Why Biological Systems Suddenly Change State: An Intuitive Guide to Freidlin–Wentzell Theory

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  Stochasticity is ubiquitous in biology and neuroscience, manifesting in various forms, including ion channel noise, synaptic variability, gene regulatory fluctuations, noisy population dynamics, and more. Many biological systems spend long periods in a stable “state” and only rarely transition to another state due to noise. For instance, a neuron typically remains inactive but may occasionally trigger a spontaneous spike. Similarly, a gene can switch from the OFF state to the ON state due to rare bursts of transcription factors. Cells can also transition out of metabolic or epigenetic states, populations might shift between different ecological equilibria, and a viral infection can fluctuate between phases of control and uncontrollability. Freidlin–Wentzell theory provides a mathematically rigorous framework to study these phenomena when noise is small but nonzero . It tells you, firstly, h ow likely rare transitions are,    secondly,   h ow fast they occ...
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R function conf_int_Accuracy

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R function


conf_int_Accuracy <- function(GMTobs, s, nDays = 2, nAnalysts = 2, nPlates = 2, nReplicates = 3, FoldDilution = 2, Threshold = 1, alpha = 0.05) {

  # INPUT:

  # GMTobs: vector of observed geometric means at each fold dilution

  # s: vector of the standard deviations of the log-transformed replicates

  # nDays: number of experimental days (default = 2)

  # nAnalysts: number of analysts performing the tests in validation (default = 2)

  # nPlates: number of plates used per analyst per day (default = 2)

  # nReplicates: number of measurements that each analyst performs per plate per day

  # FoldDilution: step of the serial dilution (default = 2)

  # Threshold: critical threshold of the log-difference between the observed and the true mean (default=1)

  # alpha: significance level (default=0.05)

  #

  # OUTPUT:

  # vector of Relative Accuracy calculated at each fold dilution and its confidence interval. In addition, a plot of Relative Accuracy vs. fold dilution (log2 units)

 

  # calculate the expected geometric mean vector

  GMTexp <- round(GMTobs[1] * 2^(seq(0, -(length(GMTobs) - 1), -1)), 0)

 

  # Relative Accuracy (or recovery)

  RA <- log(GMTobs / GMTexp)

 

  N <- nDays * nAnalysts * nPlates * nReplicates

  n1 = n2 = N

 

  # standard deviations

  s1 <- s  # vector of standard deviations of the log-transformed replicates

  fd <- round(FoldDilution^-seq(0, length(GMTobs) - 1), 4)

  s2 <- rep(s1[1], length(GMTobs))  # fd*s1[1]

 

  # pooled variance

  sp2 = ((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)

 

  # standard error

  se <- sqrt((sp2 / n1) + (sp2 / n2))

 

  # t-statistic

  tstat <- qt(1 - alpha / 2, n1 + n2 - 2)

 

  # confidence interval

  lower <- RA - tstat * se

  upper <- RA + tstat * se

 

  # critical thresholds

  theta.low <- FoldDilution^-Threshold

  theta.up <- FoldDilution^Threshold

 

  # exponentiate to get back to original scale

  RA <- exp(RA)

  lower <- exp(lower)

  upper <- exp(upper)

 

  # plot RA versus fold dilution

  plot(log2(fd), RA, type = "o", pch = 16, lty = 2, xlab = "fold dilution (log2)", ylab = "Relative Accuracy", ylim = c(0, 3))

  lines(log2(fd), lower, col = "blue", lty = 4)

  lines(log2(fd), upper, col = "blue", lty = 4)

  grid()  

 

  # return RA and its confidence interval

  return(cbind(fd,RA, lower, upper))

}

 

This function calculates the Relative Accuracy (RA) and its confidence interval at each fold dilution and plots RA versus fold dilution (log2 units). The function takes the following inputs:

  • GMTobs: vector of observed geometric means at each fold dilution
  • s: vector of the standard deviations of the log-transformed replicates
  • nDays: number of experimental days (default = 2)
  • nAnalysts: number of analysts performing the tests in validation (default = 2)
  • nPlates: number of plates used per analyst per day (default = 2)
  • nReplicates: number of measurements that each analyst performs per plate per day
  • FoldDilution: step of the serial dilution (default = 2)
  • Threshold: critical threshold of the log-difference between the observed and the true mean (default=1)
  • alpha: significance level (default=0.05)

The function returns a matrix containing the RA and its confidence interval for each fold dilution. The confidence interval is calculated using the t-distribution with n1+n2-2 degrees of freedom and a significance level of alpha. The function also plots RA versus fold dilution (log2 units) and the confidence interval.






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