Kinetics Intro to Synthetic Biology, Copyright © 2008: Sauro

Unit Responses in Biochemical Networks Linear Hyperbolic Sigmoidal 2 Copyright © 2008: Sauro

Where does the nonlinearity come from? Hyperbolic Sigmoidal Conservation Laws Bimolecular Binding 3 Copyright © 2008: Sauro

Association or Binding Constant Ka the equilibrium constant for the binding of one molecule to another. 4 Copyright © 2008: Sauro

Dissociation Constant Kd the equilibrium constant for dissociation. 5 Copyright © 2008: Sauro

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. 6 Copyright © 2008: Sauro

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 7 Copyright © 2008: Sauro

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 8 Copyright © 2008: Sauro

Rapid Equilibrium Approximation Let Kd be the dissociation constant: 9 Copyright © 2008: Sauro

Rapid Equilibrium Approximation The equilibrium concentration of ES can be found: 10 Copyright © 2008: Sauro

Rapid Equilibrium Approximation The rate of reaction is then: 11 Copyright © 2008: Sauro

Fractional Saturation Rearrange the equation: Substitute Kd 12 Copyright © 2008: Sauro

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. 13 Copyright © 2008: Sauro

Brigg-Haldane Approach Steady State Assumption Simulation 14 Copyright © 2008: Sauro

Brigg-Haldane Approach Steady State Assumption Simulation 15 Copyright © 2008: Sauro

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. 16 Copyright © 2008: Sauro

Steady State Assumption Km Vmax 17 Copyright © 2008: Sauro

Enzyme Catalysis 18 Copyright © 2008: Sauro

Reversible Enzyme Catalysis 19 Copyright © 2008: Sauro

Haldane Relationships 20 Copyright © 2008: Sauro

Haldane Relationships Thermodynamic TermSaturation 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. 21 Copyright © 2008: Sauro

Parts of a Rate Law Thermodynamic TermSaturation Term Sometime people will also emphasize the separation of the rate equation in to two parts, a thermodynamic and a saturation part. 22 Copyright © 2008: Sauro

Product Inhibition without Reversibility Removes a further parameter from the equation 23 Copyright © 2008: Sauro

Multisubstrate Kinetics ABPQ Ordered Reaction A BPQ Random Order Reaction APBQ Ping-pong Reaction A BPQ 24 Copyright © 2008: Sauro

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: Copyright © 2008: Sauro

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. 26 Copyright © 2008: Sauro

Sigmoidal Responses 27 Copyright © 2008: Sauro

Sigmoid responses are generally seen in multimeric systems. Phosphofructokinase Tetramer of identical subunits 28 Copyright © 2008: Sauro

Sigmoid responses arise from cooperative interactions Binding at one site results in changes in the binding affinities at the remaining sites. 29 Copyright © 2008: Sauro

Many proteins are multimeric The Ecocyc database reports 774 multimeric protein complexes out of 4316 proteins. 30 Copyright © 2008: Sauro

Many Proteins are Multimeric The Ecocyc database reports 774 multimeric protein complexes out of 4316 proteins. Hexokinase Phosphoglucose Isomerase Phosphofructokinase Fructose 1,6-bisphosphate Aldolase Dimer Tetramer 31 Copyright © 2008: Sauro

Hill Equation – Simplest Model Assuming Rapid Equilibrium We assume that the ligands bind simultaneously (unrealistic!): 32 Copyright © 2008: Sauro

Hill Equation Hill Coefficient Some researchers feel that the underlying model is so unrealistic that the Hill equation should be considered an empirical result. 33 Copyright © 2008: Sauro

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 34 Copyright © 2008: Sauro

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. 35 Copyright © 2008: Sauro

Alternative Models: Sequential Binding S S S S S S SS S 36 Copyright © 2008: Sauro

Alternative Models: Sequential Binding S S S 37 Copyright © 2008: Sauro

Alternative Models: Sequential Binding S S S Association Constants: 38 Copyright © 2008: Sauro

Alternative Models: Sequential Binding S S S 39 Copyright © 2008: Sauro

Alternative Models Sequential Binding S S S 40 Copyright © 2008: Sauro

Alternative Models Sequential Binding S S S Adair Model 41 Copyright © 2008: Sauro

