1. Steady states:
• For general drift/diffusion
• For population with mutations/drift/selection
2. The bacward Kolmogorov equation:
• Steady states with absorbing states
• Derivation of the backward Kolmogorov equation
3. Fixation:
• General solution of the backward Kolmogorov equation
• Fixation probability with selection
• Fixation probability of a new mutation
4. Mean first passage times:
• Mean time to fixation
• Mean time to loss
• Mean survival time

1. Sequence comparison
1. Protein sequences: amino acids and their properties
2. Why do we need to compare sequences?
   • Find dissimilarities
   • Similar sequences = similar properties
   • Similar sequences = common ancestral genes
3. Mutations -> Gapless alignments
4. Mutations, insertions and deletions -> Gapped alignment
2. Sequence alignments
1. Score = edit distance
2. Examples of scores
3. Length of alignment, number of alignment
4. Substitution Matrices (PAM)
3. Algorithms for sequence alignments
1. Dynamics programming
2. Example
3. Global and Local
4. Advanced Dynamic Programming Tutorial
5. BLAST
1. How it works
2. Examples

1. Sequence alignment (Wikipedia applet)
• Inputs:
  • Explicit- Two sequences {a₁, a₂, ...., a_m} and {b₁, b₂, ...., b_n} (e.g. query and database)
  • Implicit- A scoring procedure, e.g. pairwise scores  s(a_i,b_j) and gap costs
• Alignment algorithm: global, local, gapped, gapless (dynamic programming, applet)
• Output:  matching (sub)sequences with an overall score  S . (example)
• Significance: What is the probability of getting a score S by chance?
2. Statistics of gapless local alignments:
• Random walks of scores as a function of matching length
• Extreme Value Distributions, the Gumbel distribution,
• Tails of distribution in a gapless alignment
• The Karlin-Dembo formula
3. Gapped alignments and Statistical Physics
• Transfer Matrix method for Directed Paths in Random Media
• Dynamic Programming method for Sequence Alignment (Example)

[Lecture Notes]

[Exercises]

Related links

• Tutorial by Jennifer Sierchio
• Tutorial by Ralf Bundschuh on Sequence Alignment (and associated paper)
• Wikipedia applet for Needleman-Wunsch alignment
• Lecture by Prof. Mirny
• Paper by Miyazawa