Chemical reaction networks and diffusion

Transcription

1 FYTN05 Fall 2012 Computer Assignment 2 Chemical reaction networks and diffusion Supervisor: Erica Manesso (Office: K217, Phone: , erica.manesso@thep.lu.se) Deadline: October 23, 2012 This document and the java program can be downloaded from erica/teaching.html 1 Introduction In this exercise we will start by modeling simple chemical reactions, and then continue to larger molecular reaction networks exhibiting more complex dynamics. Thereafter, we will also include the effect of diffusion and investigate the spatial organization that can arise from such systems. Before turning to the exercises, some key concepts will be reviewed. You can also refer to your lecture notes or the course book for help on solving the problems. 1.1 Chemical Reactions Chemical reactions describe processes that transform set of substances, reactants to another set of substances, products. In biological context most of reactions take place at constant temperature, pressure, and in aqueous solutions. A useful thermodynamic potential for description of such systems is Gibbs free energy. A change in G reflects the change in the world s entropy S tot = G/T. (1) Thus G is minimized at equilibrium and the sign of G specifies directionality of the reaction. For constant temperature and pressure the change in G becomes simply G = µ i dn i. (2) i Lets consider now a general chemical reaction ν 1 X ν k X k ν k+1 X k ν m X m. (3) 1

2 For a single forward step of this reaction we have G = ν 1 µ 1... ν k µ k + ν k+1 µ k ν m µ m, (4) where µ i T ds dn i E,Nj =i = k B T ln(c/c 0 ) + µ 0 (ε i, T) are chemical potentials for all the reactants and products of the reaction. Chemical potential specifies the rate of the change of entropy due to the change in number of molecules and can be separated to the part dependent on the concentration c and standard chemical potential µ 0 defined for a given substance at the temperature T and with respect to the reference concentration c 0. Given the definition of G we can conclude that for G < 0 reaction proceeds forwards, G > 0 reaction proceeds backwards, G = 0 reaction is in equilibrium. Similarly to the µ 0 we can define standard free energy change G 0 which is the concentration-independent part of G. With this definition it is easy to see (which means that you should work it out if you haven t done so already) that condition for equilibrium becomes [X k+1 ] ν k+1 [Xm ] ν m [X 1 ] ν 1 [Xk ] ν k = K eq e G0 /k B T. (5) Above expression is known as Mass Action rule and is equivalent to statement that reaction rates are proportional to the product of reactants concentrations raised to the power of stoichiometric coefficient (show it!). Another (out of many) factor that reaction rate depends on is activation barrier G which describes the difference in free energy that has to be overcome along reaction coordinate for reaction to proceed. According to Transition state theory the rate constant is given by k = k BT h e G /k B T. (6) Some reactions rates can be significantly increased by use of the enzymes. Enzymes are molecules that catalyze reactions by lowering the activation barrier G for given reaction. Enzymes are not used up in the reaction and at the end of the process return to their original state. The standard free energy change G 0 remains unaltered too. Kinetics of enzymatic reactions can be described by Michaelis-Menten rule. This rule predicts that rate of the reaction will be saturated at large substrate concentrations due to the finite amount of the enzyme (look up the equations in the lecture notes). 2

