMP (right) to a
step increase of GTP amount (from 1.5?106 to
5?106 molecules) at time 1500.
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Simulation results
Effect of different GTP input values on cAMP
accumulation (left) and on Ras2?GTP and PKA
activity (right).
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Ras2-GTP kinetics
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Work in progress

• Simulation tools
• DPP optimization
• integration of Genetic Algorithms
• topological distribution of molecular species in
  distinct cellular regions and/or presence of
  large signalling complexes localized in internal
  membranes
• Analysis
• communicating classes and beyond
• role of noise in Molecular Dynamics-Computing

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Work in progress

• Biological Systems Modelling
• Implementation of Gpr1/Gpa2 glucose sensing
  system (that requires t-DPP)
• Introduction of multiple feed-back levels (Ira,
  Cdc25, Ras)
• Extensive sensitivity analysis, stability
  /instability behaviour etc.
• Biofilms formation in Pseudomonas aeruginosa and
  Escherichia coli colonies

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References

• I. I. Ardelean, D. Besozzi, M. H. Garzon, G.
  Mauri, S. Roy
• P System Models for Mechanosensitive Channels
• In "Applications of Membrane Computing,
  Springer, 2005
• D. Pescini, D. Besozzi, C. Zandron, G. Mauri
• Analysis and Simulation of Dynamics in
  Probabilistic P Systems
• Proc. DNA 11, LNCS 3892, Springer, 2006
• D. Besozzi, P. Cazzaniga, D. Pescini, G. Mauri
• Seasonal variance in P system models for
  metapopulations
• First Int. Conf. on Bio-Inspired Computing
  Theory and Applications

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References

• P. Cazzaniga, D. Pescini, D. Besozzi, G. Mauri
• Tau leaping stochastic simulation method in P
  Systems
• Proc. WMC 7, LNCS 4361, Springer, 2006
• E. Martegani et al.
• Identification of an intracellular signalling
  complex for Ras/CAMP pathway in yeast
  experimental evidences and modelling
• 25th International Specialised Symposium on
  Yeasts, 2006
• P. Cazzaniga, D. Besozzi, E. Martegani, S.
  Colombo, G. Mauri
• Stochastic modelling of the Ras/cAMP signal
  transduction pathway in yeast
• Journal of Biotechnology, to appear

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The research team
61
Conclusions
Thanks!