Simulating the Dynamics of Complex Biological Systems Maria J Schilstra Biocomputation Research...
Transcript of Simulating the Dynamics of Complex Biological Systems Maria J Schilstra Biocomputation Research...
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Simulating the Dynamics of Complex Biological Systems
Maria J SchilstraBiocomputation Research Group
University of Hertfordshire, Hatfield, UK
Stephen R. MartinPhysical Biochemistry
MRC National Instritute for Medical Resaerch, London, UK
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Computer simulation
• “An attempt to model a real-life or hypothetical situation on a computer, so that it can be studied to see how the system works. By changing variables, predictions may be made about the behaviour of the system.”(Wikipedia)
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Running example
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Microtubule Dynamic Instability
• Michison & Kirschner (1984):“Microtubules in vitro coexist in growing
and shrinking populations which interconvert rather infrequently”
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Cartoon
“Catastrophe”
“Rescue”
G-end(net growth) S-end
(rapid shrinkage)
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Microtubule Dynamic Instability
• Michison & Kirschner (1984):“microtubules in vitro coexist in growing and shrinking
populations which interconvert rather infrequently”
• Horio & Hotani (1986):– Average lifetime of growing microtubules: 3 min– Growth rate: 0.6 m/min (16 subunits/s)– Average lifetime of shrinking microtubules: 18 s– Shrinkage rate: 0.13 m/s (220 subunits/s)
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Model Notation
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Model notation
G Growing endS Shrinking endTuT Tubulin-GTPTuD Tubulin-GDPn Number of Tu in MT (TuM)MT Microtubule
Gn + TuT Gn+1
Gn Sn
Sn Sn-1 + TuD
Sn + 3 TuT Gn+3
Chemical Reaction Notation “Petri-net” notation
G S
TuT TuD
3
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Petri-net?
A B
A B“Place”(pool, store, state)Contains “items”
(tokens, molecules, particles)
“Transition”(reaction, process)Can “fire”; has an average firing rate
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Transition firing?
A B
A B
X
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Transition firing?
2A B
A B
A B2
A B
A+X B
A
B
A
A B+X
A
B
A
2 2
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Transition firing rate?
A B
J = k.[A] (mol/(L.s))A Bk (1/s)
J = k.nA (items/s)nA = [A].NA.Vol
A
B
A
A+X Bk (L/(mole.s))
J = k.[A][X] mol/(L.s))
J = k.nA.[X] (items/s)nA = [A].NA.Vol
J = rate at which B appears
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Microtubule dynamic instability model
G S
TuT TuD
3
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Transition firing
G S
TuT TuD
3
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Transition firing
G S
TuT TuD
3
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Counting TuM (bound Tu)
G S
TuT TuD
3
TuM
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Counting TuM (bound Tu)
G S
TuT TuD
3
TuM
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Firing rates
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
J = kGG nG[TuT]
J = kSG nS[TuT]3
J = kSS nS
J = kGS nG
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Parameter values Initial conditions
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
J = kGG [G][TuT]
J = kSG [S][TuT]3
J = kSS [S]
J = kGS [G]kGG 1.6x106 M-1 s-1
kGS 0.0056 s-1
kSS 220 s-1
kSG 5.6x1013 M-3s-1
Vol 10x10-15 L
amount Conc (M)
G 1 0.16x10-9
S 0 0
TuT 300 50x10-9
TuD 0 0
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Stochastic Simulation
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Stochastic?
• “Random or probabilistic but with some direction. For example the arrival of people at a post office might be random but average properties (such as the queue length) can be predicted.” (http://www.cs.ucl.ac.uk/staff/W.Langdon/gpdata/glossary.html)
• Stochastic simulation: uses a random number generator to produce one or more possible time courses.
