vs. 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | - - PowerPoint PPT Presentation

Boolean versus continuous dynamics on small and large model networks

vs.

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 1

Biological background: Gene regulatory networks

Replication Transcription Translation

DNA mRNA Protein

based on: D. Del Vecchio & E. Sontag Dynamics and Control of Synthetic Bio-molecular Networks Proceedings of Americal Control Conference, 2007 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 2

Biological background: Gene regulatory networks

Replication Transcription Translation

DNA mRNA Protein DNA mRNA

based on: D. Del Vecchio & E. Sontag Dynamics and Control of Synthetic Bio-molecular Networks Proceedings of Americal Control Conference, 2007 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 2

Boolean Networks

• Toy-model: on-off states
• Parallel update
• Deterministic dynamics

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 3

Boolean Networks

• Toy-model: on-off states
• Parallel update
• Deterministic dynamics

Dynamics of individual nodes depends on update functions: In

F C1 C2 R

0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 3

Boolean and continuous dynamics for gene regulatory networks

Boolean model

σi = {0, 1} σi(t + 1) = Fi(σ(t))

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 4

Boolean and continuous dynamics for gene regulatory networks

Boolean model

σi = {0, 1} σi(t + 1) = Fi(σ(t))

Continuous model

mRNAi

Pi

mRNAi

P mRNA P mRNA P mRNA P mRNA P mRNA P mRNA P mRNA P mRNA P mRNA P mRNA

m ˙ RNAi = Fi(P) − αmRNAi ˙ Pi =

βmRNAi − δPi

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 4

Hill function

Regulation by single gene

Pi

mRNAi

m ˙ RNAi = Fi(P) − αmRNAi

1 0.5 1 1.5

f+(P) P

n = 1 n = 3 n = 10

Fi(P) = f +(P) = Pn Pn + kn Generalization to more inputs: Hill cubes

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 5

Hill cubes

Standardized method for converting any Boolean function into a continuous function

genea geneb

• utput

1 1 1 1 1

• D. Wittmann et al.

Transforming boolean models to continuous models: Methodology and application to t-cell receptor signaling. BMC Systems Biology, 3 (1) (2009)

F(Pa, Pb) = f +(Pa) · f −(Pb)

0.5 1 0 0.5 1 1

a AND NOT b

Pa Pb

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 6

Comparison:

Fixed points and oscillations

Boolean

• Fixed points

000...

• Cycles

100... 011...

Continuous

• Fixed points

time

concentration

• Oscillations

time

concentration 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 7

Generalized modelling approach

• Steady-state concentrations: mRNAi

∗, Pi ∗

• Normalized state variables: ri =

mRNAi mRNAi ∗ , pi = Pi Pi ∗ and functions: ˜

fj(pi) = Fj(Pi

∗pi)

Fj(P∗

i ) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 8

Generalized modelling approach

• Steady-state concentrations: mRNAi

∗, Pi ∗

• Normalized state variables: ri =

mRNAi mRNAi ∗ , pi = Pi Pi ∗ and functions: ˜

fj(pi) = Fj(Pi

∗pi)

Fj(P∗

i )

JN=2 =

    α α β β          −1 −1

• ∂˜

fa ∂pa ∂˜ fa ∂pb ∂˜ fb ∂pa

• 1

1

• −1

−1

• 

   

• α

β ≡ λ : ratio of time scales between mRNA and protein dynamics

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 8

Generalized modelling approach

• Steady-state concentrations: mRNAi

∗, Pi ∗

• Normalized state variables: ri =

mRNAi mRNAi ∗ , pi = Pi Pi ∗ and functions: ˜

fj(pi) = Fj(Pi

∗pi)

Fj(P∗

i )

JN=2 =

    α α β β          −1 −1

• ∂˜

fa ∂pa ∂˜ fa ∂pb ∂˜ fb ∂pa

• 1

1

• −1

−1

• 

   

• α

β ≡ λ : ratio of time scales between mRNA and protein dynamics

• ∂˜

fj ∂pi ≡ ˜

fjpi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

• T. Gross, U. Feudel

Generalized models as a universal approach to the analysis of nonlinear dynamical systems Physical Review E 73 (1) (2006) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 8

Simple Loops

• Even loops: even number of inhibitors
• Odd loops: odd number of inhibitors

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 9

Simple Loops

Example: Three gene network

Even loop: Odd loop:

f3p2 f1p3 ~ f2p1 ~ ~

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 10

Simple Loops

Example: Three gene network

Even loop: Odd loop:

f3p2 f1p3 ~ f2p1 ~ ~

even even even

• dd
• dd
• dd

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 10

Simple Loops

Example: Three gene network

Even loop: Odd loop:

f3p2 f1p3 ~ f2p1 ~ ~

even even even

• dd
• dd
• dd

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 10

Loops with an additional input

x = 1

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 11

Loops with an additional input

x = 3

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 11

Loops with an additional input

Example: Two gene network

genea geneb

• E. Gehrmann, B. Drossel

Boolean versus continuous dynamics

• n simple two-gene modules

Physical Review E 82 (4) (2010)

