From Wells to Cells

Growth vs. Gene Expression in Platereader Experiments

Rainer Machne

Plan

# Experiment Theory
1 Using R packages: platexpress, grofit, etc. The Basics: What is Exponential Growth?
2 The Experiment: View and Analyze Data Statistics: Measurement Errors
3 The Result: Summary Plots The Model: Growth vs. Expression Rates


https://github.com/raim/platexpress

Goals: learn some R for data analysis - handling large data sets, &
gain better quantitative understanding of cell growth VS. gene regulation.

  1. Use R to Analyze Growth & Expression Data
    • Installing R packages
    • Prepare & load data
  2. Use R base
    • Linear regression with lm(log(X) ~ time)
    • Non-linear fit with nls(X ~ X0*exp(mu*time))
  3. Use R packages
    • grofit, growthcurver or growthrate
    • Logistic, Gompertz, non-parametric growth models
  4. Growth vs. Gene Expression
    • Proteins/Cell: normalize fluorescence & compare
    • Monod equation: growth vs. gene expression
    • Towards the riboswitch model

From Data Hell to Model Heaven : Why Platereader?

Tube vs. Platereader Cultures

  1. Data Analysis
    • many replicates, many conditions
    • different statistics required
  2. Endpoint vs. Kinetics
    • steady-state vs. dynamics

Are tube and platereader cultures comparable?
Why (not)?

From Data Hell to Model Heaven and back.

  1. Inspect Data Files
    • Amend if required
  2. Load & Inspect Data
    • Get overview, check data: roughly Ok?
    • Clean data: subtract blanks, cut time, rm outliers etc.
    • Inspect replicates: are observations replicated, what are our measurement errors?
  3. Construct Your Analysis Pipeline
    • What do we need?
    • What were our assumptions, and are they justified?
    • Do we see additional unsuspected results?
  4. Calculate Results & Statistics over Replicates

Do we have what we need for Monday?

Installing R Packages from cran, bioconductor & github

install.packages(c("grofit","growthcurver")) # at CRAN

source("https://bioconductor.org/biocLite.R") # at bioconductor
biocLite("cellGrowth")

install.packages("devtools") # R development tools
library(devtools)
install_github("raim/platexpress") # at github

https://github.com/raim/platexpress

Do you speak git? Try to install from your local copy:
git clone git@github.com:raim/platexpress.git
... and use R CMD build & sudo R CMD INSTALL

How to use a new R package?

## load the package & explore
library(platexpress)

?platexpress # VIEW HELP FILES
vignette("platexpress") # READ THE VIGNETTE
demo("demo_ap12") # RUN THE DEMO
getData # SEE WHAT A FUNCTION DOES: just type without brackets

## APPLY TO YOUR DATA:
setwd("~/work/hhu_2015/uebung_201612/cellgrowth_20161214/praktikum_201612/test2/")
plate <- readPlateMap("161130_Praktikum_RAJ11_Test_2_layout.csv")
files <- c("161130_Praktikum_RAJ11_Test_2.csv")
raw <- readPlateData(files, type="Synergy",skip=44,sep=";",
                     time.format="%H:%M:%S",time.conversion=1/3600)
viewPlate(raw)

Why Growth Rates?

  • Injection of IPTG into one column (7 wells + 1 blank) every 100 minutes.
  • Slower response with later injections.

\( \Rightarrow \) Faster growth rate \(\mu\) with later or no induction.

Why Growth Rates?

  • Injection of IPTG into one column (7 wells + 1 blank) every 100 minutes.
  • Slower response with later injections.

\( \Rightarrow \) Faster growth rate \(\mu\) with later or no induction.

\(\mu = k \frac{\text{ribosomes}}{\text{proteins}}\)

Schaechter, Maaloe & Kjeldgaard, J Gen Microbiol 1958: Dependency on medium and temperature of cell size and chemical composition during balanced growth of Salmonella typhimurium.
Koch, Can J Microbiol 1988: Why can't a cell grow infinitely fast?
Neidhardt, J Bacteriol 1999: Bacterial growth: constant obsession with dN/dt.

Why Growth Rates?

