Performance Limitations in Autocatalytic Pathways · 2015-11-19 · Performance Limitations in...

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

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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

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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

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Pyr

Oxa Cit

ACA


PEP

Gly3p 13BPG 3PG 2PG

NADH2X

ATP

5

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Pyr

Oxa Cit

ACA


PEP

Gly3p 13BPG 3PG 2PG

NADH2X

ATP

AutocatalysisCell Consumption of

ATP

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Pyr

Oxa Cit

ACA


PEP

Gly3p 13BPG 3PG 2PG

NADH2X

ATP

AutocatalysisCell Consumption of

ATP

PFKHK

Catalyzing Enzymes

Allosteric Regulation

PK

5


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F6P F1-6BPGly3p 13BPG 3PG

PFK

ATP

Cell Consumption of ATP

PK

6


PFK

ATP


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PK


7

A minimal Autocatalytic Pathway Model

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s x'PFK

y


xPK

8


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PFK

y

xPK

Equilibrium point:

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PFK

y

xPK

10


7/24

PFK

y

xPK

Feedback designed by Nature

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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

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Previous Works

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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

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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


PFK

ATP


PK

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PFK

ATP

x1 x2 x3 xnPK

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PFK

ATP

x1 x2 x3 xnPKK1 K2

(GGP)

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Hard limits on L2-gain Disturbance Attenuation

Theorem:

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Ex. The L2-gain disturbance attenuation of autocatalytic pathways, and the obtained hard limit based on our Theorem.


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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.

Performance Limitations in Autocatalytic Pathways · 2015-11-19 · Performance Limitations in...

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