Rules for deriving rate laws for simple systems 1. Write reactions involved in forming P from S2. Write the conservation equation expressing the

distribution of total enzyme concentration [E]total

among the various species 3. Write the velocity dependence equation, summing

all the catalytic rates constants multiplied by the concentration of the respective product-forming species.

4. Divide the velocity dependence equation by the conservation equation.

5. Express the concentration of each enzymic species in terms of free enzyme concentration & substitute

6. Substitute & algebra

1. Write reactions involved in forming P from S

E + S ES E + Pk1 k3

k2

For a simple 1 step reaction, no inhibitor

2. Write the conservation equation expressing the distribution of total enzyme concentration [E]total among the various species

[E]T = [E] + [ES]

3. Write the velocity dependence equation, summing all the catalytic rates constants multiplied by the concentration of the

respective product-forming species.

v = k3[ES]

4. Divide the velocity dependence equation by the enzyme conservation equation.

[E]T [E] + [ES] v = k3[ES]

5. Express the concentration of each enzymic species in terms of free enzyme concentration & substitute

[ES] = k1[S] [E] - k2[ES] - k3[ES]

k1 [S] [E] = (k2 + k3)[ES]

d[ES] dt

@ steady state = 0, rate of formation = rate of breakdown

[ES] = k1[S] [E] - (k2+ k3)[ES]Collecting terms:

k1[S][E] k2 + k3

[ES] =

Solve for [ES];

k2 + k3

k1

Define KM= then [S][E] Km

= [ES]

Make steady state assumption:

6. Substitute & Algebra

v = k3[ES]

[E]T= [E] + [ES]

[S][E] Km

v = k3

[E]T= [E]+ [S][E] Km

Divide by [E]

[S] Km

v = k3

[E]T= 1+ [S] Km

[S][E] Km

= [ES]

Substitute & Algebra (cont’d)

Multiply by Km

[S] Km

v = k3

[E]T= 1+ [S] Km

[S]v = k3

[E]T= KM+ [S]

Define Vmax= k3[ET]

[S]v = Vmax

KM+ [S]

Expressing With Measurable Expressing With Measurable Quantities Quantities

The total concentration of enzyme [E]T can be determined, but it is very difficult to measure [E] and [ES].

Steady state rate equation: v = k1[E][S] = k-1[ES] + k2 [ES]

By defining the Michaelis Constant (KM) = (k-1 + k2)/k1 then

rearranging and substituting, [ES] can be expressed as:

[ES] = [E]T [S]/(KM + [S])

The initial velocity (vo) = k2 [ES] = k2 [E ]T [S]/(KM + [S])

The key point is that [E]T and [S] are measurable.

Importance of Initial VelocityImportance of Initial Velocity

In reality, vo is velocity measured before 10% of S is consumed.

Before [ES] has built up- can assume reverse reaction is negligible.

Working with vo minimizes complications Reverse reactions Inhibition of reaction by product

Implications of M-M EquationImplications of M-M EquationVo = Vmax [S] / (KM + [S])

1. From the M-M equation, when [S] = KM, vo = Vmax/2. This means that low values of KM imply the enzyme

achieves maximal catalytic efficiency at low [S].2. The catalytic constant, kcat = Vmax / [E]T

kcat is also called the turnover number of the enzyme, i.e. the number of reaction processes (turn-overs) that each active site catalyzes per unit time.

Turnover numbers vary over many orders of magnitude for different enzymes in accord with need.

Implications of M-M Equation Implications of M-M Equation (cont.)(cont.)

3. When [S] << KM, very little ES is formed. Under these conditions: vo ~ (kcat/Km)[E][S] The kcat/Km term is a measure of the enzyme’s catalytic efficiency. The upper limit to kcat/Km is k1, I.e. the decomposition of ES to E + P can occur no more frequently than ES is formed. The most efficient enzymes have kcat/Km values near the diffusion-controlled limit of 108-109 M-1 s-1. They will catalyze a reaction almost every time a substrate is bound in the active site- catalytic perfection!!!

Practical Summary- Analysis of Practical Summary- Analysis of KineticsKinetics

When [S] << KM, the reaction increases linearly with [S]; I.e. vo = (Vmax / KM ) [S]

When [S] = KM, vo = Vmax /2 (half maximal velocity); this is a definition of KM: the concentration of substrate which gives ½ of Vmax.

When [S] >> Km, vo = Vmax

Practical SummaryPractical Summary- - Vmax and Km

KM gives an idea of the range of [S] at which a reaction will occur. The larger the KM, the WEAKER the binding affinity of enzyme for substrate.

Vmax gives an idea of how fast the reaction can occur under ideal circumstances.

Kcat / KM gives a practical idea of the catalytic efficiency, i.e. how often a molecule of substrate that is bound reacts to give product.

Inhibition of EnzymesInhibition of Enzymes•Enzyme Inhibitor: a molecule that reduces the effectiveness of an enzyme.

•Mechanisms for enzyme inhibition

Competitive inhibition- substrate analogs bind in the active site reducing availability of free enzyme

Uncompetitive inhibition- inhibitor binds to the substrate-enzyme complex and presumably distorts the active site making the enzyme less active

Mixed or non-competitive inhibition- combination of competitive and uncompetitive inhibition

Inactivator- irriversible reaction with enzyme

Enzyme Inhibitors in MedicineEnzyme Inhibitors in Medicine•Many current pharmaceuticals are enzyme inhibitors (e.g. HIV protease inhibitors for treatment of AIDS- see pages 336-337).

• An example: Ethanol is used as a competitive inhibitor to treat methanol poisoning.

Methanol is metabolized by the enzyme alcohol dehydrogenase producing highly toxic formaldehyde.

Ethanol competes for the same enzyme.

Administration of ethanol occupies the enzyme thereby delaying methanol metabolism long enough for clearance through the kidneys.