FYTN05
Fall 2016
Computer Assignment 2
Chemical reaction networks and diusion
Supervisor: Adriaan Merlevede
Oce: K336-337, E-mail: [email protected]
1 Introduction
In this exercise session, we will model and simulate various (bio)chemical reactions.
This document contains a short repetition of the relevant theory, some theoretical
questions, and simulation exercises, which you should solve and present in a report. The exercises are designed for the Java ODE simulator supplied in the course
material. The appendix contains extra information on how to use this simulator,
as well as an introduction to Gnuplot, which you may use to plot solutions to the
exercises. More gnuplot documentation, other scientic plotting software, and more
theoretical background can be found on the internet and in the course literature.
We will start by modelling simple chemical reactions, and continue by including
enzymatic catalysis. We will also look at diusion and its eects in a spatial reaction
system.
1.1 Thermodynamics in biochemistry
In a biological context, most reactions take place inside a cell at constant temperature, pressure, and in aqueous solutions. The relevant state function in such systems
is the Gibbs Free Energy, G. The free energy may only decrease in a closed system,
and is minimal when the system is in equilibrium.
When a reaction occurs at constant temperature and pressure, the change in G
denes which reactions will occur; exactly those reactions with ∆G > 0 occur, and
the system is in equilibrium when ∆G = 0 for all reactions. ∆G is given by
∆G =
X
i
1
µi ∆Ni
where ∆N is the dierence in number of molecules X after and before the reaction,
and µ are the chemical potentials for the molecular species involved in the reaction.
The chemical potential is dened as the rate of change in entropy S upon change in
the number of molecules N . The chemical potential is dependent on the concentration, and is often separated into a constant standard chemical potential µ at some
reference concentration c , and a concentration-dependent part.
i
i
i
i
0
0
dS µi ≡ −T
dNi E,Nj6=i
µi = kB T ln (ci /c0i ) + µ0 (i , T )
Following from the above equations, ∆G can also be described as a sum of concentrationdependent and concentration-independent (∆G ) terms. At equilibrium, we have
∆G = 0, and we can derive (you should do so)
0
Keq ≡
Y
i
[Xi ]∆Ni = exp (−∆G0 /kB T )
eq
where [X ] is the concentration of the molecules X . This expression denes the
reaction constant K . Note that the equality is equivalent to the law of mass
action, which states that reaction rates are proportional to the product of reaction
concentrations raised to the power of their respective stoichiometric coecient (show
this).
In the law of mass action, the proportionality constant of the reaction rate and the
product of reagent concentrations is called the rate constant k. It can be derived
using transition state theory that the rate constant is given by
i
i
eq
k = A exp (−∆‡ G/kB T ) ;
A=
kB T
h
where G is the activation energy of the reaction (the highest dierence in G between the substrate and the reaction intermediates), A is the Arrhenius factor, and
h is Planck's constant. In biochemistry, many reactions are catalyzed by specializd
proteins called enzymes. As catalysts, they are not used up in the reaction, and
return to their original state after the reaction is complete. Enzymes speed up a
reaction by stabilizing the reaction intermediates, lowering the activation energy.
Many enzymes are so ecient that the rate of the uncatalysed reaction is insignicant compared to the enzymatic reaction. For an enzymatic reaction to occur, the
products have to interact not only with each other, but also with the enzyme, forming an intermediate activated enzyme-substrate complex. This alters the reaction
kinetics, and is captured in the Michaelis-Menten equation.
‡
2
1.2 Diusion - Patterns
The mass action law and Michaelis Menten describe reaction kinetics in a spacially
uniform system. Many biological systems are not spatially uniform; molecules are
often produced in one place and then transported. Molecules can also be transported
between cells.
The simplest case of transport is diusion. Each molecule individually makes a
random walk through the available space. As a result, the concentration of molecules
decreases in highly concentrated regions, and increases in lowly concentrated regions,
until the concentration is spatially uniform.
But diusion can also lead to the formation of patterns. Such patterns were rst
analysed by Brittish mathematician Alan Turing, who proposed that diusion can
be a destabilizing factor for states that are otherwise locally stable. Systems that
are based on an interplay between local chemical reactions and diusion spreading
substances is called a reaction-diusion system. We will look at the spatio-temporal
patterns formed by a Brusselator system.
2 Problems
2.1 A two-state reaction (18/100)
Let us consider a simple two-state reaction:
k+
A
B
k−
Write down the dierential equation for the time evolution of the
concentration [A]. Give an expression for the equilibrium constant
K .
1.a)
eq
Assume that values for the standard chemical potentials and the forward activation
barrier are µ = 8 kJ/mol, µ = 3 kJ/mol, and ∆G = 75 kJ/mol.
0
A
‡
0
B
Calculate ∆G , K , k and k . What will be the ratio between
the concentrations [B] and [A] at equilibrium? How does it depend
on the activation barrier ∆G ?
