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Performance Limitations in Autocatalytic Pathways
Milad Siami Department of Mechanical Engineering
Joint work with: Nader Motee
Gentian Buzi, Bassam Bamieh, Mustafa Khammash, and John C. Doyle
IMA Workshop: Biological Systems and NetworksNovember 20, 2015
Robustness
2
1/24
Overview
● Networks with Autocatalytic Structure
● Minimal Autocatalytic Pathway Model
● Characterization of Hard Limits● Hard Limits on Disturbance Attenuation ● Hard Limits on Output Energy
● Autocatalytic Pathways with Multiple Intermediate Metabolite Reactions
● Ongoing Work
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Networks with Autocatalytic Structure
Ex. Biological, engineered, economic networks
Autocatalytic feedbacks are
● Positive feedbacks● Destabilizing
The network’s product (output) is necessary to power and catalyze its own production
4
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Motivating Examples: Glycolysis Pathway
Glycolysis pathway is a central energy producer in a living cell TCA
Pyr
Oxa Cit
ACA
Gly G1P G6P F6P F1-6BP
PEP
Gly3p 13BPG 3PG 2PG
NADH2X
5
3/24
Motivating Examples: Glycolysis Pathway
Glycolysis pathway is a central energy producer in a living cell TCA
Pyr
Oxa Cit
ACA
Gly G1P G6P F6P F1-6BP
PEP
Gly3p 13BPG 3PG 2PG
NADH2X
ATP
5
3/24
Motivating Examples: Glycolysis Pathway
Glycolysis pathway is a central energy producer in a living cell TCA
Pyr
Oxa Cit
ACA
Gly G1P G6P F6P F1-6BP
PEP
Gly3p 13BPG 3PG 2PG
NADH2X
ATP
AutocatalysisCell Consumption of
ATP
5
3/24
Motivating Examples: Glycolysis Pathway
Glycolysis pathway is a central energy producer in a living cell TCA
Pyr
Oxa Cit
ACA
Gly G1P G6P F6P F1-6BP
PEP
Gly3p 13BPG 3PG 2PG
NADH2X
ATP
AutocatalysisCell Consumption of
ATP
PFKHK
Catalyzing Enzymes
Allosteric Regulation
PK
5
Motivating Examples: Glycolysis Pathway
4/24
F6P F1-6BPGly3p 13BPG 3PG
PFK
ATP
Cell Consumption of ATP
PK
6
F6P F1-6BPGly3p 13BPG 3PG
PFK
ATP
Motivating Examples: Glycolysis Pathway
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PK
Cell Consumption of ATP
7
A minimal Autocatalytic Pathway Model
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s x'PFK
y
Cell Consumption of ATP
xPK
8
A minimal Autocatalytic Pathway Model
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PFK
y
xPK
Equilibrium point:
9
A minimal Autocatalytic Pathway Model
7/24
PFK
y
xPK
10
A minimal Autocatalytic Pathway Model
7/24
PFK
y
xPK
Feedback designed by Nature
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A minimal Autocatalytic Pathway Model
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y
xPK
u
u
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Previous Works
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Previous Works
They use a linearized two-state model of glycolysis pathway
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Linearized Model:
Previous Works
The plant has a RHP zero, which imposes hard limits on performance
14
Previous Works
15
Chandra, et al. showed that oscillation in glycolysis is due to the existence of autocatalytic feedback
Our goal is to characterize hard limits due to autocatalytic structures in networks by using nonlinear models of such networks.
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Characterization of Hard Limits
We interpret fundamental limitation of feedback by using
● Hard limits (lower bounds) on L2-gain disturbance attenuation of the system
● Hard limits (Lower bounds) on L2-norm of the output of the system
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Hard Limits on L2-gain Disturbance Attenuation
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Theorem:
Hard Limits on L2-gain Disturbance Attenuation
(GP)
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MS, G. Buzi & N. Motee, “Characterization of Hard Limits on Performance of Autocatalytic Pathways,” ACC2013.
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Tradeoff Between Robustness and Efficiency
The glycolysis mechanism is more robust efficient if k and g are large
large k requires either a more efficient or a higher level of enzymes, and large g requires a more complex controlled PK enzyme
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Hard Limits on L2-Output Energy
Cheap optimal control problem:
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Hard Limits on L2-norm Output Energy
Theorem:
PFK
y
xPK
21
MS, G. Buzi & N. Motee, “Characterization of Hard Limits on Performance of Autocatalytic Pathways,” ACC2013.
16/24
Hard Limits on L2-norm Output Energy
Theorem:
PFK
y
xPK
21
MS, G. Buzi & N. Motee, “Characterization of Hard Limits on Performance of Autocatalytic Pathways,” ACC2013.
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Tradeoff Between Fragility and Production
Increasing q (number of ATP molecules invested in the pathway), increases fragility of the network to small disturbancesand it can result in undesirable transient behavior
Increasing q, increases the production of the pathway
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Multiple Intermediate Metabolite Reactions
F6P F1-6BPGly3p 13BPG 3PG
PFK
ATP
Cell Consumption of ATP
PK
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Multiple Intermediate Metabolite Reactions
PFK
ATP
x1 x2 x3 xnPK
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Multiple Intermediate Metabolite Reactions
PFK
ATP
x1 x2 x3 xnPKK1 K2
(GGP)
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Hard limits on L2-gain Disturbance Attenuation
Theorem:
25
MS, G. Buzi & N. Motee, “Characterization of Hard Limits on Performance of Autocatalytic Pathways,” ACC2013.
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Ex. The L2-gain disturbance attenuation of autocatalytic pathways, and the obtained hard limit based on our Theorem.
Multiple Intermediate Metabolite Reactions
26
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Conclusion
● We characterize fundamental limits on robustness and performance measures of autocatalytic pathways
● We explicitly derive hard limits on the performance of the autocatalytic pathways with intermediate reactions
● We generalize our results to higher dimensional model of autocatalytic pathways
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Related Work
● Robustness analysis of autonomous cyclic networksM. Siami and N. Motee, "Robustness and Performance Analysis of Cyclic Interconnected Dynamical Networks," The SIAM Conference on Control and Its Application, San Diego, CA, USA, 2013.
● Fundamental limitations of feedback control laws in cyclic dynamical networks
M. Siami and N. Motee, "On Existence of Hard Limits in Autocatalytic Networks and Their Fundamental Limitations, " The 3rd IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys'12), Santa Barbara, CA, USA, 2012.
M. Siami and N. Motee, "Fundamental Limits on Performance of Autocatalytic Pathways with Chain Topologies," The Mediterranean Conference on Control and Automation, Platanias-Chania, Crete, Greece, 2013.
M. Siami, N. Motee and G. Buzi, "Characterization of Hard Limits on Performance of Autocatalytic Pathways," In Proceedings of American Control Conference, Washington, DC, USA, 2013.
• Generalize our results for a class of directed networks
• Unification of network performance analysis
Ongoing Work
• M. Siami and N. Motee, "Network Sparsification with Guaranteed Systemic Performance Measures,” NecSys, 2015. • M. Siami and N. Motee, "Performance Analysis of Linear Consensus Networks with Structured Stochastic Disturbance
Inputs,” ACC, 2015. • M. Siami and N. Motee, "Systemic Measures for Performance and Robustness of Large–Scale Interconnected
Dynamical Networks, “ CDC, 2014.• M. Siami and N. Motee, "Schur–Convex Robustness Measures in Dynamical Networks, " ACC, 2014. • M. Siami and N. Motee,"Fundamental Limits on Robustness Measures in Networks of Interconnected Systems," CDC,
2013.