rainer breitling, groningen, nl david gilbert, london, uk monika … · 2009-06-21 · breitling /...
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From cell biology to Petri nets
Rainer Breitling, Groningen, NLDavid Gilbert, London, UK
Monika Heiner, Cottbus, DE
Breitling / 2
Biology = Concentrations
Breitling / 3
The simplest chemical reaction
A B
irreversible, one-molecule reactionexamples: all sorts of decay processes, e.g. radioactive, fluorescence, activated receptor returning to inactive stateany metabolic pathway can be described by a combination of processes of this type (including reversible reactions and, in some respects, multi-molecule reactions)
Breitling / 4
The simplest chemical reaction
A Bvarious levels of description:
homogeneous system, large numbers of molecules = ordinary differential equations, kineticssmall numbers of molecules = probabilistic equations, stochasticsspatial heterogeneity = partial differential equations, diffusionsmall number of heterogeneously distributed molecules = single-molecule tracking (e.g. cytoskeleton modelling)
Breitling / 5
Kinetics Description
Imagine a box containing N molecules.How many will decay during time t? k*N
Imagine two boxes containing N/2 molecules each.How many decay? k*N
Imagine two boxes containing N molecules each.How many decay? 2k*N
In general:
)(*)( tndt
tdn λ=−
Main idea: Molecules don’t talk
teNtn λ−= 0)(⇔differential equation (ordinary, linear, first-order)
exact solution (in more complex cases replaced by a numerical approximation)
Breitling / 6
Kinetics Description
If you know the concentration at one time, you can calculate it for any other time! (and this really works)
Breitling / 7
Probabilistic Description
Probability of decay of a single molecule in some small time interval:
Probability of survival in Δt:
Probability of survival for some time t:
Transition to large number of molecules:
tp Δ= λ1
tpp Δ−=−= λ11 12
tx
xe
xtp λλ −
∞→=−= )1(lim
teNtn λ−= 0)(
)()(0 tneN
dttdn t λλ λ −=−= −
or
Main idea: Molecules are isolated entities without memory
Breitling / 8
Probabilistic Description – 2
Probability of survival of a single molecule for some time t:
Probability that exactly x molecules survive for some time t:
Most likely number to survive to time t:
tx
xe
xtp λλ −
∞→=−= )1(lim
⎟⎟⎠
⎞⎜⎜⎝
⎛−= −−−
xN
eep xNtxtx
00)1()( λλ
tx eNpx λ−= 0)|max(
Breitling / 9
Probabilistic Description – 3
Decay rate: Probability of decay:Probability distribution of n
surviving molecules at time t:Description:Time: t -> wait dt -> t+dt
Molecules:n -> no decay -> nn+1 -> one decay -> n
λntn =Λ ),(dttnp ),(Λ=
),( tnP
]),(1)[,(),1(),1(
),(
dttntnPdttntnP
dttnP
Λ−++Λ+
=+
Final Result (after some calculating): The same as in the previous probabilistic description
Markov Model (pure death!)
Breitling / 10
Petri Net representation
?
Breitling / 11
Some (Bio)Chemical Conventions
Concentration of Molecule A = [A], usually in units mol/litre (molar)
Rate constant = k, with indices indicating constants for various reactions (k1, k2...)
Therefore:A B
][][][1 Ak
dtBd
dtAd
−=−=
Breitling / 12
Reversible, Single-Molecule Reaction
A B, or A B || B A, or Differential equations:
][][][
][][][
21
21
BkAkdtBd
BkAkdtAd
−=
+−=forward reverse
Main principle: Partial reactions are independent!
Breitling / 13
Reversible, single-molecule reaction – 2
Differential Equation:
Equilibrium (=steady-state):
equiequi
equi
equiequi
equiequi
Kkk
BA
BkAkdtBd
dtAd
==
=+−
==
1
2
21
][][
0][][
0][][
][][][
][][][
21
21
BkAkdtBd
BkAkdtAd
−=
+−=
Breitling / 14
Irreversible, two-molecule reaction
A+B CDifferential equations:
]][[][
][][][
BAkdtAd
dtCd
dtBd
dtAd
−=
−==
Underlying idea: Reaction probability = Combined probability that both [A] and [B] are in a “reactive mood”:
]][[][][)()()( *2
*1 BAkBkAkBpApABp ===
The last piece of the puzzle
Non-linear!