\(\mu = k \frac{\text{ribosomes}}{\text{proteins}}\)

Brauer et al., Mol Biol Cell 2008: Coordination of growth rate, cell cycle, stress response, and metabolic activity in yeast.
Slavov et al., Mol Biol Cell 2011: Coupling among growth rate response, metabolic cycle, and cell division cycle in yeast.
Scott et al., Science 2010: Interdependence of cell growth and gene expression: origins and consequences.
Weiße et al., PNAS 2015: Mechanistic links between cellular trade-offs, gene expression, and growth.

Scott et al., Mol Syst Biol 2014: Emergence of robust growth laws from optimal regulation of ribosome synthesis.


Expression of large gene groups correlates with \(\mu\).


Even at constant \(\mu\) cells are not in steady-state!

Slavov et al., Cell Rep 2014: Constant growth rate can be supported by decreasing energy flux and increasing aerobic glycolysis.

The concept of "balanced growth" is flawed, yet a central assumption of many quantitative models.

Cited at Wikipedia:

. . that's how important it is (or perhaps it was his dad :p ).

[1] Slavov et al., Cell Rep 2014: Constant growth rate can be supported by
decreasing energy flux and increasing aerobic glycolysis.

DNA Supercoiling Varies during Exponential Phase (constant \(\mu\)).

Interested in a bachelor/master project?

  • Working on platexpress and/or quantitative models of growth vs. gene expression.
  • At the interface of three projects: Ribonets, Coil-seq & Yeast Oscillations.

Fulcrand et al., Sci Rep 2016: DNA supercoiling, a critical signal
regulating the basal expression of the lac operon in Escherichia coli.

DNA Supercoiling Varies during Exponential Phase (constant \(\mu\)).

\(\Leftarrow \) © Kai, today

Interested in a bachelor/master project?

  • Working on platexpress and/or quantitative models of growth vs. gene expression.
  • At the interface of three projects: Ribonets, Coil-seq & Yeast Oscillations.

Fulcrand et al., Sci Rep 2016: DNA supercoiling, a critical signal
regulating the basal expression of the lac operon in Escherichia coli.

Growth & Gene Expression in E. coli : exponential growth

\[ \begin{equation*} \begin{aligned} \frac{\text{d}X(t)}{\text{d}t} =& \mu X(t)\\ X(t) =& X(0) e^{\mu t}\\ \end{aligned} \end{equation*} \]

Can you derive the formula
for exponential growth?

time <- seq(0,10,0.1) # hours
mu <- 0.3 # specific growth rate, hour^-1
x0 <- 0.01 # the inoculum: cell density, cells liter^-1
xt <- x0 * exp(mu*time)
par(mai=c(.75,.75,.1,.1),mgp=c(1.5,.5,0),cex=1.2)
plot(time, xt,
     xlab="time, h",ylab=expression(X[0]*e^(mu*t)))

plot of chunk unnamed-chunk-3

Growth & Gene Expression in E. coli : growth rate

\[ \begin{equation*} \begin{aligned} \frac{\text{d}X(t)}{\text{d}t} =& \mu X(t)\\ X(t) =& X(0) e^{\mu t}\\ \ln \frac{X(t)}{X(0)} =& \mu t \end{aligned} \end{equation*} \]

Which important parameter
can we calculate from here?

time <- seq(0,10,0.1) # hours
mu <- 0.3 # specific growth rate, hour^-1
x0 <- 0.01 # the inoculum: cell density, cells liter^-1
xt <- x0 * exp(mu*time)
par(mai=c(.75,.75,.1,.1),mgp=c(1.5,.5,0),cex=1.2)
plot(time, log(xt/x0),
     xlab="time, h",ylab=expression(ln(X(t)/X[0])))

plot of chunk unnamed-chunk-4

Growth & Gene Expression in E. coli : doubling time

\[ \begin{equation*} \begin{aligned} \frac{\text{d}X(t)}{\text{d}t} =& \mu X(t)\\ X(t) =& X(0) e^{\mu t}\\ \frac{\ln 2}{\mu} = & t_D \end{aligned} \end{equation*} \]

\(t_D\) is the average CULTURE doubling time.
In which cases is it also the average CELL doubling time?

