Quasispecies

Key Concepts

  • HIV Infection:

    • HIV has a short genome (~10,000 bases) and a very high mutation rate (~\(3 \times 10^{-5}\) per base per replication).
    • Rapid adaptation due to short generation time, large population sizes, and strong selective pressures (e.g., immune response, antiretroviral therapy).

  • Sequence Space:

    • Sequence Space: Describes the set of all possible genetic sequences of a given length \(L\).
    • For DNA, the sequence is made from bases \(A, C, G, T\), and for proteins, the sequence is based on 20 amino acids.
    • Binary Sequences: Simplified representation using \(0\) and \(1\). For a binary sequence of length \(L\), there are \(2^L\) possible sequences.
    • Hamming Distance: Used to measure the difference between two sequences based on the number of mismatched positions.

  • Fitness Landscapes:

    • Definition: A fitness landscape is a mapping from genotype space to fitness, \(f: G \rightarrow \mathbb{R}\), representing how well an organism can survive and reproduce.
    • Epistasis: Interactions between loci (positions on the genome) where the effect of one gene depends on the presence of others, influencing fitness.

  • Quasispecies Equation:

    • Quasispecies: A population of genetically diverse organisms that evolves under mutation and selection pressure.
    • Mathematical Representation: \[ x'(t) = x(t) \cdot Q \cdot f(t) - x(t) \cdot \phi \]
      • \(x(t)\): Genotype frequency vector.
      • \(Q\): Mutation matrix.
      • \(f\): Fitness landscape vector.
      • \(\phi\): Average population fitness.
    • The equilibrium solution of this equation predicts the balance between mutation and selection in a population.

  • Properties of the Quasispecies Equation:

    • If replication is error-free (\(Q = I\)), the equation simplifies to the selection equation.
    • If the mutation matrix \(Q\) is irreducible (strongly connected), there exists a single globally stable equilibrium \(x^*\).
    • Mutation reduces overall population fitness compared to what would be expected from selection alone.

  • Adaptation:

    • A population adapts by localizing around a peak in the fitness landscape, but mutation can cause individuals to drift away from these peaks.
    • Error Threshold: A critical mutation rate above which genetic information cannot be maintained, leading to the "mutational meltdown" phenomenon.

  • Error Threshold and HIV:

    • For HIV, with a genome length \(L = 10^4\) and mutation rate \(u = 3 \times 10^{-5}\), the probability of an error-free genome copy is only 0.74.
    • With 1 billion new viruses produced each day, a vast number of mutations occur daily.

Key Equations and Models

  • Quasispecies Equation: \[ x'(t) = x(t) \cdot Q \cdot f(t) - x(t) \cdot \phi \]

    • Describes the dynamics of a population under mutation and selection.

  • Hamming Distance:

    • Measures the difference between two sequences by counting mismatched positions.

  • Error Threshold:

    • Describes the mutation rate at which the population cannot maintain genetic information. The condition \(uL > 1\) leads to mutational meltdown.

Summary Points

  • HIV's high mutation rate and short generation time lead to extreme evolutionary dynamics, allowing rapid adaptation to selective pressures.
  • Sequence space and fitness landscapes provide a framework for understanding how populations evolve under mutation and selection.
  • The quasispecies equation models the balance between mutation and selection in populations, particularly for RNA viruses like HIV.
  • The concept of an error threshold indicates a mutation rate beyond which genetic information cannot be preserved, leading to population collapse.
  • Mutation-selection balance is central to quasispecies theory, explaining viral diversity and resistance to treatment.