Breitling / 15
A simple metabolic pathway
A B C+DDifferential equations:
-k3[C][D]+k2[B][D]=
-k3[C][D]+k2[B][C]=
+k3[C][D]-k2[B]+k1[A][B]=
-k1[A][A]=
reverseforwarddecayd/dt
Breitling / 16
Metabolic Networks as Bigraphs
A B C+D
-k3[C][D]+k2[B][D]
-k3[C][D]+k2[B][C]
+k3[C][D]-k2[B]+k1[A][B]
-k1[A][A]
reverseforwarddecayd/dt
A BC
Dk1 k2 k3
-110D
-110C
1-11B
00-1A
k3k2k1
Breitling / 17
Petri nets
Breitling / 18
Petri nets
http://www-dssz.informatik.tu-cottbus.de/web_animation/pn_demos_flat-nets.html
Breitling / 19
Qualitative Petri-Net Modelling & Analysis
Graphicalrepresentation -Snoopy
• Qualitative analysis Charlie– Unbounded, live & reversible
– Covered by T invariants
– P invariants
Breitling / 20
Biological description bigraph differential equations
KEGG
Breitling / 21
Biological description bigraph ODEs
EC 1.1.1.2
substance A substance B
A Bk1
Breitling / 22
Biological description bigraph ODEs
EC 1.1.1.2
substance A substance B
A Bk
EA EBk*k1 k2
E
Breitling / 23
A special case: enzyme reactions
In a quasi steady state, we can assume that [ES] is constant. Then:
If we now define a new constant Km (Michaelis constant), we get:
Breitling / 24
A special case: enzyme reactions
Substituting [E] (free enzyme) by the total enzyme concentration we get:
Hence, the reaction rate is:
Breitling / 25
A special case: enzyme reactions
Underlying assumptions of the Michaelis-Menten approximation:
• Free diffusion, random collisions of infinite number of molecules
• Irreversible reactions
• Quasi steady state
In cell signaling pathways, all three assumptions will be frequently violated:
• Reactions of rather rare molecules happen at membranes and on scaffold structures
• Reactions happen close to equilibrium and both reactions have non-zero fluxes
• Enzymes are themselves substrates for other enzymes, concentrations change rapidly, d[ES]/dt ≈ d[P]/dt
Breitling / 26
Cell signaling pathways
Fig. courtesy of W. Kolch
Breitling / 27
Metabolic pathways vs. Signaling Pathways
E1
(initial substrate)S
S’
E2
E3
S’’
S’’’(final product)
Metabolic
S1
Input SignalX
P2S2
S3 P3Output
Signaling cascade
P1
Product become enzyme at next stageClassical enzyme-product pathway
(can you give the mass-action equations?)
Breitling / 28
Metabolic pathways vs. Signalling Pathways
Breitling / 29
Cell signaling pathways
Breitling / 30
Cell signaling pathways
Breitling / 31
Cell signaling pathways
Breitling / 32
Cell signaling pathways
Common components:Receptors binding to ligands
R(inactive) + L RL(active)
Proteins forming complexesP1 + P2 P1P2-complex
Proteins acting as enzymes on other proteins (e.g., phosphorylation by kinases)
P1 + K P1* + K
Breitling / 33
MA1: Mass action for enzymatic reaction
A: substrate B: productE: enzymeE|A substrate-enzyme complex
E+Ak2
← ⎯ ⎯
k1⎯ → ⎯ E | A k3⎯ → ⎯ E + B
A B
E
Breitling / 34
MA2 model
EBEBEAEAk
k
k
k
k +⎯→⎯+ ⎯⎯→⎯
⎯⎯⎯←
⎯→⎯
⎯⎯←
1'
2'
31
2
|| '
Breitling / 35
MA3 model
EBEBEAEAk
k
k
k
k
k
++ ⎯⎯→⎯
⎯⎯⎯←
⎯⎯→⎯
⎯⎯⎯←
⎯→⎯
⎯⎯←
1'
2'
3'
4'
1
2
||
Breitling / 36
Cell signaling pathways – feedback loops
Breitling / 37
Cell signaling pathways – feedback loops
Fig. courtesy of W. Kolch
Breitling / 38
Feedback loops in Petri Nets
RpR
S1
RR
P1
P2
RRpRRp
RpR
S1
RR
P1
P2
RRp
RpR
S1
RR
P1
P2
RRp
RpR
S1
RR
P1
P2
(a)
(c)
(b)
(d)
Breitling / 39
Feedback loops in Petri Nets
Breitling / 40
Feedback loops in Petri Nets
Breitling / 41
…and added inhibitor
Breitling / 42
Many PN modelling challengings remain…
Lack of parametersQualitative vs. Continuous PN
Small molecule numbersDeterministics vs. Stochastic models
Spatial heterogeneity???
