Basics of biochemical systems theory

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Basics of biochemicalsystems theory

• I. Graphical representation of biochemical
  networks

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Material flow
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Information flow
FDP
-

HK
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Networks
Glc
G6P
F6P
FDP
-
-

PGI


ATP
ADP
ATP
ADP
PFK
HK
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Fundamental types of variables

• State variables any quantities that may directly
  influence a process and whose values might change
  (eg. concentrations, temperature)
• Flux variables reaction rates

v2
Glc
G6P
F6P
FDP
-
-
v4

v3
v1
PGI


ATP
ADP
ATP
ADP
PFK
HK
v5
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Other types of variables
Independent variables
fixed or isolated variables
Glc
G6P
F6P
FDP
-
-

PGI


ATP
ADP
ATP
ADP
PFK
HK
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Other types of variables
Dependent variables
Dynamic variables
Glc
G6P
F6P
FDP
-
-

PGI


ATP
ADP
ATP
ADP
PFK
HK
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Other types of variables
Aggregated variables
Linear combinations variables
Glc
G6P
F6P
FDP
-
-

PGI


ATP
ADP
ATP
ADP
PFK
HK
ATP ADP ANPtot
v2f - v2r v2
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Other types of variables
Constrained variables
Variables whose values depends on the values of
other variables not through the interactions of
the various processes but through imposed
algebraic relationships (moiety conservation,
equilibria)
Glc
G6P
F6P
FDP
-
-

PGI


ATP
ADP
ATP
ADP
PFK
HK
ATP ANPtot ADP AMP
F6P Keq G6P
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Other types of variables
Implicit variables
Variables that, in a given context, can be
assumed constant or not influencing subsystem
behavior, and are thus neglected
X
Glc
G6P
F6P
FDP
-
-

PGI


ATP
ADP
ATP
ADP
PFK
HK
Y
Z
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Symbology
State variables Xi - Dependent variables -
Independent variables - Aggregated
variables Flux variables vij
v12
X6
X1
X2
X3
X7
v21
v37
-
-

v23
v61
X9


X5
X4
X5
X4
X10
X8
v45
X5 X4 X11
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Strategy for developinga graphical representation

• Sketch graphic in familiar terms
• Potential variables
• Important processes
• Principal interactions
• Make conversion table
• Redraw graph in symbolic terms
• Analyze and refine model

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Common errors and ambiguities
Failure to explicitly account forconsumption/dilu
tion of components
Non-obvious point expansion fluxes dilution
of components owing to volume expansion (e.g.
cell growth).
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Common errors and ambiguities
Confusing convergent processeswith a
multi-substrate reaction
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Common errors and ambiguities
Failure to account for the actual molecularity of
each species in a given reaction
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Common errors and ambiguities
Confusion between material and information flow
-
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System and environment
X7
X4
X5
-
-
X3
X1
X2
X6
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Realism and accuracy vs. feasibility
We should make things as simple as possible, but
not simpler. Albert Einstein
Any intelligent fool can make things bigger,
more complex, and more violent. It takes a touch
of genius and a lot of courage to move in the
opposite direction. Albert Einstein
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Sources of simplification

• Spatial or topological features
  (compartmentalization, channeling)
• Time-scale separation (evolutionary,
  developmental, biochemical, biomolecular) and
  quasi-steady-state approximation
• Functional simplifications (operating ranges and
  linearizations, non-linearity)

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Basics of biochemicalsystems theory

• II. Mathematical representation of processes

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Rate laws
Mathematical functions representing the
instantaneous rate of a process as an explicit
function of all the state variables that have a
direct influence on the process
Mass action rate laws in a well-mixed
homogeneous environment, the rate of an
elementary reaction a1X1 a2X2 anXn ?
is given by
Kinetic orders
Rate constant
Mass action rate laws may also approximate well
the kinetics ofsome more-complex reaction
mechanisms
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Rate laws
Mathematical functions representing the
instantaneous rate of a process as an explicit
function of all the state variables that have a
direct influence on the process
Michaelis-Menten rate laws initial rates of
enzyme-catalyzed reactions in absence of
cooperative effects.
Maximal rate
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Rate laws
Mathematical functions representing the
instantaneous rate of a process as an explicit
function of all the state variables that have a
direct influence on the process
Michaelis-Menten rate laws initial rates of
enzyme-catalyzed reactions in absence of
cooperative effects.
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Rate laws
Mathematical functions representing the
instantaneous rate of a process as an explicit
function of all the state variables that have a
direct influence on the process
Hill rate laws approximation to initial rates of
enzyme-catalyzed reactions with cooperativity.
Maximal rate
Hill number
Km Half-saturation constant
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Rate laws
Mathematical functions representing the
instantaneous rate of a process as an explicit
function of all the state variables that have a
direct influence on the process
Many other rate laws possible
Could we find a convenient (even if only
approximated) generic representation?
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Taylor series
Linear approximation
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First-order (linear) Taylor series approximation
for multiple variables
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Advantages and limitationsof linearized
representation

