Stochastic Analysis of Chemical Reaction Networks using Linear Noise
Approximation
Luca Cardell iβ1,2β , Marta Kwiatkowsk πβ1β, Luca Laurentπβπβ β1β Department of Computer Science, University of Oxford
β2β Microsoft Research
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β’β― Motivationβ’β― Backgroundβ’β― Linear Noise Approximation (LNA)β’β― Stochastic Evolution Logic (SEL)β’β― Experimental Results
Summary
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Motivation Γβ―Biochemical systems are generally analysed considering
deterministic models. However, deterministic models are accurate only when the molecular population is large
Γβ―When the interacting entities are in low numbers there is a
need of considering a stochastic model
Γβ―Existing methods for analysis of discrete state space stochastic processes are not scalable and highly dependent on the initial number of molecules
β’β― Question: Can we derive a formal method to analyse the stochastic semantics of biochemical systems that is scalable and independent of the initial number of molecules?
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Chemical Reaction Networks (CRNs)
β’β― A CRN πͺ=(π²,πΉ) is a pair of sets
β’β― Ξ is a finite set of species { πβ1β, πβ2β,β¦, πβ|Ξ|β}
β’β― π is a finite set of reactions { πβ1β, πβ2β,β¦, πβ|π |β} β’β― π.π. πβπβ : πβ1β+ πβ2β ββπβ πβ3ββ’β― π is the rate constant
Γβ―A configuration or state of the system, π₯β πββ|Ξ|β, is given by the number of molecules of each species in that configuration
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Γβ―Set of autonomous polynomial ODEs:πΞ¦/ππ‘β=F(t)Ξ¦(π‘) Ξ¦(0)= π₯β0β/πβ β’β― π=ππππ’ππβ πβπ΄β is the Volumetric factor of the
system
β’β― Ξ¦(π‘)β π β|Ξ|β represents the species concentration at time t
β’β― πΉ(π‘) is determined by mass action kineticsβ’β― Number of differential equations equals number of speciesβ’β― Valid only for high number of moleculesβ’β― Does not take into account the stochastic nature of
molecular interactions 5
Deterministic Semantics of CRNs
β’β― It is a continous time Markov process ( πβπβ(π‘),π‘β₯0) with discrete state space π and infinitesimal generator matrix π determined by the reactions
β’β― The transient evoloution of πβπβ is described by the Chemical Master Equation (CME)oβ―Assuming π(π₯,π‘)=ππππ{πβπβ(π‘)=π₯ | πβπβ(0)= π₯β0β}
then the CME can be written as
ππ(π₯,π‘)/ππ‘β=π(π₯,π‘)π
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Stochastic Semantics of CRNs
β’β― One differential equation in the CME for any reachable stateβ’β― S highly dependent on the initial number of moleculesβ’β― Set of reachable states can be huge or even infinite
β’β― Solution: USE LINEAR NOISE APPROXIMATION !
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State Space Explosion Problem
Γβ―Not possible to solve the CME for large molecular populations and/or large CRNs
β’β― Technique pioneered by Van Kampen in his CME espansion
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β’β― Ξ¦(π‘) solution of the deterministic semanticsβ’β― π(π‘) is a Gaussian Process independent of π
oβ― πΈ[π(π‘)]=0 πππ π‘β₯0oβ― ππΆ[π(π‘)]/ππ‘β= π½βπΉβ(Ξ¦(π‘))πΆ[π(π‘)]+πΆ[π(π‘)]π½βπΉβπβ(Ξ¦(π‘))+πΊ(Ξ¦(π‘))β’β― π½βπΉβ(Ξ¦(t)) Jacobian of πΉ(Ξ¦(π‘)) β’β― πΊ= 1/πββπβπ ββπβπβπβπβπβπΌβπββ
Linear Noise Approximation (LNA)
πβπβ(π‘)βπβπβ(π‘)=πΞ¦(π‘)+βπβπ(π‘)
β’β― For any CRN, assuming mass action kinetics, the LNA is always accurate at least for a limited time (it is enough to increase π)
β’β― Independence of the initial number of moleculesoβ― The number of differential equations depends only on the
number of species
β’β― Number of differential equations quadratic in the number of species
