Flux Exponent Control
Predicts Metabolic Dynamics
from Network Structure
Fangzhou Xiao, Jingshuang (Lisa) Li, John C Doyle
UCSD, Caltech
American Control Conference
202305
Bioengineering: engineer cells as biomachines.
Biocontrol: How to formulate cells as controllable machines?
Essential foundation for a mature bioindustry.
Analyze and design cells like cars and power grids.
David Goodsell
Capture uniquely biological properties, not borrowing existing formulations from other disciplines.
Biocontrol: formulate cells as a controllable machine.
Gas laws
Transistors, band gap…
Electronics, radiation, amplifier...
Ohm’s law, …
Systems theory
Structures of interaction
Mechanical components
Mechanical machine
Newtonian,
Mass & Force
Steam engine
Thermo-
dynamics
Computer
Turing machine
Electrical circuits
Linear i/o systems
Communication network
Information channels
Components
Machines
Components
Machines
Lagrangian
Cells are metabolic machines.
Biocontrol: formulate cells as a controllable machine.
Molecules IN
Molecules OUT
Systems theory
Structures of interaction
Biomolecular reactions
Metabolic machines
Theory of
Flux exponent control (FEC)
Binding & Catalysis
Tool: Reaction order polyhedra (ROP)
David Goodsell
Components
Machines
Components
Machines
Specific goal: model metabolism dynamics.
Metabolism dynamics is only sparsely known ⇒ hard to model.
Lack knowledge about enzyme regulation.
substrate
(total)
product
(total)
You see this:
wikipedia
Metabolism dynamics is only sparsely known ⇒ hard to model.
You see this:
Lack knowledge about enzyme regulation.
More generally:
x, metabolite conc., variable
S, stoichiometry, known.
v, flux, unknown.
substrate
(total)
product
(total)
wikipedia
Flux control: cells are metabolic machines that control fluxes.
Stoichiometry, Known
Flux,
Unknown
Flux control: cells are metabolic machines that control fluxes.
Machine architecture:
Stoichiometry.
Machine control actions:
Flux control.
Flux control: cells are metabolic machines that control fluxes.
Lack knowledge about enzyme regulation.
You see this:
Machine architecture:
Stoichiometry.
Machine control actions:
Flux control.
More generally:
x, metabolite conc., variable
S, stoichiometry, known.
v, flux, unknown.
Constraint-based methods to model metabolism fill in unknown via optimization.
Specify detailed mechanisms.
Completely known.
Choose
biologically feasible fluxes.
Constraint-based methods to model metabolism fill in unknown via optimization.
e.g. Growth
Biological fluxes
Specify detailed mechanisms.
Completely known.
Choose
biologically feasible fluxes.
Flux control has too few constraints for dynamic fluxes.
Flux control has few constraints for dynamic fluxes.
Choose
stoichiometry-compatible fluxes.
Biological fluxes
Choose
biologically feasible fluxes.
Flux control has few constraints for dynamic fluxes.
Choose
stoichiometry-compatible fluxes.
e.g. Growth
Biological fluxes
Choose
biologically feasible fluxes.
Flux control (FC)
In practice, only steady state fluxes has enough constraints.
Choose
stoichiometry-compatible fluxes.
e.g. Growth
Biological fluxes
Flux control can only capture s.s. metabolism, because it ignores intrinsic dynamics.
Flux control (FC), incl. FBA
Flux control for steady state fluxes is flux balance analysis (FBA).
Choose static fluxes.
Dynamics in life, e.g. glycolysis.
Autocatalysis (e.g. glycolysis) is a core part of life,
From energy generation to growth.
Positive feedback.
Intrinsically unstable.
Require active regulation.
Glycolysis generates ATP from glucose.
Consume ATP.
Produce ATP.
Chance, Schoener, Elsaesser,
PNAS, 1964
Dano, Sorensen, Hynne,
Nature, 1999
Dynamics in life, e.g. glycolysis.
Autocatalysis (e.g. glycolysis) is a core part of life,
From energy generation to growth.
Glycolysis generates ATP from glucose.
Flux control: Trivial plant, ignores intrinsic dynamics
Consume ATP.
Produce ATP.
Chance, Schoener, Elsaesser,
PNAS, 1964
Dano, Sorensen, Hynne,
Nature, 1999
Flux control ignores metabolic fluxes are catalyzed by enzymes.
