📘 Lecture 6 – Video 1: Protein Electrostatics (Fun & Deep Dive Summary)

This lecture introduces protein electrostatics — how charges interact within and around proteins, and how we visualize and calculate these interactions in structural biology.

We move from basic physics (Coulomb’s law ⚡) all the way to advanced computational methods (Poisson–Boltzmann equation 🧮). Let’s break it down clearly and thoroughly.

⚡ 1. Coulomb’s Law – The Foundation of Electrostatics

Everything begins with Coulomb’s law:

• There is a force between two charges.
• The force:
  • Is proportional to the product of the charges
  • Is inversely proportional to the square of the distance
  • Acts along the line connecting the charges

So:

• Opposite charges → attract
• Same charges → repel
• Greater distance → much weaker force (because of 1/r² dependence)

Also:

• ε₀ = permittivity of vacuum (a physical constant)
• The force is a vector, meaning it has direction.

Key idea: Electrostatic interactions drop off very fast with distance.

🧲 2. Electric Field – Force Per Unit Charge

Instead of asking:

What is the force between A and B?

We ask:

What force would a charge feel at any point in space?

This leads to the electric field (E):

E = rac{F}{q}

• It is the force per unit charge.
• Independent of the charge placed there.
• A vector field (has magnitude and direction).

Important conceptual shift:

If a −1 ion feels force F, a −2 ion feels 2F, but the electric field is the same.

🟢 Field Around a Single Positive Charge

• Field lines radiate outward.
• A negative test charge moves inward.
• A positive test charge moves outward.

The field lines always show: The direction a positive test charge would move.

➕➖ 3. Multiple Charges – Superposition Principle

If multiple charges exist:

E_ = E_1 + E_2 + E_3 + ...

The total electric field is the sum of all individual fields.

Same idea applies to electrostatic potential.

🧬 4. Example: Glycine (Zwitterion at pH 7)

At pH 7:

• NH₃⁺ → positive
• COO⁻ → negative

Glycine is a zwitterion (both + and − charges).

Field lines:

• Go from positive to negative regions.

This can be visualized in PyMOL.

Electrostatics already becomes biologically meaningful here: Even the simplest amino acid has structured electrostatic behavior.

🔋 5. Electrostatic Potential – Energy Perspective

Instead of force, we now focus on energy.

Imagine:

• A fixed charge A.
• You place charge B nearby.

If B has opposite charge:

• It would naturally move toward A.
• To keep it there, you must supply energy.

That energy is the electrostatic potential energy.

Key idea:

ext{Potential energy} = ext{Force} imes ext{Distance}

Even more important:

The electrostatic potential at a point is the energy required to move a charge from that point to infinity.

Why infinity?

Because:

• At infinite distance → no force → no interaction → zero potential.

So potential is always measured relative to infinity.

🔁 6. Moving Charges Between Points

Energy required to move a charge:

Delta U = U_2 - U_1

• If you move to lower potential → you gain energy.
• If you move to higher potential → you must supply energy.

This is exactly like gravitational potential.

📏 7. Units: kT/e

Electrostatic potential in proteins is often expressed in:

rac{kT}{e}

Where:

• k = Boltzmann constant
• T = temperature
• e = elementary charge

At room temperature:

kT/e ≈ 0.0257 ext{ J/C}

Interpretation:

If potential = +1 kT/e at a point:

• Removing a singly charged ion from that point requires 1 kT of energy.

If doubly charged → 2 kT.

This makes electrostatics directly comparable to thermal energy.

Very powerful concept.

🎨 8. Iso-Potential Surfaces

You can visualize regions where:

Potential = constant value

Example:

• 1 kT/e surface
• 2 kT/e surface
• 10 kT/e surface

Closer to charges:

• Higher energy
• Stronger attraction/repulsion

Important geometric insight:

Electric field lines are perpendicular to iso-potential surfaces.

🧬 9. Electrostatic Surface Potential of Proteins

Showing iso-surfaces around whole proteins is messy.

Instead, we do something smarter:

1. Calculate protein surface.
2. Compute electrostatic potential at each surface point.
3. Color the surface.

This gives:

🔵 Negative regions 🔴 Positive regions

Color scale example:

• −1 to +1 kT/e
• Or −4 to +4 kT/e

Interpretation:

If a region is +1 kT/e:

• Attracts negative ions
• Repels positive ions

If a region is −1 kT/e:

• Attracts positive ions
• Repels negative ions

This directly tells us:

How proteins interact with ligands, ions, or other proteins.

This is biologically very important.

🌊 10. What Is the Protein Surface?

This is not trivial.

The most accepted definition:

Solvent Accessible Surface (SAS)

Method:

• Take a sphere of radius 1.4 Å (size of water molecule).
• Roll it over the protein.
• All points the sphere can touch define the surface.

Why?

Because proteins function in solution. So what matters is: What solvent can access.

Every point water can contact → part of surface.

This is the biologically relevant surface.

🧮 11. Computational Challenges

Proteins are complicated:

• Many charges.
• Non-uniform dielectric constant:
  • Inside protein → low dielectric (apolar-like).
  • Outside in water → high dielectric.
• Mobile ions in solution.
• Screening effects.

This makes electrostatic calculation difficult.

📐 12. Poisson–Boltzmann Equation

The most accurate way to calculate electrostatic potential in proteins.

It accounts for:

• Charge distribution
• Dielectric differences
• Ionic strength

But:

❗ It is slow.

Used when:

• You need accurate static structure analysis.

⚡ 13. Particle Mesh Ewald (PME)

Used in molecular dynamics.

• Faster
• Slightly less exact
• Good for repeated calculations

Essential for simulations.

🧠 14. Why Electrostatics Matters in Proteins

Electrostatic potential helps explain:

• Ligand binding
• Protein–protein interactions
• Enzyme active site behavior
• Ion selectivity
• pH dependence
• Salt effects

Ligands often carry charge. Electrostatic complementarity is often key for binding.

🧩 Big Conceptual Summary

🎯 Core Takeaways

1. Electrostatics is fundamentally about force and energy between charges.
2. Potential is more useful than force for protein analysis.
3. We care most about electrostatic potential on the protein surface.
4. Visualization helps predict biological interactions.
5. Real calculations require solving complex equations due to:
   • Dielectric differences
   • Ionic strength
   • Complex geometry