Force Fields and Energy Minimization

A molecular force field models the potential energy of a conformation as a sum of bonded and non-bonded terms:

\[E_\text{total} = \sum_\text{bonds} k_b(r - r_0)^2 + \sum_\text{angles} k_\theta(\theta - \theta_0)^2 + \sum_\text{torsions} \frac{V_n}{2}[1 + \cos(n\phi - \delta)] + \sum_{i<j}\left[\frac{A_{ij}}{r_{ij}^{12}} - \frac{B_{ij}}{r_{ij}^6} + \frac{q_i q_j}{\epsilon r_{ij}}\right]\]

Minimization proceeds via gradient descent on the $3N$-dimensional conformational space, where $N$ is the number of atoms.

Ramachandran Plot

Backbone geometry is constrained by steric clashes. For each residue, the allowed $(\phi, \psi)$ dihedral angles form a Ramachandran distribution. Under a probabilistic model:

\[P(\phi, \psi \mid \text{residue type}) \propto \exp\!\left(-\frac{E(\phi,\psi)}{k_B T}\right)\]

Most residues cluster in $\alpha$-helix and $\beta$-sheet regions, serving as a quality metric for predicted structures.

AlphaFold Distance Map Prediction

AlphaFold2 frames structure prediction as a distribution over inter-residue distances $d_{ij}$ and orientations. It learns:

\[P(d_{ij} \mid \text{MSA}, \text{templates}) = \text{softmax}(\mathbf{W}\,\mathbf{z}_{ij})\]

where $\mathbf{z}_{ij}$ are pair representations updated by triangle multiplicative attention. Final coordinates are produced by an equivariant structure module operating on $SE(3)$ frames per residue.

Homology Modelling

When a homologous template exists with sequence identity $\geq 30\%$, coordinates are transferred and refined. Model quality is estimated by the TM-score:

\[\text{TM-score} = \frac{1}{L}\sum_{i=1}^{L_\text{ali}} \frac{1}{1 + (d_i/d_0)^2}, \quad d_0 = 1.24\,(L-15)^{1/3} - 1.8\]

A TM-score above $0.5$ generally indicates the same fold.