Cellerator: A System for Simulating Biochemical Reaction Networks

Bruce E Shapiro

Jet Propulsion LaboratoryCalifornia Institute of Technologybshapiro@jpl.nasa.gov

From: Kohn (1999) Molecular interaction map of the mammalian cell cycle control and DNA repair systems. Mol Biol Cell 10:2703-2734

Part of a Biochemical Network

Biochemical Networks Are...

• Complex• Mutually interacting• Large

– Number of reactions grows exponentially with number of states

• Best understood pictorially • Best described quantitatively by a large

system of differential equations (ODEs)

Need to translate pictures to ODEs

http://www.genome.ad.jp/kegg/

Online network databases exist ...

... but mathematical simulations of these networks are hopelessly naive...

Solver

Output Canonical FormSystem of ODEs

Input Canonical FormBiochemical Notation

Concentrationsvs. Time

Activity(e.g., Cell Division)

A

BC

Caltech ERATO* Simulator ArchitectureA

BC

Application

Application

Application

Application

Application

TextTransfer Protocol

XML based protocol

GUI and Modeling meta-language

*Exploratory Research for Advanced Technology (Japan Science &

Technology Corporation)

http://www.systems-biology.org

A simpler network for cell division

Goldbeter, A (1991) A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. PNAS 88:9107-9111

C=Cyclin: enzyme that gets things going

M=MPF promoting factor. M>Threshold induces cell division

X=Cyclin Protease: enzyme that breaks down C

Equations and prections of Goldbeter Mitotic Oscillator

Cellerator canonical form for input

STN = {{reaction, rate-constants},

{reaction, rate-constants},…};

interpret[STN];

Simulation = predictTimeCourse[STN,

options];

Reactions are input with a biochemical based notation

Prints out ODES

Returns tables of values as a function of time, with optional plots

Cellerator input/output for Goldbeter Mitotic Oscillator

The Basis of Cellerator: Chemical Reactions• Simple

• Cooperative

• Conversion

• Creation, Degradation

• Enzymatic

• Reversible Enzymatic

S+E ↔ SE→ P +E

A→ B

φ→ A,A→ φ

A+B→ C

A+nB→ C

• Transcription (Gene RNA)• Post-transcriptional Processing• Translation (RNA Protein)• Diffusion• and more ...

S+E ↔ SE→ P +E (e.g.E =kinase)

P + ′ E ↔ P ′ E → S+ ′ E (e.g. ′ E =phosphatase)

Translation of Biochemical Formula to ODE

• Law of Mass Action

• Two-way Reaction

• Complex reactions built from simple reactions

A+nB→k

C is described bydCdt

=kABn

rate constantConcentrations

A+Bkf ⏐ → ⏐kr

← ⏐ ⏐ C dCdt

=kf AB−krCis described by

Similar ODE’s can be written for

B and C

Enzyme Kinetic (Catalytic) Reaction

• Enzyme E catalyzes the production of product P from substrate (source) S

• Write more compactly as

E +S→a

ES

ES→d

E +S

ES→k

E +P

⎫

⎬

⎪ ⎪

⎭

⎪ ⎪

or E +Sa ⏐ → ⏐d

← ⏐ ⏐⎧ ⎨ ⎩

ES→k

E +P

S⇒E

P

3 Reactionswritten two different ways

Rate constants

Explicit

HiddenCellerator syntax for this set of reactions

Two-way catalytic reaction

• A second enzyme F catalyzes the reverse reaction

• Total of Six Elementary Reactions

• Write more compactly as

E +Sa ⏐ → ⏐d

← ⏐ ⏐ ES→k

E +P

F +P′ a ⏐ → ⏐′ d

← ⏐ ⏐ FP→′ k F +S

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

S⇔F

EP

Rate constants

Explicit

Hidden

Cellerator syntax for this set of reactions

Canonical Forms for Translation: Chemical reactions

• Input Canonical Form for Chemical Reaction

• Output Canonical Form: Terms in an ODE

XiX i∈ ′ S ⊂S

∑ k ⏐ → ⏐ YiYi∈ ′ ′ S ⊂S

∑

τi˙ X i = ciα Xj

niαj

j∏

α∑

Cellerator Arrows: Law of Mass Action

Reaction Syntax ODE Interpretation

{S → ,P }k ′ S = − ′ P = − kS

{A + B → C, k } ′ A = ′ B = − ′ C = − kAB

{A + B

n

→ C, k } ′ A = ′ B = − ′ C = − kAB

n

{A F B, kf, kr } ′ A = − ′ B = − k

fA + k

rB

{A + B F C, kf, kr } ′ A = ′ B = − ′ C = − k

fAB + k

rC

{ ∅ → A, k } ′ A = k

{B → ∅ , k } ′ B = − kB

Cellerator Arrows: Catalytic ReactionsReaction Syntax ODE interpretation

{ S F P

E

, a, d, k }

′ S = − a ⋅ E ⋅ S + d ⋅ S

′ P = k ⋅ ( SE )

