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 slopeRatio

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 The slopeRatio function calculates the relative potency of two samples based on their linear regression slopes.

It also returns the confidence interval for the relative potency

The function slopeRatio takes four arguments: x, y0, y1, and alpha

  • x is the independent variable (e.g., dose, time, ...)
  • y0 is the standard sample
  • y1 is the test sample
  • alpha is the significance level for the confidence interval (default is 0.05)


slopeRatio uses the R function lm to fit linear models for y0 and y1 as functions of x. It then extracts the slopes and standard errors of the models and computes the ratio of the slopes (R) and its standard error using a formula from Finney (1978).

It also checks if the denominator slope (B0) is significantly different from zero using a g-value.

If g > 1, then the use of the ratio is not sensible for the data.

It then calculates the lower and upper bounds of the confidence interval for R. 

It returns a list with four elements: Slope, St.Err, RelativePotency, and CI, where:

Slope is a vector of two values, which are the slopes of the regression lines for y0 and y1 against x.

St.Err is a vector of two values, which are the standard errors of the slopes.

RelativePotency is a scalar value, which is the ratio of the slopes of y1 and y0.

CI is a vector of two values, which are the lower and upper bounds of the confidence interval for the relative potency. 


slopeRatio <- function(x,y0,y1,alpha=0.05) {

#y0: Standard sample

#y1: Test sample

#x: independent variable (e.g., dose, time, ...)

N <- length(x)

nY <- 2

DF <- data.frame(cbind(x,y0,y1))

B <- sapply(2:(nY+1), function(i) summary(mod <- lm(DF[,i] ~ x,data=DF))$coef[2]) #slopes of regression lines

B0 <- B[1]

B1 <- B[2]

SE <- sapply(2:(nY+1), function(i) summary(mod <- lm(DF[,i] ~ x,data=DF))$coef[[4]])#St.Err of slopes

SE0 <- SE[1]

SE1 <- SE[2]

R <- B[2]/B[1] #ratio of the slopes

mod0 <- lm(y0 ~ x)

mod1 <- lm(y1 ~ x)

Cov <- abs(summary(lm(y1 ~ y0))$cov.unscaled[1,2])

T <- qt(1-alpha/2,N-1)

g <- (T^2)*(SE0^2)/(B0^2)#if g>1 the denominator, B0, is not significantly different from 0 and the use of the ratio is not sensible for those data

CI.low <- (R-(g*Cov)/(SE0^2)-(T/B0)*sqrt((1-g)*(SE1^2))+(R^2)*(SE0^2)-2*R*Cov +g*(Cov^2)/(SE0^2))/(1-g)

CI.up <- (R-(g*Cov)/(SE0^2)+(T/B0)*sqrt((1-g)*(SE1^2))+(R^2)*(SE0^2)-2*R*Cov +g*(Cov^2)/(SE0^2))/(1-g)

return(list(Slope=B,St.Err=SE,RelativePotency=R,CI=c(CI.low,CI.up)))

}


#EXAMPLE

x<- seq(-6,0,-1)

yS<-c(7.92600, 6.72275, 6.07275, 5.33000, 4.35925, 3.36875, 2.30125)

yT<- c(6.913, 6.018, 5.505, 4.966, 3.440, 3.154, 1.802)

result<-slopeRatio(x,yS,yT)

print(result$Slope)

[1] -0.9034196 -0.8259286

print(result$St.Err)

[1] 0.02979541 0.05703906

print(result$RelativePotency)

[1] 0.9142247

print(result$CI)

[1] -0.6217680 -0.9317596



Since both the bounds of the confidence interval of the relative potency (absolute value) are less than 1, it means that the standard sample is more potent than the test preparation.

References

Grimm, H. (1979), FINNEY, D. J.: Statistical Method in Biological Assay. 3. ed. Charles Griffin & Co., London and High Wycombe 1978. VII, 508 S., £ 19 net. Biom. J., 21: 689-690. https://doi.org/10.1002/bimj.4710210714



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