These are notes from lecture 3 of Harvard Extension’s biochemistry class.

enzyme kinetics

Enzymes are usually named after what they catalyze:

  • oxidoreductase – redox
  • transferase – transfer of functional groups
  • hydrolase – hydrolysis reactions
  • lyases – eliminate groups to form double bonds
  • isomerases – isomerization
  • ligases – covalent bond formation coupled with ATP hydrolysis

Enzyme-substrate interactions have geometric and electronic complementarity and are also stereospecific – enzymes usually won’t work on an enantiomer of their substrate.  The earliest model (1894) was the “lock and key model” which held that static geometry dictated enzyme-ligand specificity.  The more recent model (1950s) is the “induced fit” model in which the ligand induces conformational change in the enzyme.

Enzymes often require cofactors which can be either metal ions or coenzymes.  Coenzymes in turn can be either cosubstrates or prosthetic groups.  Prosthetic groups (e.g. heme) are permanently bound, while cosubstrates transiently float in and out of the enzyme’s active site.

Reactions can be thought of in terms of substrates to products with an intermediate transition state.

S → [TS]‡ → P

TS usually has higher energy than S. It is but a brief point in time at the top of an energy hill before you reach P which is lower energy than S.

ΔG‡ is the symbol for activation energy. It equals ΔGTS – ΔGS. This quantity determines how fast the reaction will proceed.

ΔGreaction is the free energy of the reaction. It equals ΔGP – ΔGS.

If ΔGreaction is negative the reaction is spontaneous. ΔGreaction can only tell you whether a reaction is spontaneous. It cannot tell you its speed. ΔG‡ tells you the speed. Enzymes lower ΔG‡. They do not affect ΔGreaction and do not affect the position and direction of equilibrium, only the speed with which equilibrium is reached.

An enzyme can speed both directions of an S ↔ P equilibrium, but cannot change the position of equilibrium, i.e. the favored direction from a given starting state. So in a situation where GP > GS, where the reaction wants to run backwards, the enzyme would make the reaction run backwards faster.

Enzymes lower ΔG‡ by stabilizing (lowering the energy of) the transition state (TS), making it easier to reach. The “lock and key” model does not capture this because it does not depict the lock (enzyme) changing shape. The enzyme needs to adopt a shape consistent with the transition state in order to stabilize the transition state.

Examples of how an enzyme can stabilize a transition state:

  1. Acid catalysis. The conversion of ketone to enol. An enzyme (H-A) donates its proton, reducing the unfavorable character of the transition state.
  2. Base catalysis. Here the enzyme accepts a proton in order to make the transition state more favorable.
  3. Covalent catalysis. A transient covalent bond is formed between the catalyst and substrate. A-B + X: → A-X + B → A+ X: + B. This gives you two transition states, denoted X‡1 and X‡2.
  4. Metal ion catalysis. This relies on Fe2+, Fe3+, Cu2+, Mn2+ and Co2+ and is one of the reasons you need trace metals in your diet. They mediate redox reactions through reversible changes in the metal’s oxidation state, orient the substrate for the reaction, or stabilize or shield negative charges.
  5. Catalysis through proximity and orientation. Just bringing substrates close together and orienting them properly.

Amino acids that can act as acid or base catalysts: D, E, H, K, C, Y.  H can act as either acid or base depending on the local pH.

The microenvironment around the active site can affect the protonation state of amino acids. For instance, asparate (D) has a pKa of 3.9, therefore at physiological pH it should by rights be deprotonated. However aspartate in a hydrophobic active site will remain protonated to avoid having a negative charge and offending the hydrophobic amino acids around it. Two aspartates together may remain protonated to avoid repelling each other with their negative charges.

Catalysis of RNase A. H12 and H119 act as base and acid respectively to promote nucleophilic attack and bond cleavage. Both water and RNA are substrates.

Carbonic anhydrase is a metalloenzyme. It dissocates water with a Zn2+, binds the OH- and then uses the O- for a nucleophilic attack on carbon dioxide to yield H+ and something.

Digestive enzymes that cleave peptide bonds. Chymotrypsin handles aromatic and large hydrophobic residues, trypsin cleaves at basic residues R and K, and elastase handles small hydrophobic residues A and G. This specificity arises from differences in the binding pocket of each enzyme. Chymotrypsin has a deep enough pocket to bind large aromatic R groups. Trypsin is lined with negatively charged residues. Elastase has bulky amino acids to only allow small R groups in.

Zymogens are proenzymes – larger inactive precursors of enzymes which need to be cleaved to be activated. For instance trypsinogen is cleaved by enteropeptidase or trypsin to yield active trypsin. The mechanism for the prodomain suppressing enzyme activity is usually based on conformation: cleavage brings about conformational change that brings the active site residues into contact with each other in the right proximity and orientation to perform their function.

Consider this reaction:

E + S ↔ ES → E + P

Here, the first step is rate limiting – in the second step, the backwards rate is negligible so it’s written as just →. If you plot substrate concentration [S] on the x axis vs. reaction velocity V on the y axis, and hold [E] constant, you get a hyperbolic curve as eventually the available enzyme is saturated with S.

Km is a measure of an enzyme’s affinity. Km = [S] | V = ½Vmax. In other words it is the substrate concentration at which the reaction proceeds at half its maximum velocity. The higher an enzyme’s affinity the lower the Km – the enzyme is good at binding what little substrate is available.

Every enzyme has its own Km, units of concentration of substrate. You can look these up for any enzyme. Vmax on the other hand depends on the enzyme concentration. If you reduce an enzyme’s concentration by half, the whole hyperbolic curve shifts down as Vmax is lowered, and the Km point on the x axis still refers to a point on the curve reaches half its maximum.

