## 640:338:01 Discrete and Probabilistic Models in Biology COURSE NOTES by Daniel Ocone

This page contains links to a text in probabilistic and dynamic programming models for applications to biology. They are made available here free for private study. Otherwise the author reserves copyright privileges. Please send any comments or corrections to ocone@math.rutgers.edu.

1. Chapter 1, Heredity, Genes, and DNA; 25 page pdf file, last revised, June, 2016.

A brief introduction to relevant biological concepts. Contents:

• 1.1 Mendelian Genetics
• 1.2 Genes, Chromosomes and Sexual Reproduction
• 1.3 Genotypes, Phenotypes, and Polymorphisms
• 1.4 Heredity at the Molecular Level
• 1.5 Notes and References

2. Chapter 2, Probability Theory; pdf file, revised January 2014

A review of basic probability, with a focus on what is used in the text and on examples relevant to biological models.

• 2.1 Probability Spaces
• 2.2 Random Variables
• 2.3 Expectation and Its Application
• 2.4 Central Limit Theorem

3. Chapter 3, Population Genetics for Large Populations; revised January, 2014

• 3.1 Difference Equations
• 3.2 Modeling principles for population genetics
• 3.3 Models with no selection
• 3.4 An infinite population model with selection
• 3.5 Notes and References

4. Markov Chains and Applications to Population Genetics; revised, March 2014

• 4.1 Markov Chains
• 4.2 Computation with Markov Chains
• 4.3 Limit Theory of Markov Chains

5. Chapter 5, Probabilistic Analysis of Sequencing Problems; 34 page pdf file, March 2014.

• 4.1 Shotgun Sequencing
• 4.2 Restriction Enzyme Digests; Discrete Models
• 4.3 Poisson Models and Restriction Enzyme Digests

6. Chapter 6, Maximum Likelihood Estimation and Hypothesis Testing ; 24 page pdf file, March 27, 2006

Corrections to Chapter 6, April 18, 2005. (These corrections are already made on the version posted March 27, 2006.)

• 6.1 Probability models and hypotheses
• 6.2 Likelihood functions
• 6.3 Maximum Likelihood Estimation
• 6.4 Exercises
• 6.5 Hypothesis testing
• 6.6 Testing independence versus Markov chain dependence in sequence models
• 6.7 Scoring sequenc alignments for homology
• 6.8 Exercises

7. Chapter 7, Sequence Alignment by Dynamic Programming; Sections, 7.1, 7.2.1-7.2.4. 22 page pdf file; Sections 7.2.5--7.2.8 19 page pdf file.

8. Chapter 8, Hidden Markov Models; 26 page pdf file, April 18, 2005.

• 8.1 Definition and basic theory
• 8.2 Graphical representation of hidden chain paths
• 8.3 The forward algorithm
• 8.4 Maximum probability paths
• 8.5 Profile HMM's for protein families