time <- seq(0,10,0.1) # hours
mu <- 0.3 # specific growth rate, hour^-1
x0 <- 0.01 # the inoculum: cell density, cells liter^-1
xt <- x0 * exp(mu*time)
par(mai=c(.75,.75,.1,.1),mgp=c(1.5,.5,0),cex=1.2)
plot(time, log(xt/x0),
     xlab="time, h",ylab=expression(ln(X(t)/X[0])))

plot of chunk unnamed-chunk-5

Growth & Gene Expression in E. coli : growth rate

\[ \begin{equation*} \begin{aligned} \frac{\text{d}X(t)}{\text{d}t} =& \mu X(t)\\ X(t) =& X(0) e^{\mu t}\\ \ln(X(t)) =& \mu t + \ln(X(0)) \end{aligned} \end{equation*} \]

par(mai=c(.75,.75,.1,.1),mgp=c(1.5,.5,0),cex=1.2)
plot(time, log(xt), xlab="time, h",ylab=expression(ln~X(t)))
x1 <- .05; idx1 <- which(abs(xt-x1)==min(abs(xt-x1)))
x2 <- .1; idx2 <- which(abs(xt-x2)==min(abs(xt-x2)))
lines(x=time[c(idx1,idx2)], y=log(xt[c(idx1,idx1)]),col=2,lwd=5)
text(time[idx2],mean(log(xt[c(idx1,idx2)])),expression(Delta~X),pos=4,col=2)
lines(x=time[c(idx2,idx2)], y=log(xt[c(idx1,idx2)]),col=2,lwd=5)
text(mean(time[c(idx1,idx2)]),log(xt[idx1]),expression(Delta~t),pos=1,col=2)
arrows(x0=0,x1=0,y0=log(xt[1]),y1=-1.4,col=2,lwd=5);
text(x=0,y=-3,expression(ln(X(0))),pos=4,col=2)

plot of chunk unnamed-chunk-6

Load Your Data

library(platexpress)

dpath <- "~/work/hhu_2015/uebung_201612/Praktikum-M4452_20161207/ecoli_ts_20161014"

plate <-readPlateMap(file.path(dpath,"20161014_platemap.csv"), fsep=";",
                     fields=c("strain","IPTG","blank"))
files <- c("20161014_20161014 IPTG mVenus Injection  1_Absorbance.CSV",
           "20161014_20161014 IPTG mVenus Injection  1_Fluorescence.CSV")
raw <- readPlateData(file.path(dpath,files), type="BMG", time.conversion=1/60)

## Parsing file ~/work/hhu_2015/uebung_201612/Praktikum-M4452_20161207/ecoli_ts_20161014/20161014_20161014 IPTG mVenus Injection  1_Absorbance.CSV 
##  found data 584 
## Parsing file ~/work/hhu_2015/uebung_201612/Praktikum-M4452_20161207/ecoli_ts_20161014/20161014_20161014 IPTG mVenus Injection  1_Fluorescence.CSV 
##  found data 485/Em520 
## Interpolating all data to a single master time.

## Warning in listAverage(data, "time"): time : max. SD within timepoint is 70.5 % of median difference between time points.

What does the warning mean?

 
showSpectrum() # try: findWavelength(3)

plot of chunk unnamed-chunk-8

## re-name and color data
raw <- prettyData(raw, dids=c(OD="584",mVenus="485/Em520"),
                  colors=c("#000000",wavelength2RGB(600)))

vp <- viewPlate(raw,xlim=c(0,1800),xscale=TRUE)

## x-axis: Time 
## plotting OD;mVenus

plot of chunk unnamed-chunk-9

Linear and Non-linear Regression: analyzing a single data set

## GET A SINGLE DATASET
od <- getData(raw,"OD") # what is `od` ?
TIME <- raw$Time        # what does the $ do?
Xt <- od[,"A8"]
plot(TIME, Xt)

plot of chunk unnamed-chunk-10

 
## cut to growth range
rng <- TIME<1500   # what is `rng` ?
Xt <- Xt[rng]
TIME <- TIME[rng]

## look at data
par(mfcol=c(1,2), mai=c(.75,.75,.1,.1), mgp=c(1.5,.5,0), cex=1.2)
plot(TIME, Xt)
plot(TIME, log(Xt)) # log it - default `log` in R is the natural logarithm, ln

plot of chunk unnamed-chunk-11

 
## cut to linear range of growth
rng <- TIME>600 & TIME < 900
xt <- Xt[rng]
time <- TIME[rng]

## look again at data
par(mfcol=c(1,2))
plot(time, xt)
plot(time, log(xt))

plot of chunk unnamed-chunk-12

 
## DO LINEAR REGRESSION
## ln(X(t)) = mu * t + ln(X(0))
lfit <- lm(log(xt) ~ time) # what is `~` ?

## check quality of fit
summary(lfit)

## 
## Call:
## lm(formula = log(xt) ~ time)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.021694 -0.006298 -0.001451  0.006707  0.023853 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.840e+00  1.122e-02  -164.0   <2e-16 ***
## time         7.663e-04  1.488e-05    51.5   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01001 on 58 degrees of freedom
## Multiple R-squared:  0.9786, Adjusted R-squared:  0.9782 
## F-statistic:  2652 on 1 and 58 DF,  p-value: < 2.2e-16

