Section 1: A Control Theoretic Approach to Metabolic Control

Analysis

Metabolic Control Analysis (MCA)

S1 S2v

X2X1v v1 2 3

1E E E2 3

MCA investigates the relationship between the variables and parameters in a biochemical network.

Variables

1. Concentrations of Molecular Species

2. Fluxes

Parameters

1. Enzyme Levels

2. Kinetics Constants

3. Boundary Conditions

Stoichiometry Matrix:

Biochemical Systems

s1

s2

v3v2v1

Biochemical Systems

Rates:

Biochemical Systems

System dynamics:

Steady State

Steady State Sensitivity

Slope of secant describes rate of change (i.e. sensitivity) of s1 with respect to p1

As p1 tends to zero, the secant tends to the tangent, whose slope is the derivative of s1 with respect to p1, measuring an “instantaneous” rate of change.

Steady State Sensitivity

slope:

Responses (system sensitivities):

Species Concentrations:

Reaction Rates (Fluxes):

Scaled Sensitivities

measure relative (rather than absolute) changes:-- makes sensitivities dimensionless-- permits direct comparisons

Equivalent to sensitivity in logarithmic space:

This is the approach taken in Savageau's Biochemical Systems Theory (BST)

Sensitivity Analysis

Asymptotic Response

????

Perturbation

Input-Output Systems

The input u may include

• a reference signal to be tracked (e.g. input to a signal transduction network)

• a control input to be chosen by the system designer (e.g. given by a feedback law)

• a disturbance acting on the system (e.g. fluctuations in enzyme level)

The output y is commonly a subset of the components of the state

The output may represent

• the ‘part’ of the state which is of interest

• a measurement of the state

Input-Output Systems

The system dynamics

Can be linearized about

Biochemical systems:

Species concentration as output:

Reaction rates as output:

Sensitivity Analysis

Asymptotic Response

????

Perturbation

Two key properties of Linear Systems

1. Additivity

2. Frequency Response

systeminput output

Additivitysum of outputs = output of sum

allows reductionist approach

Reductionist approach can be used with a complete family of functions:

arbitrary function = weighted sum

monomials: 1, t, t2, …

etc.

sinusoids: sin(t), sin(2t), …

etc.

Expression in terms of sinusoids:Periodic functions: Fourier Series

Frequency Domain: Fourier Transform

Time Domain description

Frequency content description

Nonperiodic functions: Fourier Transform

Asymptotic Response

????

Perturbation

sum of sinusoids u1 + u2 + u3 + ...

sum of responses y1 + y2 + y3 +...

y1 + y2 + y3 +...

???

Frequency Response

The asymptotic response of a linear system to a sinusoidal input is a sinusoidal output of the same frequency.

This input-output behaviour can be described by two numbers for each frequency: • the amplitude (A)

• the phase ()

system

Frequency Response The input-output behaviour of the system can be

characterized by an assignment of two numbers to each frequency:

system

input output

These two numbers are conveniently recorded as the modulus and argument of a single complex number:

Plotting Frequency ResponseBode plot: modulus and argument plotted separately

log-log

semi-log

Calculation of Frequency Response

Through the Laplace transform:

Frequency response:

derived from the transfer function.

Recall: response to step inputSpecies:

Response to sinusoidal input

Recall: response to step input

Response to sinusoidal input

Fluxes:

Example: positive feedback in glycolysis

input u

glc

output y

feedback gain

strong feedback

weak feedback

Example: positive feedback in glycolysis

Example: negative feedback in tryptophan biosynthesis

Model of Xiu et al., J. Biotech, 1997.

input u

output y

feedback gain

mRNA

tryptophan

enzyme

active repressor

Example: negative feedback in tryptophan biosynthesis

weak feedback

strong feedback

Example: integral feedback in chemotaxis signalling pathway

Model of Iglesias and Levchenko, Proc. CDC, 2001.

input u

output y

Methylation: linear (integral feedback) or nonlinear (direct feedback)

Example: integral feedback in chemotaxis signalling pathway

direct feedback

integral feedback

Conclusion

• recovers standard sensitivity analysis at =0

• provides a complete description of the response to periodic inputs (e.g. mitotic, circadian or Ca2+ oscillations, periodic action potentials)

• provides a qualitative description of the response to 'slowly' or 'quickly' varying signals (e.g. subsystems with different timescales)

Sensitivity analysis in the frequency domain

Summation Theorem

Summation Theorem -- Example

Connectivity Theorem

Connectivity Theorem -- Example

Summation Theorem

If p is chosen so that is in the nullspace of N:

Proof: gives

Connectivity Theorem

Proof: gives

flux:

Example: Illustration of Theorems

Example: Illustration of Theorems: Summation Theorem

v1, v2, v3

Example: Illustration of Theorems: Summation Theorem

s1 s2

Example: Illustration of Theorems: Connectivity Theorem

s1

s2

Example: Illustration of Theorems: Connectivity Theorem

v2

v3