• Good enough approximation in many cases (small
  operating ranges, linear by design)
• Strong theory and methods available
• However
• Biochemical processes are strongly nonlinear

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Power-law representation
Taylor series in logarithmic space
First-order truncation
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Power-law representation
Taylor series in logarithmic space First-order
truncation for n variables
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Power-law representation
Meaning of the kinetic orders
gi
With all other conc. constant
gigt(lt)0 ? v increases (decreases) with Xi gi x
? a 1 change in Xi causes a x change in v
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Ranges of values of kinetic orders
Kinetic orders for mass action kinetics
v
Log-log plot
X
v k Xc ? g c
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Ranges of values of kinetic orders
Kinetic orders for substrates under
Michaelis-Menten kinetics
v/Vmax
X/KM
0 g 1
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Ranges of values of kinetic orders
Kinetic orders for inhibitors under
Michaelis-Menten kinetics
v/v(X0)
g-0.5
X/KI
-1 g 0
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Ranges of values of kinetic orders
Kinetic orders for substrates under cooperative
kinetics
v/Vmax
ggt1
X/KM
0 g (4)
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Parameter estimation

• From general considerations
• From experimentally characterized rate expression
• Top-down modeling

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Basics of biochemicalsystems theory

• III. Mathematical representation of networks

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Mass balance
Chemical processes do not destroy or create
matter
Instantaneous rate of X1 accumulation
Instantaneous rate of X4 conversion into X1
(Instantaneous rate of X1 conversion into X2
Instantaneous rate of X1 conversion into X3)
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Steady state
No net accumulation or dilution
O Instantaneous rate of X4 conversion into X1
(Instantaneous rate of X1 conversion into X2
Instantaneous rate of X1 conversion into X3)
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Mass balance equationsfor a network
v51
-
X5
X1
v12
v14
v40
v13
X4
v23
X2
X3
v20
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Mass balance equationsfor a network
General form
vijk unidirectional rate of utilization of Xi
for the production of Xj via the kth
parallel process ?jik number of molecules of Xj
created in each reaction ?jik number of
molecules of Xi consumed in each reaction
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Flux aggregation
Major types
Emphasis
Fluxes
Pools
Strategies
Parallel
Antiparallel
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Flux aggregation
Power-law representation under various types of
aggregation
Emphasis
Fluxes
Pools
Strategies
Parallel
Antiparallel
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Generalized Mass Action systems
Power-law representation on the basis of flux
aggregation emphasizing processes
More simply
Maintains separate identity of fluxes at branch
points
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Flux aggregation
Power-law representation under various types of
aggregation
Emphasis
Fluxes
Pools
Strategies
Parallel
Antiparallel
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Synergistic systems
Power-law representation on the basis of flux
aggregation emphasizing pools
X1, ..., Xn Dependent variables
Xn1, ..., Xnm Independent variables
Kinetic orders
Rate constants
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Some advantages of S systems

• More compact representation
• Fewer parameters
• More mathematically tractable
• Frequently more accurate

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Ranges of values of kinetic orders
Relationship between kinetic orders for the S
systems representation and those for the GMA
representation
V1 ? v211 2v212 v213
v211
X2
v212
X1
v213
X3
For unit enzyme-catalyzed steps, gij is normaly
-4?gij ?4
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Setting up an S system
1. For each dynamic metabolite, aggregate the
production fluxes and the removal fluxes
separately.
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Setting up an S system
2. Identify the variables that influence each
aggregated flux.
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Setting up an S system
3. Represent each aggregated flux as a
power-product of its influencing variables.
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How we got to love the S-system...
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Log gains
Log gains for concentrations
Xi (i1, ..., n) not fixed in this case. The log
gains are systemic properties.
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Log gains
Log gains for aggregated fluxes
Log gains for F factors (turnover numbers,
)
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IX. The distinction between kinetic orders and
log gains of fluxes
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Sensitivities
How parameters influence state variables
For an extensive list of relationships see M. A.
Savageau and A. Sorribas. J.Theor.Biol. 141
(1)93-115, 1989.
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Example
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Stability
Lyapunov stability is derived from the
linearization of the ODEs
For S systems
The analysis proceeds through evaluation of the
Lyapunov exponents as usual
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Recasting
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Further reading

• M. A. Savageau. Biochemical Systems Analysis A
  Study of Function and Design in Molecular
  Biology, Reading, Mass.Addison-Wesley, 1976.
• E. O. Voit. Computational analysis of biochemical
  systems. A practical guide for biochemists and
  molecular biologists, CambridgeCambridge
  University Press, 2000.
• M. A. Savageau. Design principles for elementary
  gene circuits Elements, methods, and examples.
  Chaos 11 (1)142-159, 2001.

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