Protein structure and purification, exploring PDB structures, enzyme kinetics and inhibition, binding and thermodynamics, spectroscopy, structural biology, molecular simulation and research career routes.

Biochemistry & Biophysics

## Why it matters

Enzymes speed up reactions by factors of millions or more, and how fast they work at different substrate concentrations tells you how they behave in cells, how a drug might block them, and whether an engineered variant is better. The **Michaelis–Menten** model is the starting point for nearly every enzyme assay, inhibitor screen and biocatalysis project. Fitting its two parameters from data is a routine task, and one where small mistakes in method give badly wrong answers.

## The model

For a single-substrate enzyme, E binds S to form a complex ES, which turns over to give product P:

```text
E + S  ⇌  ES  →  E + P
```

Under steady-state conditions (ES forms as fast as it breaks down) and with substrate in large excess over enzyme, the initial rate v0 follows:

```text
v0 = Vmax · [S] / (Km + [S])
```

- **Vmax** is the maximum rate, reached when the enzyme is saturated. It is proportional to enzyme concentration: `Vmax = kcat · [E]total`.
- **kcat** (the turnover number, s⁻¹) is how many substrate molecules one active site converts per second at saturation.
- **Km** (the Michaelis constant, in concentration units) is the substrate concentration that gives **half of Vmax**. A lower Km roughly means the enzyme reaches high rates at lower substrate levels.
- **kcat / Km** (the specificity constant, M⁻¹·s⁻¹) compares how well an enzyme uses different substrates at low concentration.

The curve rises almost linearly at low [S] (v0 ≈ (Vmax/Km)·[S]) and levels off at high [S].

## Measuring initial rates

1. Run the reaction at several substrate concentrations spanning roughly 0.2 Km to 5 Km or more (you do not know Km yet, so start wide).
2. For each, measure product formation over time, usually by absorbance or fluorescence, and take the slope of the **early, linear part**: the initial rate, before substrate depletion or product inhibition bend the curve.
3. Convert signal per minute to concentration per minute using an extinction coefficient or standard curve.
4. Run replicates and controls without enzyme (background rate).

## Fitting Vmax and Km

The best practice is a **nonlinear least-squares fit** of v0 against [S] directly to the Michaelis–Menten equation. Software such as SciPy's `curve_fit` (Python) or GraphPad Prism does this. The idea is simple: choose the Vmax and Km that minimise the sum of squared differences between measured and predicted rates.

The older **Lineweaver–Burk** (double-reciprocal) plot rearranges the equation into a straight line:

```text
1/v0 = (Km/Vmax) · (1/[S]) + 1/Vmax
```

It is useful for **visualising** inhibition patterns (next lesson), but it is a poor way to estimate parameters: taking reciprocals hugely magnifies the error in the smallest rates at the lowest [S], so those noisy points dominate the line.

## Worked example

Assay data (substrate in mM, initial rate in µM/min):

```text
[S]  (mM):     0.25   0.5    1      2      4      8      16
v0 (µM/min):   9.06   16.61  28.51  45.07  61.38  73.91  87.06
```

A nonlinear least-squares fit gives **Vmax ≈ 99 µM/min** and **Km ≈ 2.5 mM**. Sanity checks:

- At [S] = 2 mM (close to Km), v0 = 45 µM/min, a bit under half of Vmax, as expected for [S] slightly below Km.
- At 16 mM (more than 6 × Km), v0 is about 88 % of Vmax, not yet fully saturated. Data at even higher [S] would pin Vmax down more tightly.

If the enzyme concentration was 0.01 µM, then kcat = Vmax / [E] = 99 µM·min⁻¹ / 0.01 µM ≈ 9,900 min⁻¹ ≈ 165 s⁻¹.

## In Python

In your own projects with SciPy installed you would write something like:

```python
from scipy.optimize import curve_fit

def mm(s, vmax, km):
    return vmax * s / (km + s)

params, cov = curve_fit(mm, s_data, v_data, p0=[max(v_data), 1.0])
```

The Do-tab activity uses only the standard library: because the best Vmax for any fixed Km has a simple formula, a fine search over Km gives the same least-squares answer.

## Common mistakes

- **Using late time points** instead of initial rates.
- **Too narrow a substrate range,** especially no points above Km, which makes Vmax uncertain.
- **Fitting a Lineweaver–Burk line** to get the final parameters.
- **Mixing units:** Km in mM, rates in µM/min, enzyme in nM. Write units on every column.
- **Ignoring substrate solubility or inhibition** at very high [S].

## Recap

- `v0 = Vmax [S] / (Km + [S])`; Km is the [S] at half of Vmax.
- `kcat = Vmax / [E]`, and kcat/Km is the specificity constant.
- Fit untransformed data by nonlinear least squares; use double-reciprocal plots only to visualise.

## Try it

On the Do tab, write the Michaelis–Menten rate equation in Python and fit Vmax and Km to the assay data.


