Hill Kinetics Meets P Systems A Case Study on Gene Regulatory - - PowerPoint PPT Presentation

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Hill Kinetics Meets P Systems

A Case Study on Gene Regulatory Networks as Computing Agents in silico and in vivo Thomas Hinze1 Sikander Hayat2 Thorsten Lenser1 Naoki Matsumaru1 Peter Dittrich1

{hinze,thlenser,naoki,dittrich}@minet.uni-jena.de s.hayat@bioinformatik.uni-saarland.de

1Bio Systems Analysis Group

Friedrich Schiller University Jena www.minet.uni-jena.de/csb

2Computational Biology Group

Saarland University www.zbi-saar.de Eight Workshop on Membrane Computing

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Outline

Hill Kinetics Meets P Systems

Introduction

• Research Project, Motivation, Intention
• Biological Principles of Gene Regulatory Networks (GRNs)
• Modelling Approaches, Transformation Strategies, Comparison

Hill Kinetics

• Definition and Discretisation
• P Systems ΠHill
• Dynamical Behaviour
• Introductory Example

Case Study: Computational Units

• Inverter
• NAND Gate
• RS Flip-Flop and Its Validation in vivo

Discussion, Conclusion, Further Work

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Normalised concentration Time scale Input1: 1 Input2: 0 Input1: 1 Input2: 1 Input1: 0 Input2: 1 Input1: 0 Input2: 0 Output sensor

• utput

regulatory circuit signal pTSM b2 pCIRb PL* Ptrc PL* PL* Plux Lux I pAHLb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 x*x/(x*x+25) 1-x*x/(x*x+25)

z y x z x y z

NAND gate 1 1 1 1 1 1 1

&

x y a b 50% Θ Θ h+ h−− m = 2 = 5 normalised output concentration h input concentration x

EffGene RegGeneY RegGeneX complex formation AHL lux I lux R lac I lac I cl857 gfp

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

ESIGNET – Research Project

Evolving Cell Signalling Networks in silico

European interdisciplinary research project

• University of Birmingham (Computer Science)
• TU Eindhoven (Biomedical Engineering)
• Dublin City University (Artificial Life Lab)
• University of Jena (Bio Systems Analysis)

Objectives

• Study the computational properties of GRNs
• Develop new ways to model and predict real GRNs
• Gain new theoretical perspectives on real GRNs

Computing Facilities

• Cluster of 33 workstations

(two Dual Core AMD OpteronTM 270 processors)

• Use of Dresden BIOTEC laboratories for in vivo studies

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

ESIGNET – Research Project

Evolving Cell Signalling Networks in silico

European interdisciplinary research project

• University of Birmingham (Computer Science)
• TU Eindhoven (Biomedical Engineering)
• Dublin City University (Artificial Life Lab)
• University of Jena (Bio Systems Analysis)

Objectives

• Study the computational properties of GRNs
• Develop new ways to model and predict real GRNs
• Gain new theoretical perspectives on real GRNs

Computing Facilities

• Cluster of 33 workstations

(two Dual Core AMD OpteronTM 270 processors)

• Use of Dresden BIOTEC laboratories for in vivo studies

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

ESIGNET – Research Project

Evolving Cell Signalling Networks in silico

European interdisciplinary research project

• University of Birmingham (Computer Science)
• TU Eindhoven (Biomedical Engineering)
• Dublin City University (Artificial Life Lab)
• University of Jena (Bio Systems Analysis)

Objectives

• Study the computational properties of GRNs
• Develop new ways to model and predict real GRNs
• Gain new theoretical perspectives on real GRNs

Computing Facilities

• Cluster of 33 workstations

(two Dual Core AMD OpteronTM 270 processors)

• Use of Dresden BIOTEC laboratories for in vivo studies

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Motivation and Intention

Exploring Dynamical Behaviour of Gene Regulatory Networks

• Understanding biological reaction networks: essential task in

systems biology

• Many coexisting approaches:

analytic, stochastic, algebraic

• Each specifically emphasises

certain modelling aspects

• Emulating dynamical system behaviour

based on reaction kinetics = ⇒ often key to network functions

• Reaction kinetics mostly specified for

analytic models based on ODE

• Combining advantages of approaches:

transformation strategies, model shifting

• Example: Transformation of

Hill Kinetics to P Systems

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

sensor

• utput

regulatory circuit signal pTSM b2 pCIRb PL* Ptrc PL* PL* Plux Lux I pAHLb

1 1 1 1 1 1 1 1 1 1 1 1 1

? ? !