V0 is defined as the velocity measured before 10% of S is consumed, usually 60 seconds or less after starting the reaction. The Michaelis-Menten equation dictates that:

V0 = Vmax[S] / (Km + [S])

This can be arranged into a Lineweaver-Burke Plot:

1 / V0 = (Km/Vmax) * (1/[S]) + 1/Vmax

And then you plot 1/[S] on the x axis and 1/V0 on the y axis and you will find that 1/Vmax is the y-intercept.

kcat is the catalytic constant or turnover number. It describes how quickly an enzyme acts, i.e. how fast the ES complex proceeds to E + P. It represents the number of substrate molecules transformed to product in a given unit of time by a single enzyme molecule. kcat is a constant you can look up for any given enzyme.

kcat = Vmax / [E]T

[E]T is the total concentration of enzyme catalytic sites, i.e. including both ligand-bound and unbound enzymes.  When an enzyme is saturated with substrate, all of it is bound to substrate and therefore [E] = 0 (no unbound enzyme) and [ES] = [E]T.

kcat / Km is called the catalytic efficiency – tells you how rapidly the enzyme binds and how quickly it turns over.

The catalytic speed of enzymes is limited both by (1) how fast atoms and electrons can be rearranged in the catalysis and (2) how often they encounter a substrate in a “productive collision”.

For “catalytically perfect enzymes”, (1) is no object – it’s basically instant. The only limiting factor is (2).

Sometimes two different substrates for the same enzyme can have very different kcat if one of them stabilizes the transitions state.

So far we’ve talked about single substrate reactions; most reactions have multiple substrates and multiple products.

Bisubstrate reactions are often labeled:

A + B ↔ P + Q

Bisubstrate reactions have two types:

1. Sequential: all substrates must combine with E before a reaction can occur. Sometimes a particular binding order is required – example: lactate dehydrogenase converting NADH and pyruvate to lactate and NAD+ (in that order). Other enzymes allow substrates to bind in any order – example: creatine kinase can bind creatine and ATP in either order and then makes phoshocreatine.

2. Ping pong: E can encounter substrates at different times. for instance:

E + A → EA → E* + P → then add B and then E* + B → E*B → E + Q

This most often occurs with transferases where a functional group is removed, leaving an inactive intermediate.

enzyme inhibitors

  • Irreversible inhibitors or “suicide substrates” enter the active site and fail to undergo the complete reaction, becoming stuck in the active site. Then that enzyme molecule is inactive and will eventually be broken down – it never recovers.
  • Competitive inhibitors have almost the geometric shape and structure of the substrate and compete with the substrate for the active site. This can be overcome by having a very high concentration of the desired substrate. Competitive inhibitors increase Km, effectively decreasing the enzyme’s affinity for its substrate – with the inhibitor present, it takes a higher [S] to reach ½Vmax. Note that Vmax stays the same – if you added enough S – maybe a huge amount – you could still get to the original maximum velocity. In the Linebacker-Straywl plot this means a different slope but same y-intercept.
  • Uncompetitive inhibitors bind only to an enzyme-substrate complex, preventing it from completing the reaction. They don’t prevent the substrate from binding in the first place. This decreases both Vmax and decreases Km. The decrease in Km is counterintuitive: the binding of I to ES depletes the available ES complex, thus causing the E + S ↔ ES to run more forward as if the product was gone.
  • “Mixed” or “noncompetitive” inhibitors bind the enzyme at a different site than the substrate does, and can bind before or after the substrate binds. This decreases Vmax and may increase or decrease Km depending on whether I has greater affinity for E or for ES. In this class we are supposed to assume it usually increases Km, i.e. has greater affinity for E.
  • Pure noncompetitive inhibitors have no effect on the ability of E to bind S. Thus Km is unchanged; only Vmax is decreased.

Here is a table of what inhibitors do to the parameters of enzyme kinetics:

pure noncompetitive

*These bind irreversibly, so if [I] > [E], then Km is effectively increased to infinity.
**Mixed inhibitors can raise or lower Km depending on whether they have greater affinity for E (in which case they _might_ raise Km if they also reduce its affinity for S) or ES (in which case they might lower Km like an uncompetitive inhibitor does).

Cells control the activity of their enzymes through:

  • Enzyme availability: synthesis vs. degradation, localization
  • Enzyme activity – changes in affinity or catalytic efficiency: inhibitors, allosteric effects, covalent modification, ionic signals (e.g. change pH).

Allosteric enzymes. Allosteric effectors are small molecules that bind an enzyme at sites different than the substrate binding site and can increase or decrease the enzymatic activity. There are allosteric activators and allosteric inhibitors. This is common in multi-subunit proteins with multiple active sites, and very common in proteins that regulate metabolic pathways.

Homotrophic regulation is when the substrate itself is also the allosteric modulator. Heterotrophic is when there is one substrate and one allosteric modulator. (Note: here she uses “substrate” and “ligand” interchangeably.)

The binding of a ligand to one protein subunit may alter the other subunits’ activity – this is cooperativity.

Heterotrophic allosteric enzymes are said to have a catalytic subunit and a regulatory subunit. The allosteric modulator binding to the regulatory subunit affects the catalytic subunit’s activity. Allosteric enzymes are likely to have sigmoidal, rather than hyperbolic, saturation curves.

The term Km is reserved for Michaelis-Menten enzymes with hyperbolic curves. For allosteric enzymes, which have sigmoidal curves, we use the term K0.5, though as far as I can tell the concept is identical.

The most common covalent modification that affects enzyme activity is phosphorylation of S, H, Y or T.