Alternative Models Sequential Binding S S S K1 = 0.2; K2 = 10 S f 42 Copyright © 2008: Sauro

Other Models – MWC Model MWC Model or concerted model (Monod, Wyman, Changeux) S S S 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 43 Copyright © 2008: Sauro

Other Models – MWC Model MWC Model or concerted model (Monod, Wyman, Changeux) S S S Where L = 44 Copyright © 2008: Sauro

Other Models – MWC Model MWC Model or concerted model (Monod, Wyman, Changeux) S S S 45 Copyright © 2008: Sauro

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. S S S I A 46 Copyright © 2008: Sauro

Modifiers in Sigmoid Kinetics S S S I A Inhibitors Activators S f 47 Copyright © 2008: Sauro

Gene Expression TF = Transcription Factor 48 Copyright © 2008: Sauro

Molecular Details - Activation RNA Poly Weak Promoter TF Binding of a TF can increase the apparent strength of the Promoter Gene 49 Copyright © 2008: Sauro

Molecular Details - Inhibition RNA Poly TF binding overlaps RNA polymerase binding site. Prevents RNA polymerase from binding. RNA Poly Strong Promoter RNA PolyTF 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 50 Copyright © 2008: Sauro

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 A G G GA Gene G = Operator Site 51 Copyright © 2008: Sauro

Single TF Binds and Activates Expression A G A Equilibrium Condition 52 Copyright © 2008: Sauro

Single TF Binds and Activates Expression A G A Fractional Saturation 53 Copyright © 2008: Sauro

Single TF Binds and Activates Expression A G A 54 Copyright © 2008: Sauro

Single TF Binds and Activates Expression A G A 55 Copyright © 2008: Sauro

Probability Interpretation A G A 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. 56 Copyright © 2008: Sauro

Single TF Binds and Inhibits Expression A G A The active state is when the TF is not bound. 57 Copyright © 2008: Sauro

Single TF Binds and Inhibits Expression A G A The active state is when the TF is not bound. 58 Copyright © 2008: Sauro

Single TF Binds and Inhibits Expression A G A 59 Copyright © 2008: Sauro

Summary Activation Repression 60 Copyright © 2008: Sauro

Single TF Binds, Activates with Cooperativity 1) G G G 2) 3) G TG TG2 61 Copyright © 2008: Sauro

Single TF Binds, Activates with Cooperativity G TG TG2 62 Copyright © 2008: Sauro

Single TF Binds, Activates with Cooperativity G TG TG2 63 Copyright © 2008: Sauro

Single TF Binds, Inhibits with Cooperativity T v/Vmax 64 Copyright © 2008: Sauro

Repression with n binding sites: T f n = 4 NOT Gate P 65 Copyright © 2008: Sauro

In the literature you’ll often find then following variants: Activation Repression 66 Copyright © 2008: Sauro

Two Activating TFs Compete for the Same Site A G A B G B 67 Copyright © 2008: Sauro

Two TFs Compete for the Same Site 68 Copyright © 2008: Sauro

Two Activating TFs Compete for the Same Site OR Gate 69 Copyright © 2008: Sauro

Two Activating TFs Bind to two different sites A A B B G 70 Copyright © 2008: Sauro

Two Activating TFs Bind to two different sites..…but there is a slight complication G GA GB GAB Two different ways to make GAB 71 Copyright © 2008: Sauro

Two Activating TFs Bind to two different sites You only need to use one of These relations. But which one? Or does it matter? 72 Copyright © 2008: Sauro

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 for the upper and lower routes must therefore be equal. 73 Copyright © 2008: Sauro

Two Activating TFs Bind to two different sites G GA GB GAB Using: This is: the equations are identical 74 Copyright © 2008: Sauro

Comparing the two: OR Gates 75 Copyright © 2008: Sauro

AND Gates: Active only when both are bound 76 Copyright © 2008: Sauro

NOR Gates: Active when neither are bound G G A A G B B G B BA A 77 Copyright © 2008: Sauro

XOR Gate G G A A G B B G B BA A Input AInput BOutput Copyright © 2008: Sauro

Other Configurations P A B A G B G When A binds it activates. If B is bound, then A is unable to bind. B therefore acts as an inhibitor of A. 79 Copyright © 2008: Sauro