###### tags: `Learning Process`

# **Enzymes**


## **Introduction and Enzyme Vocabulary**
Enzymes are globular proteins that speed up reactions by lowering the activation energies of a reaction without being consumed (an important exception to the globaular protein rule is **Ribozymes, which are made up of RNA**).

## **Cofactors**

Cofactors are the "activation keys" for some enzymes, transforming them from **apo enzymes** to enzymes. The mapping for cofactors is shown below:
$$
\text{Cofactor Activation:} \ \text{Apo Enzyme} + \text{Cofactor} \rightarrow \text{Enzyme}
$$

Each cofactor may be grouped into one of two categories:
* **Coenzymes** (small organic molecule)
* **Cosubstrate**: Enters and leaves the active site of an enzyme along with the substrate (e.g., $NAD+$)
* **Prosthetic**: Bound to the enzyme and will need to be regenerated if the enzyme is damaged (e.g., $FAD$)
* **Metal ions** (small metal ion)

## **Classes of Enzymes**

The function of an enzyme is an **assay which evaluates the activity of protein by testing the product of a reaction with its given substrate (reactants).**

> In this reaction shown below, we test the enzyme Lactate Dehydrogenase by evaluating the presence of $NADH$. From the double arrows, we can tell Lactate dehydrogenase catalyzes the forward and the reverse reaction.

After learning the general function of an enzyme, we may classify each enzyme into one of 7 classes. An example of how to identify an enzyme is shown below:

$$
EC \ 4.2.1.1
$$
> The first number, $4$, indicates the class of enzyme. The second number, $2$, indicates the type of chemistry (see above in functions) that is done by the enzyme. The last two numbers indicate the isoform of the enzyme (sequence of amino acids).

| Enzyme Class | Reaction | Description |
| -------- | -------- | -------- |
| $EC \ 1: \ \text{Oxidoreductase}$ | $\cee{A_{red} + B_{ox} <=> A_{ox} + B_{red}}$ | Catalyzes a redox reaction. |
| $EC \ 2: \ \text{Transferase}$ | $\cee{AB + C \rightarrow A + BC}$ | Exchanges groups from molecules (e.g. kinase adding phosphate gp)|
| $EC \ 3: \ \text{Hydrolase}$ | $\cee{AB + H_2O \rightarrow HA + BOH}$ | Accelerates hydrolysis |
| $EC \ 4: \ \text{Lyase}$ | $\cee{AB <=> A + B}$ | Promotes the removal of a group from the substrate|
| $EC \ 5: \ \text{Isomerase}$ | $\cee{ABC <=> ACB}$ | Creates isomers|
| $EC \ 6: \ \text{Ligase}$ | $\cee{A + B + ATP \rightarrow AB + ADP + P_{i}}$ | Catalyzes synthesis using ATP|

:::info
==**Enzyme Class Question**==
**Hexokinase is an example of which class of enzyme?**
$A) \ \text{Oxidoreductase}$
$B) \ \text{Transferase}$
$C) \ \text{Hydrolase}$
$D) \ \text{Lyase}$
$E) \ \text{Isomerase}$
$F) \ \text{Ligase}$
:::spoiler **Answer**
$B) \ \text{Transferase}$
Hexokinase is an example of **transferase** since it is transfering the $-PO_3^{2-}$ from the $ATP$ molecule.
:::

:::info
==**Enzyme Class Question**==
**Pyruvate Decarboxylase is classified in which class?**
$A) \ \text{Oxidoreductase}$
$B) \ \text{Transferase}$
$C) \ \text{Hydrolase}$
$D) \ \text{Lyase}$
$E) \ \text{Isomerase}$
$F) \ \text{Ligase}$
:::spoiler **Answer**
$D) \ \text{Lyase}$
Pyruvate Decarboxylase is an example of **Lyase** since it is removing the $CO_2$ from the pyruvate substrate.
:::

## **Free Energy (G)**

In biochemistry, we denote the free energy of a system under "standard conditions" as $\Delta G^{\circ '}$. These conditions are:
* $T=298^{\circ}K \ \text{or} \ 25^{\circ}C$
* $P = 1 \ \text{atm} \ \text{or} \ 760 \ \text{torr/mmHg}$
* $M = 1 \ \text{M}$
* $pH = 7$

We use these conditions to calculate the free energy of a reaction:
$$
\Delta G = \Delta G^{\circ '} + -RTln(\frac{[Products]}{[Reactants]}) \quad \alpha \quad \Delta G^{\circ '} + -RTln(Q)
$$
Thus when $K=Q$, our reaction is at equilibrium.

