Download - 20080620 Formal systems/synthetic biology modelling re-engineered

formalsystems / synthetic biology

modellingre-engineered

Jonathan Blakes1st year PhD student

2008-06-20

systems biology modelling

• systems biology operates at molecular, cellular, tissue/colony, organism/population levels

• model molecular reactions• model cellular organisation

– compartments (dynamic)

• model cellular interactions• development (growth/division/differentiation)

• observe emergent behaviour• generate/test hypotheses in silico

synthetic biology modelling

• synthetic biology operates at molecular level

• (re)programming cellular chassis with new genes

• model molecular reactions– gene expression/regulation– protein interactions– metabolite turnover (I/O)

• modular assembly (orthogonality)• prototype organism design

medical / commercialpotential

Petri nets P systems π-calculus

reactions interactions

formalisms

lasI S100

[lasI] → [lasI + LasI][LasI + S] → [LasI + 3OC12]

[ 3OC12 ] → 3OC12 [ ]

LasR rhlR

3OC12 [ ] → [ 3OC12 ][LasR+3OC12] → [LasR.3OC12]

[LasR.3OC12 + LasR.3OC12]→ [LasR.3OC122]

[LasR.3OC122 + rlhR] →[LasR.3OC122 + rlhR]

3OC1250

bacteria

environment

bacteria

reaction-based formalism equivalence

P system rules can be flattened to a Petri net:

Sedwards S. Cyto-Sim Example Models: Oscillators. 2006

Petri nets P systems π-calculusmolecular species

place symbol symbol

molecule

token object process

population of molecules

marking of net multiset processes

reactions

transitions rewriting rules communication

biological mapping

Petri nets P systems π-calculusdiscrete (mechanistic)

concurrent

non-deterministic (uniform time steps)

stochastic variants (realistic time steps)

SPN MCG, DPP Sπ

compartments

distinct places:Xnucleus Xcytoplasm

membranesSπ@

BioAmbientsBrane calculi

properties

why is stochasticity important?

Gilmore S. A Beginner's Guide to Stochastic Simulation.Uni. Edinburgh, Systems Biology Club talk, 16/11/2005

Gillespie algorithm

• generates a statistically correct trajectory (simulation) of a stochastic system

• ensembles of simulations average to ODE• limited to 2 reactants (higher order reactions

modelled as sequence of binary reactions)• assumes

– constant temperature and pressure– no electrostatic, H-bonding, Van der Waals

forces– well-mixed volume

110

100

Gillespie algorithm

Direct Method:1. for each reaction calculate propensity:

propensity (ai) = k ∙ hazard function

hazard = # distinct combinations of reactants

2. generate uniform random number r = 0 ≤ 13. selected reaction = r ∙ Σ propensities4. sample τ (time to wait) from negative

exponential distribution with parameterΣ propensities (a0)

a + b → ck

2 stochastic P system approaches

•Multi-Compartmental Gillespie (MCG) algorithm– ‘exact’ SSA algorithm running in each membrane– rule with lowest τ (smallest time step) in all

membranes executed - inherently linear

•Dynamical Probabilistic P systems (DPP)– rules selected à la Gillespie, objects assigned to

selected rule, repeated until all objects assigned– all rules in all membranes applied simultaneously– maximal parallelism, bounded by mute rules: a → a– uniform time steps – qualitative– parallel implementation using MPI C library

three ideas based on observations from literature review

1. to increase performance– τ-leaping simulation algorithm facilitates

parallelisation of multi-compartment models

2. to increase realism– molecular volumes can enable correct

simulation of cell growth and division dynamics

3. to increase knowledge– combining static information (reaction

topology) with stochastic execution in a visualisation can aid reasoning about system




2. to increase realism– molecule volumes can efficiently enable

correct simulation of cell growth and division dynamics



parallelising stochastic P systems

• P systems with exact SSA inside each membrane are parallel at the level of membranes but sequential at the level of rules

• one rule selected in each membrane, different waiting time for each

• different time lines - difficult to synchronise

• can we synchronise waiting times and speed up algorithm?

tau-leaping

• exact stochastic simulation can be very slow as it tracks every reaction event in system

• tau-leaping tracks groups of reactions instead• approximate which events occur in a period:

– leap condition ε provides propensities do not change dramatically

– safe τ computed– reactions drawn using Poisson distribution of τ– one/none critical reactions fired, many non-

critical reactions fired also

a faithful approximation

Gilmore S.“Beginner’s Guide to Stochastic Simulation”

University of Edinburgh Systems Biology Club talk 16/11/05

tau-leaping in parallel DPP

• tau leaping is a form of bounded parallelism because it restricts the number of reactions