Machine architecture
Metabolic machines that control fluxes.
Metabolic fluxes have intrinsic dynamics without regulation!
They are catalyzed by enzymes!
Fluxes are controlled via binding’s regulation of enzyme activity.
Fluxes are controlled via binding’s regulation of enzyme activity.
How to formulate this mathematically?
Binding regulates fluxes’ exponents.
Cells control fluxes by binding;
Cells control fluxes’ exponents!
(Flux exponent control, FEC)
+
=
Cells control flux exponents, not fluxes themselves.
Machine architecture
Metabolic machines that control fluxes.
Metabolic machines that control fluxes’ exponents.
Idea of exponent regulation dates back to Michael Savageau’s S systems in 1970s.
Metabolic fluxes have intrinsic dynamics without regulation!
They are catalyzed by enzymes!
How to formulate flux exponent control mathematically.
Flux exponent control (FEC)
Flux control (FC)
Reference flux magnitude
Unregulated reaction dynamics
Regulation of flux exponents by binding
How to formulate flux exponent control mathematically.
Reference flux magnitude
Passive reaction dynamics
Regulation of flux exponents by binding
Intrinsic dynamics of metabolism: Autocatalysis.
Autocatalysis is a core part of life,
From energy generation to growth.
Positive feedback.
Intrinsically unstable dynamics.
Require active regulation.
Glycolysis: cells eating sugar.
Consume ATP.
Produce ATP.
Intuition: autocatalysis is like stick balancing.
Autocatalysis is a core part of life,
From energy generation to growth.
Positive feedback.
Intrinsically unstable dynamics.
Require active regulation.
Consume ATP.
Produce ATP.
Direct control of all fluxes is like no inertia.
Autocatalysis is a core part of life,
From energy generation to growth.
Positive feedback.
Intrinsically unstable dynamics.
Require active regulation.
Machine architecture
Machine control actions
Metabolic fluxes have unmodifiable intrinsic dynamics!
How to capture this?
FEC as a constraint-based approach.
In vector form:
Regulates flux exponents (reaction order) for static controller:
FBA
FEC
Flux exponent control (FEC) can constrain dynamic fluxes.
Biological fluxes
Choose
dynamic control of fluxes’ exponents.
(Correspond to regulation in binding networks)
Choose
stoichiometry-compatible fluxes.
Flux control (FC), incl. FBA
Flux exponent control (FEC) can constrain dynamic fluxes.
Choose
dynamic control of fluxes’ exponents.
(Correspond to regulation in binding networks)
Choose
stoichiometry-compatible fluxes.
Biological fluxes
Flux exponent
control
Hard limits.
(Control theory tools, e.g. conservation of robustness)
Flux control (FC), incl. FBA
Flux exponent control (FEC) can constrain dynamic fluxes.
Choose
dynamic control of fluxes’ exponents.
(Correspond to regulation in binding networks)
Choose
stoichiometry-compatible fluxes.
Biological fluxes
Flux exponent
control
Flux control (FC), incl. FBA
Flux exponent control (FEC) can constrain dynamic fluxes.
Choose
dynamic control of fluxes’ exponents.
(Correspond to regulation in binding networks)
Choose
stoichiometry-compatible fluxes.
Biological fluxes
Flux exponent
control
e.g. Growth
Flux control (FC), incl. FBA
Flux exponent control (FEC) capture oscillation in glycolysis!
Dano, Sorensen, Hynne,
Nature, 1999
Flux exponent control (FEC) capture oscillation in glycolysis!
Dano, Sorensen, Hynne,
Nature, 1999
FEC captures cell growth arrest under stress.
Cell growth arrest under stress is another hallmark of biological adaptation.
e.g. diauxic growth.
Case study:
Cell grown on glycolysis with sudden increase in maintenance cost.
Jacques Monod, PhD Thesis, 1942.
Jing Shuang (Lisa) Li
(Doyle group)
FEC captures cell growth arrest under stress.
Cell growth arrest under stress is another hallmark of biological adaptation.
e.g. diauxic growth.
Case study:
Cell grown on glycolysis with sudden increase in maintenance cost.
Jacques Monod, PhD Thesis, 1942.
Jing Shuang (Lisa) Li
(Doyle group)
FEC comes from plant-controller splits on
the layered architecture of metabolism.