Breitling / 43
Cell signaling pathways
Fig. courtesy of W. Kolch
Breitling / 44
5
X
Stochastic vs. Continuous
5
X
Breitling / 45
Stochastic model checkingTwo Reaction Model
First a simple model of two reactions:
A BC D
Assess property:P=?[ A = $X { A = D } ]
“What is the probability that, when A and D first equal each other, they both have $X number of molecules?”
0.01
0.1
Breitling / 46
Two Reaction ModelCo
ncen
tration
TimeD
A
Property:P=?[ A = $X { A = D } ]
*
Breitling / 47
Two Reaction Model
Set reactants to 10 molecules (model bound to 10 molecules)
Simulate with Gillespie 1,000 times and model check each output
Number of simulations which are true over total number of simulations is the probability.
Also checked the continuous model and the answer is the solid line.
Breitling / 48
Two Reaction Model
Breitling / 49
Spatial heterogeneity
concentrations are different in different places, n = f(t,x,y,z)diffusion superimposed on chemical reactions:
partial differential equation
diffusion)()(
±−=∂
∂xyz
xyz tnttn
λ
Breitling / 50
Spatial heterogeneity
one-dimensional case (diffusion only, and conservation of mass)
xn(t,x)K
xxxtnK
xt
xtn
∂∂
−=
∂Δ+∂
−=
−=Δ∂
∂
inflow
),(outflow
outflowinflow),(
Δx
inflow outflow
Breitling / 51
Spatial heterogeneity – 2
)()()(:reaction chemicaln with Combinatio
),,,(),,,(:dimensions for three Shorthand
),(),(:equationdiffusion get oequation t aldifferenti toTransition
),(),(),(
2
2
2
2
tnKtnttn
zyxtnKt
zyxtn
xxtnK
txtn
xxtnK
xxxtnKx
txtn
∇+−=∂
∂
∇=∂
∂
∂∂
=∂
∂
∂∂
−∂
Δ+∂=Δ
∂∂
λ
Breitling / 52
Acknowledgements
David Gilbert, Brunel University, London
Monika Heiner, Cottbus University, Germany
Robin Donaldson, Glasgow University
The Groningen Bioinformatics Centre (Netherlands) is expanding its young and successful team.
Several PhD and Postdoc positions are available for creative bioinformaticians with an interest in Systems Biology, Metabolomics, Proteomics, Quantitative Genetics, Network Reconstruction, Dynamic Modelling…
For more information and to apply……visit www.rug.nl/gbic…e-mail r.breitling@rug.nl…talk to Rainer Breitling at Petri Nets 2009
Recent GBiC papers: Breitling R et al. New surveyor tools for charting microbial metabolic maps Nature Reviews Microbiology (2008). Fu J et al. MetaNetwork: a computational protocol for the genetic study of metabolic networks Nature Protocols (2007). Swertz MA et al. Beyond standardization: dynamic software infrastructures for systems biology Nature Reviews Genetics(2007). Keurentjes JJB et al. The genetics of plant metabolism Nature Genetics (2006). Hoeller D et al. Regulation of ubiquitin-binding proteins by monoubiquitylation Nature Cell Biology(2006). Bystrykh L et al. Uncovering regulatory pathways that effect hematopoietic stem cell function using ’genetical genomics’ Nature Genetics (2005).
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