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Comparing Discrete and Piecewise Affine Models

• f Gene Regulatory Networks

Shahrad Jamshidi, Heike Siebert, Alexander Bockmayr Article in Biosystems, 2013 PhD S. Jamshidi, April 2013 Porquerolles, June 2013

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i.

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

xb xa Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

xb xa θa θ

1 b

θ

2 b

domain

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x0), Fb(x0))

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x1), Fb(x1))

x1 Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x1), Fb(x1))

x1

focal point

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x1), Fb(x1))

x1 x2

(Fa(x2), Fb(x2))

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x1), Fb(x1))

x1 x2

(Fa(x2), Fb(x2))

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x1), Fb(x1))

x1 x2

(Fa(x2), Fb(x2)) (Fa(x3), Fb(x3))

x3 Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

x0

(Fa(x1), Fb(x1))

x1 x2

(Fa(x2), Fb(x2)) (Fa(x3), Fb(x3))

x3 Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Piecewise Affine Differential Equations (PADE)

A variable xi represents the protein concentration expressed by gene i. Each variable has thresholds to indicate activation/inhibition

A B

˙ xa = λa(Fa(x) − xa) ˙ xb = λb(Fb(x) − xb) piecewise-affine differential equations

??

Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i.

Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

A B

Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12

fcI(00)fcro(00)=

Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Thomas formalism

• R. Thomas (1973) introduces discrete variables, qi to describe

different expression levels of gene i. Each variable updates by the function fi(q). qaqb fa(q) fb(q) 00 1 2 01 1 2 02 1 10 11 1 12 1

00 10 01 11 02 12

The state transition graph (STG) describes discrete dynamics.

Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model.

Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Discretising a PADE

Snoussi (1989) discretises a PADE into a Thomas model. Assign a discrete state to each domain. A domain and its focal point is like a state and its update.

00 10 01 11 02 12 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

Overview

Thomas PADE STG

Snoussi

Model Dynamics

Discrete DEs

(Proteins) (Genes)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds.

??

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds.

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds.

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics.

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics.

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Threshold Dynamics

De Jong et al (2004) showed that solutions can continue on thresholds. The intersection of two threshold hyperplanes has its own dynamics. All PADE dynamics can be represented discretely in a qualitative transition graph (QTG).

De Jong, H. et al; Qualitative Simulation of Genetic Regulatory Networks Using Piecewise-Linear Models. (2004)

Overview

Thomas PADE STG QTG

d e J

• n

g Snoussi

Model Dynamics

?

Discrete DEs

(Proteins) (Genes)

Overview

Thomas PADE STG QTG

d e J

• n

g Snoussi

Model Dynamics

?

Discrete DEs

(Proteins) (Genes)

Can the discrete model describe the behaviour on the thresholds?

PADEfying Thomas

Creating a PADE from a Thomas model:

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value.

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states.

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states. A state and its update is like a domain and its focal point.

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states. A state and its update is like a domain and its focal point.

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states. A state and its update is like a domain and its focal point.

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states. A state and its update is like a domain and its focal point.

00 10 01 11 02 12

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states. A state and its update is like a domain and its focal point.

PADEfying Thomas

Creating a PADE from a Thomas model: Each state takes its continuous value. Introduce thresholds that lie between these states. A state and its update is like a domain and its focal point.

Overview

Thomas PADE STG QTG

d e J

• n

g Snoussi PADEfy

Model Dynamics

?

Discrete DEs

(Proteins) (Genes)

One system provides two dynamics. What is their relationship?

Theorem

Local transitions can be mapped to each other

01 11 02 12 STG QTG Jamshidi S., Siebert H., Bockmayr, A.: Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks. Biosystems 112/2, 171-79, 2013.

Theorem

Local transitions can be mapped to each other

01 11 02 12 STG QTG Jamshidi S., Siebert H., Bockmayr, A.: Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks. Biosystems 112/2, 171-79, 2013.

Theorem

Local transitions can be mapped to each other

01 11 02 12 STG QTG Jamshidi S., Siebert H., Bockmayr, A.: Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks. Biosystems 112/2, 171-79, 2013.

Theorem

Local transitions can be mapped to each other

01 11 02 12 STG QTG Jamshidi S., Siebert H., Bockmayr, A.: Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks. Biosystems 112/2, 171-79, 2013.

Theorem

Local transitions can be mapped to each other

01 11 02 12 STG QTG Jamshidi S., Siebert H., Bockmayr, A.: Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks. Biosystems 112/2, 171-79, 2013.

Theorem

Local transitions can be mapped to each other

01 11 02 12 STG QTG Jamshidi S., Siebert H., Bockmayr, A.: Preservation of dynamic properties in qualitative modeling frameworks for gene regulatory networks. Biosystems 112/2, 171-79, 2013.

