Asymptotic Techniques in Enzyme Kinetics Presented By: – Dallas Hamann – Ryan Borek – Erik Wolf – Carissa Staples – Carrie Ruda

Outline Compartmental Analysis Chemical Reactions Law of Mass Action Enzyme Reactions The Equilibrium Approximation The Quasi-Steady-State Approximation Enzyme Inhibition

Compartmental Diagrams Compartmental Diagrams are visual models of a physical, biological, or a biochemical system or process.

Compartmental Diagram Elements Represents an amount of homogenous material in the process. Labeled with a variable name. Alcohol Compartments

Compartmental Diagram Elements Represents flow into or out of a compartment. Labeled by amount of variable change per unit of time. Inflow of Alcohol/second Outflow of Alcohol/second Arrows

Compartmental Diagram Governing Principle Rate of change in X = inflow rate – outflow rate X Using this principle allows formation of a system of differential equations using a compartmental diagram. A differential equation can be formed for each compartment.

Physiological Perspective of Compartmental Diagrams Variables represent amounts of biological substances in a physiological system. Compartments represent these physiological systems.

Fundamental Questions To understand the distribution of biological substances amongst various components of a physiological system. To understand how the distribution of biological substances change in a system.

Chemical Reactions and Law of Mass Action Suppose C represents a chemical, [C] which denotes its concentration. Now suppose two chemicals A,B react upon collisions to form a product C. where k is the rate constant of the reaction.

Law of Mass Action A Model = rate of accumulation of product Which depends on: - the energy of the collision - geometrical shapes and sizes of the reactant molecules

Law of Mass Action A Model Is directly proportional to reactant concentrations i.e.

Compartmental Diagram for Law of Mass Action

Reverse Reactions Biochemical reactions are typically bi-directional. So, we get the differential equations: i) ii)

Equilibrium A state when concentrations are no longer changing. In equilibrium, So, Solving for (1)

Equilibrium Now, in a closed system (no other reactions are going on) (constant) (conservation of matter equation) Substituting in (1),

Equilibrium Solving for

Equilibrium Where,(equilibrium constant) has concentration units

Reactions Elementary Enzyme

Elementary Reactions Proceed directly from collision of reactants Follows the Law of Mass Action directly

Enzymes An enzyme is a substance that acts as a catalyst for some chemical reaction. Enzymes act on other molecules (called substrates), helping convert them into products. Enzymes themselves are not changed by the reaction. Enzymes work by lowering the “free energy of activation” for the reaction.

Enzyme Reactions Enzyme reactions do not follow the Law of Mass Action directly If they did, theory would predict: but it has been shown that this is not the case. So, what do they look like?

Michaelis – Menten Model (1913) Idea Chemical Reaction Scheme Compartmental Diagram System of Differential Equations

Idea 2-step process Step 1: Enzyme E first converts the substrate S into complex C Step 2: Complex then breaks down into a product P, releasing the enzyme E in the process

Chemical Reaction Scheme is the dissociation constantNote:

Compartmental Diagram

Differential Equations From the compartmental diagram, we get the following differential equations:

So What Can We Do With All Of This?

The Equilibrium Approximation Michaelis – Menten (1913) Dr. Maud MentenLeonor Michaelis Picture Not Available

Purpose To estimate the reaction velocity of an enzyme reaction. i.e. To estimatefor

Assumption Substrate is in “instantaneous equilibrium” with the complex. i.e. Consider : or

note: Solving for c letting So, by substitution

Solving for c (Continued) thus,

Derive an expression for V Recall from previous differential equations: Substituting in the new value for c:

Observations 1) The maximum reaction velocity occurs when C is biggest ( ). i.e. When all enzyme is complexed with substrate. will be limited by the amount of enzyme present and the dissociation constant so, is called “rate limiting” for this reaction.

Observations(cont.) 2) For large substrate concentrations, we will rewrite V as: i.e. the reaction rate saturates

Observations(cont.) 3) If, then

Is that the only approximation technique for V? Clearly, that would not suffice! Hold on to your seats for another exciting method, because it gets even better.

Quasi-Steady-State Approximation

Idea Step 1: Use nondimensionalization to redefine rates. Step 2: Apply the Briggs –Haldane Assumption to those rates. J.B.S. Haldane George Edward Briggs Picture Not Available

Assumption The rates of formation and the rate of breakdown of the complex are equal. i.e.