## get parameters from linear regression
x0.1 <- exp(coefficients(lfit)[1]) # e^ln(X(0)) = ?
mu.1 <- coefficients(lfit)[2] # mu

## plot
par(mai=c(.75,.75,.1,.1),mgp=c(1.5,.5,0),cex=1.2)
plot(TIME,log(Xt))
lines(TIME, mu.1*TIME + log(x0.1), col="red")

plot of chunk unnamed-chunk-14

 
## DO NON-LINEAR REGRESSION, using
## the results of the linear fit as initial parameter guesses
dat <- data.frame(time=time, xt=xt)
start <- list(mu=mu.1,x0=x0.1)
nlfit <- nls(xt ~ x0*exp(mu*time),data=dat,start=start) # TRY MORE COMPLEX MODELS, see a few slides ahead

## check quality of fit
summary(nlfit)

## 
## Formula: xt ~ x0 * exp(mu * time)
## 
## Parameters:
##     Estimate Std. Error t value Pr(>|t|)    
## mu 7.660e-04  1.474e-05   51.99   <2e-16 ***
## x0 1.588e-01  1.792e-03   88.63   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.002792 on 58 degrees of freedom
## 
## Number of iterations to convergence: 1 
## Achieved convergence tolerance: 2.42e-06

## get parameters & plot results
mu.2 <- coefficients(nlfit)[1]
x0.2 <- coefficients(nlfit)[2]
par(mfcol=c(1,2),mai=c(.75,.75,.1,.1),mgp=c(1.5,.5,0),cex=1.2)
plot(TIME,Xt,ylim=c(0.2,.4))
lines(TIME, x0.2 * exp(TIME*mu.2),col="green", lty=2,lwd=5)
lines(TIME, x0.1 * exp(TIME*mu.1),col="red",lwd=2)
legend("bottomright",legend=c("data","lin.reg.","non-lin."),
       col=c(1,2,3),pch=c(1,NA,NA),lty=c(NA,1,2),lwd=3)
plot(TIME, x0.2 * exp(TIME*mu.2),col="green", lty=2,lwd=5)
points(TIME,Xt)
lines(TIME, x0.1 * exp(TIME*mu.1),col="red",lwd=2)

plot of chunk unnamed-chunk-16

How many linear phases can you detect in your experiments? Why are there several?

Fitting Growth Models

  • Initial Cell Density: \(X(0) \)
  • Lag Phase: \(\lambda \)
  • Exponential Phase: growth rate \(\mu \)
  • Stationary Phase: capacity \(A \)

https://cran.r-project.org/package=grofit

  • Logistic Equation:
    \(X(t) = \frac{A}{1+e^{\frac{4 \mu}{A}(\lambda - t)+2}} \)
  • Gompertz:
    \(X(t) = A e^{-e^{\frac{\mu e}{A}(\lambda -t)+1}} \)
  • Modified Gompertz:
    \(X(t) = A e^{-e^{\frac{\mu e}{A}(\lambda -t)+1}} + A e^{\alpha(t-t_{shift})} \)
  • Richard's generalized logistic model:
    \( X(t) = A (1 + \nu e^{1+ \nu + \frac{\mu}{A}(1+\nu )^{1+\frac{1}{\nu}}(\lambda -t)})^{-\frac{1}{\nu}} \)

as implemented in R package grofit

Try these equations with nls.
Do we need packages if nls works well?

