Interactive Graphs

Introduction

Interactive graphs of equations allow one to visualize mathematical concepts and show how changing the assumptions changes the predicted outcomes. They can be used to both understand protein engineering concepts and to make predictions for your own projects.

Brief explanations and still images of each graph are shown below. To view the interactive graph, click the links to download it; then view it with Wolfram's Computable Document Format Viewer, available here.

Michaelis-Menten equation

Many enzyme catalyzed reactions follow the kinetic equation derived by Michaelis and Menten in 1912.

rate = ([E]*k_cat*[S])/(K_M + [S]) where [S] is the substrate concentration.

Download this interactive graph and see how changes in k_cat and K_M affect the initial rate of the reaction.

Number of possible substitution variants

The number of variants generated by making k amino acid substitutions in a protein, V_k, is given by the product of the location possibilities and the substitution possibilities.

V_k = location possibilities * replacement possibilites

The number of location possiblities is combinations without repetition, where L is the number of amino acids in the chain. See text for a derivation of the this term.

L!/(k!(L-k)!) = (total number of permuations)/(order not important * number of permutations not used)

The number of replacement possiblities is 19^k, so

V_k = L!/(k!(L-k)!)19^k (1)

For two substitutions anywhere in a 300-aa protein:

V_k = 44,850 * 361 = 16,190,850

Download this calculator and adjust the values to see how quickly the numbers rise to impossibly large numbers.