Layered architecture is shared in control and biology.
Summary
Systems theory
Structures of interaction
FEC enables dynamic constraint-based modeling of metabolism, upgrading state-of-the-art method FBA.
Biocontrol: how cells regulate their behaviors,
viewed as a controllable machine.
Biomolecular reactions
Metabolic machines
Flux exponent control (FEC)
Binding & Catalysis
Tool: Reaction order polyhedra (ROP)
Components
Machines
Components
Machines
Unique application & challenge for MPC!
State and control constraints;
Network sparsity and controller locality;
(localized and distributed MPC via system level synthesis SLS)
Designed controllers ⇔ biological mechanisms.
Funding and collaborators
Collaborators on this project
Funding in part supported by
Army Research Office (ARO) MURI (Contract W911NF-17-1-0402)
This project spans my journey from Caltech to UCSD to Westlake University.
Jing Shuang (Lisa) Li
(Caltech -> Michigan)
John C Doyle
(Caltech CDS)
I thank the inspirations from my mentors.
Erik Winfree
Richard Murray
John Doyle
Rob Phillips
Lior Pachter
Me…
Thank you! Questions?
Imaginations about the
future of biocontrol.
from Vantage Films, by Denis Sibilev, Andrei Myshev, Dmitry Medvedev
To formulate biocontrol, we need theories for components and machines.
Components have measurable properties.
Simply exists.
A purpose or objective.
Machines are functional, specified by architecture.
Systems theory
Structures of interaction
Architecture limits performance.
image: Vecteezy.com
image: Flaticon.com
Components
Machines
Components
Machines
These properties define their interactions.
Examples of interaction structures and systems theories.
Systems theory
Structures of interaction
Mechanical components
Mechanical machine
Lagrangian
Newtonian,
Mass & Force
Components
Machines
Components
Machines
Examples of interaction structures and systems theories.
Gas laws
Transistors, band gap…
Electronics, radiation, amplifier...
Ohm’s law, …
Systems theory
Structures of interaction
Mechanical components
Mechanical machine
Newtonian,
Mass & Force
Steam engine
Thermo-
dynamics
Computer
Turing machine
Electrical circuits
Linear i/o systems
Communication network
Information channels
Components
Machines
Components
Machines
Lagrangian
To formulate biocontrol, we need theories for components and machines.
Systems theory
Structures of interaction
????
Components
Machines
Components
Machines
????
??? Biological components ???
??? Biological machines ???
To formulate biocontrol, we need theories for components and machines.
Systems theory
Structures of interaction
Components
Machines
Components
Machines
Biomolecular reactions
??? Biological machines ???
Binding & Catalysis
Tool: Reaction order polyhedra (ROP)
????
Biomolecular reactions: binding regulates catalysis.
Binding
Catalysis
E. Coli, by David Goodsell.
Biomolecular reactions: binding regulates catalysis.
Binding
Determines the direction of change.
Catalysis
enzyme
substrate
product
total substrate
total product
E. Coli, by David Goodsell.
Biomolecular reactions: binding regulates catalysis.
Binding
Catalysis
enzyme
substrate
product
total substrate
total product
Determines the direction of change.
E. Coli, by David Goodsell.
E. Coli, by David Goodsell.
Biomolecular reactions: binding regulates catalysis.
Binding
Catalysis
enzyme
substrate
product
total substrate
total product
Determines the direction of change.
Determines the rate, regulates catalysis flux.
E. Coli, by David Goodsell.
E. Coli, by David Goodsell.
Biomolecular reactions: binding regulates catalysis.
Binding
Catalysis
enzyme
substrate
product
total substrate
total product
Determines the direction of change.
Determines the rate, regulates catalysis flux.
Slow.
Fast.
Biomolecular reactions: binding regulates catalysis.
Binding
Catalysis
enzyme
substrate
product
total substrate
total product
Determines the direction of change.
Determines the rate, regulates catalysis flux.
Bioregulation
New formulation, not new problem. Only solved under restrictive assumptions.
Slow.
Fast.
Regulation profile of a binding network is hard to solve.
Simplify
Regulation profile of a binding network is hard to solve.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Totals are variables changed by catalysis.
i.e. as a function of .
Goal: how catalysis flux is regulated.
Regulation profile of a binding network is hard to solve.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Totals are variables changed in catalysis.
i.e. as a function of .
