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

Asymptotic Techniques in Asymptotic Techniques in Enzyme KineticsEnzyme Kinetics

Presented By:– Dallas Hamann– Ryan Borek– Erik Wolf– Carissa Staples– Carrie Ruda

OutlineOutline

Compartmental AnalysisChemical ReactionsLaw of Mass ActionEnzyme ReactionsThe Equilibrium ApproximationThe Quasi-Steady-State ApproximationEnzyme Inhibition

Compartmental Compartmental DiagramsDiagrams

Compartmental Diagrams are visual models of a physical, biological, or a

biochemical system or process.

Compartmental Diagram Compartmental Diagram ElementsElements

Represents an amount of homogenous material in the process.

Labeled with a variable name.

Alcohol

Compartments

Compartmental Diagram Compartmental Diagram ElementsElements

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 Compartmental Diagram Governing PrincipleGoverning 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 Physiological Perspective of Compartmental DiagramsCompartmental Diagrams

Variables represent amounts of biological substances in a physiological system.

Compartments represent these physiological systems.

Fundamental QuestionsFundamental 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 Chemical Reactions and Law of Mass ActionAction

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.

CBA k

Law of Mass ActionLaw of Mass ActionA ModelA Model

= rate of accumulation of product

Which depends on:

- the energy of the collision

- geometrical shapes and sizes of the reactant molecules

dt

Cd ][

Law of Mass ActionLaw of Mass ActionA ModelA Model

dt

Cd ][ Is directly proportional to reactant concentrations

]][[][

BAkdt

Cdi.e.

Compartmental Diagram for Compartmental Diagram for Law of Mass ActionLaw of Mass Action

A B

C

]][[1 BAk

][1 Ck

Reverse ReactionsReverse Reactions

Biochemical reactions are typically bi-directional.

CBAk

k

1

1

So, we get the differential equations:

i)][]][[

][11 CkBAk

dt

Cd

ii)]][[][

][11 BAkCk

dt

Ad

EquilibriumEquilibriumA state when concentrations are no longer changing.

In equilibrium, 0

][

dt

Ad

So, 0]][[][ 11 BAkCk

Solving for

eqeqeq BAk

kC ][][][

1

1

(1)

EquilibriumEquilibriumNow, in a closed system (no other reactions are going on)

eqeqeq BCAk

kC

CAA

ACA

])[][(][

][][

][][

01

1

0

0

(constant)

(conservation of matter equation)

Substituting in (1),

EquilibriumEquilibriumSolving for

eqC][

eqeqeq

eqeqeqeq

eqeqeq

eqeqeqeq

BAk

kB

k

kC

BAk

kBC

k

kC

BCk

kBA

k

k

BCBAk

kC

][)][1(][

][][][][

][][][

)][][][(][

01

1

1

1

01

1

1

1

1

10

1

1

01

1

EquilibriumEquilibrium

eqeq

eqeq

eq

eqeq

eq

eq

eq

BK

BAC

Bkk

BAC

Bkk

BAkk

C

][

][][

][

][][

][1

][

][

0

1

1

0

1

1

01

1

Where, (equilibrium constant) has concentration units1

1

k

kKeq

ReactionsReactions

• Elementary • Enzyme

Elementary ReactionsElementary Reactions

Proceed directly from collision of reactantsFollows the Law of Mass Action directly

CBA k

EnzymesEnzymes

•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 ReactionsEnzyme Reactions

Enzyme reactions do not follow the Law of Mass Action directly

If they did, theory would predict:

CESk

k

1

1

but it has been shown that this is not the case.

So, what do they look like?

Michaelis – Menten Model Michaelis – Menten Model (1913)(1913)

Idea Chemical Reaction Scheme Compartmental Diagram System of Differential Equations

IdeaIdea

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 Chemical Reaction SchemeScheme

EPCES k

k

k

2

1

1

is the dissociation constant2kNote:

Compartmental DiagramCompartmental Diagram

ck 1

sek1

ck2

Differential EquationsDifferential EquationsFrom the compartmental diagram, we get the following differential equations:

sekckdt

ds11 sekckk

dt

de121 )(

ckksekdt

dc)( 211

ckdt

dp2

So What Can We Do With All So What Can We Do With All Of This?Of This?

The Equilibrium The Equilibrium ApproximationApproximation

Michaelis – Menten (1913)

Dr. Maud MentenLeonor Michaelis

Picture Not Available

PurposePurpose

To estimate the reaction velocity of an enzyme reaction.

dt

dpV i.e. To estimate

EPCES k

k

k

2

1

1

for

AssumptionAssumptionSubstrate is in “instantaneous equilibrium” with the complex.

0dt

dsi.e.

Consider : 011 sekck

sekck 11

or

sekck 11

note: cee o

SolvingSolving for cfor c

1

1

k

kks

seck

k

1

1 letting

secks

So, by substitution )( cesck os

sekck 11 SolvingSolving for cfor c(Continued)(Continued)

os secksc

os seskc )(

sk

sec

s

o

scseck os

thus,

Derive an expression for VDerive an expression for V

Recall from previous differential equations:

ckdt

dpV 2

Substituting in the new value for c:

sk

sekV

s 02

ObservationsObservations1) The maximum reaction velocity occurs when C is biggest ( ).

maxV0ec

i.e. When all enzyme is complexed with substrate.

022max ekckdt

dpV

will be limited by the amount of enzyme present and the dissociation constant so,

is called “rate limiting” for this reaction.

02max ekV 2k

EPC k 2

Observations(cont.)Observations(cont.)

2) For large substrate concentrations, we will rewrite V as:

sk

sVV

s max

1

max

skV

s

s

s 1

max

s

kV

s

maxmax

1V

V

i.e. the reaction rate saturates

Observations(cont.)Observations(cont.)