AHL lux I lux R lac I lac I cl857 gfp

# time

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Motivation and Intention

Exploring Dynamical Behaviour of Gene Regulatory Networks

• Understanding biological reaction networks: essential task in

systems biology

• Many coexisting approaches:

analytic, stochastic, algebraic

• Each specifically emphasises

certain modelling aspects

• Emulating dynamical system behaviour

based on reaction kinetics = ⇒ often key to network functions

• Reaction kinetics mostly specified for

analytic models based on ODE

• Combining advantages of approaches:

transformation strategies, model shifting

• Example: Transformation of

Hill Kinetics to P Systems

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

sensor

• utput

regulatory circuit signal pTSM b2 pCIRb PL* Ptrc PL* PL* Plux Lux I pAHLb

1 1 1 1 1 1 1 1 1 1 1 1 1

? ? !

AHL lux I lux R lac I lac I cl857 gfp

# time

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Biological Principles of Gene Regulation

Intercellular Information Processing of Spatial Globality within Organisms

no/few gene product gene expression gene expression genomic DNA activation pathway signalling substances gene product signalling substances (inducers) can weak repression can amplify activation repression pathway transcription factor enables gene expression transcription factor inhibits gene expression genomic DNA

Inhibition (negative gene regulation) Activation (positive gene regulation)

regulator gene regulator gene effector gene effector gene

Feedback loops: gene products can act as transcription factors and signalling substances forming gene regulatory networks

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Modelling Approaches and Transformation Strategies

wetlab experimental data verification simulation modelling

stochastic algebraic analytic

grammars P systems X machines cellular autom. Petri nets pi−calculus ambient calc. master eqn. Markov chains Gillespie

about discrete signal carriers, hierarchical and modular compositions, identifying functional units considering randomness and probabilities to study ranges of possible scenarios predicting dynamical behaviour of species equations difference

transitions correspond to recursion steps discretise with respect to time and/or space

equations differential

parameters of reaction kinetics correspond normalised to probabilities

term rewriting systems

differentiate / molecular configurations whose trace inter− states correspond to pretable as process

state−based machines / automata

• resp. intermediate

terms correspond sentential forms to states

process calculi / models

processes or constraints; edges correspond to depen− nodes correspond to dencies between processes

networks Bayesian

cumulate / extract statistically case studies

stochastic simulation algorithms evaluating structural information phenotypic representation

• f GRNs

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Comparison of Approaches

Specific Advantages and Preferred Applications Analytic approaches

• Primarily adopted from chemical reaction kinetics
• Macroscopic view on species concentrations
• Differential equations from generation and consumption rates
• Continuous average concentration gradients
• Deterministic monitoring of temporal or spatial system behaviour

Stochastic approaches

• Aspects of uncertainty: incorporating randomness and probabilities
• Ranges of possible scenarios and their statistical distribution
• Facilitating direct comparison with wetlab experimental data
• Statistics: discovering correlations between network components

Algebraic approaches

• Discrete principle of operation
• Embedding/evaluating structural information
• Modularisation, hierarchical graduation of provided system information
• Molecular tracing
• Flexible instruments regarding level of abstraction

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Comparison of Approaches

Specific Advantages and Preferred Applications Analytic approaches

• Primarily adopted from chemical reaction kinetics
• Macroscopic view on species concentrations
• Differential equations from generation and consumption rates
• Continuous average concentration gradients
• Deterministic monitoring of temporal or spatial system behaviour

Stochastic approaches

• Aspects of uncertainty: incorporating randomness and probabilities
• Ranges of possible scenarios and their statistical distribution
• Facilitating direct comparison with wetlab experimental data
• Statistics: discovering correlations between network components