Prepare Data: blanks, cuts, etc.

raw2 <- cutData(raw, rng=c(0,1550), mid="Time")
raw3 <- correctBlanks(raw2, plate,dids="OD",max.mid=1500) 

## blanking OD 
## blanking TRUE : 84 wells, using 12 blank wells
## OD 
##  time bin: 1 1 - 309 skipping 10 bins at 1500 
##  blank: 0.2435858298247

raw4 <- correctBlanks(raw3,plate,dids="mVenus",by=c("strain","IPTG"),
                      mbins=length(raw$Time)/5,verb=FALSE)
data <- adjustBase(raw4, base=0, add.fraction=.001, 
                   wells=unlist(groups),xlim=c(1,which(raw$Time>1500)[1]),
                   each=TRUE, verb=FALSE) # set verb to TRUE

What happens in correctBlanks and cutData?
What does adjustBase do, and when could we need it?

Get Replicate Groups

groups <- getGroups(plate, c("strain"), verb=FALSE) # SET TO TRUE!
groups2 <- getGroups(plate, c("strain","IPTG"), verb=FALSE) 
viewGroups(data, groups=groups, groups2=groups2, verb=FALSE)

plot of chunk unnamed-chunk-18

What are the lines and areas?
What is the structure of the groups item? Can you make your own groupings?
Try different groupings and parameters to viewGroups.

Use Package grofit to Fit Growth Data

raw2 <- cutData(raw,rng=c(200,1500))
grodat <- data2grofit(raw2, did="OD", plate=plate, 
                      wells=groups[["pUC18mV"]])
library(grofit)
fitparams <- grofit.2.control(interactive=FALSE, plot=TRUE) # SET ALL TO TRUE!!
pdf("growthrates.pdf")
fits <- gcFit.2(time=grodat$time, data=grodat$data, control=fitparams)

## 
## 
## = 1. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 2. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... OK

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 3. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... OK

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 4. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 5. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 6. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 7. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 8. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... OK

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 9. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 10. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 11. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 12. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## Warning in grofit::gcFitModel(acttime, actwell, gcID, control): gcFitModel:
## Unable to fit this curve parametrically!

## 
## 
## = 13. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 14. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 15. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... OK

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 16. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 17. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 18. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 19. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 20. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 21. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 22. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 23. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 24. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 25. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 26. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 27. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 28. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 29. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 30. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 31. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 32. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 33. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 34. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 35. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 36. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 37. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 38. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 39. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 40. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 41. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 42. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 43. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 44. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 45. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 46. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 47. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 48. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 49. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 50. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 51. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 52. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 53. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 54. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 55. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 56. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 57. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 58. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 59. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 60. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 61. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 62. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 63. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 64. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 65. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 66. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 67. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 68. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 69. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 70. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 71. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## Warning in grofit::gcFitModel(acttime, actwell, gcID, control): gcFitModel:
## Unable to fit this curve parametrically!

## 
## 
## = 72. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 73. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 74. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 75. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 76. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

## 
## 
## = 77. growth curve =================================
## ----------------------------------------------------
## --> Try to fit model logistic

## ....... OK

## --> Try to fit model richards

## ....... ERROR in nls(). For further information see help(gcFitModel)

## --> Try to fit model gompertz

## ....... OK

## --> Try to fit model gompertz.exp

## ... ERROR in nls(). For further information see help(gcFitModel)

dev.off()

## png 
##   2

Use Package grofit to Fit Growth Data

table <- grofitGetParameters(fits, p=c("AddId","mu.spline"))
boxplot(table[,"mu.spline"] ~ table[,"AddId"], las=2, ylim=c(1.9e-4,5e-4))

plot of chunk unnamed-chunk-20

Try different well groups instead of table[,"AddId"].
Are the fits Ok? How to get more information on the fits?

Gene Expression

  • Measured: total fluorescence per well
  • Wanted: protein level per cell!

Assumptions:

  1. Fluorescence is linear with protein level
  2. OD is linear with cell number

\( \Rightarrow \) Fluor./OD ∼ proteins/cell

Gene Expression: Normalized Fluorescence - FL/OD

viewGroups(data, groups=groups, groups2=groups2, xid="OD", lwd.orig=.5, verb=FALSE)

plot of chunk unnamed-chunk-21

Gene Expression: Normalized Fluorescence - FL/OD

fl <- getData(data,"mVenus")
od <- getData(data,"OD")
data <- addData(data, ID="mV/OD", dat=fl/od, col="#0095FF")
viewGroups(data, groups=groups, groups2=groups2, 
           dids=c("OD","mV/OD"),ylims=list("mV/OD"=c(0,1e4)),emphasize.mean=T,verb=F)

plot of chunk unnamed-chunk-22

Gene Expression: Normalized Fluorescence - FL/OD

viewGroups(data, groups=groups, groups2=groups2, dids="mV/OD", xid="OD", lwd.orig=.5, 
           ylim=c(0,1e4), g2.legend=FALSE, verb=FALSE)

plot of chunk unnamed-chunk-23

What could the slope \(\frac{mV}{OD} \sim OD \) (where it's linear) mean?
Can we use it?

