In this report we present the enzymatic reaction from an unknown enzyme. We will make every attempt to understand the results that we obtained.

For this first report we will attempt to analyse the data with the enzyme kinetics method of Michaelis-Menten

Enzyme was from **unknown** source and tested in the laboratory under standard conditions.

The analysis method is derived from the script from Professor Rob Beynon in the U.K.

In particular we will calculate the ** Vmax** and

The data was obtained from an experiement performed by ** Freddie Buck** on:

The substrate data was collected and written in the lab notebook.

A vector ** S** can be created to contain the substracte concentrations:

`S <- c(0,1,2,5,8,12,30,50)`

Similarly vector ** v** will contain the velocity results for each

`v <- c(0,11.1,25.4,44.8,54.5,58.2,72.0,60.1)`

The data can be combined and presented in simple tabular form.

There are 8 observations for ** S** and 8 for

(Inside note: the number of observations for each vector is derived from an `R`

calculation with the `length()`

function.)

```
# Combine data in a dataframe:
kinData <- data.frame(S,v)
# Print the data:
kinData
```

```
## S v
## 1 0 0.0
## 2 1 11.1
## 3 2 25.4
## 4 5 44.8
## 5 8 54.5
## 6 12 58.2
## 7 30 72.0
## 8 50 60.1
```

We can explore the data with a simple first plot describing the reaction.

`plot (S,v)`

It seems that the plot is consistent with an enzymatic reaction and matches the Michaelis-Menten equation.

This enzymatic data plot indicates that the Michaelis-Menten equation can be used.

We can use a statistical method to create a model and calculate constant parameters ** Km** and

For this we will use standard statistical functions built-in in `R`

base installation: `nls()`

and `predict()`

described as follows by `R`

help:

`nls()`

: *Determine the nonlinear (weighted) least-squares estimates of the parameters of a nonlinear model.* `nls()`

is non-linear least squares optimiser that we can use to solve the ** Km** and

`predict()`

will be used to predict a theoretical curve based on the `nls()`

calculations and a series of hypothetical ** S** dilutions.

The Michaelis-Menten equation:

```
# "velocity = Vmax times S divided by (Km plus S)", stored in MMcurve
MMcurve <- formula(v ~ Vmax*S/(Km+S))
```

By inspecting the plot we can determine approximate values for ** Km** and

`nls()`

in order to compute values closer to the truth:** Vmax** is about

`50`

** Km** is about

`2`

(value of [S] when Vmax is half.)```
bestfit <- nls(MMcurve, kinData, start=list(Vmax=50,Km=2))
# print the results
bestfit
```

```
## Nonlinear regression model
## model: v ~ Vmax * S/(Km + S)
## data: kinData
## Vmax Km
## 73.261 3.437
## residual sum-of-squares: 156.4
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 5.971e-06
```

We can therefore estimate the final values of ** Km** and

** Vmax** = 73.26

** Km** = 3.44

We can use the model above to build a theoretical curve: we first create a list of hypothetical concentrations of the substrate ** S** and then use the

`predict()`

funtion with the `bestfit`

data obtained by `nls()`

to compute theoretical values for velocity `v`

:```
# Hypothetical oncentration ranges for S:
SconcRange <- seq(0,50,0.1)
# Calculate the predicted velocities using the predict() function:
theorLine <- predict(bestfit,list(S=SconcRange))
```

This will be used in the final plot.

This final plot contains all the data points and the theoretical curve, with the values of ** Vmax** and

```
# Now plot the data, the best fit line, and put the best fit coefficients in the plot
plot (kinData,
xlab="Subtrate (mM)",
ylab="Velocity (nmol/s)",
title(main="Fitted MM data"),
pch=17, col="blue", cex=1.5)
# Now add the theoretical line with the lines() function:
lines(SconcRange,theorLine,col="red")
# Add text with the values:
text(28,30, "Vmax = ")
text(35,30,round(coef(bestfit)[1],2))
text(29,25, "Km = ")
text(36,25,round(coef(bestfit)[2],2))
```

In conclusion we can report that this enzyme follows a typical Michaelis-Menten pattern and the equation can be solved with ** Vmax** = 73.26 and

`sessionInfo()`

```
## R version 3.3.3 (2017-03-06)
## Platform: x86_64-apple-darwin13.4.0 (64-bit)
## Running under: OS X El Capitan 10.11.6
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## loaded via a namespace (and not attached):
## [1] backports_1.0.5 magrittr_1.5 rprojroot_1.2 tools_3.3.3
## [5] htmltools_0.3.5 yaml_2.1.14 Rcpp_0.12.10 stringi_1.1.5
## [9] rmarkdown_1.4 knitr_1.15.1 stringr_1.2.0 digest_0.6.12
## [13] evaluate_0.10
```