The Continuous -Calculus An Algebra for Biochemical Modelling - - PowerPoint PPT Presentation

Introduction The Calculus Example Conclusions

The Continuous π-Calculus

An Algebra for Biochemical Modelling Marek Kwiatkowski

School of Informatics University of Edinburgh

14 Oct 2008, CMSB

joint work with Ian Stark

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Outline

1

Introduction: ODEs and Process Algebras

2

The Continuous π-Calculus

3

Example: the KaiABC circadian clock

4

Future work and conclusions

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Ordinary Differential Equations

S + E C P + E P ∅

k1 k2 k−1 k3

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Ordinary Differential Equations

S + E C P + E P ∅

k1 k2 k−1 k3 d[S] dt

= −k1[S][E] + k−1[C]

d[E] dt

= −k1[S][E] + k−1[C] + k2[C]

d[C] dt

= k1[S][E] − k−1[C] − k2[C]

d[P] dt

= k2[C] − k3[P]

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Ordinary Differential Equations

S + E C P + E P ∅

k1 k2 k−1 k3 d[S] dt

= −k1[S][E] + k−1[C]

d[E] dt

= −k1[S][E] + k−1[C] + k2[C]

d[C] dt

= k1[S][E] − k−1[C] − k2[C]

d[P] dt

= k2[C] − k3[P]

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Process Algebras

S + E C P + E P ∅

k1 k2 k−1 k3

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Process Algebras

S + E C P + E P ∅

k1 k2 k−1 k3

S

△

= a(x, y).(x.S + y.P) E

△

= (ν u)(ν r)a(u, r).(u.E + r.E) P

△

= τ.0 S | ... | S | E | ... | E

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Process Algebras

S + E C P + E P ∅

k1 k2 k−1 k3

S

△

= a(x, y).(x.S + y.P) E

△

= (ν u)(ν r)a(u, r).(u.E + r.E) P

△

= τ.0 S | ... | S | E | ... | E

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

ODEs vs PAs

ODEs: PAs:

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

ODEs vs PAs

ODEs: continuous PAs: discrete

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

ODEs vs PAs

ODEs: continuous deterministic PAs: discrete non-deterministic/stochastic

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

ODEs vs PAs

ODEs: continuous deterministic monolithic PAs: discrete non-deterministic/stochastic modular (compositional)

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

ODEs vs PAs

ODEs: continuous deterministic monolithic specify dynamics PAs: discrete non-deterministic/stochastic modular (compositional) specify interactions

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

ODEs vs PAs

ODEs: continuous deterministic monolithic specify dynamics very popular PAs: discrete non-deterministic/stochastic modular (compositional) specify interactions relatively unknown

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Syntax: species and processes

Species: A, B :: = 0 | π1.A1 + · · · + πn.An D( a) | A | B | (νM)A

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Syntax: species and processes

Species: A, B :: = 0 | π1.A1 + · · · + πn.An D( a) | A | B | (νM)A Processes: P, Q :: = c · A | P Q

c∈R≥0

(thus P is an element of RS)

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Syntax: affinity networks

Names represent protein interaction sites.

a b c d e f k1 k2 k3 k4 k5

An affinity network gives their interaction structure.

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Semantics

dP dt : immediate behaviour

element of RS equivalent to an ODE system

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Semantics

dP dt : immediate behaviour

element of RS equivalent to an ODE system ∂P: interaction potential element of RS×C×N equivalent to a transition system

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Semantics

dP dt : immediate behaviour

element of RS equivalent to an ODE system ∂P: interaction potential element of RS×C×N equivalent to a transition system ∂(P Q)

△

= ∂P + ∂Q d(P Q) dt

△

= dP dt + dQ dt + ∂P ∂Q

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Semantics

dP dt : immediate behaviour

element of RS equivalent to an ODE system ∂P: interaction potential element of RS×C×N equivalent to a transition system ∂(P Q)

△

= ∂P + ∂Q d(P Q) dt

△

= dP dt + dQ dt + ∂P ∂Q 1A

x

→F 1B

y

→G △

= Aff (x, y)(F · G − A − B)

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Example: a simple chemical reaction network

S + E C P + E P ∅

k1 k2 k−1 k3

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Example: a simple chemical reaction network

S + E C P + E P ∅

k1 k2 k−1 k3

S

△

= a(x, y).(x.S + y.P) E

△

= (ν M)bu, r.act.E P

△

= τ@k3.0 cE · E cS · S

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Example: a simple chemical reaction network

S + E C P + E P ∅

k1 k2 k−1 k3

S

△

= a(x, y).(x.S + y.P) E

△

= (ν M)bu, r.act.E P

△

= τ@k3.0 cE · E cS · S

act u r a b k−1 k2 k1

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Example: a simple chemical reaction network