Algebraic approaches

• Discrete principle of operation
• Embedding/evaluating structural information
• Modularisation, hierarchical graduation of provided system information
• Molecular tracing
• Flexible instruments regarding level of abstraction

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Comparison of Approaches

Specific Advantages and Preferred Applications Analytic approaches

• Primarily adopted from chemical reaction kinetics
• Macroscopic view on species concentrations
• Differential equations from generation and consumption rates
• Continuous average concentration gradients
• Deterministic monitoring of temporal or spatial system behaviour

Stochastic approaches

• Aspects of uncertainty: incorporating randomness and probabilities
• Ranges of possible scenarios and their statistical distribution
• Facilitating direct comparison with wetlab experimental data
• Statistics: discovering correlations between network components

Algebraic approaches

• Discrete principle of operation
• Embedding/evaluating structural information
• Modularisation, hierarchical graduation of provided system information
• Molecular tracing
• Flexible instruments regarding level of abstraction

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Hill Kinetics – Sigmoid-Shaped Threshold Functions

• Model cooperative and competitive aspects of interacting

gene regulatory units dynamically and quantitatively

• Homogeneous and analytic
• Formulate relative intensity of

gene regulations by sigmoid- shaped threshold functions h+, h− : R × R × N → R

• x ≥ 0: input concentration of

transcription factor activating/ inhibiting gene expression

• Θ > 0: threshold (50% level)
• m ∈ N+: degree of regulation

activation (upregulation) h+(x, Θ, m) =

xm xm+Θm

inhibition (downregulation) h−(x, Θ, m) = 1 − h−(x, Θ, m)

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 x*x/(x*x+25) 1-x*x/(x*x+25) 50% Θ Θ h+ h−− m = 2 = 5 normalised output concentration h input concentration x

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Hill Kinetics – Network Composition

• Several interacting (competing) transcription factors

influence gene expression

• Activators Ai, inhibitors Ij and proportional factor c1 > 0:

determine production rate of a gene product

• Additional assumption of

linear spontaneous decay rate c2 · [GeneProduct] with c2 > 0

• Differential equation for

corresponding gene product: d [GeneProduct ] dt = ProductionRate − c2[GeneProduct ] = c1 · h+(A1, ΘA1, m) · . . . · h+(An, ΘAn, m) · (1 − h+(I1, ΘI1, m) · . . . · h+(Ip, ΘIp, m)) −c2 · [GeneProduct]

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

A n A 1 I 1 I p

Gene GeneProduct

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Hill Kinetics – Discretisation

• Discretisation with respect to value and time =

⇒ homologous term rewriting mechanism

• Large but finite pool of particles (multiset)
• Reaction: remove number of reactant particles,

simultaneously add all products

• Time-varying reaction rates =

⇒ variable stoichiometric factors

• Gene: limiting resource
• Reaction conditions: presence of

A1, . . . , An, absence (¬) of I1, . . . , Ip

• ∆τ ∈ R+: step length between

discrete time points t and t + 1 s Gene − → s GeneProduct + s Gene ˛ ˛

A1,...,An,¬I1,...,¬Ip

where s = ⌊∆τ · c1 · [Gene]· h+(A1, ΘA1, m) · . . . · h+(An, ΘAn, m)· ` 1 − h+(I1, ΘI1, m) · . . . · h+(Ip, ΘIp, m) ´ ⌋ u GeneProduct − → ∅ where u = ⌊∆τ · c2 · [GeneProduct ]⌋

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

A n A 1 I 1 I p

Gene GeneProduct

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

P Systems ΠHill – Definition of Components

ΠHill = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, . . . , rk, f1, . . . , fk, ∆τ, m)

