1. Why the Negative Binomial Distribution is Mandatory

Technical replicate counts follow a Poisson distribution, where the variance equals the mean ($\sigma^2 = \mu$). However, biological replicates in agricultural and plant studies incorporate genetic, micro-environmental, and developmental stochasticity. Biological variance exceeds the mean: this phenomenon is called overdispersion.

The Negative Binomial distribution models count data with an overdispersion parameter $\alpha_i$: \[ K_{ij} \sim \text{NB}(\mu_{ij}, \alpha_i), \quad \text{Var}(K_{ij}) = \mu_{ij} + \alpha_i \mu_{ij}^2 \] As $\alpha_i \to 0$, the Negative Binomial converges to the Poisson distribution. DESeq2 uses an Empirical Bayes shrinkage procedure that shrinks individual gene dispersion estimates toward a smooth mean-dispersion trend curve.

2. Executing Differential Expression with DESeq2

library(DESeq2)
library(apeglm)

# Run complete DESeq2 pipeline
# (estimates size factors, dispersions, and fits Generalized Linear Models)
dds <- DESeq(dds)

# Inspect dispersion plot to verify biological shrinkage
plotDispEsts(dds, main = "Solanum tuberosum Dispersion Estimates")

# Extract contrast: Tolerant Genotype Severe Drought (D7) vs Control (D0)
res_tol_d7 <- results(
  dds,
  contrast = c("Drought_Stage", "D7_Severe", "D0_Control"),
  alpha = 0.05
)

# Apply apeglm LFC shrinkage to prevent false high fold-changes in low-count genes
res_tol_d7_shrunk <- lfcShrink(
  dds,
  coef = "Drought_Stage_D7_Severe_vs_D0_Control",
  type = "apeglm"
)

# Summarize results
summary(res_tol_d7_shrunk)