4 2 2 4

f

• apa

4 2 2 4

f

• bpa

4 2 2 4

f

• apb

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 12

Loops with an additional input

Example: Two gene network

genea geneb

• E. Gehrmann, B. Drossel

Boolean versus continuous dynamics

• n simple two-gene modules

Physical Review E 82 (4) (2010)

4 2 2 4

f

• apa

4 2 2 4

f

• bpa

4 2 2 4

f

• apb

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 12

Loops with an additional input

Example: Two-gene network

genea geneb

+

• E. Gehrmann, B. Drossel

Boolean versus continuous dynamics

• n simple two-gene modules

Physical Review E 82 (4) (2010)

˜ fjpi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

fapa fapb ~ ~

Oscillations

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

Loops with an additional input

Example: Two-gene network

genea geneb

+

• E. Gehrmann, B. Drossel

Boolean versus continuous dynamics

• n simple two-gene modules

Physical Review E 82 (4) (2010)

˜ fjpi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

fapa fapb ~ ~ a N O R b

Boolean cycle Oscillations

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

Loops with an additional input

Example: Two-gene network

genea geneb

+

• E. Gehrmann, B. Drossel

Boolean versus continuous dynamics

• n simple two-gene modules

Physical Review E 82 (4) (2010)

˜ fjpi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

fapa fapb ~ ~ a A N D b a N O R b

Boolean cycle Oscillations

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

Loops with an additional input

Example: Two-gene network

genea geneb

+

• E. Gehrmann, B. Drossel

Boolean versus continuous dynamics

• n simple two-gene modules

Physical Review E 82 (4) (2010)

˜ fjpi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

fapa fapb ~ ~ a A N D N O T b ( N O T a ) A N D b a A N D b a N O R b

Boolean cycle Oscillations

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

N-gene networks with self-input or crosslink

x = 1

• E. Ackermann, E.M. Weiel, T. Pfaff, B. Drossel

Boolean versus continuous dynamics im modules with two feedback loops In preparation

fi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

fxeff fNeff

HB HB SNB

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 14

N-gene networks with self-input or crosslink

x = 3

• E. Ackermann, E.M. Weiel, T. Pfaff, B. Drossel

Boolean versus continuous dynamics im modules with two feedback loops In preparation

fi =

∈ [ 0, n]

if protein i is an activator

∈ [−n, 0]

if protein i is an inhibitor

fxeff fNeff

SNB HB HB

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 14

Hamming distance

Example: Two-gene network with F = a NOR b

genea geneb

+ Boolean cycle

1 2 3

states

1

node

01 10 11 00 00

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 15

Hamming distance

Example: Two-gene network with F = a NOR b

genea geneb

+ Boolean cycle

1 2 3

states

1

node

01 10 11 00 00

mRNAa Pa mRNAb Pb 0.0 0.5 1.0 1.5 5 10 15 20 25 30

concentration time

01 10 11 00 00

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 15

Hamming distance

Example: Three-gene network with F = NOT b AND c

genea geneb genec

Boolean cycle

1 2 3

states

1 2

node

100 101 010 110 001 011 111 000

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 16

Hamming distance

Example: Three-gene network with F = NOT b AND c

genea geneb genec

Boolean cycle

1 2 3

states

1 2

node

100 101 010 110 001 011 111 000

mRNAb Pb mRNAa Pa mRNAc Pc

Time

Concentration

Continuous oscillations

100 101 010 110 001 011

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 16

Conclusions Part I

• Boolean vs. continuous dynamics
• Conditions for oscillations in terms of
• regulating functions’ signs
• steepness of response functions

⇒ Not size and topology, but dynamical features of a network are relevant

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 17

Conclusions Part I

• Boolean vs. continuous dynamics
• Conditions for oscillations in terms of
• regulating functions’ signs
• steepness of response functions

⇒ Not size and topology, but dynamical features of a network are relevant

• Hamming distance = 1: Cycle found in Boolean dynamics are in continuous model
• Hamming distance > 1: Intermediate states must not coincide with fixed point
• Assumption: For entirely reliable trajectories the Boolean description reflects

continuous dynamics

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 17

Entirely reliable trajectories

Hamming distance h = 1 between to subsequent states: Only one nodes flips per time step

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 18

Three general types of dynamics under random update schedule

Stochastic dynamics “Checkpoint” states Entirely reliable trajectory

Our interest is best possible case: Entirely reliable trajectories with Hamming distance h = 1

• T. P

. Peixoto, B. Drossel Boolean networks with reliable dynamics. Physical Review E 80 (5) (2009) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 19

Entirely reliable trajectories Hamming distance h = 1 states Boolean node

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 20

Entirely reliable trajectories Hamming distance h = 1 states Boolean time Continuous Concentration

1

node node

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 20

Entirely reliable trajectories Hamming distance h = 1 states Boolean node time Continuous Concentration

1

node

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 20

Method

• 1. Variables

Number of nodes: N Length of trajectory: L Hamming distance: h

• 2. Trajectory
• 4. Continuous model
• 5. Time series
• 6. Comparison
• 3. Topology & Functions

Fi = F(σj,σk,...)