S + E C P + E P ∅

k1 k2 k−1 k3

S

△

= a(x, y).(x.S + y.P) E

△

= (ν M)bu, r.act.E P

△

= τ@k3.0 cE · E cS · S

act u r a b k−1 k2 k1

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The KaiABC circadian clock of Synechococcus elongatus

C0 C1

· · ·

C6 ˜ C6

· · ·

˜ C1 ˜ C0

kps kps kps f6 ˜ kdps ˜ kdps ˜ kdps b0

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model

Ci

△

= (νMi)(τ@kps.Ci+1+τ@fi. ˜ Ci+τ@kdps.Ci−1+aiacti.(ui.Ci+ri.Ci+1)) ˜ Ci

△

= τ@˜ kps.˜ Ci+1+τ@bi.Ci+τ@˜ kdps.˜ Ci−1+bi.b′.B ˜ Ci B ˜ Ci

△

= τ@˜ kps.B ˜ Ci+1+τ@kBb

i

.(˜ Ci|B|B)+τ@˜ kdps.B ˜ Ci−1+˜ ai.˜ a′.AB ˜ Ci AB ˜ Ci

△

= τ@˜ kps.AB ˜ Ci+1+τ@˜ kAb

i

.(B ˜ Ci|A|A)+τ@˜ kdps.AB ˜ Ci−1 A

△

= a(x).x.A+˜ a.0 B

△

= b.0 P

△

= cA·A cB·B cC·C0

a a0 a6 · · · kAf kAf ˜ a ˜ a0 ˜ a6 · · · ˜ a′ ˜ kAf ˜ kAf 6 kvf b b0 b6 · · · b′ kBf kBf 6 kvf

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: no autonomous phosphorylation

Ci

△

= (νMi)(τ@kps.Ci+1+τ@fi. ˜ Ci+τ@kdps.Ci−1+aiacti.(ui.Ci+ri.Ci+1)) ˜ Ci

△

= τ@˜ kps.˜ Ci+1+τ@bi.Ci+τ@˜ kdps.˜ Ci−1+bi.b′.B ˜ Ci B ˜ Ci

△

= τ@˜ kps.B ˜ Ci+1+τ@kBb

i

.(˜ Ci|B|B)+τ@˜ kdps.B ˜ Ci−1+˜ ai.˜ a′.AB ˜ Ci AB ˜ Ci

△

= τ@˜ kps.AB ˜ Ci+1+τ@˜ kAb

i

.(B ˜ Ci|A|A)+τ@˜ kdps.AB ˜ Ci−1 A

△

= a(x).x.A+˜ a.0 B

△

= b.0 P

△

= cA·A cB·B cC·C0

a a0 a6 · · · kAf kAf ˜ a ˜ a0 ˜ a6 · · · ˜ a′ ˜ kAf ˜ kAf 6 kvf b b0 b6 · · · b′ kBf kBf 6 kvf

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: no autonomous phosphorylation

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: no autonomous phosphorylation

Ci

△

= (νMi)(τ@kps.Ci+1+τ@fi. ˜ Ci+τ@kdps.Ci−1+aiacti.(ui.Ci+ri.Ci+1)) ˜ Ci

△

= τ@˜ kps.˜ Ci+1+τ@bi.Ci+τ@˜ kdps.˜ Ci−1+bi.b′.B ˜ Ci B ˜ Ci

△

= τ@˜ kps.B ˜ Ci+1+τ@kBb

i

.(˜ Ci|B|B)+τ@˜ kdps.B ˜ Ci−1+˜ ai.˜ a′.AB ˜ Ci AB ˜ Ci

△

= τ@˜ kps.AB ˜ Ci+1+τ@˜ kAb

i

.(B ˜ Ci|A|A)+τ@˜ kdps.AB ˜ Ci−1 A

△

= a(x).x.A+˜ a.0 B

△

= b.0 P

△

= cA·A cB·B cC·C0

a a0 a6 · · · kAf kAf ˜ a ˜ a0 ˜ a6 · · · ˜ a′ ˜ kAf ˜ kAf 6 kvf b b0 b6 · · · b′ kBf kBf 6 kvf

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: weaker KaiA binding

Ci

△

= (νMi)(τ@kps.Ci+1+τ@fi. ˜ Ci+τ@kdps.Ci−1+aiacti.(ui.Ci+ri.Ci+1)) ˜ Ci

△

= τ@˜ kps.˜ Ci+1+τ@bi.Ci+τ@˜ kdps.˜ Ci−1+bi.b′.B ˜ Ci B ˜ Ci

△

= τ@˜ kps.B ˜ Ci+1+τ@kBb

i

.(˜ Ci|B|B)+τ@˜ kdps.B ˜ Ci−1+˜ ai.˜ a′.AB ˜ Ci AB ˜ Ci

△

= τ@˜ kps.AB ˜ Ci+1+τ@˜ kAb

i

.(B ˜ Ci|A|A)+τ@˜ kdps.AB ˜ Ci−1 A

△

= a(x).x.A+˜ a.0 B

△

= b.0 P

△

= cA·A cB·B cC·C0

a a0 a6 · · · kAf 0 ↓ kAf 6 ↓ ˜ a ˜ a0 ˜ a6 · · · ˜ a′ ˜ kAf ˜ kAf 6 kvf b b0 b6 · · · b′ kBf kBf 6 kvf

