Picture of an enzymatic reaction

Velocity = P/t or -S/tP

rodu

ct

Time

Rate constants are defined for reactions

• V = P/t = -S/t = k1[S]

• k1 is called the rate constant and has units of s-1

• If k1 is small, the reaction rate is slow, if large the reaction is fast

• A k of 0.03 s-1 indicates that 3% of the available S will be converted to P in 1 sec

Relationship between V and [S]

Molecular parameters from reaction rates

Assume the conversion of ES to E + P is non-reversible, then the rate of product formation or reaction velocity is dependent solely on [ES] and k2

E + S ES E + Pk1

k-1

k2

v = d[P]/dt = k2[ES] (1)

If we could measure v and [ES] then we could determine k2, however[ES] is not usually measurable. We can measure substrate (or product)concentrations and the total concentration of enzyme [E]t.

[E]t = [E] + [ES] = free enzyme + enzyme in complex with substrate (2)

Thus, we want to express the rate, v, in terms of substrate concentration[S], and total enzyme concentration [E]t.

Ks = k-1/k1 = [E][S]/[ES]

E + S ES E + Pk1

k-1

k2

From this equation:

Under certain circumstances (if k-1 >>k2), E and S are in equilibrium with ES, with an equilibrium dissociation constant Ks.

However, this assumption is not always valid, thus it is of more general use to introduce the concept of the steady state.

In steady state, the rates of formation and breakdown of [ES] are equal:

k1[E][S] = k-1[[ES] + k2 [ES]

Rearrange to give [ES] = (k1/k-1+k2)[E][S]

Define a constant Km = (k-1+k2/ k1)

Km[ES] = [E][S] (3)

Recall we want to get a formula with measurable quantities [S] and [E]t

Rearrange equation 2 (solve for [E]) and plug into 3 to get:

Km[ES] = [E]t[S] – [ES][S]

Transfer second term on right side to left side to get:

[ES](Km + [S]) = [E]t[S]

Rearrange to

[ES] = [E]t[S]/(Km + [S])

Using equation 1 we can finally solve for v, velocity

v = k2[E]t[S]/(Km + [S]) (4)

This formula is referred to as the Michaelis-Menten equation

Consider a graph that we can construct from the measurable quantities v and [S]

v =

cha

nge

in p

rodu

ct

c

hang

e in

tim

e

Increasing [substrate]At high substrate concentrations, the reaction reaches aMaximum velocity Vmax, because the enzyme molecules aresaturated; every enzyme is occupied by substrate and carryingout the catalytic step

[S] = Km

From these relationships, consider the following:

What is Km and what does it mean?

Km is a ratio of rate constants:

Km = (k-1+k2/ k1)

Thus in our catalyzed reaction, if k2 is much smaller than k-1, Km= k-1/k1 = Ks, the equilibrium constant for [ES] formation.In this case, a large Km means k-1 >>k1, thus the enzyme bindsthe substrate very weakly. However, in a separate instancea large k2 can have a similar effect on Km.

Thus, what is the utility of Km?

The most useful way to think of Km is reflected in the plotof a reaction that follows the Michaelis-Menten equation

In this plot, Km is numerically equal to the substrateConcentration at which the reaction velocity equals half of

its maximum value.

Where [S] = Km, the Michaelis-Menton equation simplifies to

v = Vmax/2

Thus, an enzyme with a high Km requires a higher substrate concentration to achieve a given reaction velocity than an enzyme with a low Km.

What are some enzyme’s Km’s

In considering Vmax mathematically, by making [S] muchlarger than Km the Michaelis-Menten equation simplifies to:

Vmax = k2[E]t

Thus, another way of writing the Michaelis-Menten rateEquation is:

v = Vmax[S] / (Km + [S])Typically, all of this is an oversimplification, and enzyme-Mediated catalysis looks more like:

E + S ES EP E + P k1

k-1

k2 k3

In this more complex system, k2 must be replaced with a more general constant, called kcat

v = kcat [E]t [S]/ (Km + [S])

In the two step reaction we considered first, kcat = k2. For more complex reactions, kcat is a combination of rate constants for all reactions between ES and E + P.

kcat is a rate constant that reflects the maximum number of molecules of substrate that could be converted to product each second per active site. Because the maximum rate is obtained at high [S], when all the active sites are occupied with substrate, kcat (the turnover number) is a measure of how rapidly an enzyme can operate once the active site is filled.

kcat = Vmax/[E]t

What are some kcat values?

