Compute estimates of the extrinsic noise.

Description

This function computes several estimates of the extrinsic noise (unscaled by the mean). The estimators, described in Fu and Pachter (2016), include the original estimators developed in Elowitz et al. (2002), the unbiased estimator, an min-MSE estimator, and an asymptotic estimator for large sample sizes.

Usage

computeExtrinsicNoise(reporter1, reporter2)

Arguments

Value

Four (unscaled) estimates of extrinsic noise: the original estimators developed in Elowitz et al. (2002), the unbiased estimator, an min-MSE estimator, and an asymptotic estimator for large sample sizes.

Author(s)

Audrey Q. Fu

References

Fu, A. Q. and Pachter, L. (2016). Estimating intrinsic and extrinsic noise from single-cell gene expression measurements. arXiv:1601.03334. Elowitz, M. B., A. J. Levine, E. D. Siggia, and P. S. Swain (2002) Stochastic gene expression in a single cell. Science, 297, 1183-1186.

See Also

computeIntrinsicNoise, simulateSC. See estimates for data elowitz_data and yang_nl10.

Compute estimates of the extrinsic noise using a known correlation in the min-MSE estimator.

Description

This function is similar to computeExtrinsicNoise, and computes several estimates of the extrinsic noise (unscaled by the mean). The estimators, described in Fu and Pachter (2016), include the original estimators developed in Elowitz et al. (2002), the unbiased estimator, an min-MSE estimator, and an asymptotic estimator for large sample sizes. The only difference between this function calculates the min-MSE estimate using a given correlation.

Usage

computeExtrinsicNoiseKnownCor(reporter1, reporter2, true.cor)

Arguments

Value

Four (unscaled) estimates of extrinsic noise: the original estimators developed in Elowitz et al. (2002), the unbiased estimator, an min-MSE estimator (using the given correlation), and an asymptotic estimator for large sample sizes.

Author(s)

Audrey Q. Fu

References

Fu, A. Q. and Pachter, L. (2016). Estimating intrinsic and extrinsic noise from single-cell gene expression measurements. arXiv:1601.03334. Elowitz, M. B., A. J. Levine, E. D. Siggia, and P. S. Swain (2002) Stochastic gene expression in a single cell. Science, 297, 1183-1186.

See Also

computeExtrinsicNoise, simulateSC.

Compute estimates of the intrinsic noise.

Description

This function computes several estimates of the intrinsic noise (unscaled by the mean). The estimators, described in Fu and Pachter (2016), include the original estimators developed in Elowitz et al. (2002), unbiased estimators with and without assuming equal mean of the two reporters, min-MSE estimators with and without assuming equal mean, and asymptotic estimators for large sample sizes with and without assuming equal mean.

Usage

computeIntrinsicNoise(reporter1, reporter2)

Arguments

Value

Six (unscaled) estimates of intrinsic noise: the original estimators developed in Elowitz et al. (2002), unbiased estimators with and without assuming equal mean of the two reporters, min-MSE estimators with and without assuming equal mean, and asymptotic estimators for large sample sizes with and without assuming equal mean.

Author(s)

Audrey Q. Fu

References

Fu, A. Q. and Pachter, L. (2016). Estimating intrinsic and extrinsic noise from single-cell gene expression measurements. arXiv:1601.03334. Elowitz, M. B., A. J. Levine, E. D. Siggia, and P. S. Swain (2002) Stochastic gene expression in a single cell. Science, 297, 1183-1186.

See Also

computeExtrinsicNoise, simulateSC. See estimates for data elowitz_data and yang_nl10.

Select a random subset of data for noise estimation.

Description

This function randomly selects a subset of cells (rows) from the data set, computes multiple estimates of intrinsic and extrinsic noise, as well as their mean and standard deviation.

Usage

computeNoiseForSubset(data, sample.size, n.iter)

Arguments

Value

A list that consists of the following components:

Author(s)

Audrey Q. Fu

References

Fu, A. Q. and Pachter, L. (2016). Estimating intrinsic and extrinsic noise from single-cell gene expression measurements. arXiv:1601.03334.

See Also

computeIntrinsicNoise, computeExtrinsicNoise, elowitz_data, yang_nl10.

Examples

data(yang_nl10)

# quantile normalization on log2 transformed data
# install bioconductor package for quantile normalization
# source('http://bioconductor.org/biocLite.R')
# biocLite('preprocessCore')
library(preprocessCore)

# ignore a few values that are negative
yang_nl10.pos <- yang_nl10[-which (yang_nl10[,1]<0),]
yang_nl10.pos.log2.quant <- normalize.quantiles (as.matrix (log2 (yang_nl10.pos[,c(1,3)])))

# subset the data and compute noise estimates
yang.50 <- computeNoiseForSubset (yang_nl10.pos.log2.quant, sample.size=50, n.iter=1000)
summary (yang.50)

Expression of reporter genes in the D22 and M22 E. coli cells from Elowitz et al (2002).

