hill kinetics meets p systemshinze/slides-wmc2007.pdf · introduction hill kinetics case study:...

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

Hill Kinetics Meets P SystemsA 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}@[email protected]

1Bio Systems Analysis GroupFriedrich Schiller University Jenawww.minet.uni-jena.de/csb2Computational Biology GroupSaarland Universitywww.zbi-saar.de

Eight Workshop onMembrane Computing

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


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

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Nor

mal

ised

con

cent

ratio

n

Time scale

Input1: 1Input2: 0

Input1: 1Input2: 1

Input1: 0Input2: 1

Input1: 0Input2: 0

Output

sensor

output regulatory circuit

signal

pTSM b2pCIRbPL*

Ptrc

PL*

PL*

PluxLux I

pAHLb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 5 10 15 20

x*x/(x*x+25)1-x*x/(x*x+25)

z yx zx

yz

NAND gate

0

1

0

0

10

1 1

1

1

1

0

&x

y

a b

50%

Θ

Θh

+

h−−

m = 2= 5

no

rmal

ised

ou

tpu

t co

nce

ntr

atio

n h

input concentration x

EffGeneRegGeneYRegGeneX

complex formation

AHL

lux I lux R lac I

lac Icl857gfp


ESIGNET – Research ProjectEvolving 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



Motivation and IntentionExploring Dynamical Behaviour of Gene Regulatory Networks

• Understanding biological reaction networks: essential task insystems biology

• Many coexisting approaches:analytic, stochastic, algebraic

• Each specifically emphasisescertain modelling aspects

• Emulating dynamical system behaviourbased on reaction kinetics=⇒ often key to network functions

• Reaction kinetics mostly specified foranalytic models based on ODE

• Combining advantages of approaches:transformation strategies, model shifting

• Example: Transformation ofHill Kinetics to P Systems


sensor

output regulatory circuit

signal

pTSM b2pCIRbPL*

Ptrc

PL*

PL*

PluxLux I

pAHLb

1

10

0

00

0

00

1

1

11

1

1

0

0

00

011

1

0 001 1

? ?

!

AHL

lux I lux R lac I

lac Icl857gfp

#

time


Biological Principles of Gene RegulationIntercellular Information Processing of Spatial Globality within Organisms

no/few gene product

gene expression

gene expression

genomic DNA

activation pathwaysignalling substances

gene product

signalling substances (inducers)can weak repression

can amplify activation

repression pathway

transcription factorenables gene expression

transcription factorinhibits 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 andsignalling substances forming gene regulatory networks



Modelling Approaches and Transformation Strategies

wetlab experimental data

verification

simulation

modelling

stochastic

algebraicanal

ytic

grammarsP systems

X machinescellular autom.

Petri netspi−calculusambient calc.

master eqn.Markov chainsGillespie

about discrete signal carriers, hierarchical

and modular com

positions, identifying functional units

considering randomness and probabilities to study

ranges of possible scenarios

pred

ictin

g dy

nam

ical

beh

avio

ur o

f spe

cies

equations

difference

transitions

correspond to

recursion steps

discretisewith respect to

time and/or space

equations

differential

parameters of reactionkinetics correspond

normalised

to probabilities

termrewritingsystems

differentiate /

molecular configurationswhose trace inter−

states correspond to

pretable as process

state−basedmachines /automata

resp. intermediateterms correspond

sentential forms

to states

processcalculi /models

processes or constraints;edges correspond to depen−

nodes correspond to

dencies between processes

networks

Bayesiancumulate /

extract

statistically

case studies

stochasticsimulationalgorithms

evaluating structural information

phenotypic

representation

of GRNs



Comparison of ApproachesSpecific 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 – 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 functionsh+, h− : R × R × N → R

• x ≥ 0: input concentration oftranscription 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)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 5 10 15 20

x*x/(x*x+25)1-x*x/(x*x+25)

50%

Θ

Θh

+

h−−

m = 2= 5

no

rmal

ised

ou

tpu

t co

nce

ntr

atio

n h

input concentration x


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 ratec2 · [GeneProduct ] with c2 > 0

• Differential equation forcorresponding 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 ]


A n

A 1

I 1

I p

Gene

GeneProduct


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,...,¬Ipwhere

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 ]⌋


A n

A 1

I 1

I p

Gene

GeneProduct


P Systems ΠHill – Definition of ComponentsΠHill = (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) 7→ 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



P Systems ΠHill – Dynamical BehaviourIteration 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 =kU

i=1(Ei,t+1 ∩ Lt )

Productst =kU

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 toa′ (reactants) and from b to b′ (products) according to functions fi

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



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 + ΘmA1

· . . . ·|Lt ∩ {(An,∞)}|m

|Lt ∩ {(An,∞)}|m + ΘmAn

·

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+


A n

A 1

I 1

I p

Gene

Gen

ePro

du

ct


Case Study: InverterGene Regulatory Networks as Computational Units

Input: concentration levels of transcription factor xOutput: concentration level of gene product y

y yx

x y

NOT gate

0 1

1 0&

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 < 301 for 10 ≤ t < 20; 30 ≤ t < 40

