8.592J | Spring 2011 | Graduate

Statistical Physics in Biology

Lecture Notes

Instructors

MK: Prof. Mehran Kardar
LM: Prof. Leonid Mirny

SES # TOPICS INSTRUCTORS
1 Introduction; Master equation MK
2 Elements of population genetics LM
3 Evolving populations (forward Kolmogorov) MK
4 Population genetics (fitness, selection, …) LM
5 Fixation (backward Kolmogorov) MK
6 Introduction to sequence alignment LM
7 Algorithms for sequence alignment LM
8 Significance of aligments (extreme value distributions) MK
9 Ionic interactions MK
10 Polymer theory 1, entropy elasticity, collapse MK
11 Polymer theory 2, collapse, Random energy model MK
12 Protein folding 1: elements of structure and forces LM
13 Protein folding 2: evolved (designed) foldable proteins LM
14 Protein folding 3: kinetics LM
15 Protein folding 4: kinetics LM
16 DNA (melting) MK
17 RNA (secondary structure) MK
18 Protein-DNA interactions 1: specific and non-specific binding LM
19 Protein-DNA interactions 2: sequence-dependence & kinetics LM
20 Microtubules MK
21 Molecular motors & cell motility MK
22 Introduction to networks MK
23 Dynamics on networks 1 MK
24 Dynamics on networks 2 LM
25 Packing of DNA inside cells LM
26 Cooperativity: hemoglobin and transcription factors LM
  1. Substitution Matrices
    1. PAM
  2. BLAST
    1. How it works
    2. Examples
  1. Macroscopic properties of DNA; packaging into chromatin (eukaryotes)
  2. Pray, L. Discovery of DNA Structure and Function: Watson and Crick. Nature Education 1, no. 1 (2008).
    Microscopic structure of DNA; the four nucleotides; binding energies
  3. Denaturation, hybridization (similarity); renaturation (zippering)
  4. (UV absorption) Melting curves for medium and long chains; general features and theoretical issues
  5. The pair-stacking model; (Poland, D., and H. A. Scheraga. Theory of Helix-Coil Transitions in Biopolymers. Academic Press, 1970.)
  6. Theoretical analysis of homopolymer denaturation (presentation by David Mukamel at ITP)
  • Meltsim is a package to calculate melting temperature of specified DNA sequences.

(Detailed Lecture Notes (PDF))

  1. Dynamics on networks:
    • Node variables evolving according to inputs from connected nodes
    • Examples—Chemical flux balance equations, Neural networks
    • Possible collective outcomes: fixed points, temporal oscillations, spatial patterns, …
  2. Time indedpendent steady-states:
  3. Stability of Fixed Points:
  4. Biochemical clocks:
  5. Synchronization:

Neural Networks

Biological Clocks

(Detailed Lecture Notes (PDF))

  1. Synchronization:
  2. Biological Patterns:
    • Morphogenesis is the process whereby a living organism develops form and structure, e.g.
    • Where do spots come from? Turing’s answer

Synchronization

Morphogenesis

(Detailed Lecture Notes (PDF))

Foundations of Molecular Evolution and Population Genetics

  1. Review of the central dogma
    1. Replication
    2. Coding sequence
      1. Gene
      2. Protein
      3. Genome
    3. Genetic code
  2. Mutations, DNA variation
    1. Synonymous
    2. Non-synonymous
  3. Foundations of genetics
    1. One locus
      1. Genotype
      2. Haploids vs diploids
      3. Homo/heterozygosis, allele frequency
      4. Dominant vs recessive (PDF): blood types
    2. Random mating: Hardy-Weinberg equilibrium
  4. Genetic Drift
    1. Dynamics in population of constants size
    2. N=1
    3. Reduction of heterozygosity
    4. N»1, Binomial sampling
    5. Probability of fixation: simulation 1, simulation 2
    6. Time to fixation
  5. Selection
    1. Absolute and relative fitness
    2. Writes equations: increase of fitness
  6. Selection and Drift
    1. Probability of fixation: (diploids, s, N, h=1/2)
    2. Time to fixation
  7. Summary
  1. The forward Kolmogorov equation:
    • Continuum limit of the Master equation
    • Drift and diffusion
    • Mutations
  2. Binomial selection:
    • Binomial rates
    • Drift/diffusion
  3. Chemical reaction mimicking selection:
    • Master equation for a binary reaction
    • Drift/diffusion coeeficients for selection
  4. Steady states:
    • For general drift/diffusion
    • For population with mutations/drift/selection

