Dynamical Dimension Flat dimension : by Karhunen-Loève expansion (PCA)
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Dynamical Dimension Flat dimension : by Karhunen-Loève expansion (PCA) Limit Simplifications A modular technique : a)define modules : set of intermediate species b)compute simple submechanisms (SS) for modules c)use idempotent algebra for the rates of SS : important complexity reduction! Example: transcriptional module of p50, one of the two proteins forming NFB. 0, interm ediatestoichiom etricm atrix 0, irreversible, 0 ism inim al 0, term inalstoichiom etricm atrix I I i i T T S S i i S S Definition of simple submechanisms Limit simplifications 32 32 0 0 33 34 69 70 ' , concentration ofFTA x m ax( , , ) prom oteractivity ism axim um w hen both FTA x and N F B are bound k k F F k k k k 32 32 0 33 70 69 34 ' m ax( , , ) prom oteractivity ism axim um w hen onlyFTA x isbound k k F k k k k 1 39 36 70 0 66 11 1 1 1 39 ( ) 37 38 39 64 7 66 11 68 k k k P k X k k k k X k X k 1 39 36 70 0 64 7 1 1 1 39 ( ) 37 38 39 64 7 66 11 68 k k k P k X k k k k X k X k P50 is downregulated by IB (X11) upregulated by NF B (X7) P50 is upregulated by IB downregulated by NF B Order conditions on non-critical parameters SS rates depend on critical parameters el Reduction and Model Comparison for NF el Reduction and Model Comparison for NF adulescu (IRMAR and IRISA, Rennes, France), Andrei Zinovyev (Institut Curie, Paris, France), Alain Lilienbaum ( contact: [email protected] Context Model reduction techniques produce small but still accurate models from larger ones. There are many reasons for simplifying systems biology models: a) discard unnecessary complexity; b) reveal critical parameters and regulation loops; c) compare models and find common patterns; d) study robustness. The transcription factor NFB has been discovered 20 years ago and is still in the spotlights. It is involved in a wide diversity of domains such as immune and inflammatory responses, cell survival and apoptosis, cellular stress and neuro-degenerative diseases, cancer and development. Starting with a rather complex model of NFB signaling pathway, we obtain its “limit simplifications”. These are compared among them and with existing models. Conclusions Limit simplifications are robust reductions of reaction mechanisms. They allow to identify critical and non-critical parameters. They also allow to compare models, by reducing them to the same reaction graph and by comparing their kinetic laws. Finding equivalent sets of parameter values for different models is made easy. Dynamical dimension determination and PCA analysis are other general tools for model comparison. All models shown here have rather similar behaviour : they oscillate with similar phase relations between main species. Nevertheless, our most complex model M(39,67,88) and the one reduced from it M(14,27,36) have more complex oscillations than Lipniacki model. This is shown by the larger dynamical dimension. These two models are also less robust (result not illustrated here) in the sense that random variations of their parameters produce relatively larger variations of the characteristic times of the oscillations (period, dumping time). Intermediate species Terminal species Reversible reactions Irreversible reactions Dynamics All models oscillate after application of a signal. Depending on parameters, oscillations can be damped, or sustained. Invariant manifold of M(14,27,29) Singular value curves are used to estimate the nonlinear dynamical dimension 4 5 7 8 99.9% 11 6 4 3 M(14,27,36) 5 4 3 2 M(6,9,13) 9 4 3 2 M(14,27,29) 15 6 4 2 M(39,67,88) 100% 99% 95% 80% Model Dimension of linear manifold embedding the limiting trajectories for various ‘‘explained’’ variance threshold Model Comparison Models are distinguished by three complexity indices (n,m,p) : numbers of species, reactions, M(39,67,88) most complex model M(6,9,13) a simplification of M(14,27,29) M(14,27,36) and M(14,27,29) have the same reaction graph, kinetic laws are different; M(14,27,29) was proposed by Lipniacki Visualization of principal components : grouped species oscillate in phase Non-conservative models: the total NFkB quantity has small amplitude oscillations Conservative models: the total NFkB quantity is constant 3 M(14,27,36) 2 M(6,9,13) 2 M(14,27,29) 5 M(39,67,88) Nonlin.dim. Model
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Context Model reduction techniques produce small but still accurate models from larger ones. There are many reasons for simplifying systems biology models: a) discard unnecessary complexity; b) reveal critical parameters and regulation loops; c) compare models and find common - PowerPoint PPT Presentation
Transcript of Dynamical Dimension Flat dimension : by Karhunen-Loève expansion (PCA)
Dynamical Dimension Flat dimension : by Karhunen-Loève expansion (PCA)
Limit Simplifications A modular technique : a) define modules : set of intermediate species b) compute simple submechanisms (SS) for modules c) use idempotent algebra for the rates of SS : important complexity reduction!
