Cellular Control Analysis
(CCA)
Also called:
Metabolic Control Analysis
Biochemical Systems Theory
Introduction to CCA
It origins:
Three groups, Edinburgh, Berlin and Michigan, simultaneously
and independently developed almost an identical theory to
describe the effects of perturbations in biochemical pathways.
What is it?
The approach aims to achieve a quantitative understand of the
relationship between genotype and phenotype, to understand
systems behavior in terms of the unit processes (enzymes)
that make up biochemical pathways.
Kacser H, Burns JA. 1973. The control of flux. Symposium of the Society for
Experimental Biology 27, 65–104. (Reprinted in Biochemical Society Transactions 23,
341–366, 1995.)
Cellular Control Analysis
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Steady state flux J
Variables
Parameters
1. Concentrations of
Molecular Species
2. Fluxes
1. Enzyme Levels
2. Kinetics Constants
3. Boundary Species
CCA investigates the relationship between the variables
and parameters in a biochemical network..
Cellular Control Analysis
E1
X1
v1
S1
E2
v2
S2
E3
v3
Steady state flux J
At steady state:
X2
A Thought Experiment
Xo
S1
S2
S3
v
S4
S5
S6
X1
Xo and X1 fixed
all reactions
reversible, assume
stable steady
state.
A Thought Experiment
50 %
Xo
S1
S2
S3
v
What happens to the steady state?
S4
S5
S6
X1
Xo and X1 fixed
all reactions
reversible, assume
stable steady
state.
A Thought Experiment
50 %
Xo
S1
S2
S3
v
S4
S5
S6
X1
What happens to the steady state?
Xo and X1 fixed
1. Steady state flux goes down.
2. All metabolites upstream go up
3. All metabolites downstream do down
all reactions
reversible, assume
stable steady
state.
Cellular Control Analysis
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Steady state flux J
If we make a change, say to the levels if E2 (change its Vmax),
we will observe a change in the steady state flux, J
Let mean a small change, therefore to investigate the influence
of changing E2 on the steady state flux J, we could measure the
ratio
Cellular Control Analysis
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Steady state flux J
Unfortunately the ratio has two undesirable aspects:
1. The ratio depends on the size of the change that we
make to E2
2. The ratio depends on the units that are used to
measure E2 and J.
Cellular Control Analysis
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Steady state flux J
To avoid these issues, we scale the ratio to remove the units and
we also use infinitesimal changes rather than small finite
changes, as a result we get a new formula:
Cellular Control Analysis
Arithmetical interpretation:
Cellular Control Analysis
Geometrical interpretation:
~0.0
0.5
1.0
Cellular Control Analysis
This is called the flux control coefficient
Cellular Control Analysis
Since changing the level of an enzyme activity can also change
the metabolite levels, there is a similar control coefficient for
the metabolites:
This is called the concentration control coefficient
Cellular Control Analysis
Computing control coefficients using Jarnac
p = defn cell …
J1: S1 -> …..
end;
println p.cc (<p.J1>, p.Vmax1);
println p.cc (<p.S1>, p.Vmax1);
Cellular Control Analysis
Some characteristics of flux control coefficients.
In a linear metabolic network, the value of any particular flux
control coefficient is bounded between zero and one.
This condition applies to a linear chain
Cellular Control Analysis
How can you measure control coefficients?
1. Changing gene expression and measuring the effect on the
system.
Cellular Control Analysis
How can you measure control coefficients?
1. Changing gene expression and measuring the effect on the
system.
2. Using inhibitors to change an enzyme’s activity and
measuring the effect on the system.
Cellular Control Analysis
How can you measure control coefficients?
1. Changing gene expression and measuring the effect on the
system.
2. Using inhibitors to change an enzyme’s activity and
measuring the effect on the system.
3. Building a computer model and using the computer to
compute the coefficients
Cellular Control Analysis
The Summation Theorems
Cellular Control Analysis
The Summation Theorems
1. The summation theorem implies that the enzymes of a pathway
share the control of flux.
2. Changes in one control coefficient result in changes in other
control coefficients.
This means that a control coefficient is a system property and
not an intrinsic property of the enzyme alone.
Rate-Limiting Steps
Where does the concept of the rate-limiting step fit into this picture?
From CancerWeb: rate-limiting step The slowest step in a
metabolic pathway.
Is the rate-limiting step the step with a flux control coefficient of
one?
If so then there can only be one.
OR Is the rate limiting step the step with the highest flux control
coefficient?
Rate-Limiting Steps
Another complication:
In branched and cyclic pathways, individual flux control coefficients
can exceed 1.0, this means that some other steps may have flux
control coefficients < -1.0.
Does this mean that these steps are hyper-rate-limiting?
Hyper-Rate-Limiting Steps
Rate-Limiting Steps
The answer is of course to drop the idea of the rate-limiting
step and instead to quote, where possible, the value of the flux
control coefficient which will give a quantitative measure of
the limiting-ness of the step.
Why they don’t exist
Why rate-limiting steps don’t exist
in metabolism.
The flow through cellular networks
does not behave like road traffic.
The rate at which cars move in a
traffic line is not a function of the
number of cars in the line.
Why they don’t exist
The flow through cellular
networks does not work like
checkout lines.
The rate at which customers
move through the checkout
line is not a function of the
length of the line.
Why they don’t exist
The rate at which a substrate is
catalyzed by an enzyme (reaction
flow) is a function of the
concentration of substrate.
The “longer the substrate line”
the faster the enzyme works
Hexokinase
http://mgl.scripps.edu/people/goodsell/pdb/p
db50/pdb50_2.html
Simple Example
Xo
R1
S
R2
Xo
R1
Flux Control on R1
Flux Control on R2
Product Insensitive
?
?
Product Sensitive
?