Fit Michaelis-Menten parameters Vmax and Km to initial-rate data and compute kcat.

Enzyme kinetics

Steady states and the Michaelis Menten equation | Biomolecules | MCAT | Khan Academy

Biochemistry | Michaelis-Menten Equation

## Why it matters

Most drugs are enzyme inhibitors: statins block HMG-CoA reductase, many HIV and hepatitis C drugs block viral proteases, and antibiotics such as penicillin inactivate bacterial cell-wall enzymes. Cells also regulate their own pathways by inhibiting enzymes. In drug discovery and basic research you will often get rate data with and without an inhibitor and need to say **how** it inhibits, because that tells you where it binds and how it will behave at the substrate levels in a cell.

## Reversible inhibition types

Reversible inhibitors bind and unbind; irreversible ones (covalent inactivators) permanently disable the enzyme and are analysed differently. For reversible inhibition at inhibitor concentration [I] with inhibition constant Ki, define α = 1 + [I]/Ki.

**Competitive:** the inhibitor binds the free enzyme at the active site, competing with substrate.

- Apparent Km increases: `Km,app = α · Km`.
- Vmax is unchanged: enough substrate outcompetes the inhibitor.
- Lineweaver–Burk lines cross on the **y-axis**.

**Uncompetitive:** the inhibitor binds only the ES complex.

- Both fall by the same factor: `Vmax,app = Vmax / α'` and `Km,app = Km / α'`, where α' = 1 + [I]/Ki' and Ki' is the inhibitor's dissociation constant from ES.
- Lineweaver–Burk lines are **parallel**.

**Noncompetitive (pure):** the inhibitor binds E and ES equally well, at a site other than the active site.

- Vmax falls: `Vmax,app = Vmax / α`. Km is unchanged.
- Lines cross on the **x-axis**.

**Mixed:** the inhibitor binds E and ES with different affinities. Vmax falls and Km changes (up or down). Lines cross off both axes. Pure noncompetitive inhibition is really a special case of mixed inhibition.

## Reading the parameters

The cleanest approach is to fit Michaelis–Menten curves to each data set (no inhibitor, and one or more inhibitor concentrations), then compare the apparent parameters:

```text
Vmax,app ≈ Vmax   and   Km,app > Km          → competitive
Vmax,app < Vmax   and   Km,app ≈ Km          → noncompetitive
Vmax,app < Vmax   and   Km,app < Km, same factor  → uncompetitive
anything else with Vmax,app < Vmax           → mixed
```

"≈" needs a tolerance, because real fits have error. A good analysis also fits a **global model** (all data sets at once, with Ki as a parameter) and compares models statistically, but comparing apparent parameters is the right first step.

## Worked example

The same enzyme as the last lesson, assayed with and without 1 mM of three different compounds:

```text
                 Vmax,app (µM/min)   Km,app (mM)
no inhibitor     99.2                2.48
compound X       104.9               8.20
compound Y       49.9                2.50
compound Z       50.0                1.31
```

- **X:** Vmax about the same (within the fit's error), Km up more than threefold → **competitive**.
- **Y:** Vmax halved, Km unchanged → **noncompetitive**.
- **Z:** Vmax and Km both roughly halved (ratios 0.50 and 0.53) → **uncompetitive**.

For competitive compound X, Ki follows from `Km,app = Km (1 + [I]/Ki)`:

```text
Km,app / Km = 8.20 / 2.48 = 3.31
Ki = [I] / (3.31 − 1) = 1 mM / 2.31 ≈ 0.43 mM
```

## IC50 is not Ki

Screening assays often report **IC50**, the inhibitor concentration that halves the rate under the assay conditions. For a competitive inhibitor, IC50 depends on how much substrate you used: `IC50 = Ki (1 + [S]/Km)` (the Cheng–Prusoff relation). Two labs with different substrate levels can report different IC50s for the same compound, so compare Ki values, or state the assay conditions.

## Common mistakes

- **Over-reading small differences.** A 5 % change in Vmax is often within fit error. Use replicates and confidence intervals.
- **Calling every Vmax decrease "noncompetitive"** without checking Km.
- **Testing only one inhibitor concentration** for a publication-level claim. Use several.
- **Ignoring tight-binding inhibitors.** When Ki is close to the enzyme concentration, the simple equations fail.

## Recap

- Competitive: Km up, Vmax same. Noncompetitive: Vmax down, Km same. Uncompetitive: both down together. Mixed: Vmax down, Km changes.
- Classify by comparing fitted apparent parameters, with sensible tolerances.
- Ki is a constant of the inhibitor; IC50 depends on assay conditions.

## Try it

On the Do tab, finish a Python function that classifies inhibition type from fitted Vmax and Km values.


Identify competitive, uncompetitive, noncompetitive and mixed inhibition from fitted parameters and estimate Ki.