Example: transcriptional module of p50, one of the two proteins forming NFB.
0, intermediate stoichiometric matrix
0, irreversible, 0 is minimal
0, terminal stoichiometric matrix
I I
i i
T T
S S
i i
S S
Definition of simple submechanisms
Limit simplifications
32 32 0 0
33 34 69 70
' , concentration of FTAx
max( , , )
promoter activity is maximum when both
FTAx and NF B are bound
k k F F
k k k k
32 32 0
33 70 69 34
'
max( , , )
promoter activity is maximum when
onlyFTAx is bound
k k F
k k k k
1
39 36 70 0 66 111 1 139 ( )
37 38 3964 7 66 11 68
k k k P k X
k k kk X k X k
1
39 36 70 0 64 71 1 139 ( )
37 38 3964 7 66 11 68
k k k P k X
k k kk X k X k
P50 is downregulated by IB (X11) upregulated by NF B (X7)
P50 is upregulated by IB downregulated by NF B
Order conditions on non-critical parameters
SS rates depend on critical parameters
Model Reduction and Model Comparison for NFModel Reduction and Model Comparison for NFB SignalingB SignalingOvidiu Radulescu (IRMAR and IRISA, Rennes, France), Andrei Zinovyev (Institut Curie, Paris, France), Alain Lilienbaum (CNRS URA 2115, Paris, France)
contact: [email protected]
Context Model reduction techniques produce small but still
accurate models from larger ones. There are many reasons for simplifying
systems biology models: a) discard unnecessary complexity; b) reveal critical
parameters and regulation loops; c) compare models and find common
patterns; d) study robustness.
The transcription factor NFB has been discovered 20 years ago and is still in
the spotlights. It is involved in a wide diversity of domains such as immune
and inflammatory responses, cell survival and apoptosis, cellular stress and
neuro-degenerative diseases, cancer and development. Starting with a rather
complex model of NFB signaling pathway, we obtain its “limit simplifications”.
These are compared among them and with existing models.
Conclusions Limit simplifications are robust reductions of reaction mechanisms. They allow to identify critical and non-critical parameters. They also allow to compare models, by reducing them to the same reaction graph and by comparing their kinetic laws. Finding equivalent sets of parameter values for different models is made easy. Dynamical dimension determination and PCA analysis are other general tools for model comparison. All models shown here have rather similar behaviour : they oscillate with similar phase relations between main species. Nevertheless, our most complex model M(39,67,88) and the one reduced from it M(14,27,36) have more complex oscillations than Lipniacki model. This is shown by the larger dynamical dimension. These two models are also less robust (result not illustrated here) in the sense that random variations of their parameters produce relatively larger variations of the characteristic times of the oscillations (period, dumping time).
Intermediate speciesTerminal species
Reversible reactionsIrreversible reactions
DynamicsAll models oscillate after application of a signal. Depending on parameters, oscillations can be damped, or sustained.
Invariant manifold of M(14,27,29)
Singular value curves are used to estimate the nonlinear dynamical dimension
4
5
7
8
99.9%
11643M(14,27,36)
5432M(6,9,13)
9432M(14,27,29)
15642M(39,67,88)
100%99%95%80%Model
Dimension of linear manifold embedding the limiting trajectories for various ‘‘explained’’ variance threshold
Model ComparisonModels are distinguished by three complexity indices (n,m,p) : numbers of species, reactions, parameters.
M(39,67,88) most complex model
M(6,9,13) a simplification of M(14,27,29)
M(14,27,36) and M(14,27,29) have the same reaction graph, kinetic laws are different; M(14,27,29) was proposed by Lipniacki
Visualization of principal components : grouped species oscillate in phase
Non-conservative models: the total NFkB quantity has small amplitude oscillations
Conservative models: the total NFkB quantity is constant
3M(14,27,36)
2M(6,9,13)
2M(14,27,29)
5M(39,67,88)
Nonlin.dim.Model
![MOX–Report No. 59/2014 · (e.g. through a modal decomposition exploiting a Karhunen-Loève expansion [24] or suitable greedy algorithms [23]), several techniques have emerged in](https://static.fdocuments.us/doc/165x107/5f854594113f663402623a08/moxareport-no-592014-eg-through-a-modal-decomposition-exploiting-a-karhunen-love.jpg)