β’β― Still good approximation for a large class of CRNs even for quite small molecular populations oβ― Not able to handle multinomial distributions
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Linear Noise Approximation (LNA)
LNA Also Known as Gaussian Approximation
β’β― π΅β πββ|Ξ|β, the linear combination of species π΅βπβπβπβ(π‘) is still Gaussian
oβ― πΈ[π΅βπβπβπβ(π‘)]= π΅βπβπΈ[πβπβ(π‘)]=π(π΅βTβΞ¦(π‘))
oβ― πΆ[π΅βπβπβπβ(π‘)]=π΅πΆ[πβπβ(π‘)]π΅βπβ=π΅πΆ[π(π‘)]π΅βπβ
β’β― Probability is calculated by solving Gaussian integrals
β’β― π(π‘) is a Gaussian process
Γβ― πβπβ(π‘)=πΞ¦(π‘)+βπβπ(π‘) is a Gaussian process
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β’β― π={ π π’ππ, ππππ,π π’ππΈ,ππππΈ} β’β― πΌ set of closed disjoint intervals β’β― π΅β πββ₯0 β|Ξ|β
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Stochastic Evolution Logic (SEL)
Γβ― πββΌπβ[π΅,πΌ ]β[π‘β1β, π‘β2β]β : probabilistic operator
Γβ― π π’ππ/ππππββΌπ£β[π΅]β[π‘β1β, π‘β2β]β : supremum/infimum of variance operators
Γβ― π π’ππΈ/ππππΈββΌπ£β[π΅]β[π‘β1β, π‘β2β]β : supremum/infimum of expected value operators
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Γβ―We use the LNA for a numerical approximate model checking algorithm of SEL
oβ― πβπβ is approximated by the Gaussian Process πβπβ
oβ― The probability that π΅βπβπβπβ is within the interval [π,π] at time t is:
β«πβπβππ₯ πΈ[π΅βπβπβπβ(π‘)],πΆ[π΅βπβπβπβ(π‘)]βππ‘β where π(π₯|πΈ,πΆ) is the Gaussian distribution with
expected value πΈ and variance πΆ.
oβ― πΈ[π΅βπβπβπβ(π‘)] and C[π΅βπβπβπβ(π‘)] are obtained by solving the LNA for the given initial condition
Approximate Model Checking
Property to check:
Species = {πΏ1,πΏ1π,πΏ2,πΏ2π,πΏ3,πΏ3π,π΅} πβ1β: πΏ1+π΅ βββπβ1ββπ΅+πΏ1π πβ2β: πΏ1π+πΏ2 ββπβ2ββ πΏ1+πΏ2π πβ3β: πΏ2π+πΏ3 ββπβ2ββ πΏ2+πΏ3π πβ4β: πΏ3πββπβ3ββ πΏ3
Comparison with standard Uniformization
Initial Condition: xβ0β(πΏ1)= π₯β0β(πΏ2)= π₯β0β(πΏ3)=πΌπππ‘; π₯β0β(π΅)=3β πΌπππ‘; where πΌπππ‘ is a variable with values in π
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Phosphorelay Network
β’β― CRN composed of more than 50 reactions and species!oβ― Initial condition such that all species with non zero concentration
have 105 moleculesoβ― Exploration of state space infeasibleoβ― Simulations are time consuming for such a biochemical system
π π’ππΈβ=?β[#π ππ:πΉπ π2]β[π,π]β π π’ππβ=?β[#πππ:πΉπ π2]β[π,π]β For πβ[0,8000]
Comparison of SEL and single stochastic simulation
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FGF Pathways
β’β― We have presented SEL with an approximate model checking algorithm based on the LNA
β’β― Our method can be useful for a fast stochastic characterization of biochemical systems or for stochastic analysis of systems too large to be checked with standard techniques.
β’β― Increasing the number of molecules the LNA is always a valid model assuming mass action kinetics, but can be accurate even far from the thermodynamic limit for a large class of CRNs
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Conclusion
β’β― Wallace, E. W. J., et al. "Linear noise approximation is valid over limited times for any chemical system that is sufficiently large." IET systems biology 6.4 (2012): 102-115.
β’β― Van Kampen, Nicolaas Godfried. Stochastic processes in physics and chemistry. Vol. 1. Elsevier, 1992. e
β’β― Gillespie, Daniel T. Deterministic limit of stochastic chemical kinetics. The Journal of Physical Chemistry B 113.6 (2009): 1640-1644.
β’β― Luca Cardelli, Marta Kwiatkowska, Luca Laurenti. Stochastic Analysis of Chemical Reaction Networks Using Linear Noise Approximation. arXiv preprint arXiv:1506.07861 (2015).
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Some References