Gene Expression: Normalized Fluorescence - Interpolate to OD

od.data <- interpolatePlateData(data,"OD")
viewGroups(od.data, groups=groups[2], groups2=groups2,xlim=c(.01,.15),
           ylims=list("mV/OD"=c(0,1e4)),dids="mV/OD", show.ci=F,lwd.orig=0, verb=F)

plot of chunk unnamed-chunk-24

Can we smooth upstream data to get rid of the noise?

Gene Expression: The Fold Change

flod <- getData(od.data,"mV/OD")
uninduced <- rowMeans(flod[, groups2[["pUC18mV_Inj:NA"]]],na.rm=TRUE)
od.data <- addData(od.data, ID="mV/OD/uninduced", dat=flod/uninduced, col="#AAAA00")

plot of chunk unnamed-chunk-25

Gene Expression: Data Normalization

plot of chunk unnamed-chunk-26 plot of chunk unnamed-chunk-27

What's wrong with mV/OD/uninduced vs. time w/o interpolation?

Gene Expression: Data Normalization - Result

results <- boxData(od.data, did="mV/OD/uninduced", rng=.08, groups=groups2, type="bar")

plot of chunk unnamed-chunk-28

Extracts values in given range; plots by groups, and returns values for each well.

What are the error bars?
See ?cutData on how to obtain values in a given range directly.
Or try yourself with base R.

Gene Expression: Data Normalization - Result

library(platexpress)
head(results)

##   well     group mV/OD/uninduced_0.08038
## 1   A1 WT_Inj:NA               0.6856046
## 2   B1 WT_Inj:NA               0.6510172
## 3   C1 WT_Inj:NA               0.6200460
## 4   D1 WT_Inj:NA               0.7815149
## 5   E1 WT_Inj:NA               0.6994579
## 6   F1 WT_Inj:NA               0.6037127

Calculate confidence intervals and error bars yourself.

 
imgtxt <- matrix(results[,1], nrow=7, ncol=12) # MATRIX THAT LOOKS LIKE PLATE
imgdat <- matrix(results[,3], nrow=7, ncol=12)
tmp <- t(apply(imgdat, 2, rev)) # WHAT HAPPENS HERE?
par(mai=c(.5,.5,.01,.01))
image(x=1:12, y=1:7, z=log(tmp), axes=FALSE, ylab=NA, xlab=NA)
text(x=rep(1:12,7), y=rep(7:1,each=12), paste(t(imgtxt),":\n",t(round(imgdat,2)))) 
axis(1,at=1:12); axis(2, at=1:7, labels=toupper(letters[7:1]), las=2)

plot of chunk unnamed-chunk-30

In which order does image plot rows and columns of a matrix?
Can you spot any systematic plate effects?



hide: pixels in the first row, from left to right,
correspond to the first column in the matrix, bottom up

Gene Expression - Summary

  1. Growth rate \( \mu \) matters!
  2. Fluorescence per OD as a measure of proteins per cell.
    • Note noise accumulation at low values. Why?
  3. Interpolate data to a common OD.
  4. Calculate fold-ratio to control.
    • Be careful; what is your control?
  5. Get data at a given OD, or search maximum, etc.


What other measures would be interesting or
could be required for a quantitative model?

From Wells to Cells

Conversion of Carbon&Energy Source S to biomass X:

S \(\rightarrow\) y X + (1-y) P

What are substrate S and product P in our experiments?

Catalyzed by X:

\[ \begin{equation} \label{apeq:monod} \begin{aligned} \frac{\text{d}X}{\text{d}t} &= \mu X\\ \frac{\text{d}S}{\text{d}t} &= - \frac{1}{y} \mu X \\ \mu & =\mu_{max}\frac{S}{S+K} %\mu &= f(S) \end{aligned} \end{equation} \]

How can we estimate the yield \(y\) ? Does \(y\) have units? Which?
What is the steady state of the system?