• VGenes: alphabet of genes, VGenes ∩ VGeneProducts = ∅
• VGeneProducts: alphabet of gene products. Let V = VGenes ∪ VGeneProducts.
• Σ ⊆ VGeneProducts: output alphabet
• [1]1: skin membrane as only membrane
• L0 ∈ V: initial configuration, multiset over V
• ri: reaction rules with initial stoichiometric factors, i = 1, . . . , k
• Ei,0 ⊆ V × N: multiset of ri reactants at time point 0,
• Pi,0 ⊆ V × N: multiset of ri products at time point 0,
• TF i ∈ VGeneProducts: set of involved transcription factors
• ri ∈ Ei,0 × Pi,0 × P(TF i) whereas

A: all multisets over A, P(A): power set over A

• fi: corresponding function for updating stoichiometric factors
• fi : R+ × V × N+ → N with (∆τ, Lt, m) → s
• ∆τ ∈ R+: time step between discrete time points t and t + 1
• m ∈ N+: degree of all sigmoid-shaped functions h+ and h− used in fi

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

P Systems ΠHill – Definition of Components

ΠHill = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, . . . , rk, f1, . . . , fk, ∆τ, m)

• VGenes: alphabet of genes, VGenes ∩ VGeneProducts = ∅
• VGeneProducts: alphabet of gene products. Let V = VGenes ∪ VGeneProducts.
• Σ ⊆ VGeneProducts: output alphabet
• [1]1: skin membrane as only membrane
• L0 ∈ V: initial configuration, multiset over V
• ri: reaction rules with initial stoichiometric factors, i = 1, . . . , k
• Ei,0 ⊆ V × N: multiset of ri reactants at time point 0,
• Pi,0 ⊆ V × N: multiset of ri products at time point 0,
• TF i ∈ VGeneProducts: set of involved transcription factors
• ri ∈ Ei,0 × Pi,0 × P(TF i) whereas

A: all multisets over A, P(A): power set over A

• fi: corresponding function for updating stoichiometric factors
• fi : R+ × V × N+ → N with (∆τ, Lt, m) → s
• ∆τ ∈ R+: time step between discrete time points t and t + 1
• m ∈ N+: degree of all sigmoid-shaped functions h+ and h− used in fi

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

P Systems ΠHill – Definition of Components

ΠHill = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, . . . , rk, f1, . . . , fk, ∆τ, m)

• VGenes: alphabet of genes, VGenes ∩ VGeneProducts = ∅
• VGeneProducts: alphabet of gene products. Let V = VGenes ∪ VGeneProducts.
• Σ ⊆ VGeneProducts: output alphabet
• [1]1: skin membrane as only membrane
• L0 ∈ V: initial configuration, multiset over V
• ri: reaction rules with initial stoichiometric factors, i = 1, . . . , k
• Ei,0 ⊆ V × N: multiset of ri reactants at time point 0,
• Pi,0 ⊆ V × N: multiset of ri products at time point 0,
• TF i ∈ VGeneProducts: set of involved transcription factors
• ri ∈ Ei,0 × Pi,0 × P(TF i) whereas

A: all multisets over A, P(A): power set over A

• fi: corresponding function for updating stoichiometric factors
• fi : R+ × V × N+ → N with (∆τ, Lt, m) → s
• ∆τ ∈ R+: time step between discrete time points t and t + 1
• m ∈ N+: degree of all sigmoid-shaped functions h+ and h− used in fi

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

P Systems ΠHill – Dynamical Behaviour

Iteration scheme

• Updating system configuration Lt and
• Stoichiometric factors of reaction rules ri ∈ Ei,t × Pi,t × P(TF i)
• Starting from initial configuration L0

Lt+1 = Lt ⊖ Reactantst ⊎ Productst with Reactantst =

k

U

i=1

(Ei,t+1 ∩ Lt) Productst =

k

U

i=1

(Pi,t+1 ∩ Lt) Ei,t+1 = ˘ (e, a′) | (e, a) ∈ Ei,t ∧ a′ = fi(∆τ, Lt, m) ¯ Pi,t+1 = ˘ (q, b′) | (q, b) ∈ Pi,t ∧ b′ = fi(∆τ, Lt, m) ¯ Informally:

• Specification of Ei,t+1 and Pi,t+1: all reactants e and products q remain

unchanged over time, just their stoichiometric factors updated from a to a′ (reactants) and from b to b′ (products) according to functions fi Computational output