′ E = − a ⋅ E ⋅ S + ( d + k ) ⋅ ( SE ) = − ( SE ′ )

{ S F P

F

E

, ,a ,d ,k

1,a 1,d 1}k

′ S = k1

⋅ ( PF ) − a ⋅ E ⋅ S + d ⋅ ( SE )

′ P = − a1

⋅ F ⋅ P + d1

⋅ ( PF ) + k ⋅ ( SE )

′ E = − a ⋅ E ⋅ S + ( d + k ) ⋅ ( SE ) = − ( SE ′ )

′ F = − a1

⋅ F ⋅ P + ( d1

+ k1

) ⋅ ( PF ) = − ( PF ′ )

{ S → P

E

, k }′ S = − k ⋅ E ⋅ S = − ′ P

{ S a P

E

}′ S =

( k + vE ) S

n

K

n

+ S

n

= − ′ P

Cellerator Arrows: Transcriptional Regulation

Reaction Syntax ODE Interpretation

Hill Function:

{ {A1

a B, hill[ L ]},

{A2

a B, hill[ L ]} ...} ,

′ B = r0

+

( r1

+ vi

Ai

)

n

i = 1

p

∑

K

n

+ ( r1

+ vi

Ai

)

n

i = 1

p

∑

Neura l Networ k Dynami cs(Genetic Regulatory Network):

{ {A1

a ,B [GRN L ]},

{A2

a ,B [GRN L ]}, K }

′ B =

R

1 + exp − Ti

A

i

ni

+ hii = 1

p

∑( )

Non-hierarchica l Cooperative Activation (Pseudo-MW C Dynamics):

{ {A1

a ,B [NHCA L ]},

{A2

a ,B [NHCA L ]}, K }

′ B =

1 + ( T

i

+

A

i

ni

)

m

i = 1

p

∑

k ( Ti

−

A

i

ni

)

m

i = 1

p

∑ + ( T

i

+

A

i

ni

)

m

i = 1

p

∑

NHCA with Cooperativ e Binding:

{ ⟨ A1

, A2

, K , Ap

⟩ a ,B

[NCHA TPLUS → {T1

+

, T2

+

, K }, K ] }

′ B =

1 + ( T

i

+

A

i

ni

)

m

i = 1

p

∑

k ( Ti

−

A

i

ni

)

m

i = 1

p

∑ + ( T

i

+

A

i

ni

)

m

i = 1

p

∑

MAP Kinase Cascade

INPUT

OUTPUT

MAP Kinase in Scaffold

σ*K4K3K2K1Scaffold(MAPK)(MAPKK)(MAPKKK)OUTPUTINPUT

K3⇒K4

K3*

K2⇒K3

*

K2*⇒

K3*

K2**

K1 ⇒K2

**

K1* ⇒

K2**

K1**

The combinatoric explosionNumber of Reactions

10

100

1000

10000

100000

2 3 4 5 6

Slots

Single PhosphorylationDouble Phosphorylation

IP3 Calcium Receptor

IP3 Calcium Receptor (continued)

Repressilator

RepressilatorXYZPZPXPY

Object Oriented Implementation:“Domains” and “Fields”

• Domain: object• Field: function that maps domains to R• Field of Domains: maps domain elements to domains• Example

– graphDomain: represents tissue– node Domains: cells– neighbors[g,n] returns a list of nodeDomains that are neighbors

of node n n in graph g

Multicellular Organisms

Myogenesis: Collaboration with Laboratory Dr. Barbara Wold (Chris Hart), Caltech

Plant Growth: Collaboration with Laboratory Dr. Elliot Meyerowitz, Caltech

Secondary Leukemia: Collaboration with City of Hope National Medical Center (NASA/BSRP)

Focus: Pathogenesis of myelodysplasia & acute myeloid leukemia following high-dose chemo/radiotherapy and autologous peripheral blood stem cell transplantation for treatment of Hodgkin’s disease and non-Hodgkin’s lymphoma

aPBSCT

Pre-BMT Day-100 6-months 1 year 2-years 3-years

JPL Collaborations using Cellerator

• Effects of microgravity during space flight on bone and muscle development (Caltech, JSC, and UCI)

• Development of childhood leukemias (Caltech, Children’s Hospital of LA, and UC, Irvine)

• Description of “core” signal transduction units (Johns Hopikins)

• Improving algorithms for micro-array data analysis (Caltech, Harvey Mudd)

• Systems Biology Workbench (Caltech, JST/Erato)

Acknowledgements

• Eric Mjolsness* - UC, Irivine• Andre Levchenko* - Johns Hopkins University• Barbara Wold - Caltech• Elliot Meyerowitz - Caltech

* Original Developers