Moran Process Calculator

About the Moran Process Calculator

The Moran Process Calculator models evolutionary dynamics in finite populations using the classic Moran process — a fundamental model in mathematical biology. It computes fixation probability (the chance a mutant type takes over the entire population) and expected fixation time under frequency-dependent or constant selection. That makes it a useful bridge between qualitative evolutionary intuition and the quantitative behavior of finite populations.

In the Moran process, at each time step one individual is chosen to reproduce (proportional to fitness) and one random individual dies. This birth-death process models genetic drift and selection in populations of constant size N. A single mutant with fitness r in a population of N-1 wild-type individuals fixes with probability (1 - 1/r) / (1 - 1/rᴺ) for r ≠ 1.

Enter the population size, initial mutant count, relative fitness, and explore how selection strength and population size affect evolutionary outcomes. Compare neutral drift (r=1) to advantageous, deleterious, and strongly selected mutations.

When This Page Helps

Use this calculator when you want fixation probability and takeover time from a finite-population model instead of relying on qualitative intuition alone. It is useful for population genetics, evolutionary game theory, and mathematical biology work where drift and selection need to be compared explicitly. That makes the long-run behavior easier to discuss in a quantitative way.

How to Use the Inputs

1. Enter the total population size (N).
2. Enter the initial number of mutant individuals (i).
3. Enter the relative fitness of the mutant type (r).
4. View fixation probability and expected fixation time.
5. Compare different fitness values in the sweep table.
6. Use presets for common evolutionary scenarios.
7. Examine how population size affects fixation probability.

Example Calculation

Tips & Best Practices

• For a single mutant (i=1) with large N: ρ₁ ≈ 1 - 1/r when r > 1 (strongly selected).
• The "1/3 rule" in evolutionary game theory: cooperation is favored if benefit/cost > N/3.
• Population structure (graphs) can amplify or suppress selection compared to well-mixed populations.
• Fixation time scales as N² for neutral drift and N·ln(N) for strongly advantageous mutations.
• In cancer evolution, tumor populations are small enough that drift significantly affects which mutations fix.
• Compare neutral (r=1) to selected (r≠1) to understand the relative roles of drift and selection.

Evolutionary Dynamics Fundamentals

The Moran process is one of two canonical models for finite-population evolution (the other being the Wright-Fisher model). Unlike Wright-Fisher, which has non-overlapping generations, the Moran process uses overlapping generations with continuous birth-death events. This makes it more realistic for many biological populations.

The key insight is that even beneficial mutations don't always fix — they can be lost by random drift, especially when rare. Conversely, deleterious mutations can fix, especially in small populations. This tension between selection (deterministic) and drift (stochastic) is central to evolutionary biology.

Structured Populations and Evolutionary Graph Theory

The Moran process on graphs extends the model to structured populations where individuals interact only with neighbors. The population structure can dramatically affect fixation probabilities. "Amplifiers of selection" (like the star graph) increase the fixation probability of beneficial mutants. "Suppressors of selection" make evolution more neutral-like.

This has implications for understanding evolution in spatially structured environments such as tissues (cancer), biofilms (bacteria), and social networks (cultural evolution).

Applications in Cancer Biology

Tumors evolve by a process similar to the Moran process: cells divide, acquire mutations, and compete for space. Because tumor populations can be relatively small (especially early), genetic drift plays a significant role alongside selection. Understanding fixation probabilities helps predict resistance evolution and design treatment strategies.

Frequently Asked Questions

• What is the Moran process?

  A stochastic model of evolution in a finite population of constant size N. Each step: one individual reproduces (fitness-weighted) and one dies (random). Over time, one type eventually fixates (reaches 100%). It's used to model genetic drift and selection.

• What is fixation probability?

  The probability that a mutant type starting at frequency i/N eventually replaces all wild-type individuals. For a neutral mutation (r=1), this equals i/N (e.g., 1/N for a single mutant). Beneficial mutations fix with higher probability.

• How does fitness affect fixation?

  A mutant with relative fitness r > 1 has enhanced fixation probability: ρ₁ ≈ 1 - 1/r for large N. Even a 1% advantage (r=1.01) dramatically increases fixation probability in large populations. Deleterious mutants (r < 1) can still fix by drift, especially in small populations.

• What is neutral drift?

  When r = 1, all individuals have equal fitness, and evolution is purely random (genetic drift). A single mutant in a population of N fixes with probability 1/N. In a population of 1000, there's a 0.1% chance — small but nonzero.

• How does population size matter?

  Larger populations make selection more efficient and drift less important. In large N: beneficial mutations fix more reliably, deleterious ones are eliminated more effectively, and neutral drift takes longer. In small N: drift dominates and even harmful mutations can fix.

• What is the Moran process used for?

  Modeling evolution on graphs (structured populations), cancer tumor dynamics, antibiotic resistance evolution, game theory (evolutionary games), immune system dynamics, and cultural evolution. It's the simplest birth-death process with selection.