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Stochastic simulation
1. Assess which transitions are “enabled” (can fire)
2. Use a “weighted lottery” to determine which enabled transition will fire first (Tfirst), and when (tnext)
3. Let transition Tfirst fire at time tnext
4. Repeat from 1 as long as there are enabled transitions, and tnext < tstop
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Firing rates
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
J = kGG [G][TuT]
J = kGS [G]kGG 1.6x106 M-1 s-1
kGS 0.0056 s-1
kSS 220 s-1
kSG 5.6x1013 M-3s-1
Vol 10x10-15 L
amount Conc (M)
G 1 0.16x10-9
S 0 0
TuT 300 50x10-9
TuD 0 0
JGG = 1.6x106 x 50x10-9 x 1= 0.08 s-1
(i.e. lifetime Gn = 12.5 s)
JGS = 0.0056 x 1 = 0.0056 s-1
(i.e. lifetime G = 180 s)
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Probability that reaction has occurred(Cumulative distribution functions)
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250
t-t 0 (s)
PGn Gn+1 (lifetime Gn: 12.5 s)
G S (lifetime G: 180 s)
)( 01 ttkeP
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0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250
t-t 0 (s)
P
The weighted lottery
rGG = 0.76
rGS = 0.40
tGG = 18 s
tGS = 92 s
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G G transition fires
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
time (s)0 20 40 60
18 s
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0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250
t-t 0 (s)
P
The weighted lottery (continued)
rGG = 0.98
rGS = 0.15
tGG = 49 s
tGS = 30 s
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G S transition fires
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
time (s)0 20 40 60
18 s 30 s
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Other transitions enabled
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
time (s)0 20 40 60
18 s 30 s
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-25,000
-20,000
-15,000
-10,000
-5,000
-
5,000
10,000
15,000
20,000
25,000
0 600 1200 1800 2400 3000 3600 4200 4800
Time (s)
De
lta
n
Trajectory of TuM[TuT] constant (10 M) - 150,000 events
GG
SS
De
lta
Tu
M
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[TuT] variable2 million events
0
50
100
150
0 60 120 180
Time (min)
0
10
20
30
40
50
60
MT
leng
th (m
)
[TuT
], [
TuD
] (
M)
MT length
TuDTuT
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TuT
More than one MT end
TuD
Method 1: modelling all items individually
n
G S G S
n n
G S
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Trajectories(6 million events)
0
20
40
60
0 20 40 60
Time (min)
0
10
20
30
40
50
60
MT
leng
th (m
)
[TuT
] (
M)
MT length
[TuT]
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Improving efficiency
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“First Reaction Algorithm” (Gillespie,1976)
• After most recent transition firing at time t:– For each transition Ti:
• Compute current firing rate Ji for Ti
– Draw two numbers, r1 and r2
• Compute J = sum of all J
• Compute firing time tfire from J, r1, and t
• Construct roulette wheel (pie chart) for all J
• Spin wheel over r2 x 360º, find associated T
– Set t to tnext and fire T
J2
J1
J4
J3
J
rttnext
]1log[ 2
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Algorithms
• Accurate (no further assumptions)– First Reaction (Gillespie, 1976)– Next Reaction (Gibson & Bruck, 2000)
• Priority queue; exploits independence in system
• Approximate– Tau-leaping (Gillespie, 2001)
• Assumes that “leap condition” is satisfied (leap condition: negligible changes in firing probability over interval Tau)
• Multiple firings per step– Chemical Langevin
• Assumes that 1) leap condition is satisfied; 2) All current firing rates are much larger than 1/Tau (“Large yet small”)
• Like deterministic trajectory with superimposed noise
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Deterministic Simulation
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Deterministic?