?
Simple Example
Xo
R1
S
R2
Xo
R1
Flux Control on R1
Flux Control on R2
Product Insensitive
1.0
0
Product Sensitive
?
?
Simple Example
Xo
R1
S
R2
Xo
R1
Flux Control on R1
Flux Control on R2
Product Insensitive
1.0
0
Product Sensitive
~0.5
~0.5
Cellular Control Analysis
What about some real values?
The following information was taken from a paper by S. Thomas
et al, Biochemical Journal, 1997, 322, 119-127
Enzyme
Glycolysis in tuber
tissue of potato.
Values computed
using a combination
of calculation and
experimental work.
Enzymes, metabolites
and fluxes David A.
Fell J Exp Botany,
56(410),267-272
Flux Control Coefficient
PGM
0.029
PGI
0.139
PFK
0.132
Aldolase
0.0
TPI
0.0
GAPDH/PGK
0.001
enolase
0.005
PK
0.702
Sum
1.008
Cellular Control Analysis
Outline of metabolism in
Trypanosoma brucei
http://bip.cnrs-mrs.fr/bip10/eise2.htm
“Pharmacological Manipulation of
Metabolism”
http://bip.cnrs-mrs.fr/bip10/eise3.htm
“Computer Simulation and Drug
Design”
http://homepage.mac.com/mfield/lab/Widgets.html
Cellular Control Analysis
Metabolic control
analysis of glycolysis in
trypanosomes as an
approach to improve
selectivity and
effectiveness of drugs.
Bakker et al. Molecular
and Biochemical
Parasitology 106 (2000)
1–10
Enzyme
Flux Control Coefficient
Glucose Transport
0.90
0.08
Hexokinase
0.02
0.05
Phosphofructokiase
0.00
0.01
Aldolase
0.02
0.28
GAPDH
0.02
0.23
Phosphoglycerate
0.02
0.15
Experimental and in Silico
Pyruvate kinase
0.0
Analyses of Glycolytic Flux
Control in Bloodstream
G3P DH
0.02
Form Trypanosoma brucei,
Sum
1.0
J. Biol. Chem., Vol. 280,
Issue 31, 28306-28315,
35% increase over measured Vmax
August 5, 2005
of Glucose Transporter
0.01
0.17
0.98
What Determines the Control
Coefficients?
Elasticities
If the value of a control coefficient is a system property, what
is the relationship of the control coefficients to the individual
enzymes of the pathway?
Somehow, all the enzymes contribute to the value of a given
control coefficient, how can we describe this?
In order to answer this question we must consider another
type of coefficient, called the elasticity coefficient.
Elasticities
Michaelis-Menten Curve for an isolated enzyme
Let us define the elasticity as:
v
S
Elasticities
Michaelis-Menten Curve for an isolated enzyme
Note that substrate
elasticities are positive
and product elasticities
are negative
v
S
Elasticities
Another way of looking at an Elasticity:
We can use an elasticity to predict the change in the rate of
a reaction given a change in the substrate concentration.
Response at a Single Reaction Step
Another way of looking at an Elasticity:
Product inhibition
term
<0!
In general if there are multiple changes happening
around an enzyme we can simply sum each
contribution using the appropriate elasticity.
Response of a System
E1
X1
v1
S1
E2
v2
S2
E3
v3
Increase E3:
1. v3 Increases
2. S2 Decreases
3. v2 Increases
4. S1 Decreases
5. v1 Increases
X2
Response at a Single Reaction Step
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Response of a System
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Response of a System
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Response of a System
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Response of a System
E1
X1
v1
S1
E2
v2
S2
E3
v3
X2
Response of a System
E1
X1
S1
E2
S2
E3
X2
Extra term
Response of a System
E1
X1
S1
E2
S2
E3
X2
Response of a System
E1
X1
S1
E2
S2
E3
X2
What happens as the feedback gets stronger and stronger?
i.e
Response of a System
E1
X1
S1
E2
S2
E3
X2
Response of a System
E1
X1
S1
E2
S2
E3
X2
What does this mean?
Feedback has the following consequences:
1. All flux control moves down stream beyond the signal
leaving little or no flux control upstream. In fact, the
‘controlled’ step has very little flux control.
2. The signal molecule is locked into homoeostasis
Response of a System
E1
X1
S1
E2
S2
E3
X2
What does this mean?
The net effect of this is that feedback control creates a demand
controlled network. That is, control over the flux through the
pathway is determined largely by the demand for S2.
Important examples is this include:
1. Glycolysis
2. Amino acid biosynthesis
Response Coefficients
Measureing the effect of external factors.
Response Coefficients
The effect that a perturbation has on a system
depends on the target site and how the target site
interacts with the rest of the network.
Transfer Functions
Linearize around steady state:
Ingalls, B. P., A Frequency Domain Approach to Sensitivity Analysis of
Biochemical Systems , Journal of Physical Chemistry B, 108 (2004) pp. 11431152.
Relationship to Classical Control Theory:
Transfer Functions
This has the same structure as the linear state variable equations:
Relationship to Classical Control Theory:
Transfer Functions
Response to step input
Relationship to Classical Control Theory:
Transfer Functions
Response to sinusoidal input
Frequency Input
Relationship to Classical Control Theory:
Transfer Functions
Response to sinusoidal input
Frequency Input
Transfer Functions
Frequency Response for Simple Gene Circuit
Further Reading
185578047X, Ashgate
Publishing 1996, David Fell
3527310789, Wiley-VCH,
2005, Kipp et al.
Biochemical Systems Theory
0201067382, Addison Wesley
Longman Publishing 1977,
Michael Savageau.
Response Coefficients
Proof
Increase all rates by the same factor, alpha
Combining all local effects:
Proof
Response Coefficients
Response Coefficients