3 1.2 Diffusion - Patterns So far we haven t been concerned with detailed description of spatial distribution of molecules in chemical reactions. Substances were readily available for reactions and concentrations were spatially uniform. In biological systems that is certainly not the whole story. The molecules in the cell are often produced in one place and than transported to different place to fulfill its function. Similarly hormones and other substances can be transported between cells through cell s membrane. The simplest case of passive transport of molecules is diffusion. The molecules diffuse from high concentration region to one of low concentration by stochastic (random) motion. In fact it can be shown that random walk process, which allows particles to move with equal probability in any direction on the lattice, leads to the diffusion equation for the probability density of finding the particle at given position and time (see problem 2.3). Surprisingly diffusion can also lead to formation of the patterns. This phenomena was analyzed by british mathematician Alan Turing who proposed that diffusion can be a destabilizing factor for homogeneous stable state of two component reaction. Such interplay between local chemical reactions and diffusion spreading substances in space is called reaction-diffusion system. A prominent example of reaction-diffusion is Brusselator - a system of model reactions developed in 1970 by group of the researches in Brussels. You will study pattern forming capabilities and other properties of Brusselator system in problem Problems 2.1 A two-state reaction (18/100) Let us consider a simple two-state reaction A k + k B a) Write down the differential equation for the time evolution of the concentration of A ([A]). Define the equilibrium constant, K eq, from k + and k. Assume that values for the standard chemical potentials and the forward activation barrier have been taken from a table, such that µ 0 A = 8, µ0 B = 3, and G = 75 all measured in the unit kj/mol. 3

4 b) Define G and evaluate K eq from these values. What will the ratio between the concentrations [B] and [A] be at equilibrium? How does it depend on the activation barrier G? c) Calculate the forward and backward reaction rates and plug them into the java simulator. In the simulator, this system is called TwoStateReaction, where Y 0 refers to A and Y 1 refers to B. Simulate using different initial conditions and describe what happens. In what direction is the reaction moving? Would a change in G alter the behavior? Save data files from a couple of simulations and plot in an external application, e.g., gnuplot. 2.2 Enzyme reactions (22/100) Imagine that the forward direction in the reaction above is catalyzed by an enzyme, E. Since this is essentially a one-way reaction, keep k small, and start all simulations with [B]=0. a) Assume that the effect of the enzyme is to lower the activation barrier to G E = 65 kj/mol. How does this affect the forward rate k +? Compare two simulations with the different activation barriers. Let us explicitly include the enzyme in the reaction. First, assume a simplistic description A + E k f B + E b) Write down the differential equations for the time evolution of [A], [B], and [E]. How does the reaction rate (the number of reactions per second per volume) depends on the concentrations of A and E? What will the reaction rate be if [A]? Is it reasonable? Simulate the reaction using the SimpleEnzymatic program, where Y 0 = [A] and Y 1 =[B], and Y 2 =[E]. Discuss how your predictions compare with simulations using different initial values of [A] and [E]. Assume now instead a description of the enzyme reaction as A + E k 1 k2 AE k+ B + E 4

5 c) Write down the differential equations for all molecules. Derive the Michaelis-Menten (MM) equation by assuming a constant amount of enzyme (free+bound) and that the second part of the reaction is the rate limiting step, i.e., the first part of the reaction is in quasi equilibrium. Redo exercise b) (using the MichaelisMenten program) for the resulting MM equation, i. e. investigate analytically and numerically how the reaction depends on the concentrations [A] and [E]. What is the main difference? 2.3 Diffusion (12/100) We will introduce a diffusion-like transport of substances between cells lying on a line, i.e., cell i is neighbor with cells i 1 and i + 1, where i is the cell index. The transport is defined in the model by dc i dt = D(c i 1 2c i + c i+1 ) (7) where c i is the concentration in cell i. a) Starting from a molecular random walk process between cells in one dimension, show that Eq. 7 is a reasonable description of passive (entropic) transport between cells. How does Eq. 7 relate to the diffusion equation? b) Simulate the diffusion model starting with random initial concentrations in the cells between 0 and 10. Plot the concentrations c(x, t) as a function of position (x) and time (t). Discuss the result in comparison with Fig. 4.12A in Nelson. 2.4 The Brusselator (23/100) The Brusselator reaction network consists of the reactions A k 1 X 2X + Y k 2 3X B + X k 3 Y + C X k 4 D. 5