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: weaker KaiA binding

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: weaker KaiA binding

Ci

△

= (νMi)(τ@kps.Ci+1+τ@fi. ˜ Ci+τ@kdps.Ci−1+aiacti.(ui.Ci+ri.Ci+1)) ˜ Ci

△

= τ@˜ kps.˜ Ci+1+τ@bi.Ci+τ@˜ kdps.˜ Ci−1+bi.b′.B ˜ Ci B ˜ Ci

△

= τ@˜ kps.B ˜ Ci+1+τ@kBb

i

.(˜ Ci|B|B)+τ@˜ kdps.B ˜ Ci−1+˜ ai.˜ a′.AB ˜ Ci AB ˜ Ci

△

= τ@˜ kps.AB ˜ Ci+1+τ@˜ kAb

i

.(B ˜ Ci|A|A)+τ@˜ kdps.AB ˜ Ci−1 A

△

= a(x).x.A+˜ a.0 B

△

= b.0 P

△

= cA·A cB·B cC·C0

a a0 a6 · · · kAf 0 ↓ kAf 6 ↓ ˜ a ˜ a0 ˜ a6 · · · ˜ a′ ˜ kAf ˜ kAf 6 kvf b b0 b6 · · · b′ kBf kBf 6 kvf

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: KaiA and KaiB can dimerize

Ci

△

= (νMi)(τ@kps.Ci+1+τ@fi. ˜ Ci+τ@kdps.Ci−1+aiacti.(ui.Ci+ri.Ci+1)) ˜ Ci

△

= τ@˜ kps.˜ Ci+1+τ@bi.Ci+τ@˜ kdps.˜ Ci−1+bi.b′.B ˜ Ci B ˜ Ci

△

= τ@˜ kps.B ˜ Ci+1+τ@kBb

i

.(˜ Ci|B|B)+τ@˜ kdps.B ˜ Ci−1+˜ ai.˜ a′.AB ˜ Ci AB ˜ Ci

△

= τ@˜ kps.AB ˜ Ci+1+τ@˜ kAb

i

.(B ˜ Ci|A|A)+τ@˜ kdps.AB ˜ Ci−1 A

△

= a(x).x.A+˜ a.0 B

△

= b.0 P

△

= cA·A cB·B cC·C0

a a0 a6 · · · kAf kAf ˜ a ˜ a0 ˜ a6 · · · ˜ a′ ˜ kAf ˜ kAf 6 kvf b b0 b6 · · · b′ kBf kBf 6 kvf k˜ ab

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

The model: KaiA and KaiB can dimerize

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Future work

1

Model Checking

Very desirable Most likely intractable for large models Existing work: LTL and time series

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Future work

1

Model Checking

Very desirable Most likely intractable for large models Existing work: LTL and time series

2

Evolutionary Applications

Evolutionary trajectories (easy) Robustness and evolvability (hard) The adaptive landscape (hard)

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Future work

1

Model Checking

Very desirable Most likely intractable for large models Existing work: LTL and time series

2

Evolutionary Applications

Evolutionary trajectories (easy) Robustness and evolvability (hard) The adaptive landscape (hard)

3

Hybrid Modelling

Allows to model protein-DNA interactions

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Conclusions

1

ODEs and process algebras

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Conclusions

1

ODEs and process algebras

2

The Continuous π-Calculus

Compositional modelling Continuous semantics Flexible interaction structure

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Conclusions

1

ODEs and process algebras

2

The Continuous π-Calculus

Compositional modelling Continuous semantics Flexible interaction structure

3

Future work

Model Checking Hybrid Modelling Evolutionary Applications

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus

Introduction The Calculus Example Conclusions

Appendix

Key references: Ishiura, M., Kutsuna, S., Aoki, S., Iwasaki, H., Andersson, C.R., Tanabe, A., Golden, S.S., Johnson, C.H., Kondo, T.: Expression of a gene cluster KaiABC as a circadian feedback process in

• cyanobacteria. Science 281(5382) (1998) 1519-1523

van Zon, J.S., Lubensky, D.K., Altena, P.R.H., ten Wolde, P.R.: An allosteric model of circadian KaiC phosphorylation. PNAS 104 (2007) 7420-7425 Pictures: W. Eikrem, J.Trondhsen, IMU, Purdue Uni, N. Camelo.

• M. Kwiatkowski, I. Stark

The Continuous π-Calculus