### **Representations of G in Reaction Coordinate Diagrams**

A reaction coordinate diagram details the amount of free energy in a system according to its reactants (reagents, substrate, etc.) and products. We can represent a reaction coordinate diagram via:
* The entire system: Shows the $G$ of a system given a ratio of $\text{Products}:\text{Reactants}$
* Individual molecules: Shows the $G$ of a system for each reactant molecule as it overcomes transition states $[T*S]^{=}$ and forms the product.

#### System Representation in Coordinate Diagrams


We may represent an entire system using a reaction coordinate diagram where the $x-$axis represents the ratio between $\text{Products}:\text{Reactants}$ and the $y-$ axis represents the free energy of a system. The derivative of any given point, $Q$, presents the change in free energy for any ratio between $\text{Products}:\text{Reactants}$, in other words, $\Delta G_{Q}^{\circ '}=-RTQ$.

#### Molecular Representation in Coordinate Diagrams

Like the system representation, we may represent the energy of any given molecule in the system at time $t$ via a reaction coordinate diagram (shown above). In this diagram, our $x-$axis is the reaction progression (unitless) with the $y-$axis being the energy $G$ of a system. The difference between the energy of the reactants and the highest point in the graph (transition state $[ \quad ]^{=}$) is known as the activation energy.

### Enzymes in Reaction Coordinate Diagrams


In a reaction, there is a set distribution of energy among particles. **Enzymes "speed up" a reaction by lowering the activation energy threshold, such that more particles will meet the activation energy threshhold**. To lower the activation energy of a reaction, enzymes will gather the substrate a particular site on the enzyme (known as the active site) and promote the formation of the transition state, which will form the product.


## Active Sites


The site in which the substrates are gathered together in an enzyme is known as the **active site:** a small, three-dimensional cleft/fold in the surface of a protein that is formed by the R-groups from the primary sequence. The active site creates a unique microenvironment which induces multiple weak interactions (not bonds), helping the substrate molecule shift towards its transition state.

### Lock-Key vs Induced Fit

Before recent research, scientists and biologists proposed a theory known as the **Lock and Key** method for fitting substrates onto active sites. This theory proposed the active site of a protein was a rigid structure of which only specific substrates could fit into; however, this theory has been proven false. The modern theory suggests active sites perform an **induced fit** for each substrate such that the active site will slightly alter its shape for a substrate to bind to. This process is shown below:

> Lock and Key fit suggests the active site is static while the induced fit suggests the active site is dynamic, depending on the substrate

### Binding Energy
**Binding energy** is the free energy that is released when a substrate interacts with the enzyme. **The difference in change in free energy between the uncatalzed reaction and catalzed reaction is the binding energy.**

## Rates and Velocity
Kinetics is the study of speed with respect to chemical change (i.e., how fast products form given some reactant). We measure the speed of a reaction (known as velocity) using the following formula:
$$
\text{rate} = \frac{\Delta [P] (M)}{\Delta t (sec.)}
$$

We define **the order of a reaction as the proportion the rate changes with respect to each of the reactant**. Biological reactions are typically bimolecular (i.e., 2 reactants involved) with the form: $\text{rate}=k[R_1][R_2]$ or more generally:
$$
\text{Biochemical Reaction}: R_1 + R_2 \rightarrow P
$$

However, it has been noticed that in many biological reactions, one of the reactants do not significantly impact the formation of the product, leading to the term **"psuedo-first order"** or more generally in the form:
$$
rate = k[R_1] = k[R_1][R_2]^{0}
$$


### Enzyme Mediated Catalysis
> Idk what...(?)

In many biological reactions, we observe the substrate concentration vs reaction velocity graph forming a logarithmic shape (starting off with a high slope and falling to a slope of 0). **The limit of the velocity is known as the maximal velocity** or in other words:
$$
\lim_{[Substrate]\to\infty}(Velocity)=\text{Maximum Velocity}
$$


#### The Michaelis-Menten Equation

The Michaelis-Menten equation (M-M equation) is an equation that states the formation of products from substrate is a two step process with a slow step 1 and a fast/rate-limiting step 2 (i.e., there exists an intermediate with the formation of the products given the reactants. The second step $ES \rightarrow E + P$ is the rate limiting step. The M-M equation is shown below:
$$
\cee{E + S <=>^{k_1}_{k_{-1}} ES \rightarrow^{k_2} E + P}
$$

We ignore the reverse reaction $k_{-2}$ as we measure the enzymatic activity when $[P] \approx 0$. Moreover, the M-M equation implores the **steady-state assumption** which assumes the $[ES]$ stays constant throughout the entire reaction.