(Detailed Lecture Notes (PDF))

  1. The backward Kolmogorov equation:
    • Steady states with absorbing states
    • Derivation of the backward Kolmogorov equation
  2. Fixation:
    • General solution of the backward Kolmogorov equation
    • Fixation probability with selection
    • Fixation probability of a new mutation
  3. Mean first passage times:
    • Mean time to fixation
    • Mean time to loss
    • Mean survival time

(Detailed Lecture Notes (PDF))

  1. Elements of networks:
  2. Scale free networks:
    • Examples from the colloquium by A. L. Barabasi
      • The world wide web- description and results
      • Yeast protein network (Two hybrid assays, A. Wagner: Fig. 1, Fig.2)
      • Food web
      • Metabolic network
      • p53 network- description and results
    • The Barabasi-Albert (BA) model (Applet)
    • The degree distribution of the BA model

Lecture by Prof. Mirny on Biological Networks

(Detailed Lecture Notes (PDF))

  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 alignments
  3. Algorithms for sequence alignments
    1. Dynamics programming
    2. Example
    3. Global and Local alignments
  4. BLAST
    1. How it works
    2. Examples

Outline

TOPICS DESCRIPTION CATEGORY
I. Sequence nucleic acids in DNA/RNA; amino-acids in proteins information
II. Structure DNA double helix; RNA secondary form; protein folding interactions
III. Function building blocks, force and motion: membranes, tubules, motors… mechanisms
IV. Assembly networks: modules, patterns, collective dynamics… complexity

Evolving Probabilities

  1. Mutating Sequence:
    • Mutations & Classical genetics
    • Binary sequence of independent elements
    • Transition matrix and time dependent evolution
  2. Population Perspective:
    • Master equation for evolving population
    • Steady state (binary distribution)
    • Enzymatic analog

(Detailed Lecture Notes (PDF))

  1. Ionic solutions:
    • Why do electrolytes dissociate (ionize) in water?
    • The bare Coulomb interaction and the Bjerrum length
    • Acids, bases, and salts
    • The importance of Coulomb repulsion in biological systems
    • Macroions, counterions, salt ions
    • Restricted partition function
  2. Statistical treatments of ionic solutions:
    • ‘Mean-field’ potential and charge density via the self-consistent Poisson-Boltzmann equation
    • Screening in salt solutions
    • Dissociation from a charged membrane
      • Solution of the 1d equation
      • The Gouy-Chapman layer
    • Interaction between charged parallel plates
    • Importance of fluctuations; like charges can attract

(Detailed Lecture Notes (PDF))

  1. Filaments in the cycloskeleton; Major types:
  2. Microtubule structure:
    • Tubulin subunits
    • Protofilaments, nucleation and growth
    • Polarity
  3. Dynamic Instability [Mitchison, T., and M. Kirchner. “Microtubule Assembly Nucleated by Isolated Centrosomes.” Nature 312, (1984): 232.]
  4. Search strategy
  5. Ndlec, F.J., et al. “Self-organization of microtubules and motors.” Nature 389, (1997): 305-308.

(Detailed Lecture Notes (PDF))

Molecular Motors

  1. Molecular motors:   

  2. General scheme of motor transport:   

    • Asymmetry for directionality
    • Rectification of fluctuations by consumption of energy

  3. Motion in a noisy environment   

     

A Brownian ratchet (Image from Wikimedia Commons)

View a ratchet and pawl animation

  1. Asymmetric hopping models:
    • Fisher, M. E., A. B. Kolomeisky. “The force exerted by a molecular motor.” PNAS 96, no. 6597 (1999).
    • Multiple internal states
    • Solution with two internal state
    • Einstein force, stalling force

Cell Motion

  1. Brownian motion: 
  2. Crawling cells
  3. Swimming cells:
    • Life at Low Reynolds Number (E.M. Purcell)
    • The scallop theorem: impossibility of reciprocal motion
    • Flagellar rotary motion (efficiency)
    • Chemotaxis by runs and tumbles—why?