3) If , sks

s

s

k

kVV

2maxthen

2maxV

Is that the only approximation Is that the only approximation technique for V?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 Quasi-Steady-State ApproximationApproximation

IdeaIdea 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

AssumptionAssumptionThe rates of formation and the rate of breakdown of the complex are equal.

0dt

dci.e.

RatesRates We know these differential equations:

sekckdt

ds11

sekckkdt

de121 )(

ckksekdt

dc)( 211

ckdt

dp2

cee 0

Variables and ParametersVariables and Parameters

Independent variables: s, c

Parameters: k1, k-1, k2

By nondimensionalization, we make new variables:

os

s:

oe

cx : tek o1:

oss oxec

oekt

1

:

Step 1: ReplacementStep 1: ReplacementWe start by rewriting our differential rate equations from using

our independent variables to using the new variables.

dt

d

d

ds

dt

ds

dt

dsoo

oekdt

d1:

We know: By substitution:

d

deks

dt

dsoo 1

1) Note:dt

d

d

d

dt

d

)1( xexeecee oooo

Now substitute:

Solve for: )1(1

1 xxks

k

d

d

o

d

deksxeskxek

sekckdt

ds

ooooo 111

11

)1(

into:

Let1

1

ks

k

o

so )(

xd

d

d

deks

dt

dsoo 1

oss oxec

Previous Statement:

ckksekdt

dc)( 121

oekdt

d1:

We know: By substitution:

ddx

kedt

dco 12

2)

dt

d

d

dxe

dt

dxe

dt

dcoo

dt

d

d

dx

dt

dx

Note:

)1( xee o :

into:

We don’t want to divide by zero (in case eo is small) so we will divide the right by so to minimize parameters.

d

dxkekkxexesk

ckksekdt

dc

oooo 12

121

211

)()1(

)(

)()1(1

12

k

kkxxs

d

dxe oo

)()1(1

12

oo

o

sk

kkxx

d

dx

s

e

Previous Statement:

ddx

kedt

dco 12

Now substitute:

oss oxec

)()1(1

12

oo

o

sk

kkxx

d

dx

s

e

Let

o

o

s

e and

1

12

ks

kk

o

so

)(

)1(

xd

dx

xxd

dx

d

dx

s

e

o

o

And now our parameters are: ,

Previous Statement:

Step 2: Apply assumptionStep 2: Apply assumptionNow we will apply the assumption of Briggs-Haldane:

0dt

dxeo

We assumed:

Because of these assumptions we can conclude:

xec o

0dt

dcWhich is comparable to : 0

ddx

then 012

ddx

keo 0

d

dx

Now applying those to our rates:

)(0

xd

dx

Solve for x:

x

Substitute x into

)()(xd

d

)(

d

d

Substitute:

q

d

d)(

osk

kq

1

2

Previous Statement:

)(

d

d

Bringing it all togetherBringing it all together

0dt

dp

dt

dc

dt

dsWe know:

Assume 0dt

dc by the Quasi-Steady-State Assumption.

sodt

ds

dt

dp

nowd

deks

dt

ds

dt

dpV oo 1

substitute

q

d

d so

q

eksdt

dpV oo 1

substituteback to: osk

kq

1

2 andos

s

somks

sVV

max where oekV 2max

1

21

k

kkkm

Equilibrium vs. Equilibrium vs. Quasi-Steady-StateQuasi-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 InhibitionEnzyme 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” Enzyme “Lock & Key” StructureStructure

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” Enzyme “Lock & Key” StructureStructure

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 “LockEnzyme “Lock & Key” & Key” StructureStructure

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.

EnzymeActive Site

Allosteric Site

Inhibitor

Allosteric Inhibition

Competitive InhibitionCompetitive Inhibition

Enzyme

Active Site

Allosteric Site

Inhibitor

Reaction Scheme

S + E C1 E + Pk1

k-1

k2

E + I C2

k3

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 InhibitionAllosteric Inhibition

Enzyme

Active Site

Allosteric Site

InhibitorIn 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 Allosteric Inhibition Reaction SchemeScheme

Let:S ~ SubstrateE ~ EnzymeI ~ InhibitorC1 ~ ES complexC2 ~ EI complexC3 ~ EIS complexP ~ Product

E + S C1 E + Pk1

k-1

k2

E + I C2

k3

k-3

C2 + S C3

k1

k-1

C1 + Ik3

k-3

C3


For Allosteric Inhibition we should use “Complex Free Reaction Notation.”

In Complex Free Reaction Notation one substance is implied.


E + S C1 E + Pk1

k-1

k2

E ES E + Pk1s

k-1

k2

Former Notation:

Complex Free Notation:

Let:S ~ SubstrateE ~ EnzymeI ~ InhibitorC1 ~ ES complexC2 ~ EI complexC3 ~ EIS complexP ~ Product


E ES E + Pk1s

k-1

k2

EI EISk1s

k-1

k3i k-3 k3i k-3

Complex Free Notation:


Complex Free Compartmental DiagramLet:X = ESY = EIZ = EIS E X

ZY

Pk1se

k-1x

k2x

k1sy

k-1z

k3ie k-3y k3ix k-3z

Allosteric Inhibition Reaction Allosteric Inhibition Reaction Differential EquationsDifferential Equations

zkzksykixkdt

dz3113

. . .

ConclusionConclusion

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.

BibliographyBibliography 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:

http://www.cdnmedhall.org/Inductees/menten_98.htm

http://www.marxists.org/archive/haldane

Special Thanks to:Special Thanks to:

Dr. Jean FoleyDr. Steve DeckelmanAll of you for coming!

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

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