1.b)
0
eq
−
+
‡
3
Simulate the system in the java simulator. It is a TwoStateReaction, where Y and Y correspond to A and B, respectively. Use
dierent initial conditions and describe what happens. In what direction is the reaction moving? Would a change in ∆G alter the
behavior? Save data les from a couple of simulations and plot in an
external application, e.g., gnuplot.
1.c)
0
1
‡
2.2 Enzyme reactions (22/100)
Imagine that the forward direction in the reaction above is catalyzed by an enzyme,
E . The enzyme stabilizes certain reaction intermediates, eectively lowering the
activation energy to ∆G = 65 kJ/mol.
‡
E
What is the new value of k ? Did k change? Try to get a feel
of how small changes in the eciency of the enzyme can change the
reaction properties, and imagine the eects this can have on a cell.
2.a)
−
+
Let us now explicitly include the enzyme in our model for the reaction kinetics.
First, assume a simplistic reaction:
kf
A+E →B+E
Here, we will disregard the reverse reaction. You can simulate this system in the
program as SimpleEnzymatic, where Y = [A] and Y = [B]. Start your simulations
with [B] = 0.
0
1
Write down the dierential equations for the time evolution
of [A], [B], and [E]. How does the reaction rate (the number of
reactions per second per volume) depend on the concentrations [A]
and [E]? Is this model reasonable in a large range of concentrations,
like [A] → ∞?
2.b)
Assume now instead a description of the enzyme reaction where an intermediate
reactive complex AE is formed. This is called Michaelis Menten kinetics, a very
successful model for enzymatic reaction kinetics.
k1
k+
A + E AE → B + E
k2
4
Derive the Michaelis Menten equation, by assuming that the
rst reaction step is in equilibrium in the time scale of the second
step. Compare the result with the previous exercise. How does the
system depend now on [A], particularly for high values?
2.c)
2.3 Diusion (12/100)
We will introduce a diusion-like transport of substances between cells in a onedimensional system: cell i is neighbor with cells i − 1 and i + 1. We describe the
transport using the equation
dci
= D(ci−1 − 2ci + ci+1 )
dt
(1)
where c is the concentration in cell i.
i
Starting from a molecular random walk process between cells in
one dimension, show that Eq. 1 is a reasonable description of passive
(entropic) transport between cells. How does Eq. 1 relate to the
diusion equation?
3.a)
Simulate the diusion model with a simple system starting with
random initial concentrations between 0 and 10. Plot the concentrations c(x, t) as a function of position (x) and time (t).
3.b)
2.4 The Brusselator (23/100)
The Brusselator is a reactive system, rst studied in the 1970s in Brussels. It is
of interest because it exhibits a stochastically unstable oscillating behavior. The
Brusselator can lead to dierent kinds of patterns for dierent input parameters,
particularly when introduced in a reaction-diusion system. The reactions used are
k
A →1 X
k
2X + Y →2 3X
k
B + X →3 Y + C
k
X →4 D.
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The concentrations of A and B are considered to be in excess, forming an essentially
innite reservoir.
Write down the dierential equations describing the time evolution of [X] and [Y ].
4.a)
4.b) Run simulations using the Brusselator model for one cell and
dierent parameter values. Investigate the behavior. Can you get
the system to oscillate? Increase the simulation time if neccessary.
Increase the number of cells to 100, random initial concentrations for X and Y , and parameters leading to oscillations. Describe
the dierence between dierent cells.
4.c)
4.d)
Introduce diusion for X. What happens for the dierent cells?
Run simulations with parameter values k A = 0.1, k = 0.1,
, and k = 0.1. Use small random deviations in the initial concentrations of X and Y . Vary the two diusion parameters
and describe dierent behaviors. Note that you are now looking for
emerging spatial patterns that are hard to see in the time series plot;
you will need to plot both the time and space dimensions.
4.e)
k3 B = 0.2
1
2
4
3 Guidelines for the report
The student can choose how to organise their text. The following elements should
be presented:
• Introduction describe the problem in general terms, give background information, etc.
• Theory discuss the theory that underpins your work. This part should also
include answers to the theoretical exercises. This should be a continuous text
explaining the theory and your solutions as examples, not a list of exercises
with answers.
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Results and discussion describe your results from the dierent simulations,
discuss them and explain in your own words your understanding of the reasons
behind your results.
• Conclusions summarize what you have done, and present any general discussion or conclusions you want to include.
•
It is important to present your report in a scientic style, and written in English.
Organization and form of the report are weighted as 25/100 of the grade.
Reports should be made and handed in individually. However, students are encouraged to work together when doing the exercises.
The reports should be submitted in .pdf format (!) to the supervisor by e-mail
(see heading of this document). They may also be printed and delivered to the
supervisor's mailbox at the department of theoretical physics.
Do not forget to properly mention any work that is not originally yours, including
sources such as web pages and other students. Any non-original material that is not
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.
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