Goal: how catalysis flux is regulated.
Solve: A degree 2 polynomial.
r=1
r=2
2 s.s. Eqns, ⇒ Degree 3 polynomials.
3 conservations.
Regulation profile of a binding network is hard to solve.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Totals are variables changed by catalysis.
i.e. as a function of .
Goal: how catalysis flux is regulated.
Solve: Degree 2 polynomial.
n=1
n=2
n
Degree n+1 polynomial…
Degree 3 polynomial.
Regulation profile of a binding network is hard to solve.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Totals are variables changed in catalysis.
i.e. as a function of .
Goal: how catalysis flux is regulated.
r=r
Degree r+1 polynomials…
Solve: A degree 2 polynomial.
r=1
r=2
Numerical scan: 100 points per dim.
e.g. r=2, dim=4, 100^4 pts....
Regulation profile of a binding network is hard to solve.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Totals are variables changed in catalysis.
i.e. as a function of .
Goal: how catalysis flux is regulated.
r=r
Degree r+1 polynomials…
Solve: A degree 2 polynomial.
r=1
r=2
Numerical scan: 100 points per dim.
e.g. r=2, dim=4, 100^4 pts....
Intractable analytically or computationally to obtain the full bioregulation profile.
To make progress, people make restrictive assumptions…
Steady state equation:
Conserved quantities:
Assuming
Total substrate:
Total enzyme:
(Michaelis-Menten),
i.e. as a function of .
Goal: how catalysis flux is regulated.
Totals are variables changed in catalysis.
To make progress, people make restrictive assumptions…
Assuming
i.e. as a function of .
Goal: how catalysis flux is regulated.
Combinatorial regulation
Too restrictive for Biocontrol today!!
Highly dynamic
Antebi YE et. al. 2017 Cell
Zhu RH et. al. 2022 Science
Olsman N et. al. 2019 Cell Systems
Briat C. et al 2016 Cell Systems
(Michaelis-Menten),
To make progress, people make restrictive assumptions…
i.e. as a function of .
Goal: how catalysis flux is regulated.
Need to capture the full profile without assumptions.
Rates/fluxes are intractable to solve directly.
Find alternatives to describe bioregulation.
Assuming
(Michaelis-Menten),
Give up rates/fluxes, look at reaction orders.
i.e. as a function of .
Goal: how catalysis flux is regulated.
Assuming
(Michaelis-Menten),
Give up rates/fluxes, look at reaction orders.
i.e. as a function of .
1
0
Two regimes, with 1 or 0 as exponents / reaction orders in tS.
How to capture the full behavior?
Goal: how catalysis flux is regulated.
Assuming
(Michaelis-Menten),
Give up rates/fluxes, look at reaction orders.
i.e. as a function of .
1
0
Log derivative as continuous analog of exponent.
Goal: how catalysis flux is regulated.
Two regimes, with 1 or 0 as exponents / reaction orders in tS.
How to capture the full behavior?
Assuming
(Michaelis-Menten),
Give up rates/fluxes, look at reaction orders.
i.e. as a function of .
1
0
Log derivative as continuous analog of exponent.
Goal: how catalysis flux is regulated.
Two regimes, with 1 or 0 as exponents / reaction orders in tS.
How to capture the full behavior?
Assuming
(Michaelis-Menten),
Reaction order is an alternative representation of bioregulation.
MM formula
0
i.e. as a function of .
Goal: how catalysis flux is regulated.
Assuming
(Michaelis-Menten),
1
Reaction order is an alternative representation of bioregulation.
MM formula
0
i.e. as a function of .
Goal: how catalysis flux is regulated.
Assuming
(Michaelis-Menten),
1
Reaction order is an alternative representation of bioregulation.
(Michaelis-Menten),
MM formula
0
Assuming
i.e. as a function of .
Goal: how catalysis flux is regulated.
Relax this?
1
Reaction order is a HOLISTIC representation of bioregulation.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Directly use implicit function theorem.
MM formula
Reaction order is a HOLISTIC representation of bioregulation.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
Directly use implicit function theorem.
MM formula
Reaction order is a HOLISTIC representation of bioregulation.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
MM formula
Directly use implicit function theorem.
No assumptions, completely structural.
Directly from binding network topology.