• Function output : N → N with output(t) = |Lt ∩ {(w, ∞) | w ∈ Σ}|

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

P Systems ΠHill – Introductory Example

ΠHill,GRNunit = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, r2, f1, f2, ∆τ, m) VGeneProducts = {A1, . . . , An, ¬I1, . . . , ¬Ip, GeneProduct} VGenes = {Gene} Σ = {GeneProduct} L0 = {(Gene, g), (A1, a1), . . . , (An, an), (¬I1, i1), . . . , (¬Ip, ip)} r1 : s1 Gene − → s1 GeneProduct + s1 Gene ˛ ˛

A1,...,An,¬I1,...,¬Ip

r2 : s2 GeneProduct − → ∅ f1(∆τ, Lt, m) = ⌊∆τ · |Lt ∩ {(Gene, ∞)}| · |Lt ∩ {(A1, ∞)}|m |Lt ∩ {(A1, ∞)}|m + Θm

A1

· . . . · |Lt ∩ {(An, ∞)}|m |Lt ∩ {(An, ∞)}|m + Θm

An

· 1− |Lt ∩ {(¬I1, ∞)}|m |Lt ∩ {(¬I1, ∞)}|m + Θm

¬I1

·...· |Lt ∩ {(¬Ip, ∞)}|m |Lt ∩ {(¬Ip, ∞)}|m + Θm

¬Ip

! ⌋ f2(∆τ, Lt, m) = ⌊∆τ · |Lt ∩ {(GeneProduct, ∞)}|⌋, ∆τ ∈ R+, m ∈ N+

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

A n A 1 I 1 I p

Gene GeneProduct

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Case Study: Inverter

Gene Regulatory Networks as Computational Units

Input: concentration levels of transcription factor x Output: concentration level of gene product y

y y x x y

NOT gate 1 1

&

x a

RegulatorGene EffectorGene

Dynamical behaviour depicted for m = 2, Θj = 0.1, j ∈ {x, a},

a(0) = 0, y(0) = 0, x(t) =  0

for

0 ≤ t < 10; 20 ≤ t < 30 1

for

10 ≤ t < 20; 30 ≤ t < 40

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 Normalised concentration Time scale Input: 0 Input: 1 Input: 0 Input: 1 Output

˙ a = h+(x, Θx, m) − a ˙ y = h−(a, Θa, m) − y

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Case Study: Inverter

Definition and Simulation of Corresponding P System ΠHill,GRNinv.

ΠHill,GRNinv. = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, . . . , r4, f1, . . . , f4, ∆τ, m) VGenes = {RegulatorGene, EffectorGene} VGeneProducts = {x, y, ¬a} Σ = {y} L0 = {(RegulatorGene,rg), (EffectorGene,eg), (x,x0), (y,y0), (¬a,a0)} r1 : s1 RegulatorGene − → s1 ¬a + s1 RegulatorGene ˛ ˛

x

r2 : s2 ¬a − → ∅ r3 : s3 EffectorGene − → s3 y + s3 EffectorGene ˛ ˛

¬a

r4 : s4 y − → ∅ f1(∆τ, Lt, m) = $ ∆τ · |Lt ∩ {(RegulatorGene, ∞)}| · |Lt ∩ {(x, ∞)}|m |Lt ∩ {(x, ∞)}|m + Θm

x

% f2(∆τ, Lt, m) = ⌊∆τ · |Lt ∩ {(¬a, ∞)}|⌋ f3(∆τ, Lt, m) = $ ∆τ · |Lt ∩ {(EffectorGene, ∞)}| · 1 − |Lt ∩ {(¬a, ∞)}|m |Lt ∩ {(¬a, ∞)}|m + Θm

¬a

!% f4(∆τ, Lt, m) = ⌊∆τ · |Lt ∩ {(y, ∞)}|⌋

• Dynamical behaviour depicted for m = 2, ∆τ = 0.1, Θj = 500, j ∈ {x, ¬a}
• rg = 10, 000, eg = 10, 000, x0 = 0, y0 = 0, a0 = 0 (MATLAB, P system iteration scheme)

