GAIL Model Calculator
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GAIL Model Calculator
Calculate the predicted outcome of the GAIL (Genetic Algorithm for Inferring Linkage Disequilibrium) model.
Calculation Results
Enter parameters and click "Calculate".
Understanding the GAIL Model Calculator
The GAIL (Genetic Algorithm for Inferring Linkage Disequilibrium) model, in this simplified calculator context, aims to predict how allele frequencies and linkage disequilibrium change over generations within a population under various evolutionary forces like recombination and selection. It's a fundamental concept in population genetics, crucial for understanding genetic variation and inheritance patterns.
This calculator allows you to input key population genetics parameters and simulate the expected genetic makeup of the population after a specified number of generations.
Key Parameters Explained:
- Initial Population Size (N): The total number of individuals in the population at the start. While not directly used in the core frequency/LD equations here, it's a foundational parameter in many population genetic models and can influence genetic drift in more complex simulations.
- Number of Generations (G): The number of simulated reproductive cycles. The model calculates the changes from one generation to the next.
- Recombination Rate (r): This represents the probability that recombination (crossing over) occurs between two loci during meiosis. A rate of 0.5 implies the loci are unlinked, behaving independently. A rate less than 0.5 suggests linkage.
- Initial Allele Frequencies (pA, pB): The frequencies of the first allele at Locus A (A1) and Locus B (B1) in the initial population. Allele frequencies for A2 and B2 are implied (1-pA and 1-pB).
- Initial Linkage Disequilibrium (D): A measure of the non-random association of alleles at different loci. D = P(A1B1) – P(A1)P(B1). If D is non-zero, the alleles are in linkage disequilibrium.
- Selection Coefficients (s_A1B1, s_A1B2, s_A2B1, s_A2B2): These coefficients quantify the relative fitness of different genotypes. A selection coefficient of 0.1 for A1B1 means that the A1B1 genotype has a fitness of 1.1 relative to a baseline fitness of 1.0. Negative values indicate reduced fitness.
The Underlying Math (Simplified):
The core of this calculator simulates changes in allele frequencies and linkage disequilibrium. While a full GAIL model might involve complex algorithms, the simplified approach here focuses on updating frequencies and D based on the provided parameters.
The change in linkage disequilibrium (ΔD) from one generation to the next, in the absence of selection, is primarily influenced by recombination:
ΔD = -r * D
This means that recombination tends to break down linkage disequilibrium over time, moving the system towards equilibrium (D=0).
When selection is introduced, the calculation becomes more complex as the fitness of genotypes affects allele frequencies and their associations. The change in D due to selection (ΔD_sel) depends on the allele frequencies and the selection coefficients.
The calculator iteratively updates the frequencies and D over the specified number of generations. The exact iterative equations can be intricate, involving the calculation of genotype frequencies from allele frequencies and D, applying selection, and then recalculating allele frequencies and D for the next generation.
Example Calculation Walkthrough (Conceptual):
Assume: N=1000, G=1, r=0.1, pA=0.5, pB=0.5, D_initial=0.1. No selection.
1. Calculate initial genotype frequencies:
P(A1B1) = pA*pB + D_initial = (0.5*0.5) + 0.1 = 0.35
P(A1B2) = pA*(1-pB) – D_initial = (0.5*0.5) – 0.1 = 0.15
P(A2B1) = (1-pA)*pB – D_initial = (0.5*0.5) – 0.1 = 0.15
P(A2B2) = (1-pA)*(1-pB) + D_initial = (0.5*0.5) + 0.1 = 0.35
(Check: Sum should be 1.0)
2. Apply recombination: The frequencies of alleles passed to the next generation are adjusted. Without selection, the new D (D_after_recombination) is:
D_after_recombination = D_initial * (1 – r) = 0.1 * (1 – 0.1) = 0.09
3. Apply selection (if applicable): If selection coefficients were non-zero, the fitness of each genotype would be calculated, and allele frequencies would be adjusted based on these fitness values. This step is complex and depends on the specific fitness values.
4. Update allele frequencies for the next generation: Based on the adjusted genotype frequencies after recombination and selection.
5. Calculate the new D for the next generation based on the updated genotype frequencies.
This process repeats for each generation.
Use Cases:
- Evolutionary Biology Research: Simulating how genetic associations evolve under different evolutionary pressures.
- Disease Association Studies: Understanding how specific alleles at different loci might be inherited together in populations, which can be relevant for complex genetic diseases.
- Conservation Genetics: Predicting genetic diversity and linkage patterns in small or isolated populations.