6 a) Write down the differential equations describing the time evolution of [X] and [Y]. b) Run simulations using the Brusselator model for one cell and different parameter values (Y 0 =[X] and Y 1 =[Y]). Investigate the behavior. Can you get the system to oscillate? Increase the simulation time if neccessary. c) Increase the number of cells to 100 and use random initial concentrations for X and Y, and parameters leading to oscillations from (b). Describe the difference between different cells. d) Introduce diffusion for X. What happens for the different cells? e) Run simulations with parameter values k 1 A = 0.1, k 2 = 0.1, k 3 B = 0.2, and k 4 = 0.1. Use small random deviations in the initial concentrations of X and Y. Vary the two diffusion parameters and describe different behaviors. Note that you are now looking for emerging spatial patterns that are hard to see in the time series plot. The patterns will be more clear if you plot concentrations as a function of position (cell index). 6

7 A Guidelines for the report The report should consist of four main parts, it is up to you how you want to organize your text, but the following should be included in one way or another: Introduction describe the problem in general terms, give background information, etc. Theory discuss the theory that describes the problem, including answers to exercises. To get the highest grade this part should be a continuous text, not just a list of answers to the exercises. Results and discussion describe your results from the different simulations, discuss them and try to explain them as far as possible. If you prefer, results and discussion can be two separate sections. Conclusions summarize what you have done and present any general discussion or conclusions you want to include. Organization and form of the report are weighted as 25/100. The reports should preferrably be submitted in printed form to the supervisor s mailbox at the department. If they are submitted electronically, send them as.pdf or.ps files (not.doc). The corrected reports can be found in the bookshelf outside lecture hall D. You are encouraged to collaborate when doing the exercise but the reports should be individual. In the report you are allowed to use information from any source you like, but a reference should be given for each statement. Any material that is not originally yours, i. e. text from the internet or other students, and not correctly referenced, will be recognized and reported to the university disciplinary council. Past experience has shown that the vast majority of students understand this, but there have been exceptions. B Gnuplot gnuplot is a useful and popular program for generating plots. To start gnuplot you simply type gnuplot in a terminal window, and you should see a copyright notice followed by a gnuplot prompt, like this one: gnuplot> Actually you do not always have to start gnuplot, you can save all your plot commands in a file and type gnuplot filename directly in the terminal window, but more about that later. Here follows some examples of useful gnuplot commands. 7

8 Plot Plotting simple functions: plot x * plot sin(x) Defining and plotting a function: f(x,a) = x+a plot f(x,1) title a=1, f(x,2) title a=2 Setting ranges for the axes: set xrange [0:1] set yrange [0:*] (only lower bound set) There is a replot command, which will let you add more plots in an existing window: replot cos(x) 3D Plots To generate 3D plots you use the command splot instead of plot otherwise everything works the same way as in the 2D case, with set zrange setting the range of the z-axis. For example: splot x*x + y*y Plotting data from a file It is of course also possible to plot data from a text file. Suppose you have a file called out.data that looks like this:

9 This is typically how the outputs from the programs used in the exercises will look like. To plot the data from this file, you can type plot "out.data" using 1:3 with linespoints This will create a plot where x coordinates are taken from column 1 and y coordinates from column 3. If you don t specify the columns (i.e., you remove "using 1:3"), the first columns will be used. linespoints means that the data points will be plotted as points connected with lines. Some other useful styles are lines, points, dots and yerrorbars. With yerrorbars you need to have a column with error values, and the using clause should specify three columns (e.g. using 1:3:5). To set the plotting style for all subsequent plot commands, use set data style, e.g.: set data style linespoints Of course you are also able to splot from a text file, simply type splot "out.data" using 1:2:3 with linespoints You can transform the data in a column before plotting it, for example if you want to plot the square of the data in the second column, this will do the job plot "out.data" using 1:($2**2) The operator ** means to the power of. Title and labels To give the diagram a title and set labels on the axes, you can type: set title "Title goes here" set xlabel "My x label" set ylabel "My y label" 9