From Wells to Cells

Continuous Culture - Dilution Rate \(\; \phi\)

\[ \begin{equation} \label{apeq:monod2} \begin{aligned} \frac{\text{d}X}{\text{d}t} &= (\mu-\phi) X\\ \frac{\text{d}S}{\text{d}t} &= (S_{in}-S)\phi - \frac{1}{y} \mu X \\ \mu & =\mu_{max}\frac{S}{S+K} %\mu &= f(S) \end{aligned} \end{equation} \]

What is the steady state of above system?
What is the feedback structure of growth?
Can above system show more complex dynamics, such as
oscillations or multistablity?

From Wells to Cells: Growth

Fluorescent Protein F, total culture concentration:

\[ \begin{equation} \label{eq:Protein} \begin{aligned} % \frac{\text{d} F}{\text{d} t} = k X V_c - d F \frac{\text{d}X}{\text{d}t} &= \mu X\\ \frac{\text{d} F}{\text{d} t} &= K(?) - \delta F \end{aligned} \end{equation} \]

Intracellular concentration:

\[ \begin{equation} f = \frac{F}{X V_c} \end{equation} \]


What is \(K\), must be \(\sim X\), right ?
How can we map \(F\) from total culture to intracellular concentration \(f\) ?
Calculate \(\frac{\text{d} f}{\text{d} t}\).

From Wells to Cells : Growth

\[ \begin{equation} \begin{aligned} f =& \frac{F}{X V_c} \;;\; v_f=X V_c\\ \dot{f} =& \dot{F}\frac{1}{X V_c} - \dot{X}\frac{F}{X}\frac{1}{X V_c}\\ =& \dot{F}\frac{1}{v_f} - \dot{X}\frac{F}{X}\frac{1}{v_f}\\ =& (K - \delta f\,v_f)\frac{1}{v_f} - \mu X \frac{f v_f}{X} \frac{1}{v_f}\\ =& \frac{K}{v_f} - (\delta + \mu)f\\ K =& k X V_c \end{aligned} \end{equation} \]

What is \(X V_c\) ?
Note the term \(\delta + \mu\) . What is it, and where did the \(\mu\) come from?

From Wells to Cells : Growth

\[ \begin{equation} \label{apeq:wellmodel} \begin{aligned} \frac{\text{d}X}{\text{d}t} &= \mu X\\ \frac{\text{d}S}{\text{d}t} &= - \frac{1}{y} \mu X\\ \frac{\text{d}f}{\text{d}t} &= k - (\delta+\mu)f \end{aligned} \end{equation} \]


  • Add activation of \(f\) production by an inducer and modulation by a riboswitch \(r\).
  • Add expression of both from a plasmid (which itself increases exponentially within cells).
  • And how could we account for the effect of induced gene expression on growth rate?

. . . and find appropriate parameters (back in data hell),
start e.g. at bionumbers.
Parameter Unit Value Description Source
\(y\) g DCW / g glucose 0.5 yield factor \(\ \frac{\Delta X}{- \Delta S} \) bionumbers
\(V_c\) μm3 1 average cell volume bionumbers
\(C_c\) fg/μm3 242∓43 cell carbon content bionumbers
DCW/OD (g/L)/OD 0.36 g DCW per OD600 bionumbers
cells/OD (cells/mL)/OD 5.9-21 1e8 cells per OD600 bionumbers
\(\mu_{max}\) 1/h maximal growth rate estimate from exponential phase
\(K\) mol/L \(S\) where \(\mu = \mu_{max}/2 \) estimate from data
\(k\) mol/(L*h) protein expression rate search literature/databases
\(\delta\) 1/h protein degradation rate search literature/databases

From Wells to Cells : Riboswitch

\[ \scriptsize \begin{equation} \label{apeq:riboswitch} \begin{aligned} \frac{\text{d}sRNA}{\text{d}t} &= C P_{s} (TetR, aTc) - (\mu + \delta_s)sRNA\\ \frac{\text{d}mRNA}{\text{d}t} &= C P_{m} ( LacI,IPTG) - (\mu + \delta_m)mRNA\\ \frac{\text{d}GFP}{\text{d}t} &= (r_0 mRNA + r_1 sRNA::mRNA)\frac{m}{m+\mu+\delta_g} - (\mu + \delta_g)GFP \end{aligned} \end{equation} \]