To solve for the ratio of the rate constants $k_m$, we may set the following equation:
$$
k_{1}[E][S]=k_{-1}[ES] + k_{2}[ES]
$$
$$
\frac{[E][S]}{[ES]}=\frac{k_{-1}+k_2}{k_1}=k_m
$$

## Quiz Study Guide - Individual

### Match name of class of enzymes to the associated chemical change it catalyzes
Look at Jeffrey's cool table:

| Enzyme Class | Reaction | Description |
| -------- | -------- | -------- |
| $EC \ 1: \ \text{Oxidoreductase}$ | $\cee{A_{red} + B_{ox} <=> A_{ox} + B_{red}}$ | Catalyzes a redox reaction. |
| $EC \ 2: \ \text{Transferase}$ | $\cee{AB + C \rightarrow A + BC}$ | Exchanges groups from molecules/substrates (e.g. kinase adding phosphate gp)||
| $EC \ 3: \ \text{Hydrolase}$ | $\cee{AB + H_2O \rightarrow HA + BOH}$ | Accelerates hydrolysis |
| $EC \ 4: \ \text{Lyase}$ | $\cee{AB <=> A + B}$ | Promotes the removal of a group from the substrate|
| $EC \ 5: \ \text{Isomerase}$ | $\cee{ABC <=> ACB}$ | Creates isomers|
| $EC \ 6: \ \text{Ligase}$ | $\cee{A + B + ATP \rightarrow AB + ADP + P_{i}}$ | Catalyzes synthesis using ATP (energy released)|

### Explain significance of M-M enzyme kinetics parameters
**$K_m$** - AKA Michaelis constant, Indicates substrate (reactant) concentration at which the reaction rate is $\frac{1}{2}V_{max}$
* Describes the affinity of the substrate to the active site of the enzyme
* Smaller value means greater binding affinity (useful for determining which enzyme is more effective when [S] < $K_m$)
* Enzyme activity is high when subtrate concentration is above $K_m$

**$k_{cat}$** - AKA turnover number or $k_2$, measures rate of catalytic process
* $K_{cat} = \frac{V_{max}}{[E]_T}$
* Because this measures how many substrate molecules are transformed into products per unit time by a single enzyme, the units are $\frac{1}{time}$
* Need to know $V_0$ of the catalyzed reaction when $[S]>>[K_m]$ (or $V_{max}$) and also the $[E]_T$

**$V_0$** - Initial velocity/reaction rate, important for:
* Measuring forward reaction veolcity with negligible contribution from reverse reaction
* Measure reaction velocity with known [S] before it changes significantly over time
* $V_0 = \frac{V_{max}[S]}{K_m+[S]}=\frac{k_2[E]_T[S]}{K_m+[S]}$

**$V_{max}$** - Maximum velocity or reaction rate as substrate concentration increases
* $V_{max} = k_2[E]_T$ or $k_{cat}[E]_T$

**$Catalytic \ Efficiency$** - A measure of enzyme efficiency and substrate specificity
* Ratio = $\frac{k_{cat}}{K_m}$

### Interpret Line-Weaver Burk plot for M-M enzymes
The Line-Weaver Burk equation, which linearizes the M-M plot into the $y=mx+b$ format, is as follows:
$$
\frac{1}{v_0}=(\frac{K_m}{V_{max}}*\frac{1}{[S]})+\frac{1}{V_{max}}
$$

*Slope* = $\frac{K_m}{V_{max}}$

*Y-Axis* = $\frac{1}{v_0}$

*X-Axis* = $\frac{1}{[S]}$

*Y-Intercept* = $\frac{1}{V_{max}}$

*X-Intercept* = $\frac{-1}{K_m}$


## Quiz Study Guide - Group

### Match name of class of enzymes to the associated chemical change it catalyzes
(same as individual)

### Explain how an enzyme can change kinetics of a chem rxn, but not the thermodynamics
????? :dizzy_face:

Enzymes do not change the amount of energy or heat (enthalpy). They only lower activation energy (which can also be represented as the difference in free energy between the substrate and transition state) to speed up the rate of reaction, and do not affect reaction equilibrium. This is because reaction equilibrium is determined only between the free energy (G) difference between the products and the reactants.

Binding energy also allows for entropy reduction in the enzyme-substrate complex. ?????

### Explain "catalytic efficiency"
An enzyme is more **catalytically efficient** when it increases the rate of reaction (products are generated at a faster rate). The optimal enzyme catalyzes the most reactions in the least amount of time.

### Fully interpret Line-Weaver Burke plot
(same as individual)

### Given experimentally determined parameters from M-M experiment, comment on efficacy of enzyme
Calculate the catalytic efficiency by the following ratio:
$$Ratio = \frac{k_{cat}}{K_m}$$

A higher value for the ratio means the enzyme is more effective, since the rate of catalysis is higher.