10 Log scale Once in a while you may need to plot something with a logarithmic scale on either axis. For instance, if you want a logarithmic y axis: set logscale y And to set the scale back to linear: unset logscale y Save your plot as postscript file To print a plot, you first need to save it to a postscript file. (Postscript is a language understood by printers.) This is done by telling gnuplot to output in postscript, rather than to the screen. The commands needed for this are: set terminal postscript set output "yourfilename.ps" You will then have to re-run the plot command (or simply replot). The postscript plot will now be in the file yourfilename.ps. To print or view this file, open another terminal window. There, you can start the postscript viewer gv: gv yourfilename.ps It s possible to print from gv, or you can do it directly from the shell using lpr yourfilename.ps. Depending on what computer you are using and what printer you want to use, a different command (or a -P argument to lpr) may be needed. After generating a postscript file you ll probably want to go back to plotting to the screen: set terminal X11. Putting it all together It is often easier to save all plot commands in a file and type gnuplot file directly at the terminal (without first starting gnuplot). For example suppose you create a file called mygnuplotfile.gnplt that looks like this: 10

11 set terminal postscript set output "myoutput.ps" set title "My postscript plot" set xlabel "X" set ylabel "Y" set zlabel "Z" splot x*x + y*y with lines quit If you type gnuplot mygnuplotfile.gnplt in a terminal, a file called myoutput.ps that looks like this will automatically be created: My postscript plot x*x + y*y Z X Y Getting help gnuplot has a help command which lists all of the available commands and their options, and provides some useful examples. It may be a bit cumbersome to use, but once you get used to it it s not that bad. To get a list of all help topics, type help If you want a list of the available commands: help commands If you want help with a specific command: help plot help set key help set term post There is also help to be find online, see for example 11

12 C Using the ODE simulator For this exercise you will use a software where all the reaction networks are already defined so you do not need to write any code by yourselves. The program is pretty easy to understand and should not need any elaborate tutorial. Downloading the program The program can be downloaded from erica/teaching.html Just download the file ODE.jar and save it in some directory on your computer. Preferably you first create a directory where you want to save all your data files and plots and then download the file to that directory. To do this first open a terminal window and print: mkdir MyDirectory Download the file to that directory, and if you then go back to the terminal window and type cd MyDirectory ls you should see the filename ODE.jar printed on your screen, if you haven t already figured it out mkdir creates a directory, cd changes directory and ls list the files in the current directory. Running the program To run the ODE simulator simply type java -jar ODE.jar Now a a small window should appear on the screen. Simply choose a model, the number of cells you want to have in your model, and adjust the model parameters and initial values to your liking and click on the Start new simulation button. After some time a graph will appear on your screen displaying the concentrations as function of time. Each time you click on Start new simulation button, the output will be saved in the data file you have chosen in the output field. This allows you to create more elaborate plots using for example gnuplot. Note that the program will erase any previous data in the file you choose, so if you want to save your data files make sure you change filename before you start a new simulation. 12

13 Single cell simulations The most simple case is when you have set the number of cells to one, a typical data file will in this case look like this; The first column just enumerates the outputs from 0 to the number of outputs you have chosen in the Simulation settings, the second column displays the time, the third shows which cell we are looking at (in this case we only have one cell and thus this column is 0 everywhere) and the following columns display the concentrations at that specific time point. In gnuplot you plot it by writing for example; plot out.data u 2:4 with lines which will plot the first molecule concentration as a function of time. Simulations with multiple cells When you set the number of cells to something other then one, you will immediately see that the Initial Values panel changes. Instead of setting specific initial values you can now define a range of values. Each cell will then use a random value within that range. In this way we can simulate systems with multiple cells with different initial conditions. The output from the program looks as above but now each cell will appear on a new line for example

14 the only difference is that now each cell is printed on a new line with a different cell index in the third column. If we want to create a 3D plot with time on one the x-axis, the spatial position on the y-axis and a concentration on the z-axis, compare with the plots on page X in Nelson, simply type splot out.data using 2:3:4 with lines 14