Theorem

Given QTG(A) = (D, T ) and STG(f ) = (Q, E), let D ∈ D, D′ ⊂ ∂D. Let I (resp. I ′) be the set of singular variables in D (resp. D′). Then: (1) (D, D′) ∈ T if and only if Ψ(D) = 0 and for all i ∈ I ′ \ I there exists q ∈ H(D) and q′ ∈ H(D′) \ H(D) with qi = q′

i and (q, q′) ∈ E.

(2) (D′, D) ∈ T if and only if Ψ(D) = 0 and for all i ∈ I ′ \ I there exists q ∈ H(D) and q′ ∈ H(D′) \ H(D) such that qi = q′

i, q′ j = qj

for all j = i and (q, q′) / ∈ E. (3) Ψ(D) = 0 if and only if for all i ∈ I one of the following conditions holds (where ei = (0, . . . , 1, . . . , 0) is the i-th unit vector in Rn):

1

there exist q, q′ ∈ H(D) with q′ = q + ei such that (q, q′) ∈ E and (q′, q) ∈ E.

2

there exist q, q′ ∈ H(D) with q′ = q + ei such that (q, q′) / ∈ E and (q′, q) / ∈ E.

3

there exist q, q′ ∈ H(D) and ˜ q, ˜ q′ ∈ H(D) with q′ = q + ei, ˜ q′ = ˜ q + ei such that both (q, q′) ∈ E, (q′, q) / ∈ E, and (˜ q′, ˜ q) ∈ E, (˜ q, ˜ q′) / ∈ E.

Illustration

D q q′ (b) D q q′ (a) D q q′ (c) ˜ q ˜ q′

Overview

Thomas PADE STG QTG

d e J

• n

g Snoussi PADEfy

Model Dynamics

Discrete DEs

(Proteins) (Genes)

Do the dynamics agree?

Corollaries

Let D ∈ Dr, and let D′ ⊂ ∂D be a singular domain of order

• ne. Set q := d(D) and denote by q′ the unique element in

the set H(D′) \ H(D). Then (D, D′) ∈ T if and only if (q, q′) ∈ E, and (D′, D) ∈ T if and only if (q, q′) / ∈ E. There exists a path (D1, . . . , D2k+1) in QTG(A) with Di ∈ Dr for i ∈ {1, . . . , 2k + 1} odd and Di a singular domain

• f order one for i ∈ {1, . . . , 2k + 1} even, if and only if

(q0, q1, . . . , qk) is a path in STG(f ) such that (qj, qj−1) / ∈ E fo r all j ∈ {1, . . . , k} and qi = d(D2i+1) for all i ∈ {0, . . . , k}.

Paths and Long Term Dynamics

00 10 01 11 02 12 STG QTG

Paths and Long Term Dynamics

Some paths agree.

00 10 01 11 02 12 STG QTG

Paths and Long Term Dynamics

Some paths agree. STG paths can disagree with the QTG

00 10 01 11 02 12 STG QTG

Paths and Long Term Dynamics

Some paths agree. STG paths can disagree with the QTG QTG long term dynamics can disagree with the STG

00 10 01 11 02 12 STG QTG

?

Paths and Long Term Dynamics

STG QTG

Paths and Long Term Dynamics

STG QTG

Paths and Long Term Dynamics

Steady states in the STG agree with the QTG

STG QTG

Paths and Long Term Dynamics

Steady states in the STG agree with the QTG Steady states in the QTG can disagree with the STG

STG QTG

?

Paths and Long Term Dynamics

Steady states in the STG agree with the QTG Steady states in the QTG can disagree with the STG QTG paths can disagree with STG paths

STG QTG

?

Paths and Long Term Dynamics

Steady states in the STG agree with the QTG Steady states in the QTG can disagree with the STG QTG paths can disagree with STG paths STG long term dynamics can disagree with the QTG

STG QTG

Conclusion

Paths and long term dynamics do not always agree between the STG and QTG.

Conclusion

Paths and long term dynamics do not always agree between the STG and QTG. The dynamics can depend on the choice of the formalism.

Conclusion

Paths and long term dynamics do not always agree between the STG and QTG. The dynamics can depend on the choice of the formalism. Care must be taken when interpreting the model!

Outlook

Where do more complex dynamics in the STG and QTG agree/disagree? Incorporate other extensions of the Thomas and PADE formalisms.

Jamshidi S.; Comparing discrete, continuous, and hybrid models of gene regulatory networks. PhD thesis, Freie Universit¨ at Berlin, April 2013.