Rates We know these differential equations:

Variables and Parameters Independent variables: s, c Parameters: k 1, k -1, k 2 By nondimensionalization, we make new variables:

Step 1: Replacement We start by rewriting our differential rate equations from using our independent variables to using the new variables. We know:By substitution: 1)Note:

Now substitute: Solve for: into: Let so Previous Statement:

We know:By substitution: 2) Note:

: into: We don’t want to divide by zero (in case e o is small) so we will divide the right by s o to minimize parameters. Previous Statement: Now substitute:

Letand so And now our parameters are: Previous Statement:

Step 2: Apply assumption Now we will apply the assumption of Briggs-Haldane: We assumed: Because of these assumptions we can conclude: Which is comparable to : then

Now applying those to our rates: Solve for x: Substitute x into

Substitute: Previous Statement:

Bringing it all together We know: Assumeby the Quasi-Steady-State Assumption. so

now substituteso substitute back to: and so where

Equilibrium vs. Quasi-Steady-State These two reaction schemes are similar, but not the same. Differences include: the Equilibrium Approximation is simple to apply but has less scope, while the Quasi- Steady-State uses nondimensionalization and applies to a greater scope. In other reaction schemes the estimates are not as similar.

Enzyme Inhibition Enzyme inhibitors are substances that inhibit the catalytic action of an enzyme. Slow down or decrease enzyme activity to zero. Examples: nerve gas and cyanide are irreversible inhibitors or catalytic poisons (these examples reverse the activity of life supporting enzymes).

Enzyme “Lock & Key” Structure Enzyme Molecules can bind to two different types of sites on an enzyme. Active Site Allosteric Site Enzyme molecules are usually large protein molecules to which other molecules can bind.

Enzyme “Lock & Key” Structure Active Sites are sites on an enzyme where substrate can bind to form complex. However, when an inhibitor is bound to the active site it is called Competitive Inhibition. Enzyme Active Site Allosteric Site Competitive Inhibition Inhibitor

Enzyme “Lock& Key” Structure Enzyme “Lock & Key” Structure Allosteric Sites are secondary sites on an enzyme that regulate the catalytic activity of an enzyme. When an inhibitor is bound to the allosteric site it is called Allosteric Inhibition. Enzyme Active Site Allosteric Site Inhibitor Allosteric Inhibition

Competitive Inhibition Enzyme Active Site Allosteric Site Inhibitor Reaction Scheme S + EC1C1 E + P k1k1 k -1 k2k2 E + I C2C2 k3k3 k -3 In competitive inhibition the inhibitor “competes” with the substrate to bind to the Active Site. If the inhibitor wins there is less product formed.

Allosteric Inhibition Enzyme Active Site Allosteric Site Inhibitor In allosteric inhibition the inhibitor binds to the Allosteric Site. The binding of the inhibitor to the Allosteric Site changes the action of the enzyme. However, the substrate may also bind to the Enzyme at the Active Site. This leaves a very complicated reaction scheme. Substrate

Allosteric Inhibition Reaction Scheme Let: S ~ Substrate E ~ Enzyme I ~ Inhibitor C1 ~ ES complex C2 ~ EI complex C3 ~ EIS complex P ~ Product E + SC1C1 E + P k1k1 k -1 k2k2 E + I C2C2 k3k3 k -3 C 2 + SC3C3 k1k1 k -1 C 1 + I k3k3 k -3 C3C3

Allosteric Inhibition Reaction Scheme For Allosteric Inhibition we should use “Complex Free Reaction Notation.” In Complex Free Reaction Notation one substance is implied.

Allosteric Inhibition Reaction Scheme E + SC1C1 E + P k1k1 k -1 k2k2 EESE + P k1sk1s k -1 k2k2 Former Notation: Complex Free Notation: Let: S ~ Substrate E ~ Enzyme I ~ Inhibitor C1 ~ ES complex C2 ~ EI complex C3 ~ EIS complex P ~ Product

Allosteric Inhibition Reaction Scheme EESE + P k1sk1s k -1 k2k2 EIEIS k1sk1s k -1 k3ik3i k -3 k3ik3i Complex Free Notation:

Allosteric Inhibition Reaction Scheme Complex Free Compartmental Diagram Let: X = ES Y = EI Z = EIS EX ZY P k 1 se k -1 x k2xk2x k 1 sy k -1 z k 3 ie k -3 y k 3 ix k -3 z

Allosteric Inhibition Reaction Differential Equations...

Conclusion As you can see, asymptotic techniques in enzyme kinetics can get quite complex. However, these techniques give us vital information about the model without having to solve the differential equation directly. We will continue our study into next semester.

Bibliography Mathematical Physiology by Keener and Sneyd, Springer-Verlag 1998 Mathematics Applied to Deterministic Problems in the Natural Sciences by Lin and Segal, SIAM 1988 Mathematical Models in Biology by Leah Edelstein- Keshet, McGraw-Hill 1988 A Course in Mathematical Modeling by Mooney and Swift, MAA 1999 Sites of pictures:

Special Thanks to: Dr. Jean Foley Dr. Steve Deckelman All of you for coming!