Section 1: A Control Theoretic Approach to Metabolic Control Analysis

Metabolic Control Analysis (MCA) S 1 S 2 v X 2 X 1 vv EEE 23 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 s1s1 s2s2 v3v3 v2v2 v1v1

Rates:

Biochemical Systems System dynamics:

Steady State

Steady State Sensitivity Slope of secant describes rate of change (i.e. sensitivity) of s 1 with respect to p 1 As  p 1 tends to zero, the secant tends to the tangent, whose slope is the derivative of s 1 with respect to p 1, 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 systeminputoutput

Additivity sum 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, t 2, … 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 u 1 + u 2 + u sum of responses y 1 + y 2 + y y 1 + y 2 + y ???

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

Plotting Frequency Response Bode 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 input Species: 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, 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, 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 Ca 2+ 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 s1s1 s2s2

Example: Illustration of Theorems: Connectivity Theorem s1s1 s2s2

v2v2 v3v3