Inhibition

Lineweaver-Burk Plot and Reversible Inhibition

Biochemistry | Michaelis-Menten & Lineweaver-Burk Plot

## Why it matters

Almost everything in biology works by molecules binding each other: hormones to receptors, antibodies to antigens, transcription factors to DNA, drugs to targets. The strength of that binding, the **affinity**, decides how much complex forms at a given concentration. Drug programs rank compounds by affinity, antibody engineers optimise it, and biophysicists ask what drives it: is the binding powered by favourable contacts (enthalpy), or by released water and flexibility (entropy)? This lesson covers the dissociation constant Kd, how to estimate it from a binding curve, and how it links to thermodynamics.

## The dissociation constant

For a protein P binding a ligand L to form a complex PL:

```text
P + L  ⇌  PL          Kd = [P][L] / [PL]
```

Kd has units of concentration. A **smaller Kd means tighter binding**: an antibody might have Kd around 1 nM, a weak fragment hit around 1 mM. Kd is also the ratio of rate constants, `Kd = k_off / k_on`, so tight binders usually dissociate slowly.

When the ligand is in large excess over the protein (so binding does not deplete free ligand), the **fraction of protein bound** follows a hyperbola, like Michaelis–Menten:

```text
θ = [L] / (Kd + [L])
```

At [L] = Kd, half the protein is bound. To go from 10 % to 90 % bound, [L] must rise about 81-fold (from Kd/9 to 9·Kd), so binding curves are usually plotted against **log [L]**, where they look sigmoidal.

## Measuring binding

- **Fluorescence** (intrinsic tryptophan fluorescence, fluorescence anisotropy with a labelled ligand), **surface plasmon resonance (SPR)**, **biolayer interferometry**, **microscale thermophoresis** and **isothermal titration calorimetry (ITC)** all give binding curves.
- **SPR** also measures k_on and k_off directly.
- **ITC** measures the heat released or absorbed at each injection of ligand into protein. One experiment gives Kd, the binding enthalpy ΔH and the stoichiometry n, with no labels needed.

## Worked example: fitting Kd

A fluorescence titration gives fraction bound at several ligand concentrations (µM), with the protein concentration far below Kd:

```text
[L] (µM):  0.05   0.1    0.2    0.5    1      2      5      10     20
θ:         0.060  0.124  0.191  0.395  0.553  0.712  0.881  0.928  0.961
```

Half-binding falls between 0.5 and 1 µM. A least-squares fit of θ = [L]/(Kd + [L]) gives **Kd ≈ 0.79 µM**. Check: at 1 µM, the model predicts 1/(0.79 + 1) ≈ 0.56, close to the measured 0.553.

## From Kd to free energy

The standard free energy of binding uses Kd relative to a 1 M standard state:

```text
ΔG° = RT · ln(Kd / 1 M)
```

with R = 8.314 J/(mol·K) and T in kelvin. For Kd = 0.79 µM at 25 °C (298.15 K):

```text
ΔG° = 8.314 × 298.15 × ln(0.79 × 10⁻⁶) ≈ −34.8 kJ/mol
```

A negative ΔG° means binding is favourable. Every tenfold improvement in Kd at 25 °C is worth about 5.7 kJ/mol (RT·ln 10).

ITC also gives ΔH directly, so the entropy term follows:

```text
ΔG° = ΔH° − T·ΔS°        →   −T·ΔS° = ΔG° − ΔH°
```

If ΔH° = −50 kJ/mol, then −TΔS° = −34.8 − (−50) = +15.2 kJ/mol: binding is **enthalpy driven** and pays an entropy cost (for example, from freezing flexible groups). Two compounds with the same Kd can have very different enthalpy/entropy balances, which medicinal chemists use to guide optimisation.

## Common mistakes

- **Ignoring ligand depletion.** If the protein concentration is near or above Kd, free [L] is not the total [L] added, and the simple hyperbola gives the wrong Kd. Use the quadratic binding equation instead.
- **Not reaching saturation.** Without points well above Kd, the fit cannot pin down the plateau.
- **Mixing units** in the ΔG° formula: Kd must be in molar (M) units there.
- **Using °C** for T. Use kelvin.
- **Confusing Kd and Ka.** The association constant Ka = 1/Kd; larger Ka means tighter binding.

## Recap

- `Kd = [P][L]/[PL]`; smaller Kd is tighter binding; θ = [L]/(Kd + [L]) when ligand is in excess.
- Fit Kd by least squares on the untransformed curve.
- `ΔG° = RT ln(Kd)` (Kd in M); ITC adds ΔH and hence TΔS.

## Try it

On the Do tab, write the fraction-bound equation in Python, fit Kd to the titration data and convert it to ΔG°.


Estimate a dissociation constant from binding data and convert it to a standard binding free energy.

Binding and thermodynamics

Quick overview of Kd (equilibrium dissociation constant) & binding affinity

Principles of isothermal titration calorimetry (ITC)

Fit Michaelis-Menten parameters to assay data, identify inhibition types from fitted parameters, and estimate binding affinity and its free energy.