Boolean Continuous 0 1 1 1 1 0 … 0 0 1 1 0 0 … 1 1 1 0 0 0 …

...

Hill coefficient n

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 21

Results: Hamming distance h = 1

0.00 0.20 0.40 0.60 0.80 1.00 2.5 3.0 3.5 4.0 Proportion of trajectories in agreement

Hill coefficient n

N= 10 N= 15 N= 20 N= 30 N= 50 N=100

Variation: Number of nodes N (with L = 2N)

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 22

Results: Hamming distance h = 1

0.00 0.20 0.40 0.60 0.80 1.00 2.5 3.0 3.5 4.0 Proportion of trajectories in agreement

Hill coefficient n

N= 10 N= 15 N= 20 N= 30 N= 50 N=100

Variation: Number of nodes N (with L = 2N)

0.00 0.20 0.40 0.60 0.80 1.00 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Proportion of trajectories in agreement

Hill coefficient n

L=2N L=3N L=4N L=5N

Variation: Length of trajectory L (with N = 10)

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 22

Results: Hamming distance h > 1

0.0 0.2 0.4 0.6 0.8 1.0 1.0 3.0 5.0 7.0 9.0 Proportion of trajectories in agreement

Hill coefficient n

h=1.0 h=1.1 h=1.2 h=1.3 h=1.4 h=1.5

Variation: Hamming distance h (N = 10, L = 20)

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 23

Results: Hamming distance h > 1

0.0 0.2 0.4 0.6 0.8 1.0 1.0 3.0 5.0 7.0 9.0 Proportion of trajectories in agreement

Hill coefficient n

h=1.0 h=1.1 h=1.2 h=1.3 h=1.4 h=1.5

Variation: Hamming distance h (N = 10, L = 20)

Features of robust trajectories [1]

• “Catcher states“: Only one node

changes its state

• Activity states are kept for an

extended time

[1] S. Braunewell & S. Bornholdt Superstability of the yeast cell-cycle dynamics: Ensuring causality in the presence of biochemical stochasticity Journal of Theoretical Biology, 2007 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 23

Results: Features of consistent trajectories Duration of node states must not be too short

… 1 1 1 1 1 1 1 1 1 1 … … 0 0 0 1 1 1 1 1 1 1 … … 0 0 0 0 0 0 0 0 0 0 … … 0 1 1 1 0 1 1 1 1 0 … … 0 0 0 0 0 1 1 1 0 0 … … 0 0 0 0 0 0 0 1 1 1 … … 0 0 0 0 0 0 1 1 1 1 … … 0 0 0 0 0 0 0 0 0 0 … … 0 0 0 0 0 0 0 0 0 0 … … 1 1 0 0 0 0 0 0 0 0 ...

N = 10, L = 20, h = 1.1

Time 9 8 7 6 5 4 3 2 1

Example: Duration of node states too short and simultaneous update of 2 nodes

⇒ No oscillations in continuous model

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 24

Results: Features of consistent trajectories Duration of node states must not be too short

… 1 1 1 1 1 1 1 1 1 1 … … 0 0 0 1 1 1 1 1 1 1 … … 0 0 0 0 0 0 0 0 0 0 … … 0 1 1 1 0 1 1 1 1 0 … … 0 0 0 0 0 1 1 1 0 0 … … 0 0 0 0 0 0 0 1 1 1 … … 0 0 0 0 0 0 1 1 1 1 … … 0 0 0 0 0 0 0 0 0 0 … … 0 0 0 0 0 0 0 0 0 0 … … 1 1 0 0 0 0 0 0 0 0 ...

N = 10, L = 20, h = 1.1

Time 9 8 7 6 5 4 3 2 1

Example: Duration of node states too short and simultaneous update of 2 nodes

⇒ No oscillations in continuous model

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 24

Conclusions Part II

• Entirely reliable trajectories

⇒ Boolean description reflects continuous dynamics

• Increased Hamming distance

⇒ Agreement of continuous dynamics with Boolean dynamics

becomes worse

• Features of robust trajectories
• Catcher states
• Duration of node states are not too short
• Biological relevance: Processes in biological networks must be reliable

despite fluctuations affecting the timing of different steps

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 25

Thank you for your attention

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 26

Thank you for your attention Special Thanks to . . .

Eva Ackermann (Gehrmann)

• Dr. Tiago Peixoto (University of Bremen)

Torsten Pfaff Eva Marie Weiel

14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 26