Cellerator: A System for Simulating Biochemical Reaction Networks
Bruce E Shapiro
Jet Propulsion LaboratoryCalifornia Institute of [email protected]
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