Complex Kinetics
Systems and Synthetic Biology, 499A
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Unit Responses in Biochemical
Networks
Linear
Hyperbolic
Sigmoidal
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Where does the nonlinearity
come from?
Conservation Laws
Bimolecular Binding
Hyperbolic
Sigmoidal
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Association or Binding Constant
Ka the equilibrium constant for the binding of
one molecule to another.
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Dissociation Constant
Kd the equilibrium constant for dissociation.
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Enzyme Catalysis
The set of reactions shown below is the classic view for enzyme
catalysis. Enzyme binds to substrate to form enzyme-substrate
complex. Enzyme-substrate complex degrades to free enzyme and
product.
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Enzyme Catalysis
Unfortunately the kinetic constants that describe enzyme catalysis
are very difficult to measure and as a result researchers do not
tend to use the explicit mechanism, instead they use certain
approximations.
The two most popular approximations are:
1. Rapid Equilibrium
2. Steady State
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Rapid Equilibrium Approximation
The rapid equilibrium approximation assumes that the binding and
unbinding of substrate to enzyme to much faster than the release
of product. As a result, one can assume that the binding of
substrate to enzyme is in equilibrium.
That is, the following relation is true at all times (Kd = dissociation
constant):
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Rapid Equilibrium Approximation
Let Kd be the dissociation constant:
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Rapid Equilibrium Approximation
The equilibrium concentration of ES can be found:
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Rapid Equilibrium Approximation
The rate of reaction is then:
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Fractional Saturation
Rearrange the equation:
Substitute Kd
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Fractional Saturation
Yields:
This shows that the rate is proportional to the fraction of
total enzyme that is bound to substrate.
This expression gives us the fractional saturation.
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Brigg-Haldane Approach
Steady State Assumption
Simulation
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Brigg-Haldane Approach
Steady State Assumption
Simulation
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Steady State Assumption
It is possible to relax the constraints that ES should be in equilibrium
with E and S by assuming that ES has a relatively steady value over a
wide range of substrate concentrations.
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Steady State Assumption
Vmax
Km
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Enzyme Catalysis
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Reversible Enzyme Catalysis
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Haldane Relationships
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Haldane Relationships
The Haldane relationship puts constraints on the values of the
kinetic constants (cf. Principles of Detailed Balance Slide).
The relationship can be used to substitute one of the kinetic
parameters, eg the reverse Vmax and replace it with the more
easily determined equilibrium constant.
Thermodynamic Term
Saturation Term
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Parts of a Rate Law
Sometime people will also emphasize the separation of the rate
equation in to two parts, a thermodynamic and a saturation part.
Thermodynamic Term
Saturation Term
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Product Inhibition without Reversibility
Removes a further parameter from the equation
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Multisubstrate Kinetics
A
B
P
Q
Ordered Reaction
A
A
B
P
P
B
Q
Q
Ping-pong Reaction
B
A
Q
P
Random Order Reaction
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Multisubstrate Kinetics
Generalized Rate Laws
Irreversible:
Reversible:
Liebermeister W. and Klipp E. (2006), Bringing metabolic networks to life: convenience
rate law and thermodynamic constraints, Theoretical Biology and Medical Modeling 3:41.
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Problems with Steady State Derivations
Although deriving rate laws using the steady state assumption
is more reasonable compared to the rapid equilibrium approach,
the algebra involved in deriving the steady state equations can
rapidly become wieldy.
Graphical methods such as the King and Altman method have
been devised to assist is the derivation but the general approach
does not scale very well.
For many systems, the rapid equilibrium method is the preferred
method for deriving kinetic rate laws.
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Sigmoidal Responses
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Sigmoid responses are generally
seen in multimeric systems.
Phosphofructokinase
Tetramer of identical subunits
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Sigmoid responses arise from
cooperative interactions
Binding at one site results in changes in the binding affinities
at the remaining sites.
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Many proteins are multimeric
The Ecocyc database reports 774 multimeric
protein complexes out of 4316 proteins.
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Many Proteins are Multimeric
The Ecocyc database reports 774 multimeric
protein complexes out of 4316 proteins.
Phosphoglucose Isomerase
Hexokinase
Dimer
Dimer
Fructose 1,6-bisphosphate Aldolase
Phosphofructokinase
Tetramer
Tetramer
http://www.rcsb.org/pdb/static.do?p=education_discussion/molecule_of_the_month/pdb50_1.html
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Hill Equation – Simplest Model
We assume that the ligands bind simultaneously (unrealistic!):
Assuming Rapid Equilibrium
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Hill Equation
Some researchers feel that
the underlying model is so
unrealistic that the Hill
equation should be considered
an empirical result.
Hill Coefficient
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Hill Coefficient
The Hill Coefficient, n, describes
the degree of cooperativity.
If n = 1, the equation reverts
to a simple hyperbolic response.
n > 1 : Positive Cooperativity
n = 1 : No Cooperativity
n < 1 : Negative Cooperativity
Hill Coefficient
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Hill Equation
What is wrong with the Hill equation?
1. The underlying model is unrealistic (assuming this is important)
2. It’s a dead-end, no flexibility, one can’t add additional effectors
such as inhibitors or activators.