Under physiological conditions, enzymes usually do notoperate under saturating substrate conditions. Typically, theratio of [S] to Km is in the range of 0.01-1.0.

When Km >> [S], the Michaelis-Menten equation simplifies to:

v = kcat/Km ([E]t[S])

The ratio kcat/Km is referred to as the specificity constantwhich indicates how well an enzyme can work at low [S].The upper limit of kcat/Km is in the range of 108 to 109 dueto limits of diffusion theory.

Both kinetic parameters contribute to enzyme efficiency

Lineweaver-Burk plots are convenient for determination of Km and kcat

Lineweaver-Burk plots result from taking a double reciprocalof the Michaelis-Menten equation.

v = Vmax[S] / (Km + [S])

1/v = Km/(Vmax[S]) + 1/Vmax

Plotting 1/v on the y-axis and 1/[S] on the x-axis (both known quantities)

The slope is equal to Km/Vmax, the y-intercept is 1/Vmax

And the x-intercept is –1/Km

Kinetics of enzymes with multiple substrates

Ordered Ping-Pong

Reversible inhibition

Reversible / non-covalent

Mixed inhibitors bind both E and ES

Non-competitive is special mixed inhibition

Non-competitive

Inhibition effects on kinetic constants

Irreversible inhibition destroy enzyme function

• Suicide inactivators

Regulation of metabolic enzymes is key for the cell

• In metabolic pathways, there is at least one enzyme that sets the rate of flux through the pathway because it catalyzes the slowest or rate-limiting step

• These steps can be modulated through interactions with other cellular components leading to increased or decreased activities, allowing cells to adjust to changing metabolic conditions

Enzyme modification can alter their activity

• Types of modification– Reversible, non-covalent binding of regulatory

compounds or proteins• Enzymes modified in this manner are called

Allosteric – threonine dehydratase is an example

– Reversible, covalent modification such as phosphorylation (LHCII in chloroplasts)

– Activation via proteolytic cleavage

Allosteric enzymes exist in different “states”

Modulators can be stimulatory or inhibitory

• A stimulator or activator is often the substrate itself (homotropic)

• When the modulator is a molecule other than the substrate the enzyme is said to be heterotropic

• Note that allosteric enzymes don’t necessarily have just active sites, but include other sites for modulator binding

• Only in homotropic enzymes are active sites also regulatory sites

Enzymes can be covalently modified with a wide assortment of groups

• Phosphoryl, adenylyl, methyl, etc.

• One third to one half of all proteins in a eukaryotic cell are phosphorylated

• Tyrosine, serine, threonine, and histidine are known amino acids to accept phosphate groups from enzymes known as protein kinases

Properties of allosteric enzymes

• Sigmoidal instead of hyperbolic Michaelis-Menten plots

• Reflects cooperative interactions between multiple subunits (allosteric enzymes often contain multiple subunits)

Substrate-activity curves for allosteric enzymes

Substrate binding influences rates of activity

Cooperativity Hysteresis

Phosphorylation regulates glycogen phosphorylase

• Catalyzes the

removal of a glucose

from the polymer

glycogen in the form

of G1P

Although covalent –

reversible

Some enzymes are made as inactive precursors

• These inactive precursors are called zymogens or proproteins

• For instance, the serine proteases involved in insect immunity (Kanost) are synthesized as zymogens and are active only following cleavage

• In addition, these enzymes are also regulated by interactions with other cellular proteins

Activation by subtraction

• Naturally, biology is more complicated than one enzyme exhibiting one mode of regulation.

• Enzymes can be regulated by multiple mechanisms!