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Time (min)
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Time (min)
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Average of many trajectories converge to a smooth path, that is entirely determined by the initial conditions
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Assessing the average trajectory
• Stochastic– Many trajectories– Many particles (larger volume)
• Deterministic– Numerical integration of coupled ODEs
(ordinary differential equations)
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ODEs
• ODE: defines how system variables change with another variable
• General form:– d[X]/dt = v+(X) – v-(X)
– v+(X) = J+(X) and v-(X) = J-(X)
Rate at which X is produced
Rate at which X is consumedChange in [X]
over very small time interval dt
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Composing ODEs from firing rates
kSG
G Sk G
G
TuT TuD
kGS
k S
S
3
TuM
JGG = kGG [G][TuT]
JSG = kSG [S][TuT]3
JSS = kSS [S]
JGS = kGS [G]
v+(S) = JSS + JGS
v(S) = JSG + JSS
3]][[][
)(][
TuTSkGk
JJ
JJJJdt
Sd
SGGS
SGGS
SGSSGSSS
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The full system
][]][[][
][][
]][[3]][[][
]][[][][
][]][[][
3
3
3
SkTuTGkdt
TuMd
Skdt
TuDd
TuTSkTuTGkdt
TuTd
TuTSkGkdt
Sd
GkTuTSkdt
Gd
SSGG
SS
SGGG
SGGS
GSSG
Growing ends
Shrinking ends
Free Tu-GTP
Free Tu-GDP
Bound Tu
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Predicting average trajectories: Numerical integration
1. Set component concentrations to their initial values
e.g. [TuT] = 50.0 M, [G] =1.6 nM, [S]=[TuM]=[TuD]=0.0
2. For each component, compute concentration change over small time interval t
e.g. [TuM] = [TuM]t+t – [TuM]t = {kGS[G]t [TuT]t - kSG[S]t} x t
3. Solve all concentrations at t + te.g. [TuM]t+t = 0.0 M + (1.6x106 M-1s-1 x 50 M x 1.6 nM – 220 s-1 x 0.0 M) x 1 ms = 0.128 nM
4. Set t to t+t, and [X]t to [X]t+t, and repeat from 2 until end condition is fulfilled
e.g. t > tmax or no more significant changes in any concentration
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Result
0
20
40
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0 20 40 60
Time (min)
0
10
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30
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60
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Things to keep in mind
• Main assumption: firing rates (J) remain constant over t– Therefore: [X] used to compute J must be negligible, so that t
must be made sufficiently small– Sophisticated algorithms calculate most efficient t before each
time step– Assessment of efficient t uses “accuracy” parameter
• Approach invalid when J not constant over t (within set limits)– Causes: accuracy too low– Warning signs: wildly oscillating trajectories, negative
concentrations, very different results for greater accuracy
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Algorithms
• Forward (explicit)– Solve:
• [G]t+t = [G]t + { kSG[S]t – kGS[G]t [TuT]t3 } x t
– Examples: Euler, Runge-Kutta, Adams-Bashford– Use: for many ODE systems (easier to implement, individual
steps less computationally intensive than implicit methods)
• Backward (implicit)– Solve:
• [G]t= [G]t +t + { kSG[S]t +t – kGS[G]t +t [TuT]t +t 3 } x t
– Examples: Gear, Adams-Moulton– Use: for “stiff” systems (concentrations changing on very
different time scales)
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Stochastic vs deterministic approaches
Stochastic• Deal with finite numbers
of items• Cheap assessment of
inherent “noise”• Assessment of average
population behaviour expensive
• Easy to form mental picture
• Easy to incorporate in physical models
Deterministic• Deal with populations of
infinite size• No information about
inherent noise• Assessment of average
population behaviour computationally cheap
• More difficult to grasp and use
• Integration with other model types not trivial
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Stochastic or deterministic?
• “Well-stirred containers” – Deterministic faster, results less confusing than stochastic
• “Noise” (deviations from average) important for interpretation experimental results – Many items: deterministic with superimposed variation;
approximate stochastic (Tau-leap, chemical Langevin)– Few items: accurate stochastic (Gillespie, Gibson-Bruck)
• Firing probabilities depend on geometry that changes after each event (e.g. detailed description of microtubule ends) – Accurate stochastic, needs tailor-made software
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Software tools
• Programming languages– C/C++, C#, Delphi , Java, Python, VBasic…
• High level math/statistics software– Maple, Mathematica, MATLAB, Octave, R …
• Dynamical Systems software– Berkeley-Madonna, Dymola, ModelMaker, XPP…
• Dedicated chemical kinetics software– Deterministic: CellDesigner, DBSolve , E-Cell,
Gepasi, Jarnac/JDesigner, Promot-DIVA, PySces, Virtual Cell…
– Stochastic: Dizzy, StochKit…– Both: Copasi, NetBuilder’…
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Acknowledgement
• Peter Bayley, Justin Molloy (NIMR, London)
• The SBML Forum
• Jack Correia
• The Wellcome Trust, EPSRC, UH