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Alternative Models: Sequential Binding
S S
S
S
S S
S
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S S
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Alternative Models: Sequential Binding
S
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S S
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Alternative Models: Sequential Binding
S
S S
Association Constants:
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Alternative Models: Sequential Binding
S
S S
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Alternative Models
Sequential Binding
S
S S
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Alternative Models
Sequential Binding
S
S S
Adair Model
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Alternative Models
Sequential Binding
S
S S
K1 = 0.2; K2 = 10
f
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S
42
Other Models – MWC Model
MWC Model or concerted model (Monod, Wyman, Changeux)
1. Subunits exist in two conformations, relaxed (R)
and taut (T)
S
S
2.
S S
S S
Is disallowed
One conformation has a higher binding affinity
than the other (R)
3. Conformations within a multimer are the same
4. Conformations are shifted by binding of ligand
Taut (T) – less active
Relaxed (R) – more active
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Other Models – MWC Model
MWC Model or concerted model (Monod, Wyman, Changeux)
S
S S
Where L =
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Other Models – MWC Model
MWC Model or concerted model (Monod, Wyman, Changeux)
S
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S S
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Modifiers in Sigmoid Kinetics
Modifiers can be incorporated into the models by assuming
that the modifier molecule can bind either exclusively to the
relaxed or taut forms.
Thus inhibitors will bind to the
taut form while activators can
bind to the relaxed form. They
effectively change the L constant.
A
I
S
S S
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Modifiers in Sigmoid Kinetics
A
I
f
Activators
S
Inhibitors
S S
S
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Gene Expression
TF = Transcription Factor
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Molecular Details - Activation
Weak Promoter
RNA Poly
Gene
Binding of a TF can
increase the apparent
strength of the Promoter
TF
RNA Poly
Gene
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Strong Promoter
RNA Poly
Molecular Details - Inhibition
TF
TF binding overlaps RNA polymerase binding site.
Prevents RNA polymerase from binding.
RNA Poly
RNA Poly
TF
TF binding downstream of RNA polymerase binding site.
Prevents RNA polymerase from moving down DNA strand.
RNA Poly
TF
TF binding causes looping of the DNA, preventing
RNA polymerase from moving down DNA strand.
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The Central Equation – Fractional
Saturation
A state refers to the state of the operator site, eg is
it free or bound to a TF.
G
Gene
G
G
A
Gene
GA
G = Operator Site
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Single TF Binds and Activates Expression
G
A
A
Equilibrium Condition
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Single TF Binds and Activates Expression
G
A
A
Fractional Saturation
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Single TF Binds and Activates Expression
G
A
A
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Single TF Binds and Activates Expression
G
A
A
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Probability Interpretation
Inside a cell there are of course only a few operator binding sites.
Therefore we should strictly interpret the fractional saturation as a
probability that a TF will be bound to the operator site. The rate
of gene expression is then proportional to the probability of
the TF being bound.
G
A
A
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Single TF Binds and Inhibits Expression
G
A
A
The active state is
when the TF is not
bound.
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Single TF Binds and Inhibits Expression
G
A
A
The active state is
when the TF is not
bound.
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Single TF Binds and Inhibits Expression
G
A
A
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Summary
Activation
Repression
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Single TF Binds, Activates with Cooperativity
1)
G
G
2)
G
TG
3)
G
TG2
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Single TF Binds, Activates with Cooperativity
G
TG
TG2
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Single TF Binds, Activates with Cooperativity
G
TG
TG2
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Single TF Binds, Inhibits with Cooperativity
v/Vmax
T
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Repression with n binding sites:
f
n=4
P
NOT Gate
T
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In the literature you’ll often find then
following variants:
Activation
Repression
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Two Activating TFs Compete for the Same Site
G
A
A
G
B
B
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Two TFs Compete for the Same Site
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Two Activating TFs Compete for the Same Site
OR Gate
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Two Activating TFs Bind to two different sites
B
B
G
A
A
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Two Activating TFs Bind to two different sites
..…but there is a slight complication
Two different ways to make GAB
GA
G
GAB
GB
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Two Activating TFs Bind to two different sites
You only need to use one of
These relations.
But which one?
Or does it matter?
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Two Activating TFs Bind to two different sites
GA
G
GAB
GB
The principle of detailed balance means that the energy
change along each path must be the same since the end points
(G and GAB) are the same. The overall equilibrium constants for
the upper and lower routes must therefore be equal.
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Two Activating TFs Bind to two different sites
GA
G
GAB
GB
Using:
Using:
This is: the equations are identical
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Comparing the two:
OR Gates
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AND Gates: Active only when both are bound
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NOR Gates: Active when neither are bound
G
G
A
A
G
B
B
G
B
B
A
A
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XOR Gate
G
G
A
A
Input A
Input B
Output
1
1
0
1
0
1
0
1
1
0
0
0
G
B
B
G
B
B
A
A
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Other Configurations
When A binds it activates. If B is bound, then A is unable to
bind. B therefore acts as an inhibitor of A.
G
A
G
B
A
B
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P
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