5 10 15 20 25 30 35 40 2000 4000 6000 8000 10000 Time scale Number of molecules Input: 0 Input: 1 Input: 0 Input: 1

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Case Study: NAND Gate

Input: concentration levels of transcription factors x (inp.1), y (inp.2) Output: concentration level of gene product z

z y x z x y z

NAND gate 1 1 1 1 1 1 1

&

x y a b

EffGene RegGeneY RegGeneX complex formation

Dynamical behaviour depicted for m = 2, Θj = 0.1, j ∈ {x, y, a, b}

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Normalised concentration Time scale Input1: 1 Input2: 0 Input1: 1 Input2: 1 Input1: 0 Input2: 1 Input1: 0 Input2: 0 Output

˙ a = h+(x, Θx, m) − a ˙ b = h+(y, Θy, m) − b ˙ z = 1 − h+(a, Θa, m) · h+(b, Θb, m) − z

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Case Study: NAND Gate

Definition and Simulation of P System ΠHill,GRNnand

ΠHill,GRNnand = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, . . . , r6, f1, . . . , f6, ∆τ, m) VGenes = {RegGeneX, RegGeneY, EffGene} VGeneProducts = {x, y, z, ¬a, ¬b} Σ = {z} L0 = {(RegGeneX, rgx), (RegGeneY, rgy), (EffGene, eg), (x, x0), (y, y0), (z, z0), (¬a, a0), (¬b, b0)} r1 : s1 RegGeneX − → s1 ¬a + s1 RegGeneX ˛ ˛

x

r2 : s2 ¬a − → ∅ r3 : s3 RegGeneY − → s3 ¬b + s3 RegGeneY ˛ ˛

y

r4 : s4 ¬b − → ∅ r5 : s5 EffGene − → s5 z + s5 EffGene ˛ ˛

¬a,¬b

r6 : s6 z − → ∅ . . .

• Simulation result (MATLAB, P system iteration scheme)
• Dynamical behaviour depicted for m = 2, ∆τ = 0.1, Θj = 500, j ∈ {x, y, ¬a, ¬b}
• rgx = 10, 000, rgy = 10, 000, eg = 10, 000, x0 = 0, y0 = 0, z0 = 0, a0 = 0, b0 = 0

5 10 15 20 25 30 35 40 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time scale Number of molecules

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Case Study: RS Flip-Flop

Input: concentration levels of transcription factors S, R Output: concentration level of gene product Q

Q S R Q R S Q

low active RS flip−flop

& &

1 1 1 1 1 hold −

S a b R

EffGene RegGeneSetState RegGeneResetState

Dynamical behaviour depicted for m = 2, Θj = 0.1, j ∈ {a, b, R, S}

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Normalised concentration Time scale Set

• S: 0, -R: 1

Store

• S: 0, -R: 0

Reset

• S: 1, -R: 0

Store

• S: 0, -R: 0

Output

˙ a = 1 − h+(b, Θb, m) · h−(S, ΘS, m) − a ˙ b = 1 − h+(a, Θa, m) · h−(R, ΘR, m) − b ˙ Q = h+(b, Θb, m) · h−(S, ΘS, m) − Q

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Case Study: RS Flip-Flop

Definition and Simulation of P System ΠHill,GRNrsff

ΠHill,GRNrsff = (VGenes, VGeneProducts, Σ, [1]1, L0, r1, . . . , r6, f1, . . . , f6, ∆τ, m) VGenes = {RegGeneResetState, RegGeneSetState, EffGene} VGeneProducts = {Q, ¬S, ¬R, ¬a, ¬b} Σ = {Q} L0 = {(RegGeneResetState, rgr), (RegGeneSetState, rgs), (EffGene, eg), (Q, q0), (S, ss0), (R, rs0), (¬a, a0), (¬b, b0)} r1 : s1 RegGeneResetState − → s1 ¬a + s1 RegGeneResetState ˛ ˛

¬S,¬b

r2 : s2 ¬a − → ∅ r3 : s3 RegGeneSetState − → s3 ¬b + s3 RegGeneSetState ˛ ˛

¬R,¬a

r4 : s4 ¬b − → ∅ r5 : s5 EffGene − → s5 Q + s5 EffGene ˛ ˛

¬S,¬b

r6 : s6 Q − → ∅ . . .

