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