(Detailed Lecture Notes (PDF))

  1. Binding to multiple sites
  2. Hemoglobin
    • Slides in pdf format from 2009
    • Lecture notes on the MWC model (2009)
    • Hill 1910, where the hill equation is derived
  1. Polymers are long covalently bonded macromolecules, with N»1 monomers
    • Biological polymers: Polynucleotides, polypeptides, polysaccharides
    • Homopolymers, heteropolymers
    • Artificial polymers, e.g (-CHH-)N
  2. How straight is a polymer?
    • Thermal excitation of rotational isomores
    • Bending rigidity and Persistence length
  3. Entropic elasticity:
    • Random walks, Kuhn segments, and the central limit theorem
    • Polymer spring
  4. Interactions:

(Detailed Lecture Notes (PDF))

  1. Interacting polymers:
    • Entropy; excluded volume; and solvent-mediated interactions
    • Mean-field estimate of the partition function
  2. Swollen (coil) phase in good solvents
    • Flory exponent
    • Scaling behavior in other dimensions
  3. Compact (globular) phase in poor solvents
    • Polymer collapse, theta-point
    • Thermodynamic behavior at the transition; reduction in entropy
  4. Frozen (folded) state of heteropolymers
    • The Random Energy Model (REM)
    • The freezing transition and associated singularities
    • Designed REM as a model of protein folding

(Detailed Lecture Notes (PDF))

  1. Selection
  2. Wright’s equation
  3. Effective population size
  1. Forces—wrap-up
  2. Protein structure
    • secondary
    • 3D structure
  3. Sequence-structure mapping
    • analogous and homologous proteins
    • hydrophobic core
    • periodicity
  1. Biomolecular Forces
    • van der Waals interactions
    • Hydrogen bonds
    • Hydrophobic effect
    • Hydrophobic collapse
  2. Protein Folding Problem(s)
    • Anfinsen experiment
    • Thermodynamic hypothesis
    • Levinthal paradox
  3. Thermodynamics of folding
    • Cooperativity
  4. Energy Gap
  1. Review of protein structure
  2. Protein Folding Problem(s)
  3. Lattice Model
  4. Designed sequence
  5. Monte Carlo sequence design, detailed balance
  6. Analogy with Halden’s equation in population genetics
  1. Review of REM
  2. Chemical kinetic, Arrhenious Law
  1. Regulation of genes by protein-DNA interactions
    • Lac repressor
    • CRP activation
    • Recruitment of polymerase
  2. Structure of protein-DNA complexes
    • Specificity
    • Bonds
    • DNA bending, non-direct readout
  1. Kinetics of binding (Lecture notes)
    • Reaction Kinetics
    • Debye-Smoluchowski theory and equation
    • Estimated and observed rates
    • Berg-von Hippel theory of facilitated diffusion

(Detailed Lecture Notes (PDF))

  1. Structure and function of RNA
  2. The ‘RNA world’ hypothesis
  3. Structural levels:
  4. Representations without psuedoknots
  5. Algorithms for finding optimal secondary structure: Nussinov, R., and A.B. Jacobson. “Fast algorithm for predicting the secondary structure of single-stranded RNA.” PNAS 77, no. 11(1980): 6309-6313. “Research by Michael Zuker”.
  6. Recursive (Hartree) calculation of partition function

(Detailed Lecture Notes (PDF))

  1. Sequence alignment
    • Inputs:
      • Explicit—Two sequences {a1, a2, …., am} and {b1, b2, …., bn} (e.g. query and database)
      • Implicit—A scoring procedure, e.g. pairwise scores s(ai,bj) 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:
  3. Gapped alignments and Statistical Physics

(Detailed Lecture Notes (PDF))

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