Reaction order is a HOLISTIC representation of bioregulation.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
MM formula
Directly use implicit function theorem.
No assumptions, completely structural.
Directly from binding network topology.
Rates are not solvable.
Approximations are not holistic.
Using reaction orders, we can solve holistically.
Reaction order polyhedra (ROP) capture
full bioregulation profiles.
New holistic foundation, replacing Michaelis-Menten used for 100+ years.
Let’s celebrate this Historic Moment…. of Triangles!
Here we literally have a triangle!
Hydrogen atom of bioregulation!
Rob Phillips
This “hydrogen atom” of bioregulation is an art totem!
Displayed in Chandler cafeteria on Caltech campus.
Activator
Repressor
ROP solves bioregulation problems in full. This is powerful.
Cell fate control and multistability in
Combinatorial regulations…
Adaptation and circuit failure in
Highly dynamic scenarios…
Antebi YE et. al. 2017 Cell
Zhu RH et. al. 2022 Science
Olsman N et. al. 2019 Cell Systems
Briat C. et al 2016 Cell Systems
Glycine cleavage system
J Ren, W Wang, J Nie, W Yuan, AP Zeng, 2022
Catalysis with macromolecule substrates.
Example: adaptation to disturbance in reaction order polyhedra.
Degradation
Adapts
Fails to adapt
Fails for high conc. disturbance
Functional regimes are described in reaction order polyhedra.
Repression
Adapts
Fails to adapt
Fails for low conc. disturbance
Both “activating” and “repressing” are present in one binding step.
Activator
Repressor
Substrate acts as an …
Reaction order polyhedra capture full bioregulation profile
(Free enzyme).
Functional regimes are described in reaction order polyhedra.
Degradation
Adapts
Fails to adapt
Fails for high conc. disturbance
TAKE AWAY: Reaction order polyhedra says
Bioregulation MORE THAN saturation/MM/Hill/Monod!
“What are the behaviors for bioregulation (induction curves) in cells?”
Century-old answer: “Michaelis-Menten (MM), of course.”
This is challenged by full bioregulatory profile described by reaction order polyhedra.
Reaction order polyhedron captures full bioregulation profile.
MM formula
Overabundant substrate limit.
Exact
Reaction order polyhedron captures full bioregulation profile.
MM formula
Tight binding limit.
Exact
Overabundant substrate limit.
Reaction order polyhedra says
bioregulation >> saturation/MM/Hill/Monod!
Reaction order polyhedra says
bioregulation >> saturation/MM/Hill/Monod!
Reaction order polyhedra says
bioregulation >> saturation/MM/Hill/Monod!
Reaction order polyhedra says
bioregulation >> saturation/MM/Hill/Monod!
Reaction order polyhedra says
bioregulation >> saturation/MM/Hill/Monod!
Bioregulation has THREE archetypal behaviors.
“Q: What are the typical behaviors for bioregulation in cells?”
(1) Saturation
(2) Limiting factor
(3) Hypersensitivity
Different archetypal behaviors are cuts through different regimes.
The full bioregulation profile in one binding reaction.
Activator
Repressor
Physical interpretation of reaction orders
Physical interpretation of reaction orders
Reaction order relates internal and external chemical potentials.
Total chemical potential <-> chemical potential of each particle type
Reaction order is a change of coordinates for internal vs external.
Mathematical tools for general applications.
Computational sampling of reaction order polyhedron, applied to induced activator.
Mathematical tools for general applications.
Mathematical technique to directly obtain polyhedron, applied to induced activator.
Mathematical tools for general applications.
Combined to analyze biological behavior holistically, e.g. product inhibition of enzymes.
Mathematical tools for general applications.
Full bioregulation profile reveal hidden adaptive regimes.
Enzyme allostery
Back to our goal: formulate cells as a control system.
1
0
Biomolecular reactions
Binding & Catalysis
Biological machines ???
??
systems theory ??
Mechanical components
Mechanical machine
Lagrangian,
Applied force on mechanical structures
Newtonian,
Mass & Force
Tool: Reaction order polyhedra (ROP)
Back to biocontrol: formulate cells as a controllable machine.
Systems theory
Structures of interaction
Biomolecular reactions
Biological machines ???
Binding & Catalysis
Tool: Reaction order polyhedra (ROP)
????