with transcriptional activities:

\[ \tiny \begin{equation} \label{apeq:transcription} \begin{aligned} P_m(LacI, IPTG) &= P_m^0 \frac{1+\frac{1}{f_{lac}}(\frac{LacI}{K_{lac}(1+\frac{IPTG}{K_{IPTG}})})^{n_{lac}}}{1+(\frac{LacI}{K_{lac}(1+\frac{IPTG}{K_{IPTG}})})^{n_{lac}}}\\ P_s(TetR,aTc) &= P_s^0 \frac{1+\frac{1}{f_{tet}}(\frac{TetR}{K_{tet}(1+\frac{aTc}{K_{aTc}})})^{n_{tet}}}{1+(\frac{TetR}{K_{tet}(1+\frac{aTc}{K_{aTc}})})^{n_{tet}}}\\ \end{aligned} \end{equation} \]

What and where is sRNA::mRNA?
What are all the parameters? Can you derive equations for the steady state?

From Wells to Cells : Riboswitch

Rodrigo et al. 2012 - Steady State Solution:

\[ \scriptsize \begin{equation} \label{apeq:steadystate} GFP_{ss}(IPTG, aTc) = F_0 \frac{1 + f_1(\frac{IPTG}{K_1})^{n_{1}} + f_2(\frac{aTc}{K_2})^{n_{2}} + f_1 f_{sRNA}(\frac{IPTG}{K_1})^{n_{1}}(\frac{aTc}{K_2})^{n_{2}} }{1 + (\frac{IPTG}{K_1})^{n_{1}} + (\frac{aTc}{K_2})^{n_{2}} + (\frac{IPTG}{K_1})^{n_{1}}(\frac{aTc}{K_2})^{n_{2}}} \end{equation} \]


What are these parameters>? See Supplement of Rodrigo et al. 2012

Plot this function in R or python, using the
concentration ranges in your experiments: can you use image? Can R do 3D plots?

Is the steady state assumption justified? Can we combine it with our cell growth model?

Model

## Chaos in the atmosphere
growth <- function(t, state, parameters) {
    with(as.list(c(state, parameters)), {
        #y  <- a*S + b # variable yield, Heinzle et al. 1983
        mu <- mu_max*S/(S+K) 
        dX <-  mu*X
        dS <-  - mu*X/y
        dp <- (mu.p - mu) * p
        df <- p*k - (d+mu)*f
        list(c(dX, dS, dp, df))
    })
}

Solve

library(deSolve)

parameters <- c(mu_max = 1, K = 1, mu.p = 1, k = .1, d=.1, y=0.5)#, a=1, b=-1)
state      <- c(X = 0.001, S = 10, p = 1, f = 0)
times      <- seq(0, 20, by = 0.01)

out <- ode(y = state, times = times, func = growth, parms = parameters)

par(mai=c(.75,.75,.1,.1))
plot(out)

plot of chunk unnamed-chunk-32

Vary Parameters & Initial Conditions

ode.scan <- function(id, val) {
    outs <- rep(list(NA),length(val))
    for ( i in 1:length(val) ) {
        if ( id %in% names(parameters) )
            parameters[id] <- val[i]
        if ( id %in% names(state) )
            state[id] <- val[i]
        outs[[i]] <- ode(y = state, times = times, func = growth,
                         parms = parameters)
    }
    outs
}

Vary Parameters & Initial Conditions

par(mfcol=c(1,2),mai=c(.75,.75,.1,.1),mgp=c(1.7,.5,0))
outs <- ode.scan("K",seq(0.1,10,1))
X <- matrix(unlist(lapply(outs, function(x) x[,"X"])), ncol = length(times), byrow = TRUE)
matplot(times,t(X),type="l",col=1:nrow(X),lty=1:nrow(X))
legend("topleft",paste("K",seq(0.1,10,1)),col=1:nrow(X),lty=1:nrow(X))

outs <- ode.scan("S",seq(0.1,10,1))
X <- matrix(unlist(lapply(outs, function(x) x[,"X"])), ncol = length(times), byrow = TRUE)
matplot(times,t(X),type="l",col=1:nrow(X),lty=1:nrow(X))
legend("topleft",paste("S",seq(0.1,10,1)),col=1:nrow(X),lty=1:nrow(X))

plot of chunk unnamed-chunk-34

Fit To Data