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Kinetics
Intro to Synthetic Biology, 424
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Unit Responses in Biochemical Networks
Linear
Hyperbolic
Sigmoidal
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Where does the nonlinearity come from?
Hyperbolic Sigmoidal
Conservation Laws Bimolecular Binding
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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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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
Enzyme Catalysis
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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):
Rapid Equilibrium Approximation
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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 ApproachSteady State Assumption
Simulation
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Brigg-Haldane ApproachSteady 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
Km
Vmax
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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
Thermodynamic Term Saturation Term
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.
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Parts of a Rate Law
Thermodynamic Term Saturation Term
Sometime people will also emphasize the separation of the rate equation in to two parts, a thermodynamic and a saturation part.
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Product Inhibition without ReversibilityRemoves a further parameter from the equation
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Multisubstrate Kinetics
A B P Q
Ordered Reaction
A
B P Q
Random Order Reaction
A P B Q
Ping-pong Reaction
A
B PQ
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Multisubstrate KineticsGeneralized 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 assumptionis more reasonable compared to the rapid equilibrium approach,the algebra involved in deriving the steady state equations canrapidly become wieldy.
Graphical methods such as the King and Altman method havebeen devised to assist is the derivation but the general approachdoes not scale very well.
For many systems, the rapid equilibrium method is the preferredmethod 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 affinitiesat the remaining sites.
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Many proteins are multimericThe Ecocyc database reports 774 multimeric protein complexes out of 4316 proteins.
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Many Proteins are MultimericThe Ecocyc database reports 774 multimeric protein complexes out of 4316 proteins.
Hexokinase
Phosphoglucose Isomerase
Phosphofructokinase
Fructose 1,6-bisphosphate Aldolase
Dimer
DimerTetramer
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
Assuming Rapid Equilibrium
We assume that the ligands bind simultaneously (unrealistic!):
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Hill Equation
Hill Coefficient
Some researchers feel that the underlying model is so unrealistic that the Hill equation should be considered an empirical result.
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Hill Coefficient
Hill Coefficient
The Hill Coefficient, n, describes the degree of cooperativity.
If n = 1, the equation revertsto a simple hyperbolic response.
n > 1 : Positive Cooperativityn = 1 : No Cooperativityn < 1 : Negative Cooperativity
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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 effectorssuch as inhibitors or activators.
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Alternative Models: Sequential Binding
S S S
S S S
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
Association Constants:
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Alternative Models: Sequential Binding
S S S
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Alternative ModelsSequential Binding
S S S
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Alternative ModelsSequential Binding
S S S
Adair Model
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Alternative ModelsSequential Binding
S S S
K1 = 0.2; K2 = 10
S
f
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Other Models – MWC ModelMWC Model or concerted model (Monod, Wyman, Changeux)
S
SS
S
S S
1. Subunits exist in two conformations, relaxed (R) and taut (T)
2. 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
Is disallowed
Relaxed (R) – more active
Taut (T) – less active
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Other Models – MWC ModelMWC Model or concerted model (Monod, Wyman, Changeux)
S S S
Where L =
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Other Models – MWC ModelMWC Model or concerted model (Monod, Wyman, Changeux)
S 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 thetaut form while activators canbind to the relaxed form. Theyeffectively change the L constant.
S
S S
I A
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Modifiers in Sigmoid Kinetics
S
S S
I A
Inhibitors
Activators
S
f
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Gene Expression
TF = Transcription Factor
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Molecular Details - Activation
RNA Poly
RNA Poly
Weak Promoter
TF
Binding of a TF can increase the apparent strength of the Promoter
Gene
Gene
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Molecular Details - Inhibition
RNA Poly
TF binding overlaps RNA polymerase binding site. Prevents RNA polymerase from binding.
RNA Poly Strong Promoter
RNA Poly TF
RNA Poly TF
TF binding downstream of RNA polymerase binding site. Prevents RNA polymerase from moving down DNA strand.
TF binding causes looping of the DNA, preventing RNA polymerase from moving down DNA strand.
TF
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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
AG
G
GA
Gene
Gene
G = Operator Site
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Single TF Binds and Activates Expression
A
GA
Equilibrium Condition
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Single TF Binds and Activates Expression
A
GA
Fractional Saturation
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Single TF Binds and Activates Expression
A
GA
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Single TF Binds and Activates Expression
A
GA
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Probability Interpretation
A
GA
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.
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Single TF Binds and Inhibits Expression
A
GA
The active state is when the TF is not bound.
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Single TF Binds and Inhibits Expression
A
GA
The active state is when the TF is not bound.
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Single TF Binds and Inhibits Expression
A
GA
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Summary
Activation
Repression
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Single TF Binds, Activates with Cooperativity
1)
G
G
G2)
3)
G
TG
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
T
v/Vmax
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Repression with n binding sites:
T
f n = 4
NOT GateP
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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
A
GA
B
GB
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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
A
A
B
BG
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Two Activating TFs Bind to two different sites..…but there is a slight complication
G
GA
GB
GAB
Two different ways to make GAB
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Two Activating TFs Bind to two different sites
You only need to use one ofThese relations.
But which one?
Or does it matter?
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Two Activating TFs Bind to two different sites
G
GA
GB
GAB
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 forthe upper and lower routes must therefore be equal.
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Two Activating TFs Bind to two different sites
G
GA
GB
GAB
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
GA
A
GB
B
GB
B A
A
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XOR GateG
GA
A
GB
B
GB
B A
A
Input A Input B Output
1 1 0
1 0 1
0 1 1
0 0 0
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Other Configurations
PAB
AG
BG
When A binds it activates. If B is bound, then A is unable to bind. B therefore acts as an inhibitor of A.