Description

Expression of reporter genes CFP and YFP in over 200 E. coli cells of two strains: D22 and M22. These values are displayed in a scatterplot in Elowitz et al (2002) Fig 3a.

Usage

data("elowitz_data")

Format

The format is: List of 2 $ D22:'data.frame': 284 obs. of 2 variables: ..$ CFP: num [1:284] 3080 3082 2893 3053 2891 ... ..$ YFP: num [1:284] 2309 2394 2145 2340 2245 ... $ M22:'data.frame': 250 obs. of 2 variables: ..$ CFP: num [1:250] 2438 2316 2521 2646 2830 ... ..$ YFP: num [1:250] 1409 1391 1511 1460 1638 ...

References

Elowitz, M. B., A. J. Levine, E. D. Siggia, and P. S. Swain (2002) Stochastic gene expression in a single cell. Science, 297, 1183-1186.

Examples

data(elowitz_data)

# Normalize data such that they are 
# comparable to Fig 3a in Elowitz et al. (2002).
# Normalized data have mean 1.
D22.cfp.norm <- (elowitz_data$D22[,1]-mean (elowitz_data$D22[,1]))/sd(elowitz_data$D22[,1])/8+1
D22.yfp.norm <- (elowitz_data$D22[,2]-mean (elowitz_data$D22[,2]))/sd(elowitz_data$D22[,2])/8+1

M22.cfp.norm <- (elowitz_data$M22[,1]-mean (elowitz_data$M22[,1]))/sd(elowitz_data$M22[,1])/12+1
M22.yfp.norm <- (elowitz_data$M22[,2]-mean (elowitz_data$M22[,2]))/sd(elowitz_data$M22[,2])/12+1

# Compute noise estimates.
# Since the mean is 1, estimates with and without 
# the scaling are the same.
unlist (computeIntrinsicNoise (D22.cfp.norm, D22.yfp.norm))
unlist (computeExtrinsicNoise (D22.cfp.norm, D22.yfp.norm))

unlist (computeIntrinsicNoise (M22.cfp.norm, M22.yfp.norm))
unlist (computeExtrinsicNoise (M22.cfp.norm, M22.yfp.norm))

Simulate single-cell expression levels of two reporters

Description

This function simulates expression levels of two reporters across single cells, mimicking the two-reporter assay. The hierarchical model described in Fu and Pachter (2016) is used for simulation. We further make the simplifying assumption that intrinsic noise is the same across cells.

Usage

simulateSC(n = 1000, intrinsic = 0.7, extrinsic = 0.8, mean = 1)

Arguments

Value

A data frame of two columns and n rows. Each column contains the expression levels of a reporter. Each row is a single cell.

Author(s)

Audrey Q. Fu

References

Fu, A. Q. and Pachter, L. (2016). Estimating intrinsic and extrinsic noise from single-cell gene expression measurements. arXiv:1601.03334.

See Also

computeIntrinsicNoise, computeExtrinsicNoise.

Examples

# simulation #1

# simulate 500 data sets
n.simu <- 500

# true intrinsic and extrinsic noise
int.true <- 0.7
ext.true <- 0.8

# create matrices to hold estimated intrinsic and extrinsic noise
# using different estimators
int.simu.mtx <- matrix (0, nrow=n.simu, ncol=8)
ext.simu.mtx <- matrix (0, nrow=n.simu, ncol=4)

for (i in 1:n.simu) {
    n <- 1000
    simu <- simulateSC (n=n, intrinsic=int.true, extrinsic=ext.true, mean=1)
    
    int.simu.mtx[i,] <- c(unlist (computeIntrinsicNoise (simu[,1], simu[,2])), 
        cor (simu[,1], simu[,2]))
    ext.simu.mtx[i,] <- unlist (computeExtrinsicNoise (simu[,1], simu[,2]))
    
}

# add column names to simulation estimates
colnames (int.simu.mtx) <- c("ELSS", "unbiasedGeneral", "unbiasedEqualMean", 
    "minMSEGeneral", "minMSEEqualMean", "asymptoticGeneral", 
    "asymptoticEqualMean", "cor")
colnames (ext.simu.mtx) <- c("ELSS", "unbiased", "minMSE", "asymptotic")