Components
Machines
Components
Machines
Future work 3: application to systems and synthetic biology
Segall-Shapiro TH et. al. 2018 Nature
Full bioregulation profile reveal hidden adaptive regimes.
Future work 2: dissipative network control for multistability in biomachines
Vertices are special, correspond to monomial birth-death systems.
General behaviors are “convex combination” of vertex behaviors?
General Nonlinear
Biomolecular circuits
Lyapunov or Storage functions:
Quadratic
Entropy-like
Dissipativity: energy expenditure
Dissipativity: reaction order
Easy-to-analyze basis:
Future work 2: dissipative network control for multistability in biomachines
Bistable system.
Any fixed point in grey area has entropy-like Lyapunov function certifying regional stability.
Summary
Systems theory
Structures of interaction
NEXT: Rules of life across scales from biomachine architecture.
Biocontrol: how cells regulate their behaviors,
viewed as a control system.
Biomolecular reactions
Metabolic machines
Flux exponent control (FEC)
Binding & Catalysis
Tool: Reaction order polyhedra (ROP)
Components
Machines
Components
Machines
Overview of the talk
Structural view of biomolecular systems.
Mathematical
Foundation of Structure.
Dynamics from Structure.
Implications on metabolism.
Classical (linear) approach
Structural approach
vs
for (local) system property
System specification
(numerical, exact)
Calculate fixed point and Linearization
(unbounded, sparsity structured)
Determine system property
(parameter specific,
or scan through parameters,
or solve symbolic polynomial inequalities)
System specification
(symbolic, approximate)
Log derivative polytope
(bounded by structure, more than sparsity)
Determine system property
(structural, robust to parameter variations)
Relate structure to fixed point stability.
Derivative:
Positive system
Log derivative:
Additive deviation:
Multiplicative deviation:
Production
Degradation
Constrained structurally.
Varies with rates and conc.
Johan Paulsson et. al.
Michael Savageau et. al.
Relate structure to fixed point stability.
Derivative:
Positive system
Log derivative:
Additive deviation:
Multiplicative deviation:
Production
Degradation
Constrained structurally.
Varies with rates and conc.
Johan Paulsson et. al.
Michael Savageau et. al.
Example: heat shock response in bacteria
Non-saturating regime
From Olsman, Alonso, Doyle 2018
Calculate fixed point and linearization:
Symbolic tests for stability, e.g. Routh Hurwitz rule, need characteristic polynomial:
Log derivative is constant:
All parameters are in time scales!
LMI test:
Any positive fixed pt is structurally stable!
Example: heat shock response in bacteria
Non-saturating regime
Account for saturation effects:
From Olsman, Alonso, Doyle 2018
Calculate fixed point and linearization:
Symbolic tests for stability, e.g. Routh Hurwitz rule, need characteristic polynomial:
Any positive fixed pt, even when saturating, is structurally stable!
From local to global properties.
Vertices are special, correspond to monomial birth-death systems.
General behaviors are “convex combination” of vertex behaviors?
General Nonlinear
Biomolecular circuits
Lyapunov or Storage functions:
Quadratic
Entropy-like
One example, certifying regional multistability.
Bistable system.
Any fixed point in grey area has entropy-like Lyapunov function certifying regional stability.
Vertex of reaction order polyhedra can be scalably computed via zonotopes
Saturation regimes for multistability and oscillations.
Oscillations are hard to describe / analyze.
Repressilator.
No stable fixed point.
Bounded.
2D.
Saturation regimes for multistability and oscillations.
Repressilator:
Stable.
Saturation regimes for multistability and oscillations.
Repressilator:
Saturation regimes for multistability and oscillations.
Log derivative (reaction order) calculations
ALL variables and functions are POSITIVE.
Log vs Linear derivative:
Fold-change
Sums → Weighted average
Terms compete
for DOMINANCE of order
in calculus of positive variables.
Weight
Order
Log derivative (reaction order) calculations
ALL variables and functions are POSITIVE.
Log vs Linear derivative:
Fold-change
Sums → Weighted average
Ratio → Translation
Log derivative operator decomposition
Example:
(Simple binding)
Structure of log derivative comes from stoichiometry.
Steady state equation:
Conserved quantities:
Total substrate:
Total enzyme:
MM formula
Log derivative polytope
Conserved quantities
(log derivative operator decomposition)
Reaction stoichiometry of