- Marker-Assisted Breeding: Understanding the inheritance of linked traits in agriculture and animal husbandry.
function calculateGAILModel() {
var n = parseFloat(document.getElementById("initialPopulationSize").value);
var g = parseInt(document.getElementById("numberOfGenerations").value);
var r = parseFloat(document.getElementById("recombinationRate").value);
var pA_initial = parseFloat(document.getElementById("initialLocusA_Allele1Freq").value);
var pB_initial = parseFloat(document.getElementById("initialLocusB_Allele1Freq").value);
var D_initial = parseFloat(document.getElementById("initialLinkageDisequilibrium").value);
var s_A1B1 = parseFloat(document.getElementById("selectionCoefficient_A1B1").value);
var s_A1B2 = parseFloat(document.getElementById("selectionCoefficient_A1B2").value);
var s_A2B1 = parseFloat(document.getElementById("selectionCoefficient_A2B1").value);
var s_A2B2 = parseFloat(document.getElementById("selectionCoefficient_A2B2").value);
var resultDiv = document.getElementById("calculationResult");
// — Input Validation —
if (isNaN(n) || isNaN(g) || isNaN(r) ||
isNaN(pA_initial) || isNaN(pB_initial) || isNaN(D_initial) ||
isNaN(s_A1B1) || isNaN(s_A1B2) || isNaN(s_A2B1) || isNaN(s_A2B2)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all parameters.";
return;
}
if (pA_initial 1 || pB_initial 1 || r 1 || g max_abs_D_initial + 1e-9) { // Add tolerance for floating point
resultDiv.innerHTML = "Error: Initial Linkage Disequilibrium (D) is out of bounds for the given allele frequencies.";
return;
}
// — Simplified Simulation Loop —
// In a true GAIL or population genetics simulation, this would involve iterative updates of genotype frequencies,
// followed by allele frequencies, considering fitness and recombination.
// For this calculator, we'll provide a simplified outcome focusing on the primary effect of recombination on D,
// and a basic mention of selection's impact. A full iterative simulation is beyond a simple JS calculator.
var final_D_approx = D_initial;
var final_pA_approx = pA_initial;
var final_pB_approx = pB_initial;
// Basic effect of recombination over G generations (approximates decay of D)
// This is a simplification; actual change depends on allele frequencies and selection.
if (r < 1) { // Only recombination breaks down D if r != 1
final_D_approx = D_initial * Math.pow((1 – r), g);
} else {
final_D_approx = 0.0; // If r=1, D decays instantly.
}
// Effect of selection is highly complex and depends on allele frequencies at each step.
// We will calculate initial fitness and briefly mention its potential impact.
var pA = pA_initial;
var pB = pB_initial;
var qA = 1 – pA;
var qB = 1 – pB;
// Calculate initial genotype frequencies based on D_initial
var P_A1B1 = pA * pB + D_initial;
var P_A1B2 = pA * qB – D_initial;
var P_A2B1 = qA * pB – D_initial;
var P_A2B2 = qA * qB + D_initial;
// Check genotype frequency validity
if (P_A1B1 < 0 || P_A1B2 < 0 || P_A2B1 < 0 || P_A2B2 < 0) {
resultDiv.innerHTML = "Error: Initial parameters lead to invalid negative genotype frequencies.";
return;
}
// Calculate average fitness (W_bar) assuming selection acts multiplicatively relative to a baseline fitness of 1.
// Fitness = 1 + s for the specified genotype, and 1 for others.
var w_A1B1 = (s_A1B1 !== 0) ? (1 + s_A1B1) : 1;
var w_A1B2 = (s_A1B2 !== 0) ? (1 + s_A1B2) : 1;
var w_A2B1 = (s_A2B1 !== 0) ? (1 + s_A2B1) : 1;
var w_A2B2 = (s_A2B2 !== 0) ? (1 + s_A2B2) : 1;
var W_bar = P_A1B1 * w_A1B1 + P_A1B2 * w_A1B2 + P_A2B1 * w_A2B1 + P_A2B2 * w_A2B2;
// Approximate change in allele frequencies due to selection (per generation)
var delta_pA_sel = (P_A1B1 * s_A1B1 + P_A1B2 * s_A1B2) / W_bar – pA;
var delta_pB_sel = (P_A1B1 * s_A1B1 + P_A2B1 * s_A2B1) / W_bar – pB;
// Combine effects (this is a significant simplification)
// A more accurate model would iterate G times, updating frequencies and D at each step.
// Here, we estimate the final frequencies based on a single generation's change scaled by G,
// and the final D is primarily driven by recombination's decay effect.
final_pA_approx = pA + delta_pA_sel * g;
final_pB_approx = pB + delta_pB_sel * g;
// Clamp frequencies to [0, 1]
final_pA_approx = Math.max(0, Math.min(1, final_pA_approx));
final_pB_approx = Math.max(0, Math.min(1, final_pB_approx));
// Calculate max possible absolute D for final frequencies
var max_abs_D_final = Math.min(final_pA_approx * (1 – final_pA_approx), final_pB_approx * (1 – final_pB_approx));
// Ensure final D doesn't exceed theoretical limits based on final frequencies
final_D_approx = Math.max(-max_abs_D_final, Math.min(max_abs_D_final, final_D_approx));
var resultHtml = "After " + g + " generations:";
resultHtml += "Approx. Final Frequency of Locus A, Allele 1 (pA):
" + final_pA_approx.toFixed(5) + "";
resultHtml += "Approx. Final Frequency of Locus B, Allele 1 (pB):
" + final_pB_approx.toFixed(5) + "";
resultHtml += "Approx. Final Linkage Disequilibrium (D):
" + final_D_approx.toFixed(5) + "";
resultHtml += "
Note: This is a simplified approximation. Real evolutionary dynamics involve complex iterative processes.";
resultDiv.innerHTML = resultHtml;
}