A Mathematical Framework for Modularity�in Chemical Reaction Networks (CRNs)��Workshop on �Modularity of Biological Systems�Chicago, IL�

Joseph L Hellerstein^1,2,3, Steven S Andrews¹, Herbert M Sauro^1,3,4

¹Department of Bioengineering

²Allen School of Computer science

³eScience Institute

⁴Molecular Engineering and Sciences Institute

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University of Washington, Seattle

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April 13, 2026

Modules in Engineering�Electrical, Mechanical, Software Engineering

• Modules can be added and removed with limited impact (other than adding/removing the function that they provide).
• Modules can be tested in isolation.
• There may be many implementations of modules with the same function.
• Modules are often developed and marketed separately from the systems in which they are used.

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Many industries would be impossible without modules.

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*Roche Applied Science, © 1993 Boehringer Mannheim GmbH.

1. What is a biological module?
2. How can we discover biological modules?
3. How does a modular structure relate to biological function?
4. How are biological modules combined?
5. What are the benefits of modularity in biology?
6. How does modularity arise in evolution?

Prokaryotic and Eukaryotic Metabolic Pathways*

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* Claude, 3/29/2026 with prompting from SA Andrews, JL Hellerstein, HM Sauro, S Wiley

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Focus of this work

Elements of a CRN

S1, S2, S3 are chemical species.

k0, k1, k2, k3, k4 are constants.

x1, x2, x3 are the time-varying concentrations of S1, S2, S3

R0: -> S1; k0

R1: S1 -> S2; k1*x1

R2: S2 -> S1; k2*x2

R3: S2 -> S3; k3*x2

R4: S3 -> S2; k4*x3

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Prof. Doutor Pedro Silva,  Universidade Fernando Pessoa

glycogenesis

glycolysis

pentose phosphate pathway

citric acid cycle

Interactions between Common Metabolic Pathways

CRN Behavior (Kinetics)

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Rate laws imply differential equations.

Not stable.

Dominant eigenvalue is 0.

R0: -> S1; k0

R1: S1 -> S2; k1*x1

R2: S2 -> S1; k2*x2

R3: S2 -> S3; k3*x2

R4: S3 -> S2; k4*x3

x1(0)-x3(0)=0

k0=1;k1=1;k2=0.8

k3=0.5;k4=0.5

Not all kinetics are linear.

But they can be linearized.

(Linear Systems Theory)

Research Objectives

Multiple Input Multiple Output (MIMO) Analysis

• Stability of combined chemical networks
• Steady state of combined (stable) chemical networks.
• How does the DC gain (steady state step response) of two subsystems in isolations relate to the DC gain of the subsystems in combination?

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Relates to the following questions

1. What is a biological module?

2. How can we discover biological modules?

3. How does a modular structure relate to biological function?

4. How are biological modules combined?

5. What are the benefits of modularity in biology?

6. How does modularity arise in evolution?

Subsystems

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• A subsystem of a chemical reaction network (CRN) is a subset of the species and reactions of the CRN.
• Subsystems may have species in common (e.g., Fructose-6-P)
• A reaction is in at most one subsystem, and, if present in a subsystem, all of the reaction’s participants must be present.
• Subsystems are an analysis concept that may not have a physical realization.

glycolysis

subsystem

pentose phosphate

subsystem

Modularity without species in multiple modules.

Subsystems, Modules, Motifs

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Sauro Figure

Propopsed Here

Kinetics of Combining Subsystems

Jacobian additivity: Because subsystems have disjoint reactions, the Jacobian of their combination is the sum of their Jacobians.

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S2 is a coupling species.

S1

S2

S3

0

0

Jacobians

Gershgorin Circle Theorem (GCT) & Stability

• Common approaches to stability
• Liapunov analysis
• Calculate eigenvalues of Jacobian & check if negative

Neither helps with stability of questions related to subsystems in combination..

• Gershgorin circle theorem (GCT)
• All eigenvalues must lie in at least one “circle”

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• GCT guarantees stability if

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Stability Detection & Remediation With GCT

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R0: -> S1; k0

R1: S1 -> S2; k1*x1

R2: S2 -> S1; k2*x2

R3: S2 -> S3; k3*x2

R4: S3 -> S2; k4*x3

x1(0)-x3(0)=0

k0=1;k1=1;k2=0.8

k3=0.5;k4=0.5

Not stable: Dominant eigen is 0

Stability Analysis of Combined Subsystems

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Green subsystem

Red subsystem

DC Gain of Stable Subsystems

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DC Gain is the steady state response to a step input.

It addresses “How does a stable system behave?”

DC Gain in Combined Subsystems

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Retroactivity: Green system (Y) changes the output of red system (Z).

How coupling works

coupling species

Solving for Steady State Step Response

Overbar indicates steady state.

Analysis of Network Structures

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Join

Bi-directional Interactions

Branch

Feedback

A module is a stable, physically realizable subsystem that is unaffected by the consumption of its outputs.

Evaluating the Linearity Assumption�How Linear Are Models in BioModels?

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• BioModels has a collection of ~1,100 published simulation models in biology.
• Linearity is a good approximation if the Jacobian does not change over time.
• Calculated the maximum coefficient of variant of entries in Jacobians over 100 time points for ~750 models in BioModels

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Possible Answers to the Questions

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Acknowledgements

• Co-authors: Steve Andrews, Herbert Sauro
• Reinhart’s monthly meeting
• Colleagues at this workshop

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