• Simulation Result (MATLAB, P system iteration scheme)
• Dynamical behaviour depicted for m = 2, ∆τ = 0.1, Θj = 500, j ∈ {¬a, ¬b, ¬R, ¬S}
• rgr = 10, 000, rgs = 10, 000, eg = 10, 000, q0 = 0, ss0 = 0, rs0 = 0, a0 = 0, b0 = 0

5 10 15 20 25 30 35 40 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time scale Number of molecules

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Wetlab Implementation of GRN-Based RS Flip-Flop

Experimental Setup

• in vivo system (bistable toggle switch in Vibrio fischeri)

mimics RS flip-flop

• Encoding of all genes using two constructed plasmids
• Quantification of its performance using flow cytometry
• Presence or absence of inducers AHL and IPTG acts as

input signals, green fluorescent protein (gfp) as output

Collaboration with S. Hayat, at this time Dresden University of Technology, BIOTEC laboratories. Thanks to J.J. Collins, W. Pompe, G. Rödel, K. Ostermann, L. Brusch for their support. Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Wetlab Experimental Results

B A

5.000 10.000 30.000 5.000 10.000 15.000 12 24 36 48 60 72 1 1 30°C 42°C GFP mean (average units) Flip−flop output Setting IPTG, Resetting AHL, Reset Set Reset Store Store Store low (0) high (1) time (hrs) GFP mean: 188 units after 12 hrs GFP mean: 312 units after 24 hrs GFP mean: 32.178 units after 36 hrs GFP mean: 4.106 units after 48 hrs 644 units GFP mean: after 60 hrs GFP mean: 373 units after 72 hrs GFP mean: 14.803 units GFP mean: 4.856 units GFP mean: 1.108 units GFP mean: 601 units GFP mean: 15.621 units GFP mean: 7.073 units after 12 hrs after 24 hrs after 36 hrs after 48 hrs after 60 hrs after 72 hrs 12 24 36 48 60 72 1 1 30°C 42°C GFP mean (average units) Flip−flop output Setting IPTG, Resetting AHL, Set Reset Set Store Store Store low (0) high (1) time (hrs)

Repeated activation and deactivation of the toggle switch based on inducers and temperature. Temperature was switched every 24 hours. Cells were incubated with inducers for 12 hours, followed by growth for 12 hours without inducers, initially kept at 30◦C (A) and 42◦C (B). The cells successfully switched states thrice. Collaboration with S. Hayat, at this time Dresden University of Technology, BIOTEC laboratories. Thanks to J.J. Collins, W. Pompe, G. Rödel, K. Ostermann, L. Brusch for their support. Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Discussion and Conclusion

Discussion

• Gap between idealised models and observed wetlab data
• Consider GRN of the whole microorganism rather than

isolated part

• Granularity of simulation
• Undersatisfy problem: avoid conflicts between reactions at

low reactant amounts Conclusion

• P systems framework for applications in systems biology:
• Map reaction kinetics to P systems with dynamical

behaviour

• Exemplified by transformation of Hill kinetics for GRNs
• Computability issues addressed by logic gates as simple

GRNs

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich

Introduction Hill Kinetics Case Study: Computational Units Discussion, Conclusion, Further Work

Further Work

Theory

• Integrate structural information into ΠHill,

extend symbol objects to string objects

• Introduce matching strategies based on string objects
• Unify ΠHill with ΠCSN (presented at WMC7)

Computational Applications

• Design GRNs for solution to NP-complete problems and for

emulating behaviour of different automata (first results)

• Emerge artificial GRNs by evolutionary computation

Wetlab

• Coupling of several GRN-based computational units in vivo
• Coping with side effects (signal weakening)
• Grant application in preparation.

Interested in participation? hinze@minet.uni-jena.de

Hill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich