Simulating Biological Systems in
the Stochastic p-calculus
Andrew Phillips
&
Luca Cardelli
Microsoft Research, Cambridge UK
Biological Computing
Modelling Biology
The Human Genome project:
Scientists hoped that DNA code would help predict system behaviour
Functional meaning of the code is still a mystery
Systems Biology:
Understand and precisely describe the behaviour of biological systems
Two complementary approaches:
Look at experimental results and “infer” general system properties
Build detailed models of systems and test these in the lab
Biological Modelling:
Need tools for modelling complex parallel systems.
Should also scale up to very large systems.
The beginnings of a biological programming language...
Programming Biology
Languages for complex,
parallel computer systems:
Languages for complex,
parallel biological systems:
stochastic p-calculus
Reactions vs. Components
Traditional modelling: model
individual reactions
Large, Connected Reaction Graphs
http://www.celldesigner.org/
Reactions vs. Components
Traditional modelling: model
individual reactions
Stochastic p-calculus: model
components
Compositional Modelling
Build complex models incrementally, by direct
composition of simpler components:
Model Analysis
A mathematical programming language
A range of analysis techniques (types, equivalences,
model checking)
Could provide insight into fundamental properties of
biological systems
Related Work
Stochastic p-calculus proposed by [Priami, 1995]
Used to model and simulate a range of biological
systems:
RTK MAPK pathway [Regev et al., 2001]
Gene Regulation by positive Feedback [Priami et al., 2001]
Cell Cycle Control in Eukaryotes [Lecca and Priami, 2003]
First simulator for stochastic p-calculus [BioSPI]
A subset of p-calculus with limited choice
Compiles a calculus process to an FCP procedure
Executed by the FCP Logix platform [Silverman et al., 1987]
Project Overview
Graphical Stochastic p-calculus
Display stochastic p-calculus models as graphs
[Phillips and Cardelli, Bioconcur 2005]
Helps make the stochastic p-calculus more accessible
Defined a graphical calculus and graphical execution model
Proved equivalent to the stochastic p-calculus
[Phillips, Cardelli and Castagna, TCSB 2006]
let generate_pep(k:float,free:chan,bound:chan,c:chan,e:chan)=
delay@gpep;(pep(k,free,bound,c,e)|generate_pep(k,free,bound,
c,e))
and pep(k:float,free:chan,bound:chan,c:chan,e:chan)= (
new u@k:chan
or !bind(u,c,e);pep_bound(u,k,free,bound,c,e)
or delay@dpep
)
and
pep_bound(u:chan,k:float,free:chan,bound:chan,c:chan,e:chan)
=
?u;pep(k,free,bound,c,e)
and MHCc_pep(u:chan,c:chan,e:chan)=
do delay@open;MHCo_pep(u,c,e)
or !c
and MHCc()=
do ?bindT(uT);MHCc_TPN(uT)
or delay@open;MHCo()
and MHCo_TPN(uT:chan)=
do ?bind(u,c,e);MHCo_TPN_pep(u,c,e,uT)
or !uT;MHCo() (* *)
and MHCo_TPN_pep(u:chan,c:chan,e:chan,uT:chan)=
do delay@r*trigger;MHCt_TPN_pep(u,c,e,uT)
or !u;MHCo_TPN(uT)
and MHCt_TPN_pep(u:chan,c:chan,e:chan,uT:chan)=
do !u;delay@close;MHCc_TPN(uT)
or delay@close;MHCc_TPN_pep(u,c,e,uT)
and MHCc_TPN_pep(u:chan,c:chan,e:chan,uT:chan)=
do delay@open;MHCo_TPN_pep(u,c,e,uT)
or !uT;MHCc_pep(u,c,e)
and MHCc_TPN(uT:chan)=
do delay@open;MHCo_TPN(uT)
or !uT;MHCc() (* *)
PROVED EQUIVALENT
let TPN()= (
new uT@0.01:chan
!bindT(uT);(?uT;TPN())
)
let MHCo()=
do ?bind(u,c,e);MHCo_pep(u,c,e)
or ?bindT(uT);MHCo_TPN(uT)
and MHCo_pep(u:chan,c:chan,e:chan)=
do delay@trigger;MHCt_pep(u,c,e)
or !u;MHCo()
and MHCt_pep(u:chan,c:chan,e:chan)=
do !u;delay@close;MHCc()
or delay@close;MHCc_pep(u,c,e)
The Stochastic Pi Machine
A simulation algorithm for stochastic p-calculus
[Phillips and Cardelli, Bioconcur 2004]
Based on standard theory of chemical kinetics [Gillespie 1977]
The probability of a reaction is proportional to the rate of the reaction
times the number of reactants.
Proved correct with respect to the stochastic p-calculus.
PROVED CORRECT
The SPiM Simulator
Simulation algorithm mapped to functional code (OCaml / F#)
Used as the basis for implementing the SPiM simulator.
[Phillips, SPiM 2006]
Close correspondence between formal algorithm and functional code
Correct specification improves confidence in simulation results
Used in various research centres (UK, France, Italy, Sweden...)
http://research.microsoft.com/~aphillip/spim
Gene Networks Library
Proposed a combinatorial library of gene gates
[Blossey, Cardelli, Phillips, TCSB 2006]
Used as building blocks to model complex gene networks
Explain behaviour of controversial gene networks engineered in vivo
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!tet
!lambda
!lac
!gfp
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40
© 2000 Elowitz, M.B., Leibler. S. A Synthetic Oscillatory
Network of Transcriptional Regulators. Nature 403:335-338.
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Immune System Modelling
Model of MHC Class I Antigen Presentation
A key pathway in the immune system
[Goldstein, MPhil Dissertation 2005]
Ongoing collaboration with Immunologists at Southampton University
[Cardelli, Elliott, Goldstein, Phillips, Werner. In preparation]
© 2003 Current Opinion in Immunology, 15:75–81.
© 2003 Nature Reviews Immunology, 3(12):952–96.
Gene Networks
with
Luca Cardelli (MSR Cambridge)
Ralf Blossey (IRI Lille)
Gene with Negative Control
Neg(a,b) produces protein b and is blocked by protein a
transcribe = 0.1
unblock = 0.0001
degrade = 0.001
rate(a,b) = 1.0
val transcribe = 0.1 val degrade = 0.001
val unblock = 0.0001
new a@1.0:chan new b@1.0:chan
let Neg(a:chan,b:chan) =
do delay@transcribe; ( Protein(b) |
Neg(a,b) )
or ?a; delay@unblock; Neg(a,b)
and Protein(b:chan) =
do !b; Protein(b)
or delay@degrade
a
b
run Neg(a,b)
Neg Gate
x1
A protein b can be transcribed at rate 0.1 s-1
Neg Gate
x1
x1
Another protein b can be transcribed
Neg Gate
x1
x2
And another...
Neg Gate
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x3
A protein b can be degraded at rate 0.001 s-1
Neg Gate
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x2
Equilibrium between transcription and degradation
Neg Gate
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Level of protein b stabilises at around 100
Repressilator [Elowitz and Leibler, 2000]
A gene network engineered in live bacteria.
Modelled as a simple combination of Neg gates:
Neg(lac,tet)
| Neg(tet,lambda)
| Neg(lambda,lac)
| Neg(tet,gfp)
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!tet
!lambda
!lac
!gfp
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40
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© 2000 Elowitz, M.B., Leibler. S. A Synthetic Oscillatory
Network of Transcriptional Regulators. Nature 403:335-338.
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Repressilator: Debugging
lac
Why do we get oscillations?
Neg(lac,tet) | Neg(tet,lambda) | Neg(lambda,lac)
lambda
tet
Repressilator: Debugging (1)
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x1
Initially there is one copy of each gene
Any one of the proteins can be transcribed at rate 0.1
Repressilator: Debugging (2)
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The tet protein can block the lambda gene at rate 1.0
Repressilator: Debugging (3)
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Now no lambda protein can be transcribed.
But lac protein can still be transcribed.
Repressilator: Debugging (4)
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The lac protein can block the tet gene at rate 1.0
Repressilator: Debugging (5)
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Now no tet protein can be transcribed.
All of the tet protein eventually degrades at rate 0.001
Repressilator: Debugging (5)
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x1
Meanwhile, lots of lac protein is transcribed
Repressilator: Debugging (363)
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Represents one oscillation cycle
Eventually, the lambda gene unblocks at rate 0.0001
Repressilator: Debugging (2173)
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The lambda protein can now take over...
Repressilator: Results
Alternate oscillation of proteins: tet, lac, lambda, tet, ...
lac
lambda
lac
tet
lambda
lac
tet
lambda
tet
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!lambda
!lac
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Refined Neg Gate
Refined Neg gate with cooperative binding
Different implementation, same main program
Neg(lac,tet) | Neg(tet,lambda) | Neg(lambda,lac)
let Neg(a,b) = (
new u1@10000.0
do delay@transcribe; (Protein(b) | Neg(a,b))
or !a(u1); Blocked(a,b,u1)
)
and Blocked(a,b,u1) = (
new u2@0.0001
do ?u1; Neg(a,b)
or !a(u2); ?u2; Blocked(a,b,u1)
)
and Protein(b) = (
do delay@degrade
or ?b(u); !u; Protein(b)
)
Negp Gate with Inhibitor
transcribe = 0.1
unblock = 0.0001
degrade = 0.001
rate(a,b,r) = 1.0
val transcribe = 0.1
val unblock = 0.0001
new a@1.0:chan
new r@1.0:chan
val degrade = 0.001
new b@1.0:chan
let Negp(a:chan,b:chan,r:chan) =
do delay@t; (Proteinp(b,r) | Negp(a,b,r))
or ?a; delay@u; Negp(a,b,r)
and Proteinp(b:chan,r:chan) =
do !b; Proteinp(b,r)
or ?r
or delay@d
let Inh(r:chan) = !r; Inh(r)
r
a
b
Run Negp(a,b,r)
Bacteria Logic Gates [Guet et al., 2002]
© 2002 AAAS. Reprinted with permission from Guet et al. Combinatorial Synthesis of Genetic Networks. Science 296 (5572): 1466 - 1470
3 genes: tetR, lacI, lcI
5 promoters: PL1, PL2, PT, Pl-, Pl+
125 possible networks consisting of 3 promoter-gene units
2 inputs: IPTG (represses Tet), aTc (represses Lac)
1 output: GFP (linked to Pl-)
Bacteria Logic Gates [Guet et al., 2002]
© 2002 AAAS. Reprinted with permission from Guet et al. Combinatorial Synthesis of Genetic Networks. Science 296 (5572): 1466 - 1470
IPTG, aTc as boolean inputs, GFP as boolean output
Combinatorial Library of Genes
Model complex gene networks from simpler building blocks.
Also used simulation to investigate system behaviour.
[Blossey, Cardelli, Phillips, TCSB 2006]
D038() = tet() | lac() | cI() | gfp()
tet() = Negp(TetR,TetR,aTc)
lac() = Negp(TetR,LacI,IPTG)
cI() = Neg(LacI,lcI)
gfp() = Neg(lcI,GFP)
D038()
D038() | Inh(aTc)
D038() | Inh(IPTG)
D038() | Inh(aTc) | Inh(IPTG)
Immune System Modelling:
MHC I Antigen Presentation
with
Luca Cardelli (MSR Cambridge)
Leonard Goldstein (Cambridge University)
Tim Elliott (Southampton University)
Joern Werner (Southampton University)
MHC I Antigen Presentation
A key pathway of the immune system:
MHC I complexes can
signal when a cell is
infected.
They present peptides
(small pieces of virus or
bacteria) at the surface
of infected cells.
A way of marking
infected cells for
destruction.
© 2003 Jonathan W. Yewdell, Eric Reits, and Jacques Neefjes. Making sense of mass destruction:
quantitating MHC class I antigen presentation. Nature Reviews Immunology, 3(12):952–96.
MHC I Antigen Presentation
[© 2005 from Immunobiology, Sixth Edition by Janeway et al. Reproduced by permission of
Garland Science/Taylor & Francis LLC.]
MHC I Assembly
MHC I complexes follow an assembly line:
© 2003 Antony N Antoniou, Simon J Powis, and Tim Elliott. Assembly and export
of MHC class I peptide ligands. Current Opinion in Immunology, 15:75–81.
Peptide Selection: Flytrap Model
MHC I captures peptides like a Venus Flytrap.
Sensitive to disease peptides only.
non disease
peptide escapes
peptide enters
open MHC
disease peptide is captured
and presented at cell surface
Flytrap Model: p-calculus
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Flytrap Model: p-calculus
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Flytrap Model: p-calculus
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Flytrap Model: p-calculus
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Simulation Results
Tapasin allows disease peptides to be rapidly selected in favour of
non-disease peptides.
Always with Tapasin
No Tapasin
Transient binding to Tapasin
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Next steps:
Investigate influence of tapasin in the lab.
Incrementally build a more detailed immune system model.
Construct a minimal functional model of the system.
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Simplified Flytrap
Similar results obtained with a simpler model
Hypothesis: Tapasin is the 2nd stage of a 2-stage filter
Perhaps an explanation of system function.
Future Plans
Interface SPiM with stochastic analysis tools
Improve scalability of core SPiM algorithm
Prove correctness of multiprocessor algorithm
Extend SPiM with membranes
Stochastic p-calculus and ODEs
Improve SPiM user interface and debugging
Collaborate with medical researchers to model and
simulate a range of biological systems
Collaborate with pharmaceutical industry to design a
next-generation programming language for Systems
Biology
References
Graphical Stochastic p-calculus:
[Phillips and Cardelli, Bioconcur 2005]
[Phillips, Cardelli and Castagna, TCSB 2006]
SPiM Simulator:
[Phillips and Cardelli, Bioconcur 2004]
[Phillips, SPiM 2005]
Gene Networks:
[Blossey, Cardelli, Phillips, TCSB 2006]
Antigen Presentation:
[Goldstein, MPhil Dissertation 2005]