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

Nor

mal

ised

con

cent

ratio

n

Time scale

Input: 0 Input: 1 Input: 0 Input: 1

Output

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



Case Study: InverterDefinition 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

˛

˛

xr2 : s2 ¬a −→ ∅r3 : s3 EffectorGene −→ s3 y + s3 EffectorGene

˛

˛

¬ar4 : s4 y −→ ∅

f1(∆τ, Lt , m) =

$

∆τ · |Lt ∩ {(RegulatorGene, ∞)}| ·|Lt ∩ {(x,∞)}|m

|Lt ∩ {(x,∞)}|m + Θmx

%

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)

0 5 10 15 20 25 30 35 400

2000

4000

6000

8000

10000

Time scale

Num

ber

of m

olec

ules

Input: 0 Input: 1 Input: 0 Input: 1



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

z yx zx

yz

NAND gate

0

1

0

0

10

1 1

1

1

1

0

&x

y

a b

EffGeneRegGeneYRegGeneX

complex formation

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

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Nor

mal

ised

con

cent

ratio

n

Time scale

Input1: 1Input2: 0

Input1: 1Input2: 1

Input1: 0Input2: 1

Input1: 0Input2: 0

Output

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



Case Study: NAND GateDefinition 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˛

˛

xr2 : s2 ¬a −→ ∅r3 : s3 RegGeneY −→ s3 ¬b + s3 RegGeneY

˛

˛

yr4 : s4 ¬b −→ ∅r5 : s5 EffGene −→ s5 z + s5 EffGene

˛

˛

¬a,¬br6 : 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

0 5 10 15 20 25 30 35 400

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Time scale

Numb

er of

molec

ules



Case Study: RS Flip-FlopInput: concentration levels of transcription factors S, ROutput: concentration level of gene product Q

Q

S

RQRS Q

low active RS flip−flop

&

&

0

0

1

1

1

0

0

0

1

1

hold

−

S

a

b

R

EffGeneRegGeneSetStateRegGeneResetState

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

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Nor

mal

ised

con

cent

ratio

n

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) − QHill Kinetics Meets P Systems Thomas Hinze, Sikander Hayat, Thorsten Lenser, Naoki Matsumaru, Peter Dittrich


Case Study: RS Flip-FlopDefinition 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,¬br2 : s2 ¬a −→ ∅r3 : s3 RegGeneSetState −→ s3 ¬b + s3 RegGeneSetState

˛

˛

¬R,¬ar4 : s4 ¬b −→ ∅r5 : s5 EffGene −→ s5 Q + s5 EffGene

˛

˛

¬S,¬br6 : 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

0 5 10 15 20 25 30 35 400

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Time scale

Numb

er of

molec

ules



Wetlab Implementation of GRN-Based RS Flip-FlopExperimental 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 Universit y of Technology, BIOTEC laboratories.Thanks to J.J. Collins, W. Pompe, G. Rödel, K. Ostermann, L. B rusch for their support.



Wetlab Experimental Results

B

A

0

5.000

10.000

30.000

0

5.000

10.000

15.000

12 24 36 48 60 720

0

1

0

1

30°C

42°C

GF

P m

ean

(ave

rage

uni

ts)

Flip

−flo

p ou

tput

Set

ting

IPT

G,

Res

ettin

g

AH

L,

Res

et

Set

Res

et

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 60 720

0

1

0

1

30°C

42°C

GF

P m

ean

(ave

rage

uni

ts)

Flip

−flo

p ou

tput

Set

ting

IPT

G,

Res

ettin

g

AH

L,

Set

Res

et

Set

Store Store Store

low (0)

high (1)

time (hrs)

Repeated activation and deactivation of the toggle switch based on inducers and temperature. Temperaturewas switched every 24 hours. Cells were incubated with inducers for 12 hours, followed by growth for 12 hourswithout 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 Universit y of Technology, BIOTEC laboratories.Thanks to J.J. Collins, W. Pompe, G. Rödel, K. Ostermann, L. B rusch for their support.



Discussion and ConclusionDiscussion

• Gap between idealised models and observed wetlab data

• Consider GRN of the whole microorganism rather thanisolated part

• Granularity of simulation

• Undersatisfy problem: avoid conflicts between reactions atlow reactant amounts

Conclusion• P systems framework for applications in systems biology:

• Map reaction kinetics to P systems with dynamicalbehaviour

• Exemplified by transformation of Hill kinetics for GRNs

• Computability issues addressed by logic gates as simpleGRNs



Further WorkTheory

• 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? [email protected]


hill kinetics meets p systemshinze/slides-wmc2007.pdf · introduction hill kinetics case study:...

Documents