# simulation #2

# simulate 500 data sets
n.simu <- 500

# true intrinsic and extrinsic noise
int.true <- 0.7
ext.true <- 0.8

# use true correlation for the min-MSE estimates of extrinsic noise
true.cor <- ext.true / (ext.true + int.true)

# create matrices to hold estimated intrinsic and extrinsic noise
# using different estimators
int.simu.mtx <- matrix (0, nrow=n.simu, ncol=8)
ext.simu.mtx <- matrix (0, nrow=n.simu, ncol=4)
ext.simu.mtx.2 <- matrix (0, nrow=n.simu, ncol=4)

for (i in 1:n.simu) {
    n <- 50
    simu <- simulateSC (n=n, intrinsic=int.true, extrinsic=ext.true, mean=1)
    
    int.simu.mtx[i,] <- c(unlist (computeIntrinsicNoise (simu[,1], simu[,2])), 
      cor (simu[,1], simu[,2]))
    ext.simu.mtx[i,] <- unlist (computeExtrinsicNoise (simu[,1], simu[,2]))
    ext.simu.mtx.2[i,] <- c(unlist (computeExtrinsicNoiseKnownCor (simu[,1], 
      simu[,2], true.cor)))
}

# add column names to simulation estimates
colnames (int.simu.mtx) <- c("ELSS", "unbiasedGeneral", "unbiasedEqualMean", 
    "minMSEGeneral", "minMSEEqualMean", "asymptoticGeneral", 
    "asymptoticEqualMean", "cor")
colnames (ext.simu.mtx) <- c("ELSS", "unbiased", "minMSE", "asymptotic")
colnames (ext.simu.mtx.2) <- c("ELSS", "unbiased", "minMSE", "asymptotic")

# compute the MSE of estimates
computeMSE <- function (a, t) {return (mean((a-t)^2))}
apply (int.simu.mtx[,1:7], 2, computeMSE, t=int.true)
apply (ext.simu.mtx, 2, computeMSE, t=ext.true)
apply (ext.simu.mtx.2, 2, computeMSE, t=ext.true)

Expression of reporter genes in the NL010 E. coli cells from Yang et al (2014).

Description

Expression of reporter genes CFP and mCherry in over 40,000 E. coli cells. A subset of these values are displayed in a scatterplot in Yang et al (2014) Fig 3a rightmost panel.

Usage

data("yang_nl10")

Format

A data frame with 40683 observations on the following 3 variables.

CFP

a numeric vector

Venus

a numeric vector

mCherry

a numeric vector

References

Yang, S., S. Kim, Y. R. Lim, C. Kim, H. J. An, J.-H. Kim, J. Sung, and N. K. Lee (2014) Contribution of RNA polymerase concentration variation to protein expression noise. Nature Communications, 5, 4761.

Examples

data(yang_nl10)

# compute the noise estimates

# no normalization
# unscaled by mean
unlist (computeIntrinsicNoise (yang_nl10[,1], yang_nl10[,3]))
unlist (computeExtrinsicNoise (yang_nl10[,1], yang_nl10[,3]))

# scaled by mean
unlist (computeIntrinsicNoise (yang_nl10[,1], yang_nl10[,3])) / 
    mean (yang_nl10[,1]) / mean(yang_nl10[,3])
unlist (computeExtrinsicNoise (yang_nl10[,1], yang_nl10[,3])) / 
    mean (yang_nl10[,1]) / mean(yang_nl10[,3])

# quantile normalization on log2 transformed data
# install bioconductor package for quantile normalization
# source('http://bioconductor.org/biocLite.R')
# biocLite('preprocessCore')
library(preprocessCore)

# ignore a few values that are negative
yang_nl10.pos <- yang_nl10[-which (yang_nl10[,1]<0),]
yang_nl10.pos.log2.quant <- normalize.quantiles (as.matrix (log2 (yang_nl10.pos[,c(1,3)])))

# unscaled by mean
unlist (computeIntrinsicNoise (yang_nl10.pos.log2.quant[,1], yang_nl10.pos.log2.quant[,2]))
unlist (computeExtrinsicNoise (yang_nl10.pos.log2.quant[,1], yang_nl10.pos.log2.quant[,2]))

# scaled by mean
unlist (computeIntrinsicNoise (yang_nl10.pos.log2.quant[,1], yang_nl10.pos.log2.quant[,2])) / 
    mean (yang_nl10.pos.log2.quant[,1]) / mean(yang_nl10.pos.log2.quant[,2])
unlist (computeExtrinsicNoise (yang_nl10.pos.log2.quant[,1], yang_nl10.pos.log2.quant[,2])) / 
    mean (yang_nl10.pos.log2.quant[,1]) / mean(yang_nl10.pos.log2.quant[,2])