• in tau-DPP each membrane computes a safe τ• smallest safe τ used to select reactions in each• reactions applied in all membranes in parallel

and system advances in leaps of time τ• whereas DPP simulations would normally

proceed in qualitative steps, now proceed in actual time so quantitative

• tau-MCG would unify our approaches

Cazzaniga P, Pescini D, Besozzi D, Mauri, G.“Tau Leaping Stochastic Simulation Method in P Systems”

WMC 7 2006 298-313

rates and volumes

• consider a compartment (volume v)• with a sub-compartment (volume v’)• calculating reaction propensity requires rate k

[ A + B ] → [ C ] propensity = k ∙ A ∙ B

• k implicitly contains the volume

• same reaction in v and v’ kv ≠ kv’

• probability of a binary collision is inversely proportional to volume

1. measure/calculate k in reference volume2. make volume explicit propensity = (k/v) ∙ A ∙ B

v v’

Smaldon J, Blakes J, Lancet D, Krasnogor N."A Multi-scaled Approach to Artificial Life Simulation With P Systems and Dissipative

Particle Dynamics" paper accepted for GECCO 2008 Atlanta, USA.

k

affect of membrane structure

vv’

v = v – v’

v

dynamic volumes

2v

cells grow

dynamic volumes

v

cells divide

dynamic volumes

cells divide

v/2 v/2

but how to calculate volume?

• extend stochastic π-calculus syntax with compartments (Sπ@) SPiM compatible

• Sπ@: compartments or molecules have volume

• compartment volume = ∑ (speciesi

quantity x speciesivolume)

• species volume ≈ molecular weight• need to model water → calculate pH

Versari C and Busi N. “Efficient Stochastic Simulation of Biological Systems with Multiple

Variable Volumes.” Electronic Notes in Theoretical Computer Science 2007 94(3) 165-180.

Proceedings of the First Workshop "From Biology To Concurrency and back” (FBTC 2007)

visualise propensity information

• The Next Reaction Method (and Optimized Direct Method) use a dependency graph to recalculate only the propensities of the affected reactions

• use dependency graph to visualise state of system during simulation of stochastic P systempropensity of each reaction colours nodes hot or cold

visualise propensity information

example

first reaction fires

example

molecules are consumed and produced,two propensities changed

example

second, unlikely, reaction fires

example

three propensities changed,dramatic effect on next reaction

example

visualisation of discarded informationcan reveal more than simply tracking

quantities

``The specific value of visual modelinglies in tapping the potential of high bandwidth

spatial intelligence, as opposed to lexical intelligence used with textual information.”

Samek, M.“Practical Statecharts in C/C++:

Quantum Programming for Embedded Systems”CMP Books, 2002

example

visualisation of discarded informationcan reveal more than simply tracking

quantities

observations

• novel visualisation method for pathways• topology same as Petri net minus places• identify cliques – define modules• visualisation of modules only, zooming out• can use as heuristic for simulation

algorithm selection: automatically determine when to use slow-scale SSA (respond to stiffness)

• not-suitable for string rewriting rules

references

• Wilkinson D J. Stochastic Modelling for Systems Biology. “Chapter 8: Beyond the Gillespie algorithm” Chapman & Hall / CRC 2006

• Cazzaniga P, Pescini D, Besozzi D, Mauri, G. “Tau Leaping Stochastic Simulation Method in P Systems” WMC 7 2006 298-313

• Gilmore S. “Beginner’s Guide to Stochastic Simulation” University of Edinburgh Systems Biology Club talk 16/11/05

• Versari C and Busi N. “Efficient Stochastic Simulation of Biological Systems with Multiple Variable Volumes” Electronic Notes in Theoretical Computer Science 2007 94(3) 165-180. Proceedings of the First Workshop "From Biology To Concurrency and back” (FBTC 2007)

• Smaldon J, Blakes J, Lancet D, Krasnogor N. "A Multi-scaled Approach to Artificial Life Simulation With P Systems and Dissipative Particle Dynamics" paper accepted for GECCO 2008 Atlanta, USA. Cazzaniga P, Pescini D, Besozzi D, Mauri, G. “Tau Leaping Stochastic Simulation Method in P Systems” WMC 7 2006 298-313

• Gibson M A, Bruck J. “Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels” J. Phys. Chem. A 2000 1876-1889

acknowledgements

• Supervisor Dr. Natalio Krasnogor• Dr. Francisco J. Romero-Campero• Dr. Jamie Twycross

Questions?

Download - 20080620 Formal systems/synthetic biology modelling re-engineered

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